Boltzmann equation for spin-dependent transport in magnetic inhomogeneous systems

From the Dyson equation of the nonequilibrium Green's functions, we derive a Boltzmann equation for the spin-dependent electronic transport, in which the electron distribution function is in general a 2X2 off-diagonal matrix in spin space. This equation is applicable to the transport in magnetic inhomogeneous systems, such as the magnetic layered structures and granular solids, with arbitrary magnetization configurations. For the parallel transport in the magnetic multilayers, it recovers the phenomenological Boltzmann equation proposed previously. It is shown that, as a consequence of continuity of the currents. the current density distribution depends only on the voltage drop between the ends of the sample. This allows us to determine the measurable physical quantities without calculating the actual internal electric field.