A numerical method to solve the Boltzmann equation for a spin valve

TL;DR

This paper introduces a numerical algorithm for solving the Boltzmann equation in magnetic multilayer heterostructures, enabling direct calculation of spin-related quantities and validation against existing circuit theory.

Contribution

A new numerical method based on scattering matrix formalism for solving the Boltzmann equation in non-collinear magnetic multilayers.

Findings

Accurate calculation of spin-transfer torque in spin valves.

Results agree with established circuit theory.

Method applicable to various geometries.

Abstract

We present a numerical algorithm to solve the Boltzmann equation for the electron distribution function in magnetic multilayer heterostructures with non-collinear magnetizations. The solution is based on a scattering matrix formalism for layers that are translationally invariant in plane so that properties only vary perpendicular to the planes. Physical quantities like spin density, spin current, and spin-transfer torque are calculated directly from the distribution function. We illustrate our solution method with a systematic study of the spin-transfer torque in a spin valve as a function of its geometry. The results agree with a hybrid circuit theory developed by Slonczewski for geometries typical of those measured experimentally.