Modeling of transport through submicron semiconductor structures: A   direct solution to the Poisson-Boltzmann equations
  We report on a computational approach based on the self-consistent solution of the steady-state Boltzmann transport equation coupled with the Poisson equation for the study of inhomogeneous transport in deep submicron semiconductor structures. The nonlinear, coupled Poisson-Boltzmann system is solved numerically using finite difference and relaxation methods. We demonstrate our method by calculating the high-temperature transport characteristics of an inhomogeneously doped submicron GaAs structure where the large and inhomogeneous built-in fields produce an interesting fine structure in the high-energy tail of the electron velocity distribution, which in general is very far from a drifted-Maxwellian picture. 
D. Csontos and S. E. Ulloa