Quantum Semiconductor Models

Abstract

We give an overview of analytic investigations of quantum semiconductor models, where we focus our attention on two classes of models: quantum drift diffusion models, and quantum hydrodynamic models. The key feature of those models is a quantum interaction term which introduces a perturbation term with higher-order derivatives into a system which otherwise might be seen as a fluid dynamic system. After a discussion of the modeling, we present the quantum drift diffusion model in detail, discuss various versions of this model, list typical questions and the tools how to answer them, and we give an account of the state-of-the-art of concerning this model. Then we discuss the quantum hydrodynamic model, which figures as an application of the theory of mixed-order parameter-elliptic systems in the sense of Douglis, Nirenberg, and Volevich. For various versions of this model, we give a unified proof of the local existence of classical solutions. Furthermore, we present new results on the existence as well as the exponential stability of steady states, with explicit description of the decay rate.