Current-induced nonadiabatic spin torques and domain-wall motion with spin relaxation in a ferromagnetic metallic wire

Within the s-d model description, we derive the current-driven spin torque in a ferromagnet, taking explicitly into account a spin-relaxing Caldeira-Leggett bath coupling to the s-electrons. We derive Bloch-Redfield equations of motion for the s-electron spin dynamics and formulate a gradient expansion scheme to obtain nonadiabatic (higher-order) corrections to the well-known adiabatic (first-order) spin torque. We provide simple analytical expressions for the second-order spin torque. The theory is applied to current-driven domain-wall motion. Second-order contributions imply a deformation of a transverse tail-to-tail domain wall. The wall center still moves with a constant velocity that now depends on the spin-polarized current in a nontrivial manner.