Abstract
An accurate method is developed to extract the spin–orbit (SO) effective fields by analyzing harmonic Hall voltage measurements and deriving detailed analytical equations that consider both the z component of the applied magnetic field and the secondorder perpendicular magnetic anisotropy. The method is tested by analyzing the results of a macrospin simulation. The SO effective fields that are extracted from the analysis are consistent with the input SO effective fields that are used in the macrospin simulation over the entire range of the polar magnetization angle and for a wide range (0–2) of the ratio of the planar to the anomalous Hall voltage considered in this study. The accuracy of the proposed method is demonstrated via a systematic study that involves comparing its results with those of a conventional analytical method.
Introduction
Inplane currents in a nonmagnetic (NM)/ferromagnetic (FM) bilayer nanostructure can generate a torque due to the spin–orbit (SO) coupling, which is known as the SO torque (SOT). SOT is sufficient for reversing the magnetization in the FM layer.^{1} Numerous studies have been conducted to identify the principal mechanism of the SOT as either the spin Hall effect (SHE) in the NM layer^{2, 3, 4} or the interfacial SO coupling—frequently referred to as the Rashba effect—at the NM/FM interface.^{5, 6, 7, 8, 9, 10, 11, 12} In a system in which the NM/FM interface is perpendicular to the z axis and the inplane currents flow along the x axis, spin currents that are polarized along the y axis are generated in the system based on the SHE induced by a bulk SO coupling in the NM layer. The spin currents are injected into the adjacent FM layer, which causes torque to be transferred to the magnetization of the FM layer. The SHEinduced SOT generates a strong dampinglike torque (T_{DL}∝m × m × y) and a weak fieldlike torque (T_{FL}∝m × y).^{13, 14} Theoretically, the strength of the SHEinduced SOT is known to be independent of the magnetization direction of the FM layer. For an interfacial SO couplinginduced SOT, spins that are polarized along the y axis accumulate due to the broken inversion symmetry at the NM/FM interface. Direct exchange coupling between the magnetization of the FM layer and the accumulated spins generates a strong T_{FL} but a weak T_{DL}.^{7, 15, 16, 17, 18} Unlike the strength of the SHEinduced SOT, the interfacial SO couplinginduced SOT is known to depend on the magnetization direction of the FM layer.^{19, 20, 21} In these two cases, both the SHE and the interfacial SO coupling qualitatively induce the same torque on the FM layer. To identify the dominant mechanism of the SOT, a quantitative analysis of the values of T_{DL} and T_{FL} over an extensive range of magnetization angles is required.^{19, 20, 21}
The harmonic Hall voltage measurement method is a useful approach for quantifying the effective fields of T_{DL} and T_{FL} that originate from the SOT.^{22} This method is particularly suitable for identifying the angular dependence of the SOT that acts on the FM layer with perpendicular magnetization.^{19, 20} Several corrections are required for an accurate analysis of the measured results, including the measured results for the planar Hall effect (PHE),^{20, 23} the outofplane component of the external magnetic field^{20} and the anomalous Nernst effect.^{20, 24} In the harmonic Hall voltage measurement, the second harmonic voltage (V^{2ω}) consists of two major components: anomalous Hall voltage and planar Hall voltage (denoted as V_{AHE} and V_{PHE}, respectively).^{20, 23} When an external magnetic field (H_{ext}) is applied along the longitudinal (x) direction, the V^{2ω} values from the AHE and PHE are proportional to T_{DL} and T_{FL}, respectively. Under a transverse (y) H_{ext}, however, these values are proportional to T_{FL} and T_{DL}, respectively. This necessitates the use of an analytical solution that is based on Cramer’s rule to separate T_{FL} and T_{DL}.^{23} This analytical solution has been successfully applied to only a system in which V_{PHE}<V_{AHE}. For a system in which V_{PHE}>V_{AHE}, such as the W/CoFeB/MgO trilayer structure,^{25} a divergence occurs in the solution, which makes it extremely difficult to analyze the measurement results.
This problem can be overcome by incorporating some necessary corrections into the analysis of the measurement results, including measurement results for the outofplane component of H_{ext}. As coherent magnetization rotation is an important requirement in the analysis of harmonic Hall voltage measurement results, H_{ext} is usually applied along the direction tilted slightly (4°−15°) from the basal plane (x–y plane).^{20} In this condition, the z component of H_{ext} has a nonzero value, although it has been neglected to simplify the analysis. This assumption is reasonable in the lowH_{ext} range, where the magnetization direction is close to the z axis and the perpendicular magnetic anisotropy (PMA) field is consequently dominant over the z component of H_{ext}.^{23} The simplifying assumption, however, is not valid in the highH_{ext} range, where the magnetization direction deviates considerably from the z axis, with a resultant reduction in the PMA field. As a result, the PMA field loses its dominance over the z component of H_{ext}. Several attempts have been made to include the z component of H_{ext}, which is obtained by repetitively solving equilibrium torque equations until the desired convergence is achieved (the recursive method).^{19, 20} However, this method is complicated, and it has not been validated for systems with V_{PHE}>V_{AHE}. Unwanted voltages, which originate from thermoelectric effects such as the anomalous Nernst effect, should be eliminated from the harmonic signals. Although several methods have been proposed for this purpose,^{20, 24} erasing all artificial signals remains difficult. Another important issue that has not been considered is the inclusion of the secondorder PMA in an analysis of the harmonic Hall voltage measurement results. The inclusion of the secondorder PMA is considered important because many PMA materials exhibit secondorder PMA, whose strength is comparable to that of the firstorder PMA in many cases.^{26, 27}
In this study, two corrections—one for the z component of H_{ext} and the other for the secondorder PMA—are considered in the analysis of the harmonic Hall voltage measurement results. All related analytical equations are described. Both conventional and refined analytical methods are used to analyze the results of the macrospin simulation, which mimics the harmonic Hall voltage measurement by numerically solving the Landau–Lifshitz–Gilbert equation.^{28, 29} The accuracy of these two analytical methods is tested by comparing the input SO effective fields for the macrospin simulation with the input SO effective fields calculated by analytical methods. To test the refined analytical method, systems are examined over a wide ratio R, which is defined as V_{PHE}/V_{AHE}. A similar comparative study that analyzes the experimental results of harmonic Hall voltage measurements for a Pt/Co/MgO structure was also performed.
Materials and methods
Analytical solutions for conventional approach
When an inplane AC current with frequency ω (I_{AC}=I_{0} sinωt) is applied to an NM/FM bilayer structure, the angle between the z axis and the magnetization of the FM layer (θ_{M}) and the angle between the x axis and the orthographic projection of the magnetization on the x–y plane (ϕ_{M}) oscillate as θ_{M}(t)=θ_{M}°+Δθ_{M} sinωt and ϕ_{M}(t)=ϕ_{M}°+Δϕ_{M} sinωt. Here the superscript ° and the symbol Δ denote the value in the absence of I_{AC} and the amplitude of the related angles, respectively. The total energy equation for the NM/FM bilayer structure can be expressed as follows:
Here K_{1}^{eff} is the effective firstorder PMA energy density that considers the demagnetizing term, that is, K_{1}^{eff}=K_{1}−N_{d}M_{S}^{2}/2 (K_{1}, N_{d} and M_{S} are the firstorder PMA energy density, demagnetizing factor and saturation magnetization, respectively).^{30} K_{2} is the secondorder PMA energy density.^{30} m is the unit vector of magnetization. The effective magnetic field (ΔH) induced by the maximum value of the inplane AC current (I_{0}) is composed of the dampinglike effective field (ΔH_{DL}) and the fieldlike effective field (ΔH_{FL}). θ_{H} and ϕ_{H} are the polar angle and azimuthal angle of H_{ext}, respectively. Given that the inplane anisotropy is negligibly small over the PMA field, ϕ_{M}° is assumed to be identical to ϕ_{H}. The values of Δθ_{M} and Δϕ_{M} can be analytically expressed as follows (refer to Supplementary Equations (S1)–(S13) for a detailed derivation):
Here H_{K}^{eff} and H_{K1}^{eff} are the effective PMA field and the effective firstorder PMA field, respectively, and H_{K,2} is the secondorder PMA field. These parameters are defined as follows: H_{K}^{eff}≡H_{K,1}^{eff}+H_{K,2}; H_{K,1}^{eff}≡2K_{1}^{eff}/M_{S}; H_{K,2}≡4K_{2}/M_{S}. Note that Equations (5) and (6) are identical to the analytical expressions derived by Hayashi et al.^{23} when H_{K,2}=0. If the values of Δθ_{M} and Δϕ_{M} are sufficiently small, the components of the m vector can be approximated in the form m(t)=m°+(2Δm) sinωt:
Both the anomalous Hall voltage and planar Hall voltage contribute to the measured Hall voltage: V_{H}=I_{AC} R_{H}=I_{AC} R_{AHE} m_{z}+I_{AC} R_{PHE} m_{x} m_{y}.^{20, 31, 32} Here R_{AHE} and R_{PHE} are the anomalous and planar Hall resistance, respectively. With the application of I_{AC}, the m values oscillate as given in Equations (7)–(9) with the resultant expressions for Hall voltages as follows:
Here the following relations exist: V_{AHE}=I_{0} R_{AHE} and V_{PHE}=I_{0} R_{PHE}. The subscripts x and y indicate that the harmonic Hall voltages are measured at ϕ_{H}=0° and ϕ_{H}=90°, respectively. The first harmonic Hall voltage (V^{1ω}) contains information about the θ_{M}° value, and the second harmonic Hall voltage (V^{2ω}) contains information about the Δθ_{M} and Δϕ_{M} values. The conventional analytical solution considers only the case in which the magnetization direction has slightly deviated from the z axis (θ_{M}°≈0°). In this case, the z component of H_{ext} is negligibly small over the PMA field along the same direction (H_{ext} cosθ_{H} ≪ H_{K}^{eff} cosθ_{M}°), and therefore, the assumption of sinθ_{M}°=H_{ext}/H_{K}^{eff} in the conventional solution is reasonable.^{8, 23, 33} Note that the contribution due to H_{K,2} is also negligible at θ_{M}°≈0° (Equation (5)). With this assumption, V^{1ω} and V^{2ω} can be rewritten as follows:
The second harmonic Hall voltages, as given in Equations (15) and (16), are composed of two terms that contain ΔH_{DL} and ΔH_{FL}. When the R ratio is negligibly small, the values of ΔH_{DL} and ΔH_{FL} can be obtained using the following T ratios:
At R=0, the values of T_{x} and T_{y} are identical to the values of ΔH_{DL} and −ΔH_{FL}, respectively.^{8} When the R ratio becomes comparatively large, T_{x} and T_{y} should be corrected using Cramer’s rule.^{23}
In Equation (20), ΔH_{DL} and ΔH_{FL} can be calculated if the determinant B_{0}^{2}−A_{0}^{2} is not 0. If B_{0}^{2}−A_{0}^{2} is 0, it is not possible to obtain the individual values of ΔH_{DL} and ΔH_{FL}; it is only possible to obtain the relation T_{x}=T_{y}=ΔH_{DL}−ΔH_{FL} (Equations (17)–(19)).
Analytical solutions for refined approach
The assumption of H_{ext} cosθ_{H} ≪ H_{K}^{eff} cosθ_{M}° is not valid at high H_{ext} values. In this case, the θ_{H} value is not negligible, and Δθ_{M} and Δϕ_{M} (Equations (5) and (6)) must be substituted into Equations (12) and (13) to obtain the expression for V^{2ω}:
Considering that V_{x}^{1ω}=V_{y}^{1ω}=V_{AHE} cosθ_{M}° (Equation (11)), the G ratios, which correspond to the T ratios in the conventional approach, can be defined as follows:
Similar to the conventional analytical equations, the refined equations also need to be solved using Cramer’s rule (as given in Equation (20)).
Results
Conventional analysis
The conventional analytical method is used to analyze the results of the macrospin simulation. Figure 1a and b shows the results for V^{1ω} as a function of H_{ext} in two different systems: (a) H_{K,1}^{eff}=5 kOe and H_{K,2}=0 kOe; and (b) H_{K,1}^{eff}=5 kOe and H_{K,2}=−1 kOe. Two sets of results are shown in Figure 1a and b: the first set is obtained from the macrospin simulation (squares), and the second set is obtained from Equation (14), which is based on the conventional assumption of sinθ_{M}°=H_{ext}/H_{K}^{eff} (dashed lines). For the macrospin simulation, the following parameters are applied: ΔH_{DL}=−50 Oe; ΔH_{FL}=−100 Oe; θ_{H}=86°; and V_{AHE}=1 mV. Refer to Supplementary Figure S1 for a detailed description of the macrospin simulation. Agreement between the results obtained from the macrospin simulation and the results obtained from Equation (14) based on the conventional analytical method, which is the focus of this study, is only obtained in the lowH_{ext} range. In the highH_{ext} range, the deviation is very large, which indicates the limited validity of conventional solutions.
Figure 1c and d shows the analytical results for B_{0}^{2}−A_{0}^{2} calculated from Equation (19) as a function of H_{ext} at two different R values of 0.3 (red lines) and 1.75 (blue lines) (R=V_{PHE}/V_{AHE}). The results in Figure 1c pertain to the system with H_{K,1}^{eff}=5 kOe and H_{K,2}=0, and the results in Figure 1d pertain to the system with H_{K,1}^{eff}=5 kOe and H_{K,2}=−1 kOe. The H_{K}^{eff} values for both systems are also indicated in the figures. The detailed equation for B_{0}^{2}−A_{0}^{2} is rewritten for clarity:
Recalling that H_{ext}/H_{K}^{eff} is approximated with sinθ_{M}°, we note that the results for B_{0}^{2}−A_{0}^{2} at H_{ext}>H_{K}^{eff} have no physical meaning. According to Equation (28), the B_{0}^{2}−A_{0}^{2} value decreases from R^{2}−1 to −1 as the H_{ext} value increases from 0 to H_{K}^{eff}. The determinant B_{0}^{2}−A_{0}^{2} always has a negative value at R<1. At R⩾1, however, the determinant can have both positive values and negative values over the H_{ext} range of 0–H_{K}^{eff}, which indicates the occurrence of a crossover (B_{0}^{2}−A_{0}^{2}=0) at a certain H_{ext} value. This feature is visible in the results shown in Figure 1c and d. In both systems, that is, with H_{K,2}=0 and −1 kOe, the B_{0}^{2}−A_{0}^{2} value is always negative at R=0.3; at R=1.75, it initially has a positive value, after which it passes through 0 and then has a negative value. The crossovers occur at 3.3 and 2.6 kOe for the systems with H_{K,2}=0 kOe and H_{K,2}=−1 kOe, respectively. Recalling that T_{x}=T_{y}=ΔH_{DL}−ΔH_{FL} when the determinant is 0, an H_{ext} value should exist at T_{x}=T_{y} when R>1.
Figure 2a–f shows the results for V_{x}^{2ω} and V_{y}^{2ω} (a and b) and T_{x} and T_{y} (c and d) as functions of H_{ext} and the results for ΔH_{DL} and ΔH_{FL} (e and f) as functions of θ_{M}° for two different systems, that is, with H_{K,2}=0 (solid lines) and H_{K,2}=−1 kOe (dashed lines). The left (Figure 2a, c and e) and right panels (Figure 2b, d and f) show the results for R=0.3 and 1.75, respectively. The results for V^{2ω} were obtained by the macrospin simulation, and the results for T_{x} and T_{y} (Equations (17) and (18)) and ΔH_{DL} and ΔH_{FL} (Equation (20)) were analytically calculated using the simulation results. As shown in Figure 2a and b, the sign of V_{x}^{2ω} for a small R value of 0.3 is negative, whereas the sign of V_{x}^{2ω} for a large R value of 1.75 is positive. Of the two major contributions of V_{AHE} and V_{PHE} to V^{2ω}, the sign of the former is negative, but that of the latter is positive. Equations (15) and (16) predict this behavior (the opposite signs of V_{AHE} and V_{PHE}) and explain that the V_{y}^{2ω} value for R=1.75 is higher than that for R=0.3.
The results for V_{x}^{2ω} and V_{y}^{2ω} and their variation with R have a critical effect on T_{x} and T_{y}. At R=0.3, the signs of T_{x} and T_{y} are opposite because the signs of V_{x}^{2ω} and V_{y}^{2ω} are the same, indicating that there are no H_{ext} values at which T_{x}=T_{y} in both systems, that is, for H_{K,2}=0 kOe and H_{K,2}=−1 kOe (Figure 2c). These results are consistent with the results for B_{0}^{2}−A_{0}^{2} (Figure 1c and d). The values of T_{x} and T_{y} are the same at a specific H_{ext} value, at which B_{0}^{2}−A_{0}^{2}=0 (Equation (19)). For R=1.75, the signs of T_{x} and T_{y} are the same because the signs of V_{x}^{2ω} and V_{y}^{2ω} are opposite (Figure 2d). There are H_{ext} values for which T_{x}=T_{y} exist in both systems, that is, with H_{K,2}=0 kOe and H_{K,2}=−1 kOe. The positions, however, differ from those for which B_{0}^{2}−A_{0}^{2}=0. The H_{ext} values in the former case are 3.6 and 3.8 kOe for the systems with H_{K,2}=0 and H_{K,2}=−1 kOe, respectively, whereas those in the latter case are 3.3 and 2.6 kOe, respectively. These deviations occur because the determinant poorly reflects the behavior of the first harmonic.
The inappropriate determinant, that is, B_{0}^{2}−A_{0}^{2}, causes large errors in the SO effective fields, as shown in Figure 2e and f. Recalling that the input SO effective fields are ΔH_{DL}=−50 Oe and ΔH_{FL}=−100 Oe, we consider the results at R=0.3 (Figure 2e) to be reliable in the θ_{M}° range from 0° to the angles that correspond to H_{ext}=H_{K}^{eff}. These angles are 61° and 52° when H_{K,2}=0 kOe and H_{K,2}=−1 kOe, respectively. Beyond these two angles, which are indicated by the vertical and horizontal lines in Figure 2e and f, respectively, the output SO effective fields start to deviate from the input values. The indicated regions end not at 90° but at ~82° as the m vector is not completely aligned along the x or y axis, even for H_{ext}=10 kOe (Figure 1a and b). The output SO effective fields reveal a divergence, which is physically meaningless at θ_{M}°=~81° (H_{K,2}=0 kOe) and ~75° (H_{K,2}=−1 kOe). The deviations are very large at R=1.75 (Figure 2f). For the system with H_{K,2}=0 kOe, divergence occurs even at ~37°, which is not located in the region of physical insignificance (indicated by vertical lines). A similar behavior is observed for the system with H_{K,2}=−1 kOe, where the divergence occurs at ~32°. These divergences are attributed to the misallocation of the H_{ext} value, at which B_{0}^{2}−A_{0}^{2}=0 (Figure 1c and d). The occurrence of the additional divergences significantly limits the reliability of the conventional analytical method for both systems, that is, with H_{K,2}=0 kOe and H_{K,2}=−1 kOe, as shown in Figure 2f.
Refined analysis
The main reason behind the unreliable results obtained from the conventional analysis is the determinant, which poorly describes the behavior of the first harmonic Hall voltage. For an accurate evaluation of the determinant, the refined analysis begins by determining the relation between θ_{M}° and H_{ext}, which can be obtained using the equation θ_{M}°=cos^{−1}(V^{1ω}/V_{AHE}) or the total energy equation^{34} (Equation (1)). The results for V^{1ω} as a function of H_{ext} (Figure 1a and b) can be applied to obtain the relation. Figure 3a and b shows the results for the determinant B_{1}^{2}−A_{1}^{2} as a function of the H_{ext} obtained from the refined analysis (Equations (23) and (24)) for the systems with H_{K,2}=0 kOe and H_{K,2}=−1 kOe, respectively. The solid lines in Figure 3a and b indicate the results for B_{1}^{2}−A_{1}^{2}, as calculated using H_{K,2}=0 and H_{K,2}=−1 kOe from the macrospin simulation and the refined analysis, respectively. Using the relation between θ_{M}° and H_{ext} in the refined analysis enables the behavior of V^{1ω} to be duly reflected in the determinant. At R=1.75, the H_{ext} values for which B_{1}^{2}−A_{1}^{2}=0 are 3.6 and 3.8 kOe for the systems with H_{K,2}=0 and H_{K,2}=−1 kOe, respectively; these H_{ext} values are identical to those obtained at T_{x}=T_{y} (Figure 2d). To apply the behavior of V^{1ω} to the determinant, the determinant B_{1}^{2}−A_{1}^{2} needs to be calculated with a precise value of H_{K,2}, which was used in the macrospin simulation. To demonstrate the importance of the inclusion of H_{K,2}, the determinants were also calculated by disregarding H_{K,2} (even though the actual H_{K,2} value of the system is −1 kOe); these results are also shown in Figure 3b (dotted lines). The difference between the two cases of R=0.3 and 1.75 is very large, which indicates that H_{K,2} should be considered in the analysis. At R=1.75, for example, the H_{ext} value for which the determinant is 0 is misallocated from 3.8 to 3.2 kOe when H_{K,2} is disregarded; a new location that reveals a 0 value of the determinant emerges at H_{ext}=9.0 kOe.
Figure 4a and b shows the results for G_{x} and G_{y}, which correspond to T_{x} and T_{y} in the conventional analysis at R=0.3 and 1.75, respectively. The results for the systems with H_{K,2}=0 (solid lines) and H_{K,2}=−1 kOe (dashed lines) are shown. Note the H_{ext} values for which G_{x}=G_{y} are 3.6 and 3.8 kOe for the systems with H_{K,2}=0 kOe and H_{K,2}=−1 kOe, respectively. These H_{ext} values are identical to the H_{ext} values for which the determinant B_{1}^{2}−A_{1}^{2}=0 (Figure 3a and b). This finding contradicts the case of the conventional analysis, where the H_{ext} value for which the determinant is 0 differs substantially from the H_{ext} value for which T_{x}=T_{y} (Figures 1c and d, and 2d,). With the new set of results for the determinant and the G ratios, the task of calculating the SO effective fields is straightforward; these results as a function of θ_{M}° are shown in Figure 4c and d, for R=0.3 and 1.75, respectively. Two sets of results are shown: one set for the system with H_{K,2}=0 kOe (solid lines) and one set for the system with H_{K,2}=−1 kOe (dashed lines). As shown in Figure 4c and d, in both systems, the calculated values of ΔH_{DL} and ΔH_{FL} are consistent with the input values for the macrospin simulation (over the entire θ_{M}° range of 0°–~82°); this finding demonstrates the reliability of the refined analysis. At R=0.3, the agreement between the two systems is perfect to such an extent that the solid lines for the H_{K,2}=0 kOe system completely overlap with the dashed lines for the H_{K,2}=−1 kOe system over the entire θ_{M}° range. A similar behavior is observed at R=1.75, with the difference that small peaks are observed at ~43° and ~50° for the systems with H_{K,2}=0 kOe and H_{K,2}=−1 kOe, respectively, for which B_{1}^{2}−A_{1}^{2}=0.
In systems with both H_{K,1}^{eff} and H_{K,2}, the determinant B_{1}^{2}−A_{1}^{2} significantly differs if H_{K,2} is disregarded (Figure 3b). A similar difference is expected for the calculated values of ΔH_{DL} and ΔH_{FL} (using Equation (27)), which are also shown in Figure 4c and d (dotted lines). For R=0.3, the absolute values of ΔH_{DL} and ΔH_{FL} are underestimated in the θ_{M}° range of 0°–60° and overestimated in the range of 60°–~82°. This finding is reasonable based on the H_{K,2} term, which is proportional to sinθ_{M}°sin3θ_{M}° (Equation (23)). For R=1.75, the differences are significant, with two divergences at ~39° and ~80° due to the misallocated H_{ext} fields of 3.2 and 9.0 kOe, for which B_{1}^{2}−A_{1}^{2}=0 (Figure 3b). These results demonstrate that H_{K,2} should not be neglected in the analysis of the harmonic measurement results in systems with both H_{K,1}^{eff} and H_{K,2}.
Comparison of conventional and refined analyses over wide R range
Two typical R ratios of 0.3 and 1.75 have been considered. To test the refined analytical method over a wide R range, a systematic study was conducted by varying the R ratio from 0 to 2 in steps of 0.05 for the system with H_{K,2}=−1 kOe. Figure 5a and b displays contour plots that show the deviation (in %) from the input values of ΔH_{DL} (left panels) and the input values of ΔH_{FL} (right panels) as a function of θ_{M}° and R. The results calculated using the conventional analytical method are shown in Figure 5a, whereas the results calculated using the refined method are shown in Figure 5b. In the case of conventional solutions, the θ_{M}° range in which H_{ext}>H_{K}^{eff} has no physical significance is indicated in Figure 5a as inclined lines. In Figure 5a and b, the solid lines indicate a deviation of 0.8% and the white regions indicate a minimum deviation of 4%. As shown in Figure 5a, the conventional solutions are valid over very limited ranges of θ_{M}° and R. For example, the R range in which the deviations are <4% is 0.06–0.12 for ΔH_{DL} and 0.21–0.46 for ΔH_{FL} in the θ_{M}° range of 0°–52°. For R values higher than 1.1, the validity range is even more limited for both ΔH_{DL} and ΔH_{FL}; specifically, the θ_{M}° values for which the deviations are <4% are 4.5° at R=1.1 and 7.9° at R=2.0 for ΔH_{DL} and 4.5° at R=1.1 and 9.4° at R=2.0 for ΔH_{FL}. In the intermediate R range of 0.9–1.1, the deviations are always larger than 4%. The accuracy of the calculated results improves significantly with the use of the refined method, as shown in Figure 5b. With the z component of H_{ext} considered in the refined analysis, no region has physical insignificance. The predictions made in the refined analysis are highly accurate. For R<0.85, the deviations are <0.4% over the entire θ_{M}° range of 0°–82° for both ΔH_{DL} and ΔH_{FL}. Even for R>0.85, the deviations are <0.8% for both ΔH_{DL} and ΔH_{FL} over the entire region, except in the regions marked by solid lines, where the deviations are large due to the existence of divergences (zero determinants).
Analysis of experimental results
An additional test of the refined method was performed by analyzing the experimental results of the harmonic Hall voltage measurements for a stack with the following structure: Si substrate (wetoxidized)/Ta (5 nm)/Pt (5 nm)/Co (0.6 nm)/MgO (2 nm)/Ta (2 nm). Refer to Supplementary Figures S2 and S3 for Hall bar dimensions. The results were obtained at three different I_{0} values: 1.0; 1.5; and 2.0 mA. The magnetization direction was controlled by H_{ext}, which was swept from +90 to –90 kOe with two different directions: θ_{H}=85° and ϕ_{H}=0°; and θ_{H}=85° and ϕ_{H}=90°. The values of H_{K,1}^{eff} and H_{K,2}, which were extracted using the Generalized Sucksmith–Thompson method,^{35} were 33.1 and −8.1 kOe, respectively. The R ratio of the sample was measured to be 0.423. Figure 6a–d shows the results for ΔH_{DL} (a and c) and ΔH_{FL} (b and d) as functions of θ_{M}° at three different I_{0} values: 1.0 mA (black squares); 1.5 mA (red circles); and 2.0 mA (blue triangles). Both conventional (a and b) and refined analytical methods (c and d) were used to analyze the experimental results. The results from the conventional method show incorrect divergences at θ_{M}°=~60°, whereas the results from the refined method do not show this behavior over the entire θ_{M}° range. Note that both ΔH_{DL} and ΔH_{FL} depend on θ_{M}°. The values of ΔH_{DL} and ΔH_{FL} should be proportional to I_{0}, with zero values at I_{0}=0.^{15} The expectation is only satisfied for the results extracted using the refined method, as shown in Figure 6e and f, where the results for ΔH_{DL} (red circles) and ΔH_{FL} (black squares) obtained at a fixed θ_{M}° value of 55° are shown as functions of I_{0}. A large deviation from the linearity is particularly noted for the ΔH_{FL} results calculated using the conventional method. Refer to Supplementary Figures S4–S6for detailed results.
Discussion
The test of the conventional analytical method, which involves an analysis of the macrospin simulation results, indicates that its validity range is very limited in terms of θ_{M}° and R, which is primarily attributed to the singularities involved in Cramer’s rule at incorrect θ_{M}° values. This problem is overcome by the refined analytical method proposed in this study with detailed analytical equations, in which both the z component of H_{ext} and the secondorder PMA are considered. The SO effective fields that are extracted using the refined analytical method are consistent with the input SO effective fields that are used for the macrospin simulation over the entire θ_{M}° range and over a wide R range of zero to two. For R<0.85, deviations from the input SO effective fields are <0.4% over the entire θ_{M}° range of 0°–82° for both ΔH_{DL} and ΔH_{FL}. Even at R>0.85, the deviations are <0.8% for both ΔH_{DL} and ΔH_{FL} over the entire region, with the exception in some limited regions that exhibit singularities. The accuracy of the refined method is reconfirmed by an additional comparative study that involves analyzing the experimental results of harmonic Hall voltage measurements for a Pt/Co/MgO structure. An accurate analysis of the harmonic Hall voltage measurement results by the refined analytical method over very wide ranges of θ_{M}° and R will significantly contribute to the identification of a dominant mechanism of the SOT and the development of highly efficient SOT devices.
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Acknowledgements
This research was supported by the Creative Materials Discovery Program via the National Research Foundation of Korea (No 2015M3D1A1070465).
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