Spin transport for spin diffusion lengths comparable to mean free paths

In perpendicular-transport structures with magnetic and nonmagnetic layers, there is a resistance associated with changes in the value of current polarization throughout the structure. Starting from a Boltzmann equation, Valet and Fert [Phys. Rev. B 48, 7099 (1993)] derived macroscopic transport equations to describe this effect. Their derivation is formally justified in the limit that the spin-diffusion length of each material is long compared to the mean free path of that same material, but appears to agree with experiment even for spin diffusion lengths comparable to the appropriate mean free paths. Here, numerical studies of the Boltzmann equation verify that their approach is accurate in this limit. In addition, we have examined the case of anisotropic spin-flip scattering. For reasonable parameters, there is only a small difference in the resistance from the isotropic case.