Origin of spin–orbit torque in single-layer CoFeB investigated via in-plane harmonic Hall measurements

It has been shown over the last decade that the current-induced spin–orbit torque (SOT)^1–6 can be significant in heavy-metal/ferromagnetic (HM/FM) bilayers in the presence of strong spin–orbit coupling. Such a strong torque on the FM layer not only poses questions about the origin of SOT^7–9 but also enables spintronic applications such as all-electrical magnetization switching.^2–4,10 Although the origin of SOT is still in debate, the HM layer is generally considered to play a critical role in generating the SOT in HM/FM bilayers via the bulk spin Hall effect¹¹ or interfacial effects.^12–14 

Recently, it has been shown that the FM layer itself can be the source of efficient SOT generation. Taniguchi et al.¹⁵ proposed that the anomalous Hall effect (or anisotropic magnetoresistance) can be the source of spin current generation. Through the drift-diffusion derivations for a hetero-structure consisting of two FM layers separated by one nonmagnetic layer, Taniguchi et al.¹⁵ showed that the spin currents originating from one FM layer (the injecting layer) exert sizable torques on the other FM layer by properly designing the FM layer combinations. Experimental measurements^16–21 of such a SOT in FM trilayers without any HM layer showed that the overall SOT efficiency is comparable to that of HM/FM bilayers and could be used for efficient magnetization switching.^19–21 

While the FM-layer-induced SOT was confirmed in FM trilayers, its mechanism appears to be complicated because of the existence of two FM layers and multiple interfaces. More recently, SOT studies focused on the experimental determination of torques in FM single layers.^22–25 For example, the finite SOT was found in Fe_0.9Mn_0.2 single-layer films.²³ This means that that sizable SOT can be expected even without any additional nonmagnetic layer, which is quite promising for the downscaling of SOT devices along the film-normal direction. However, up until now, few studies have focused on a quantitative determination of the current-induced SOT in FM single layers, and the origin of the nontrivial total SOT is unclear. In particular, the disentanglement of such a SOT into the (anti-)damping-like (DL) and field-like (FL) components has not been demonstrated; in addition, a detailed thickness-dependent SOT study has been lacking.

In this work, we quantitatively determine and disentangle the current-induced SOT in CoFeB single-layer thin films via in-plane harmonic Hall measurements. The decomposed DL and FL torques are on the same order of magnitude but show opposite trends of thickness dependence before saturation. The converted DL torque efficiencies increase with FM thickness, while the FL torque efficiencies show almost no thickness dependence.

Amorphous thin films of CoFeB (t_FM)/AlO_y (2) (thickness in nanometers) are deposited onto thermally oxidized Si substrates (Si/SiO_x) at room temperature by utilizing an ultrahigh vacuum magnetron sputtering facility. t_FM denotes the thickness of CoFeB, which ranges from 4 nm to 20 nm. The atomic composition of the sputtering target is Co:Fe:B = 20:60:20. The deposition condition of CoFeB follows that of a previous study,²⁶ where CoFeB is in-plane-magnetized and its saturation magnetization M_s is 7.73 × 10⁵ A/m. Because t_FM is much larger than the critical thickness to induce the perpendicular magnetic anisotropy in CoFeB, it is reasonable to assume that M_s is independent of t_FM. All films are micro-fabricated into Hall bars with 25 μm in length and 10 μm in width using photo-lithography and Ar ion milling for magneto-transport measurements.

We perform the in-plane harmonic Hall measurements following a previous study²⁷ to separate the current-induced SOT and the thermoelectric signals in magnetic hetero-structures. The analysis extracts the overall DL and FL torques in the CoFeB thin films. Figure 1 shows the schematic measurement setup in this work. A constant-amplitude sinusoidal voltage is applied to the Hall bar devices, while a constant-amplitude external magnetic field ranging from 200 mT to 3000 mT is applied in the thin-film plane. Both in-phase first harmonic and 90° out-of-phase second harmonic Hall voltages are simultaneously recorded, while the samples rotate at an angle φ with respect to the x axis. Because the magnetic easy axis of CoFeB lies in the plane of the film, its magnetization is expected to follow the external magnetic field. The φ dependence of the first and second harmonic Hall resistance (⁠$Rxyω$ and $Rxy2ω$⁠) reads

where R_PHE and R_AHE are the planar Hall and anomalous Hall resistance of the sample; B_DL and B_FL are the DL and FL effective magnetic fields (or, equivalently, DL and FL torques that are used throughout this work); B_ext and B_k are the external magnetic field and the out-of-plane anisotropy field; αB_ext and $R∇TTE$ are the second harmonic resistance induced by the ordinary Nernst effect²⁸ and anomalous Nernst effect,²⁷ respectively.

Figure 2 shows the harmonic Hall results and analysis for t_FM = 4 nm. The first harmonic signals shown in Figs. 2(a) and 2(b) slightly deviate from the sin 2φ function shown in Eq. (1). This distortion is probably due to a small sample misalignment as pointed out in a previous study.²⁷ The raw data of $Rxy2ω$ and the decomposed DL term, FL term, and parasitic term with a B_ext of 200 mT and 1800 mT are shown in Figs. 2(c) and 2(d), respectively; the corresponding enlarged FL and parasitic components are shown in Figs. 2(e) and 2(f). The parasitic term corresponds to the asymmetric signal of the raw data around φ = 180°.²⁷ The DL and thermal terms of the second harmonic signal are obtained by fitting the experimental data at φ = 45°, 135°, 225°, and 315° with the cos φ function, as shown in Figs. 2(c) and 2(d). The DL torque is obtained by fitting the experimental data in Fig. 2(g) via $Rxy2ω=RAHEBDLBext+Bk+αBext+R∇TTE$ according to Eq. (2). The FL-related term is obtained by subtracting the cosine term from the φ-dependent second harmonic data, where the FL torque is obtained via the field-dependent linear fitting shown in Fig. 2(h). As presented in Figs. 2(c) and 2(d), the second harmonic signals are dominated by the cos φ term and are insensitive to B_ext, suggesting that the overall DL torque is quite small for t_FM = 4 nm. The FL term is well-fitted by the 2 cos³ φ − cos φ function, showing that it can be reasonably quantified.

Figures 3(a) and 3(b) summarize the t_FM-dependent DL and FL torques which are normalized by the current density J_FM in CoFeB. The J_FM is estimated to be (6.9–7.8) × 10⁶ A/cm² for all the samples. We note two points as shown below. First, the maximum values of the DL and FL torques are comparable and are about one tenth of those obtained from in-plane-magnetized Pt/Co thin films.^27,29 This is reasonable because of the relatively large t_FM used in this study and the absence of strong spin–orbit coupling elements in CoFeB. Such a result is also different from the SOT measurement for a FM layer in proximity to a light metal such as Ti and Cu,³⁰ where the FL torque is reported to be much stronger than the DL torque. Second, it is worth noting that the DL and FL torques show opposite t_FM dependence before saturation, i.e., the DL torque increases with t_FM, while the FL torque decreases with it in the t_FM range of 4 nm–10 nm. Such a thickness dependence discrepancy suggests that the two torques have different origins.

Figures 3(c) and 3(d) show the DL and FL torque efficiencies obtained via $τD(F)L=(2eMstFM)/ℏ×BDFL/JFM$⁠, where e is the electron charge and ℏ is the Dirac constant. As a result, τ_DL shows strong t_FM dependence, while the τ_FL is almost t_FM-independent. The t_FM-dependent τ_DL shows strong evidence that the DL torque is associated with the charge-spin conversion in the bulk of the CoFeB layer, which is caused by its intrinsic spin–orbit coupling and exchange interactions.^31–33 Such a nontrivial charge-spin conversion in CoFeB, however, does not contribute to a discernable FL torque because the τ_FL does not follow the t_FM dependence of τ_DL. Thus, it is very likely that the overall FL torque shown in Fig. 3(d) is not induced by the spin current from bulk CoFeB. Instead, the FL torque could have other origins such as interfacial effects.^8,14,29

Figures 3(e) and 3(f) show the effective longitudinal and transverse spin Hall conductivities (⁠$σxxs$ and $σxys$⁠) obtained via $σxx(xy)s=τD(F)L/ρFM$⁠, where ρ_FM is the resistivity of CoFeB, which is shown in the inset of Fig. 3(f). By considering a t_FM-independent $σxys$⁠, as discussed above, a zero-slope fitting in Fig. 3(f) yields an average value of $σxy0s=$ 17 ± 4 Ω⁻¹ cm⁻¹. Note that we do not focus on the small sign reversal of the FL term around t_FM = 10 nm in Figs. 3(d) or 3(f) because its magnitudes are too small and comparable to the corresponding error bars; instead, we consider that an average value obtained from the linear fitting is more suitable for a quantitative discussion. For the discussion on $σxxs$⁠, we first note that $σxxs$ should saturate when t_FM becomes much larger than the exchange length in CoFeB,²² which was reported to be 4.7 nm according to a previous study.³⁴ By averaging $σxxs$ in the t_FM range of 10 nm–20 nm, where $σxxs$ is expected to saturate, a saturation value of $σxx0s=$ 110 ± 26 Ω⁻¹ cm⁻¹ is obtained. As a result, $σxx0s$ is about one order of magnitude greater than $σxy0s$⁠. Thus, by taking into account the strong t_FM dependence of $σxxs$⁠, the t_FM-independent $σxys$⁠, and the ratio of $σxy0s$/$σxx0s$⁠, we suggest that the charge-spin conversion in single-layer CoFeB is spin-Hall-like,²² which is in close analogy to that in HM/FM bilayers.⁸ 

Finally, we briefly discuss the influence of the CoFeB anomalous Hall effect on the SOT in our films. Recently, Kim and Lee³³ rigorously developed the spin drift-diffusion equations in FM-layer-based hetero-structures by considering not only the longitudinal^35,36 but also the transverse spin chemical potentials in the system. They showed that both DL and FL torques generated by the anomalous Hall effect exhibit similar FM layer dependence. In our current study, however, we do not find strong t_FM dependence of the FL torque. Such a comparison implies that the anomalous Hall effect-induced charge-spin conversion might not be significant in our CoFeB films.

In conclusion, we have quantitatively disentangled the current-induced SOT in a CoFeB single-layer ferromagnet. According to the harmonic Hall analysis, the DL and FL torques exhibit very different t_FM dependence. In particular, our results suggest that the DL torque originates from a spin-Hall-like bulk charge-spin conversion, whereas the FL torque is very likely to have an interfacial origin. While both spin Hall and anomalous Hall effects are expected to play a role in the charge-spin conversion in our CoFeB thin films, the latter might not be significant according to the comparison of t_FM dependence between DL and FL torques.

The authors acknowledge Shutaro Karube for helpful discussions. This work was partially supported by the Japanese Ministry of Education, Culture, Sports, Science, and Technology (MEXT) in Grant-in-Aid for Scientific Research (Grant No. 15H05699) and the JSPS Core-to-Core program (Grant No. JPJSCCA20160005). Y.D. acknowledges the Center for Science and Innovation in Spintronics (CSIS), Tohoku University for financial support; R.T. was supported by the Graduate Program in Spintronics at Tohoku University.