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Current-induced motion of narrow domain walls and dissipation in ferromagnetic metals

Spin transport equations in a nonhomogeneous ferromagnet are derived in the limit where the sd exchange coupling between the electrons in the conduction band and those in the d band is dominant. It is shown that spin diffusion in ferromagnets assumes a tensor form. The diagonal terms are renormalized with respect to that in normal metals and enhance the dissipation in the magnetic system while the off-diagonal terms renormalize the precessional frequency of the conduction electrons and enhance the nonadiabatic spin torque. To demonstrate what additional physics is included in the theory, we show that self-consistent solutions of the spin diffusion equations and the Landau-Lifshitz equations in the presence of a current lead to an increase in the terminal velocity of a domain wall which becomes strongly dependent on its width. We also provide a simplified equation that predicts damping due to the conduction electrons.