On modifications of the Scharfetter-Gummel scheme for drift-diffusion equations with Fermi-like statistical distribution functions

Driven by applications in fields like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with general statistical distribution functions. It is important to keep the well known qualitative properties of the Scharfetter-Gummel finite volume scheme, like positivity of solutions, dissipativity and consistency with thermodynamic equilibrium. A proper generalization to general statistical distribution functions is a topic of current research. The paper presents different state-of-the-art approaches to solve this problem. Their issues and advantages are discussed, and their practical performance is evaluated for real device structures.