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

Abstract

If 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 citeSch_Gu, understood as a one-dimensional constant current approximation, is derived for strictly monotone coefficient functions in the elliptic operator $nabla cdot bal ff(v) nabla $, $v$ chemical potential, while the hole density is defined by $p=cal F(v)le e^v.$ 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 $bal ff(v)$ and $cal F(v)$.