Numerical solution of two-carrier hydrodynamic semiconductor device equations employing a stabilized finite element method

Abstract

A space-time Galerkin/least-squares finite element method was presented in [1] for numerical simulation of single-carrier hydrodynamic semiconductor device equations. The single-carrier hydrodynamic device equations were shown to resemble the ideal gas equations and Galerkin/least-squares finite element method, originally developed for computational fluid dynamics equations [16], was extended to solve semiconductor device applications. In this paper, the space-time Galerkin/least-squares finite element method is further extended and generalized to solve two-carrier hydrodynamic device equations. The proposed formulation is based on a time-discontinuous Galerkin method, in which physical entropy variables are employed. A standard Galerkin finite element method is applied to the Poisson equation. Numerical simulations are performed on the coupled Poisson and the two-carrier hydrodynamic equations employing a staggered approach. A mathematical analysis of the time-dependent multi-dimensional hydrodynamic model is performed to determine well-posed boundary conditions for electrical contacts. The number of boundary conditions that need to be specified for the hydrodynamic equations at inflow and outflow boundaries of the device are derived. Example boundary conditions that are based either on physical and/or mathematical basis are presented. Stability of the numerical algorithms is addressed. The space-time Galerkin/least-squares finite element method and the standard Galerkin finite element method for the hydrodynamic and Poisson equations, respectively, are shown to be stable. Specifically, a Clausius-Duhem inequality, a basic stability requirement, is derived for the hydrodynamic equations and the proposed numerical method automatically satisfies this stability requirement. Numerical simulations are performed on one- and two-dimensional two-carrier p-n diodes and the results demonstrate the effectiveness of the proposed numerical method.