Spin transfer torque in continuous textures:: Semiclassical Boltzmann approach

We consider a microscopic model of itinerant electrons coupled via ferromagnetic exchange to a local magnetization whose direction vector n(r,t) varies in space and time. We assume that to first order in the spatial gradients and time derivative of n(r,t), the magnetization distribution function f(p,r,t) of itinerant electrons with momentum p at position r and time t has the ansatz form f(p,r,t)=f(parallel to)(p)n(r,t)+f(1r)(p)nx del(r)n+f(2r)(p)del(r)n+f(1t)(p)nx partial derivative(t)n+f(2t)(p)partial derivative(t)n. Using the Landau-Sillin equations of motion approach (Zh. Eksp. Teor. Fiz. 33, 1227 (1957) [Sov. Phys. JETP 6, 945 (1958)]), we derive explicit forms for the components f(parallel to)(p), f(1r)(p), f(2r)(p), f(1t)(p), and f(2t)(p) in "equilibrium" and in out-of-equilibrium situations for (i) no scattering by impurities, (ii) spin-conserving scattering, and (iii) spin-nonconserving scattering. The back action on the localized electron magnetization from the out-of-equilibrium part of the two components f(1r) and f(2r) constitutes the two spin transfer torque terms.