Spin–orbit torque-assisted switching in magnetic insulator thin films with perpendicular magnetic anisotropy

As an in-plane charge current flows in a heavy metal film with spin–orbit coupling, it produces a torque on and thereby switches the magnetization in a neighbouring ferromagnetic metal film. Such spin–orbit torque (SOT)-induced switching has been studied extensively in recent years and has shown higher efficiency than switching using conventional spin-transfer torque. Here we report the SOT-assisted switching in heavy metal/magnetic insulator systems. The experiments used a Pt/BaFe12O19 bilayer where the BaFe12O19 layer exhibits perpendicular magnetic anisotropy. As a charge current is passed through the Pt film, it produces a SOT that can control the up and down states of the remnant magnetization in the BaFe12O19 film when the film is magnetized by an in-plane magnetic field. It can reduce or increase the switching field of the BaFe12O19 film by as much as about 500 Oe when the film is switched with an out-of-plane field.


. Planar Hall Effect in Pt/BaM
In addition to the ordinary Hall effect and the anomalous Hall effect (AHE), the planar Hall effect (PHE) 1,2 is also present in the Pt/BaM structure. Supplementary Figure 1 presents the PHE resistance Rxy of the same Hall bar as described in the text measured with an in-plane field as a function of the angle of the field () relative to the +x direction. One can see that the PHE resistance shown in Supplementary  Figure 1 is about one order of magnitude larger than the AHE resistances presented in the text. However, the planar Hall effect does not invalidate the use of AHE measurements to determine the magnetization status in the BaM film. This is because all of the AHE measurements in this work involved the rotation of the magnetization in the yz plane, while the planar Hall effect concerns magnetization rotation in the xy plane.

Supplementary Note 2. Origins of Spin-Orbit Torque in Pt/BaM
The experimental data presented in the main text clearly demonstrate the existence of spin-orbit torque (SOT) at the Pt/BaM interface upon the application of a direct current in the Pt layer. It is very likely that this SOT arises from the spin Hall effect (SHE)-produced spin accumulation in the Pt near the Pt/BaM interface, as discussed previously for non-magnetic heavy metal/ferromagnetic metal heterostructures. It is also possible that there exist magnetic proximity effect (MPE)-induced magnetic moments in several Pt atomic layers near the Pt/BaM interface, as indicated by the experimental data shown in Fig. 1f in the main text. In those ferromagnetic-like Pt atomic layers, the SOT can appear through the SHE and the Rashba effect, just as in ferromagnetic metals. The sections below discuss in detail these two different SOT mechanisms. The discussions consider a more general system, a non-magnetic heavy metal (HM)/magnetic insulator (MI) bi-layered structure, but are applicable to the particular Pt/BaM structure concerned in this work.

a. SHE-Associated SOT
Supplementary Figure 2 depicts the essence of the SHE-associated SOT mechanism. When there is no charge current in the HM layer, SOT is zero and the normalized magnetization m in the MI layer remains in its equilibrium state. When a charge current is passing through the HM layer, the SHE produces a pure spin current that is polarized along the x axis and flows along the z axis, as shown schematically in the Supplementary Figure 2. The net effect is the presence of spin accumulation s near the HM/MI interface. This spin accumulation can exert torques on m through the s-d exchange interaction at the interface. 3 Based on the Onsager theory, 4 one can correlate s to the spin current density J with the following spin diffusion equation where e is the electron charge,  is the electrical conductivity of the HM, SH is the spin Hall angle in the HM, and Ey is the applied electric field. Assuming that  is the spin diffusion length in the HM, where the vector coefficients C1 and C2 can be obtained using the following two boundary conditions: (ii) The spin current density at the HM/MI interface is proportional to the spin mixing conductance G and can be written The final expression for the spin accumulation at the HM/MI interface reads where (0) s  denotes the spin accumulation in the absence of the s-d exchange interaction and is given by and the coefficients a and b are given by Through the interfacial s-d exchange interaction, the spin accumulation   0 s z  μ exerts an effective field on m, which is given by where Jsd is the s-d exchange constant. The field along x gives rise to a field-like torque (FLT), while the one along  mx leads to a damping-like torque (DLT) or the so-called Slonczewski torque. 5 These two fields are referred as HFLT and HDLT, respectively, in the discussions below.
The total field Htotal on m in the MI layer can be then written as where H and Ha are the external magnetic field and the effective anisotropy field in the MI, respectively. With this total field, one can then calculate the magnetization dynamics in the MI layer by numerically integrating the Gilbert equation where || is the absolute gyromagnetic ratio and  is the Gilbert damping constant.

b. MPE-Associated SOT
The MPE-associated SOT mechanism is illustrated schematically in Supplementary Figure 3. When there is no charge current in the HM layer, the MPE-induced magnetic moment  in the HM must be collinear with the normalized magnetization m in the MI layer, as shown in Supplementary Figure 3a. As a result, there is no torque on m in the MI. Once a charge current is applied to the HM layer, the SHE and the Rashba effect give rise to a SOT field that exerts on  in the Pt and thereby tilts  away from its initial equilibrium direction. The net effect is that  in the HM is not collinear with m in the MI anymore and  exerts a torque on m, as shown schematically in Supplementary Figure 3b.
The SOT field in the ferromagnetic-like HM produced by the SHE and the Rashba effect can be written as 6 where hFLT and hDLT are the effective fields corresponding to the field-like torque and the damping-like torque, respectively. Considering that the conduction electrons in the HM respond to external stimuli much faster than the magnetization m in the MI, one can then write  in the HM as where  is the magnetic susceptibility of the HM and Jsdm denotes the interfacial s-d exchange field. One can see from Supplementary Equation (11) that  is not collinear with m due to the SOT fields. Through the s-d exchange interaction,  in the HM will also exert an effective field on m in the MI, which is Jsd. Thus, the total field Htotal on m can be written as where the term JsdH is dropped out because it is much smaller than H, and the term (Jsd) 2 m is also  (8) and (13), one can conclude that the SOT fields in the two different mechanisms actually have the exactly same symmetry. As a result, one can extract the strength of the SOT fields from the experimental data without having to know the relative contributions of the different mechanisms. The details on the estimation of the SOT fields are provided in the next section. It should be emphasized that (0) s  in Supplementary Equation (4) and hFLT and hDLT in Supplementary Equation (10) are all proportional to the charge current density Jc. This means that the strength of the SOT fields is also proportional to Jc.

Supplementary Note 3. Estimation of SOT Fields in Pt/BaM
In this section the strength of the SOT fields in the Pt/BaM structure is estimated and is compared to that in HM/ferromagnetic metal (FM) bi-layered systems. In a HM/FM system, both HFLT and HDLT are proportional to the charge current density Jc, no matter whether they originate from the SHE or the Rashba effect. 6,7 This is the same in the Pt/BaM structure (see Supplementary Note 2). This fact enables the comparison between the SOT efficiency in the Pt/BaM and that in the HM/FM systems studied in previous work.

a. Macrospin Simulations
To determine the SOT field strength, simulations are carried out that use a macrospin m to represent the magnetization in the BaM film and numerically solve Supplementary Equation (9). The simulations use the fields defined in Supplementary Equations (8) or (13). The anisotropy field Ha is perpendicular to the BaM film plane. The external field H is in the yz plane and is tilted 20 degrees away from the +z direction initially, the same as in the experiment (see Fig. 1e). The SOT fields HFLT and HDLT are unknown and will be obtained by comparing the experimental coercivity values with the values obtained from the simulations.
One first starts from calculating the coercivity Hc for the charge current density Jc=0. For this calculation, one takes HFLT=0 and HDLT=0; the strength of Ha is set in such a way that m in the BaM flips when H is pointing in a direction opposite to its initial direction and has a strength equal to the experimentally measured Hc value, which is 1.45 kOe. Then one considers the case of Jc0 and finds Hc values for given HFLT and HDLT values via numerical simulations. Upon the application of a charge current in the Pt layer, both HFLT and HDLT present and contribute to the dynamics of m. However, from the point of view of the symmetry it is clear that a flip in the direction of the HFLT field does not lead to a change in Hc because the HFLT field is orthogonal to Ha, H, and m. In contrast, a flip in the direction of the HDLT field breaks the symmetry and therefore affects Hc. For this reason, as the first step HFLT is set to zero and Hc is calculated as a function of HDLT. It is important to note that the adjustment of the anisotropy field to make the Hc value comparable to the experimental value for the Jc=0 case is done for convenience of comparison of the measured and calculated results. The rate of the change of Hc with HDLT, however, remains a quantity which is independent of this approximation. Figures 2e and 2f in the manuscript compare the calculated results with the experimental data. Figure  2e shows the experimental Hc vs. Jc data. In Figure 2f, the blue dots show the Hc vs. HDLT response calculated for HFLT=0, and the other dots are discussed shortly. One can see that the Hc vs. HDLT response shows a linear dependence, the same as the experimental Hc vs. Jc data. Specifically, Hc increases to about 2.0 kOe when HDLT is -400 Oe and decreases to about 0.95 kOe when HDLT is 400 Oe. The same change in the experimental Hc value is observed when Jc changes between -10 7 A·cm -2 and 10 7 A·cm -2 . Thus one can conclude that the strength of HDLT in the Pt/BaM is about 400 Oe at Jc=10 7 A·cm -2 . For HM/FM systems, previous work observed a HDLT field of 17 Oe for Pt(2 nm)/Co(0.6 nm)/AlOx, 8 50 Oe for Pt(3 nm)/Co(0.6 nm)/Al2O3(2 nm), 9 55-200 Oe for Pt(3 nm)/Co(0.9 nm)/Ta(0.5-4 nm), 10 50 Oe for Pt(3 nm)/Co80Fe20(0.6 nm)/MgO, 11 and 200 Oe for Ta(5 nm)/ Co80Fe20(0.6 nm)/MgO, 11 all corresponding to the same charge current density Jc=10 7 A·cm -2 . One can see that the HDLT field in the Pt/BaM is stronger than those previously reported values. Considering that (1) the BaM layer (3 nm) in this work is at least three times thicker than the FM layer (<1 nm) in the previous work and (2) in an HM/FM structure there is always a portion of the applied current flowing in the FM layer and being wasted, one can conclude that the SOT efficiency in the Pt/BaM structure is indeed higher than that in the HM/FM systems.
The above-described analysis assumed HFLT=0, and similar analysis can be carried out for HFLT0. The red and olive dots in Figure 2f show the Hc vs. HDLT responses calculated for HFLT= HDLT/2 and HFLT= HDLT, respectively. It is evident from the data in Figure 2f that the effect of HFLT is almost negligible for HFLT= HDLT/2. For HFLT= HDLT, the Hc vs. HDLT response deviates from the linear dependence for strong negative charge currents, which, however, has not been observed experimentally, indicating that HFLT is relatively small in the Pt/BaM structure.

b. Micromagnetic Simulations
The simulations described above used a macrospin to represent the magnetization m in the BaM film. In the experiment, however, the magnetization switching in the BaM film may not be realized through coherent rotation, but through domain nucleation and subsequent domain wall motion, thanks to the relatively large size of the Pt/BaM Hall bar sample. In consideration of this possibility, full micromagnetic simulations were also carried out to estimate the SOT fields in the Pt/BaM structures.
Specifically, the simulations used the well-established OOMMF code to numerically solve Supplementary Equation (9). The code is based on the Oxs_SpinXferEvolve module, which is intended for the simulation of Slonczewski-type torques. 12 The HDLT field, as defined by Supplementary Equations (8) or (13), was taken into account by an equivalent spin torque with the polarization along the x axis, enabling the modeling of the spin-orbit torque as if it was a spin-transfer torque. In order to break the symmetry and accelerate the simulations, the magneto-crystalline anisotropy field value and the anisotropy easy axis direction were randomized by a few percent and a few degrees, respectively. The simulated film size was set to 1 µm  1 µm, and the mesh size was set to 5 nm x 5nm x 3 nm. As in the experiments, the external field was in the yz plane and was tilted 20 away from the +z direction initially. The procedures for the determination of the SOT fields were the same as those described above for the macrospin simulations. Figure 2g in the main text presents the results obtained from the micromagnetic simulations in the same format as in Figure 2f. By comparing Figures 2f and 2g, one can see that the results from the two simulations are close to each other for all three different HFLT fields, confirming the accuracy of the macrospin simulation-yielded HDLT fields described above. One can also see that, for a given HDLT, the coercivity values from the micromagnetic simulation are slightly smaller (about 4%) than those from the macrospin model. This means that, for a given coercivity change, the corresponding HDLT field from the micromagnetic analysis is slightly larger than that from the macrospin analysis, suggesting that the micromagnetic simulation indicates the presence of slightly stronger SOT in Pt/BaM than the macrospin simulation.
It should be noted that Hc can depend on many extrinsic factors, such as film surface roughness, defects, and anisotropy distribution. However, for a given film sample, these factors are fixed and will not change upon the application of a current in the Pt layer, thus it is valid to examine the SOT strength by checking the effects of the strength and polarity of the charge current in the Pt film on the Hc value of the BaM film.