A hierarchy of diffusive higher-order moment equations for semiconductors

A hierarchy of diffusive partial differential equations is derived by a moment method and a Chapman-Enskog expansion from the semiconductor Boltzmann equation assuming dominant collisions. The moment equations are closed by employing the entropy maximization principle of Levermore. The new hierarchy contains the well-known drift-diffusion model, the energy-transport equations, and the six-moments model of Grasser et al. It is shown that the diffusive models are of parabolic type. Two different formulations of the models are derived: a drift-diffusion formulation, allowing for a numerical decoupling, and a symmetric formulation in generalized dual-entropy variables, inspired by nonequilibrium thermodynamics. An entropy inequality (or H-theorem) follows from the latter formulation.