# Uniform convergence to equilibrium for a family of drift-diffusion models with trap-assisted recombination and the limiting Shockley--Read--Hall model

## TL;DR

This paper proves uniform exponential convergence to equilibrium for a family of drift-diffusion-recombination models with trap-assisted recombination, including the Shockley-Read-Hall model, using entropy methods and establishing key inequalities.

## Contribution

It establishes uniform exponential convergence to equilibrium for the models and their limit, using entropy methods and proving existence and estimates of solutions.

## Key findings

- Uniform exponential convergence to equilibrium
- Entropy-entropy production inequality established
- Existence of global solutions confirmed

## Abstract

We consider a family of drift-diffusion-recombination systems, where the recombination of electrons and holes is facilitated by an intermediate energy-level for electrons in so-called trapped states. In particular, it has been proven in [GMS07] that the associated quasi-stationary state limit of an instantaneously fast trapped dynamics yields the famous Shockley--Read-Hall model for electron and hole recombination in semiconductor devices. The main result of this paper proves exponential convergence to equilibrium uniformly in the fast reaction limit for the drift-diffusion-recombination systems and the limiting Shockley-Read-Hall model. The proof applies the so-called entropy method and the key results is to establish an entropy-entropy production inequality uniformly in the fast reaction limit. Moreover, we prove existence of global solutions and show a-priori estimates, which are necessary to rigorously verify that solutions satisfy the entropy-entropy production law.