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On thermodynamic consistency of a Scharfetter-Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement
Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter-Gummel schemes.
Numerical methods for drift-diffusion models
The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume disretization scheme and recent efforts to generalize this approach to general statistical distribution functions.