Intrinsic Spin-Orbit Torque In A Single-Domain Nanomagnet

We present theoretical studies of the intrinsic spin-orbit torque (SOT) in a single-domain ferromagnetic layer with Rashba spin-orbit coupling (SOC) using the nonequilibrium Green's function formalism for a tight-binding Hamiltonian. We find that, in the case of a small electric field, the intrinsic SOT to first order in SOC has only the field-like torque symmetry and can be interpreted as the longitudinal spin current induced by the charge current and Rashba field. We analyze the results in terms of the material-related parameters of the electronic structure, such as the band filling, bandwidth, exchange splitting, and the Rashba SOC strength. On the basis of these numerical and analytical results, we discuss the magnitude and sign of SOT. Our results suggest that the different sign of SOT in identical ferromagnets with different supporting layers, e.g., Co/Pt and Co/Ta, can be attributed to electrostatic doping of the ferromagnetic layer by the support.