Theory of Intrinsic Spin Torque Due to Interface Spin-Orbit Coupling

The effect of intrinsic spin torque due to spin-orbit coupling (SOC) at the interface between thin ferromagnetic film and non-magnetic metal has attracted significant fundamental and applied research interest [1]. We report quantum theory of SOC driven spin torque (SOT) within the Rashba model of SOC and two-band tight binding (TB) Hamiltonian including s-d exchange interactions ($J$). We employ the non-equilibrium Green Function formalism and find that SOT to the first order in SOC has symmetry consistent with the earlier quasi-classical diffusive theory [2]. An obvious benefit of the proposed approach is the expression for the SOT given in terms of TB parameters which enables a physically transparent analysis of the dependencies of SOT on material specific parameters such as Rashba SOC constant, hopping integral, Fermi level and $J$. On the basis of analytical and numerical results we discuss trends in strength of SOT and its correlation with the Spin Hall conductivity.\\[4pt] [1] I. M. Miron \textit{et al}., Nature \textbf{476}, 189 (2011).\\[0pt] [2] A. Manchon and S. Zhang, Phys. Rev. B \textbf{78}, 212405 (2008).