  The large-time asymptotics of the density matrix solving a
drift-diffusion-Poisson model for the spin-polarized electron transport in
semiconductors is proved. The equations are analyzed in a bounded domain with
initial and Dirichlet boundary conditions. If the relaxation time is
sufficiently small and the boundary data is close to the equilibrium state, the
density matrix converges exponentially fast to the spinless near-equilibrium
steady state. The proof is based on a reformulation of the matrix-valued
cross-diffusion equations using spin-up and spin-down densities as well as the
perpendicular component of the spin-vector density, which removes the
cross-diffusion terms. Key elements of the proof are time-uniform positive
lower and upper bounds for the spin-up and spin-down densities, derived from
the De Giorgi-Moser iteration method, and estimates of the relative free energy
for the spin-up and spin-down densities.
