Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi–Dirac statistic functions

If in the classic van Roosbroeck system (Bell Syst Tech J 29:560–607, 1950) the statistic function is modified, the equations can be derived by a variational formulation or just using a generalized Einstein relation. In both cases a dissipative generalization of the Scharfetter–Gummel scheme (IEEE Trans Electr Dev 16, 64–77, 1969), understood as a one-dimensional constant current approximation, is derived for strictly monotone coefficient functions in the elliptic operator $\nabla \cdot { {f}(v)} \nabla$ , $v$ chemical potential, while the hole density is defined by $p={\mathcal {F}}(v)\le e^v$. A closed form integration of the governing equation would simplify the practical use, but mean value theorem based results are sufficient to prove existence of bounded discrete steady state solutions on any boundary conforming Delaunay grid. These results hold for any piecewise, continuous, and monotone approximation of ${ {f}(v)}$ and ${\mathcal {F}}(v)$. Hence an implementation based on this discretization will inherit the same stability properties as the Boltzmann case based on the Scharfetter–Gummel scheme. Large chemical potentials and and related degeneracy effects in semiconductors can be approximated. A proven, stability focused blueprint for the discretization of a fairly general, steady state Fermi–Dirac like drift–diffusion setting for semiconductors using mainly new results to extend classic ideas is the main goal