Spin transfer torque in continuous textures: semiclassical Boltzmann approach

TL;DR

This paper develops a semiclassical Boltzmann model to analyze spin transfer torque effects in continuous magnetic textures, deriving explicit electron distribution functions and torque components under various scattering conditions.

Contribution

It introduces a detailed microscopic model for spin transfer torque in textures, explicitly deriving distribution functions and torque terms considering different scattering mechanisms.

Findings

Derived explicit forms for electron distribution components in magnetic textures.

Identified how scattering types influence spin transfer torque.

Provided a framework for analyzing out-of-equilibrium spin dynamics.

Abstract

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 gradient and time derivative of n(r,t) the magnetization distribution function f(p,r,t) of itinerant electrons has the Ansatz form: f(p,r,t)=f_{parallel}(p)n(r,t)+ f_{1 r}(p) n ^ nabla_{r} n+f_{2 r}(p) nabla_{r} n+ f_{1 t}(p) n ^ partial_t n+f_{2 t}(p) partial_t n. Using then the Landau-Sillin equations of motion approach we derive explicit forms for the components f_{parallel}(p), f_{1 r}(p), f_{2 r}(p), f_{1 t}(p) and f_{2 t}(p) in "equilibrum" and in out of equilibrum situations for: (i) no scattering by impurities, (ii) spin conserving scattering and (iii) spin non-conserving scattering. The back action on the localized electron magnetization from the out of equilibrum part of the two components f_{1 r}, f_{2 r} constitutes the two spin transfer torque terms.