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Within the s-d model description, we derive the current-driven spin torque ina ferromagnet, taking explicitly into account a spin-relaxing Caldeira-Leggettbath coupling to the s-electrons. We derive Bloch-Redfield equations of motionfor the s-electron spin dynamics, and formulate a systematic gradient expansionto obtain non-adiabatic (higher-order) corrections to the well-known adiabatic(first-order) spin torque. We provide simple analytical expressions for thesecond-order spin torque. The theory is applied to current-driven domain wallmotion. Second-order contributions imply a deformation of a transversetail-to-tail domain wall. The wall center still moves with a constant velocitythat now depends on the spin-polarized current in a non-trivial manner.

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