NUMERICAL-SOLUTION OF 2-CARRIER HYDRODYNAMIC SEMICONDUCTOR-DEVICE EQUATIONS EMPLOYING A STABILIZED FINITE-ELEMENT METHOD

A space-time Galerkin/least-squares finite element method was presente d in [1] for numerical simulation of single-carrier hydrodynamic semic onductor device equations. The single-carrier hydrodynamic device equa tions were shown to resemble the ideal gas equations and Galerkin/leas t-squares finite element method, originally developed for computationa l fluid dynamics equations [16], was extended to solve semiconductor d evice applications. In this paper, the space-time Galerkin/least-squar es 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 entr opy variables are employed. A standard Galerkin finite element method is applied to the Poisson equation. Numerical simulations are performe d on the coupled Poisson and the two-carrier hydrodynamic equations em ploying a staggered approach. A mathematical analysis of the time-depe ndent multi-dimensional hydrodynamic model is performed to determine w ell-posed boundary conditions for electrical contacts. The number of b oundary conditions that need to be specified for the hydrodynamic equa tions at inflow and outflow boundaries of the device are derived. Exam ple boundary conditions that are based either on physical and/or mathe matical 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. Specifica lly, a Clausius-Duhem inequality, a basic stability requirement, is de rived for the hydrodynamic equations and the proposed numerical method automatically satisfies this stability requirement. Numerical simulat ions are performed on one- and two-dimensional two-carrier p-n diodes and the results demonstrate the effectiveness of the proposed numerica l method.