The spinor Boltzmann equation beyond gradient approximation

Abstract

In this paper, we generalize the spinor Boltzmann equation in order to describe the spin-polarized transport in magnetic multilayers beyond gradient approximation, because the usual gradient approximation, hence the quantum Boltzmann equation based on it, is only suitable for the systems whose potentials vary slowly with respect to time and position. In our derivation, we do not adopt the gradient approximation to simplify those convolutions concerning with Fourier transformations, we just deal with them by the way given by Wigner [E. Wigner, Phys. Rev. 40, 749 (1932)], which assures the final quantum Boltzmann equation can be applied to the system with rapid varying potential. Then we illustrate it by the spin-polarized transport in magnetic multilayers in which the potential have sudden jumps at the interfaces, and apply the generalized spinor Boltzmann equation to the entire magnetic multilayers, it avoids to connect the distribution functions of different layers by matching conditions as usual. We also study the quantum corrections for the distribution function, the equations satisfied by the zero-order distribution function and the first order quantum correction are exhibited.