Spinor Boltzmann equation approach to domain-wall motion driven by spin-polarized current

Abstract

Based on the nonequilibrium Green's function formalism, the spinor Boltzmann equation beyond gradient approximation is derived in a ferromagnetic metal with a single domain wall (DW). We further obtain the charge continuity equation and the spin diffusion equation by integrating over the momentum. By using the spin diffusion equation, we get a generalized spin transfer torque (STT), in which the usual STT is extended to the case beyond the gradient approximation and with inhomogeneous current. We also calculate numerically the physical observables such as charge density $n(x)$, spin accumulation $\mathbit{m}(x)$, current density $j(x)$, spin current density ${\mathbit{j}}_{s}(x)$, etc., by the use of the spinor distribution function. Along with the Landau-Lifshitz-Gilbert-Slonczewski equation that contains the above generalized STT, we can study the motion domain wall, and the critical electric field at an initial velocity of the DW is obtained in terms of the linear stability analysis method.