Angular dependence and symmetry of Rashba spin torque in ferromagnetic heterostructures

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.