  We consider in this paper the mathematical and numerical modelling of
reflective boundary conditions (BC) associated to Boltzmann - Poisson systems,
including diffusive reflection in addition to specularity, in the context of
electron transport in semiconductor device modelling at nano scales, and their
implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the
physical boundaries of the device and develop a numerical approximation to
model an insulating boundary condition, or equivalently, a pointwise zero flux
mathematical condition for the electron transport equation. Such condition
balances the incident and reflective momentum flux at the microscopic level,
pointwise at the boundary, in the case of a more general mixed reflection with
momentum dependant specularity probability $p(\vec{k})$. We compare the
computational prediction of physical observables given by the numerical
implementation of these different reflection conditions in our DG scheme for BP
models, and observe that the diffusive condition influences the kinetic moments
over the whole domain in position space.
