Angular dependence and symmetry of Rashba spin torque in ferromagnetic heterostructures

TL;DR

This paper investigates the angular dependence and symmetry of Rashba spin torque in ferromagnetic heterostructures, revealing edge effects and anisotropic relaxation rates that influence the torque's behavior, aligning with recent experimental findings.

Contribution

It provides a detailed analysis of Rashba spin torque in the diffusive regime, highlighting edge effects and anisotropic relaxation, with a general expression for the torque depending on magnetization direction.

Findings

Nonvanishing torque at device edges even with aligned magnetization and Rashba field.

Anisotropic spin relaxation rates lead to a general torque expression dependent on magnetization angle.

Results are consistent with recent experimental observations.

Abstract

In a ferromagnetic heterostructure, the interplay between a Rashba spin-orbit coupling and an exchange field gives rise to a current-driven spin torque. In a realistic device setup, we investigate the Rashba spin torque in the diffusive regime and report two major findings: (i) a nonvanishing torque exists at the edges of the device even when the magnetization and effective Rashba field are aligned; (ii) anisotropic spin relaxation rates driven by the Rashba spin-orbit coupling assign the spin torque a general expression ${\bm T}=T^y_{\para}(\theta){\bm m}\times(\hat{\bm y}\times{\bm m})+T^y_{\bot}(\theta)\hat{\bm y}\times{\bm m}+T^z_{\para}(\theta){\bm m}\times(\hat{\bm z}\times{\bm m})+T^z_{\bot}(\theta)\hat{\bm z}\times{\bm m}$, where the coefficients $T_{\para,\bot}^{y,z}$ depend on the magnetization direction. Our results agree with recent experiments.