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

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 second-order 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.