An HDG Method for Time-dependent Drift-Diffusion Model of Semiconductor Devices

TL;DR

This paper introduces an HDG finite element method for accurately solving the coupled nonlinear drift-diffusion and Poisson equations in semiconductor device modeling, with proven optimal error estimates and supporting numerical experiments.

Contribution

It develops a novel HDG approach with two different schemes for the nonlinear and linear parts, achieving optimal error estimates for the coupled system.

Findings

Optimal error estimates for the HDG method are established.

Numerical experiments confirm the theoretical convergence rates.

The method effectively handles the nonlinearity in the drift-diffusion system.

Abstract

We propose a hybridizable discontinuous Galerkin (HDG) finite element method to approximate the solution of the time dependent drift-diffusion problem. This system involves a nonlinear convection diffusion equation for the electron concentration $u$ coupled to a linear Poisson problem for the the electric potential $\phi$. The non-linearity in this system is the product of the $\nabla \phi$ with $u$. An improper choice of a numerical scheme can reduce the convergence rate. To obtain optimal HDG error estimates for $\phi$, $u$ and their gradients, we utilize two different HDG schemes to discretize the nonlinear convection diffusion equation and the Poisson equation. We prove optimal order error estimates for the semidiscrete problem. We also present numerical experiments to support our theoretical results.