Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

A Numerical Method to Solve the Boltzmann Equation for a Spin Valve



J Xiao, A Zangwill, Mark D. Stiles


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.
European Physical Journal B


Boltzmann equation, drift-diffusion, magnetic multilayer, spin current, spin transport, spin vlave, spin-transfer torque


Xiao, J. , Zangwill, A. and Stiles, M. (2007), A Numerical Method to Solve the Boltzmann Equation for a Spin Valve, European Physical Journal B (Accessed June 15, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created September 30, 2007, Updated October 12, 2021