Accurate analysis for harmonic Hall voltage measurement for spin-orbit torques

An accurate method is developed to extract the spin-orbit effective fields through analysis of the results of harmonic Hall voltage measurements by deriving detailed analytical equations, in which both the z-component of the applied magnetic field and the second-order perpendicular magnetic anisotropy are taken into account. The method is tested by analyzing the results of a macrospin simulation. The spin-orbit effective fields extracted from the analysis are found to be in excellent agreement with the input spin-orbit effective fields used for the macrospin simulation over the entire range of the polar magnetization angle and 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 more clearly via a systematic study involving a comparison of its results with those of the conventional analytical method.


IINTRODUCTION
It has recently been found that in-plane currents in a nonmagnetic (NM)/ferromagnetic (FM) bilayer nanostructure can generate a torque due to the spin-orbit (SO) coupling, known as the spin-orbit torque (SOT), which is sufficient enough to reverse the magnetization in the FM layer. 1 Numerous studies have been conducted to identify the principal mechanism of the SOT as being either the spin Hall effect (SHE) in the NM layer [2][3][4] or the interfacial spinorbit coupling (ISOC)-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 in-plane currents flow along the x-axis, spin currents polarized along the y-axis are generated in the system on the basis of the SHE induced by a bulk SO coupling in the NM layer. The spin currents are injected into the adjacent FM layer, thus causing transfer of a torque to the magnetization of the FM layer. The SHE-induced SOT generates a strong damping-like torque (T DL m  m  y) but a weak field-like torque (T FL m  y). 13,14 Theoretically, the strength of the SHE-induced SOT is known to be independent of the magnetization direction of the FM layer. In the case of the ISOC-induced SOT, spins polarized along the y-axis accumulate owing 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 SHE-induced SOT, that of the ISOC-induced 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 ISOC qualitatively induce the same torque on the FM layer. In order to identify the dominant mechanism of the SOT, a quantitative analysis of the values of T DL and T FL over a wide range of magnetization angles is required. [19][20][21] The harmonic Hall voltage measurement method is one of the useful approaches for quantifying the effective fields of T DL and T FL originating from the SOT. 22 This method is particularly suited for identifying the angular dependence of the SOT acting on the FM layer with perpendicular magnetization. 19,20 Several corrections are required to be made for an accurate analysis of measured results, including for the following: the planar Hall effect (PHE), 20,23 the out-of-plane component of the external magnetic field, 20 and the anomalous Nernst effect (ANE). 20,24 In the harmonic Hall voltage measurement, the second harmonic voltage (V 2ω ) consists of two major components: anomalous and planar Hall voltages (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 resulting 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 based on Cramer's rule in order to separate T FL and T DL . 23 This analytical solution has been successful only for a system with V PHE < V AHE . For a system with V PHE > V AHE , such as the W/CoFeB/MgO trilayer structure, 25  This problem can be overcome by making some necessary corrections in the analysis of the measurement results, including that for the out-of-plane component of H ext . Since 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 Under this condition, the z-component of H ext has a nonzero value, although it has been neglected thus far just to simplify the analysis. This assumption is reasonable in the low-H 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 no longer valid in the high-H ext range, where the magnetization direction is considerably deviated from the z-axis with a resultant reduction in the PMA field, as a result of which it loses its dominance over the zcomponent of H ext . Several attempts have been made in the past to include the zcomponent of H ext , which is obtained by solving equilibrium torque equations repetitively until a desired convergence is achieved (the recursive method). 19,20 However, this method is quite complicated; furthermore, it has not been validated for systems with V PHE > V AHE . Unwanted voltages, which originate from thermoelectric effects such as the ANE, should also be eliminated from the harmonic signals. Although several methods have been proposed for this purpose, 20,24 erasing all the artificial signals remains difficult. Another important issue that needs to be addressed is the inclusion of the second-order PMAwhich has not been taken into account thus far-in the analysis of harmonic Hall voltage measurement results. The inclusion of the second-order PMA is considered to be of great importance, because many PMA materials exhibit the second-order PMA, with its strength being comparable to that of the first-order PMA in many cases. 26,27 In the present study, two corrections-one for the z-component of H ext and the other for the second-order PMA-are considered in the analysis of the harmonic Hall voltage measurement results. All the related analytical equations are described. Both the conventional and the refined analytical methods are used to analyze the results of a macrospin simulation, which plays a role of mimicking 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 used for the macrospin simulation with those calculated by the analytical methods. To test the refined analytical method critically, systems are examined over a wide ratio R, which is defined as V PHE /V AHE . A similar comparative study was also performed that involves analyzing the experimental results of harmonic Hall voltage measurements for a Pt/Co/MgO structure.

A. Analytical solutions for conventional approach
When an in-plane 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 that 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 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: sin sin cos sin sin cos sin , Here, the following relations exist: The second harmonic Hall voltages, as given in Eqs. (15) and (16), are composed of two terms containing ∆H DL and ∆H FL . When the R ratio is negligibly small, the values of ∆H DL and ∆H FL can be obtained using the following T ratios: Note that at R = 0, the values of T x and T y are identical to those of ∆H DL and −∆H FL , respectively. 8  1 .
In Eq. (20), ∆H DL and ∆H FL can be calculated if the determinant B 0 2 − A 0 2 is not 0.

B. Analytical solutions for refined approach
The assumption of H ext cosθ  (5) and (6)] into Eqs. (12) and (13) to obtain the expression for V 2ω : (11)], the G ratios, corresponding to the T ratios used in the conventional approach, can be defined as follows: Similarly to the conventional analytical equations, the refined equations also need to be solved using Cramer's rule [as given in Eq. (20)]. 1 .

A. Conventional analysis
The conventional analytical method is used to analyze the results of the macrospin simulation.   This is because, between 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. Indeed, Eqs. kOe, as can be seen clearly from Fig. 2(f).

B. Refined analysis
The should not be neglected in the analysis of the harmonic measurement results.

C. Comparison of conventional and refined analyses over wide R range
Two typical R ratios of 0.3 and 1.75 have been considered thus far. In order to test the refined analytical method over a wide R range, a more 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. Figures 5(a)