Diffusive spin dynamics in ferromagnetic thin films with a Rashba interaction

In a ferromagnetic metal layer, the coupled charge and spin diffusion equations are obtained in the presence of both Rashba spin-orbit interaction and magnetism. The mis-alignment between the magnetization and the non-equilibrium spin density induced by the Rashba field gives rise to Rashba spin torque acting on the ferromagnetic order parameter. In a general form, we find that the Rashba torque consists of both in-plane and out-of-plane components, ie $\bm{T}=T_{\bot}\hat{\bm{y}}\times{\hat{\bm m}}+T_{\parallel}{\hat{\bm m}}\times({\hat{\bm y}}\times{\hat{\bm m}})$. Numerical simulations on a two dimensional nano-wire discuss the impact of diffusion on the Rashba torque, which reveals a large enhancement to the ratio $T_{\parallel}/T_{\bot}$ for thin wires. Our theory provides an explanation to the mechanism that drives the magnetization switching in a single ferromagnet as observed in the recent experiments.