Comments on the formula to extract current-induced torques from the harmonic Hall voltage measurements

TL;DR

The paper addresses a discrepancy in extracting current-induced spin-orbit torques from rotating-field harmonic Hall measurements by re-deriving the rotating-field response within a Hayashi-like model with $H_A=0$, clarifying torque sign conventions and anisotropy definitions, and including thermo-electric effects. It provides the correct form of the harmonic Hall resistance $R_{2\omega}$ and outlines how damping-like and field-like torques are inferred from observables, while avoiding stacking-order ambiguities. The analysis emphasizes the role of sign conventions for $\beta$ and the effective anisotropy field, and it explains a previously neglected $1/2$ factor in the $R_{2\omega}$ prefactor, which reconciles prior inconsistencies. Overall, the work improves accuracy and consistency of torque extraction from harmonic Hall data across materials and experimental conditions by delivering a complete, convention-aware framework for $R_{2\omega}$ and related efficiencies.

Abstract

We examine the formulas commonly used to estimate current-induced spin-orbit torques from harmonic Hall voltage measurements. In particular, we focus on the factor of two discrepancy among expressions employed to fit harmonic Hall signals measured under an in-plane rotating magnetic field. By explicitly deriving the relevant relations, we clarify the origin of this discrepancy and present the correct form of the fitting formula. We further discuss the determination of the sign of the field-like torque from harmonic Hall voltage measurements, which depends on the assumed form of the current-induced torques.