Accurate extraction of anisotropic spin–orbit torques from harmonic measurements

Abstract

One of the most powerful ways to manipulate spins in nanometer-scale devices is by converting a charge current to a spin current via spin–orbit coupling. The resulting spin–orbit torques (SOTs) have been investigated and utilized extensively in the past decade. Quantitatively, however, SOTs may exhibit a non-trivial angular dependence, which is not well explored. Here, we develop a nested iterative analysis to determine the magnitude of SOTs from harmonic Hall measurements. This updated method largely improves the fit quality in the full magnetic field range and accurately retrieves even higher order, anisotropic spin–orbit torque coefficients. The numerical implementation of our algorithm is fast, robust, and designed for easy integration into existing analysis schemes. We verify our code using simulated data with and without anisotropic SOTs. Accurately quantifying higher order SOT terms can be especially useful for modeling non-uniform magnetic textures such as domain walls and skyrmions and current-induced magnetization switching characteristics.