The contribution of spin torque to the spin Hall coefficient and spin motive force in a spin–orbit coupling system

We derive rigorously the relativistic angular momentum conservation equation by means of quantum electrodynamics. The novel nonrelativistic spin current and torque in the spin–orbit coupling system, up to the order of 1/c4, are exactly investigated using Foldy–Wouthuysen transformations. We find a perfect spin Hall coefficient including the contribution of the spin torque dipole. A novel spin motive force, analogous to the Lorentz force, is also obtained to help us understand the spin Hall effect.