Determination of the Spin-Orbit Torques in Ferromagnetic–Heavy-Metal Bilayers Using Harmonic Longitudinal Voltage Measurements
Bo Han, Bo Wang, Ze Yan, Tao Wang, Dezheng Yang, Xiaolong Fan, Ying Wang, Jiangwei Cao
Abstract
The exact measurement of spin-orbit torque (SOT) in magnetic heterostructures is the basis for a solid understanding of its physical mechanism. Although different SOT terms can be properly separated from harmonic Hall voltage analysis, owing to different symmetries and magnetic-field dependences, this method may be invalid for a ferromagnetic layer with a small Hall coefficient. Here, we develop a method to determine the SOT from harmonic longitudinal voltage measurements. Starting from a standard total-energy expression of a magnetic system, the equations to compute the corresponding SOT effective fields are derived and then applied to an in-plane magnetized $\mathrm{Pt}/\mathrm{Ni}\text{\ensuremath{-}}\mathrm{Fe}/\mathrm{Mg}\mathrm{O}$ system. The effective SOT fields and spin Hall angle obtained for the system are in accordance with previous results measured by other methods. Our results illustrate the applicability of harmonic longitudinal voltage measurements for studying current-induced torques in magnetic heterostructures.