Discretization scheme for drift-diffusion equations with strong diffusion enhancement

Abstract

Inspired by organic semiconductor models based on hopping transport introducing Gauss-Fermi integrals a nonlinear generalization of the classical Scharfetter–Gummel scheme is derived for the distribution function $$\mathcal F_\gamma (\eta ) = 1/(\exp (-\eta )+\gamma )$$ . This function provides an approximation of the Fermi-Dirac integrals of different order and restricted argument ranges. The scheme requires the solution of a nonlinear equation per edge and continuity equation to calculate the edge currents. In the current formula the density-dependent diffusion enhancement factor, resulting from the generalized Einstein relation, shows up as a weighting factor. Additionally the current modifies the argument of the Bernoulli functions.