Semiclassical hydrodynamics of a quantum Kane model for semiconductors

In this paper we derive a semiclassical hydrodynamic system for electron densities and currents in the two energy bands of a semiconductor. We use the semiclassical Wigner equation with a k.p Hamiltonian and a BGK dissipative term to construct the first two moment equations. The closure of the moment system is obtained using the Maximum Entropy Principle, by minimizing a Gibbs free-energy functional under suitable constraints. We prove that the constraint equations can be uniquely solved, i.e. that the local equilibrium state can be parametrized by the density and velocity field. Some BGK-like models are proposed to mimic the quantum interband migration.