Abstract
The aim of the present paper is the mathematical study of a linear Boltzmann equation with different matrix collision operators, modelling the spin-polarized, semi-classical electron transport in non-homogeneous ferromagnetic structures. In the collision kernel, the scattering rate is generalized to a hermitian, positive-definite $2\times2$ matrix whose eigenvalues stand for the different scattering rates of, for example, spin-up and spin-down electrons in spintronic applications. We identify four possible structures of linear matrix collision operators that yield existence and uniqueness of a weak solution of the Boltzmann equation for a general Hamilton function. We are able to prove positive-(semi)definiteness of a solution for an operator that features an anti-symmetric structure of the gain respectively the loss term with respect to the occurring matrix products. Furthermore, in order to obtain matrix drift-diffusion equations, we perform the diffusion limit with one of the symmetric operators assuming parabolic spin bands with uniform band gap and in the case that the precession frequency of the spin distribution vector around the exchange field of the Hamiltonian scales with order $\epsilon^2$. Numerical simulations of the here obtained macroscopic model were carried out in non-magnetic/ferromagnetic multilayer structures and for a magnetic Bloch domain wall. The results show that our model can be used to improve the understanding of spin-polarized transport in spintronics applications.
Highlights
The coupling between the spin- and the charge degree of freedom of an electron system is a growing research topic in physics and mathematics
The aim of the present paper is the mathematical study of a linear Boltzmann equation with different matrix collision operators, modelling the spin-polarized, semi-classical electron transport in non-homogeneous ferromagnetic structures
The maximum principle was verified for the anti-symmetric collision operator Q21 in Proposition 5.6
Summary
The coupling between the spin- and the charge degree of freedom of an electron system is a growing research topic in physics and mathematics. El Hajj and Ben Abdallah [10] introduced a spin-coherent collision operator in the linear BGK approximation to obtain a matrix Boltzmann equation They performed rigorous diffusion limits in various scalings to derive a number of matrix drift-diffusion models. Our goal is to set up a matrix Boltzmann equation that incorporates spin-dependent scattering rates, or more precisely, that features a collision kernel with matrixvalued transition probabilities from momentum k to k Such a kinetic equation can be viewed as a generalization of the model in [10] to spin-dependent mean-free paths. The results show that our model provides a new means for studying spin-polarized transport in arbitrary magnetic structures (e.g. non-collinear multilayers or strongly varying magnetization)
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