First-principles calculation of spin-orbit torque in a Co/Pt bilayer (vol 3, 011401, 2019)

The angular dependence of spin-orbit torque in a disordered Co/Pt bilayer is calculated using a first-principles nonequilibrium Green's function formalism with an explicit supercell averaging over Anderson disorder. In addition to the usual dampinglike and fieldlike terms, the odd torque contains a sizable planar Hall-like term (m . E)m x (z x m) whose contribution to current-induced damping is consistent with experimental observations. The dampinglike and planar Hall-like torquances depend weakly on disorder strength, while the fieldlike torquance declines with increasing disorder. The torques that contribute to damping are almost entirely due to spin-orbit coupling on the Pt atoms, but the fieldlike torque does not require it.