Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

Instationary drift-diffusion problems with Gauss–Fermi statistics and field-dependent mobility for organic semiconductor devices

Pages: 33 – 59



Annegret Glitzky (Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany)

Matthias Liero (Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany)


This paper deals with the analysis of an instationary drift-diffusion model for organic semiconductor devices including Gauss–Fermi statistics and application-specific mobility functions. The charge transport in organic materials is realized by hopping of carriers between adjacent energetic sites and is described by complicated mobility laws with a strong nonlinear dependence on temperature, carrier densities and the electric field strength.

To prove the existence of global weak solutions, we consider a problem with (for small densities) regularized state equations on any arbitrarily chosen finite time interval. We ensure its solvability by time discretization and passage to the time-continuous limit. Positive lower a priori estimates for the densities of its solutions that are independent of the regularization level ensure the existence of solutions to the original problem. Furthermore, we derive for these solutions global positive lower and upper bounds strictly below the density of transport states for the densities. The estimates rely on Moser iteration techniques.


drift-diffusion system, nonlinear parabolic system, organic semiconductor, charge transport, existence of weak solutions, Gauss–Fermi statistics

2010 Mathematics Subject Classification

35B45, 35K20, 35K55, 78A35

This work was supported by the Einstein Center for Mathematics (ECMath) via Matheon project D-SE18.

Received 19 July 2018

Accepted 5 October 2018

Published 30 May 2019