Existence analysis for a simplified transient energy-transport model for semiconductors

TL;DR

This paper analyzes a simplified transient energy-transport model for semiconductors, proving the existence and uniqueness of solutions, and illustrating the model's behavior through numerical simulations.

Contribution

It provides a rigorous mathematical analysis of a simplified semiconductor energy-transport model, including existence, uniqueness, and numerical validation.

Findings

Global-in-time bounded weak solutions are proven to exist.

Uniqueness of solutions is established under certain regularity assumptions.

Numerical simulations demonstrate the model's behavior in a ballistic diode.

Abstract

A simplified transient energy-transport system for semiconductors subject to mixed Dirichlet-Neumann boundary conditions is analyzed. The model is formally derived from the non-isothermal hydrodynamic equations in a particular vanishing momentum relaxation limit. It consists of a drift-diffusion-type equation for the electron density, involving temperature gradients, a nonlinear heat equation for the electron temperature, and the Poisson equation for the electric potential. The global-in-time existence of bounded weak solutions is proved. The proof is based on the Stampacchia truncation method and a careful use of the temperature equation. Under some regularity assumptions on the gradients of the variables, the uniqueness of solutions is shown. Finally, numerical simulations for a ballistic diode in one space dimension illustrate the behavior of the solutions.