Modeling and analysis of second-harmonic magnetoresistance with arbitrary spin polarization and unconventional spin-orbit torques

The second-harmonic method is an important approach for studying the traditional spin-orbit torques (SOTs). In recent years, research on unconventional SOTs has provided a new approach for zero-field switching of the perpendicular magnetization and thus has received extensive attention. In this work, we theoretically extend the second-harmonic method to include the unconventional SOTs. For the four common magnetic configurations, we have derived the analytical relationship between the second-harmonic Hall (longitudinal) resistance and the (un)conventional SOT coefficients. We have comprehensively taken into account the effects of the anomalous Hall effect, the planar Hall effect, anisotropic magnetoresistance, the spin Hall magnetoresistance effect, the anomalous Nernst effect, the planar Nernst effect, and thermal fluctuations. The fitting results obtained from the numerical simulation verify the correctness of the analytical formula. This work provides a theoretical framework for measuring unconventional SOTs using the second-harmonic method, which is conducive to the development of new all-electrical spintronic devices.