Eur. Phys. J. B 59, 415-427 (2007)
https://doi.org/10.1140/epjb/e2007-00004-0

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

¹ School of Physics, Georgia Institute of Technology, Atlanta, GA, 30332-0430, USA
² Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8412, USA

Corresponding author: ^a mark.stiles@nist.gov

Received: 7 September 2006
Revised: 3 November 2006
Published online: 10 January 2007

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.