Analytical expression for the harmonic Hall voltages in evaluating spin orbit torques

TL;DR

This paper derives an analytical expression for harmonic Hall voltages to improve the evaluation of spin-orbit torques in magnetic heterostructures, aiding the development of magnetization control techniques.

Contribution

It provides a new analytical formula for harmonic Hall voltages considering anomalous and planar Hall effects, applicable to different magnetized systems.

Findings

Derived a comprehensive analytical expression for harmonic Hall voltages.

Validated the formula on out-of-plane and in-plane magnetized systems.

Facilitates more accurate extraction of spin-orbit torque parameters.

Abstract

Solid understanding of current induced torques is key to the development of current and voltage controlled magnetization dynamics in ultrathin magnetic heterostructures. A versatile technique is needed to evaluate such torques in various systems. Here we examine the adiabatic (low frequency) harmonic Hall voltage measurement that has been recently developed to study current induced effective field that originate from the spin orbit effects. We analytically derive a form that can be used to evaluate the harmonic Hall voltages and extract relevant parameters in two representative systems, i.e. out of plane and in-plane magnetized systems. Contributions from the anomalous Hall and planar Hall effects are considered.