Arthur Raefsky

A finite element formulation for the hydrodynamic semiconductor device equations

Abstract A new formulation employing the Galerkin/least-squares finite element method is presented for the simulation of the hydrodynamic model of semiconductor devices. Numerical simulations are performed on the coupled Poisson and hydrodynamic equations for one carrier devices. The hydrodynamic equations for a single carrier, i.e. for the electrons or holes, resemble the compressible Navier-Stokes equations with the addition of highly nonlinear source terms and without the viscous terms. The governing equations are nondimensionalized to improve the conditioning on the resulting system of equations and the efficiency of the numerical algorithms. Furthermore, to establish the stability of the discrete solution, the system of hydrodynamic equations is symmetrized by considering generalized entropy functions. A staggered solution strategy is employed to treat the coupled hydrodynamic and Poisson equations. Numerical results are presented for one-dimensional and two-dimensional one-carrier n + - n - n + devices. The presence of velocity overshoot has been observed and it is recognized that the heat flux term plays an important role in the simulation of semiconductor devices employing the hydrodynamic model.

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.

An implementation of a generalized sparse/profile finite element solution method

Abstract This paper describes an implementation of a generalized profile/sparse solution method for finite element analysis. The solution procedure can be used as a profile solver as well as a sparse matrix solver, depending on the physical model and the ordering scheme used to number the system of equations. The data structure employed in the implementation is discussed in detail. The procedures for element stiffness assembly, symbolic and numeric factorization of the global stiffness matrix and the calculation of the displacement solution vector are described. Experimental results are presented to demonstrate the effectiveness of this profile/sparse solution scheme.

Parallelizing a PDE solver: experiences with PISCES-MP

The paper presents a methodology for adapting dusty deck PDE solvers for parallel execution. Our approach minimizes changes to existing code and data structures, thereby preserving the value captured within dusty decks. This scheme uses the single program multiple data programming paradigm on message passing distributed memory architectures. To demonstrate the viability of our methodology the commercially available, dusty deck semiconductor device modeling program, PISCES, has been adapted for parallel execution. Simulating realistic complex device structures, we have achieved excellent performance gains over high performance serial workstations. Also, the scalability of the parallel simulator allows the simulation of structures too large for our existing serial computers.<<ETX>>

FIESTA-HD: A parallel finite element program for hydrodynamic device simulation

Publisher Summary This chapter reviews the convective hydrodynamic transport model (HD) for semiconductor device simulation by employing parallel and stabilized finite element methods. The HD model is shown to resemble the compressible Euler and Navier-Stokes equations and Galerkin/least-squares finite element methods, originally developed for compuational fluid dynamics (CFD) equations. The complexity of the HD model demands enormous computational time. The chapter presents the development of a finite element formulation for the HD transport model. A parallel finite element device simulation program, FIESTA-HD, has been developed and run on distributed memory parallel computers including Intels i860, Touchstone Delta and the IBMs SP-1 systems. A SPMD programming model is used in the parallel implementation of the device simulator, FIESTA-HD. The chapter demonstrates the portability of FIESTA-HD on distributed memory parallel computers. It presents the numerical results and demonstrates the robustness and accuracy of the numerical schemes. Taking advantage of the advances in parallel computers with stable numerical schemes, simulations are performed with more complex and realistic device models.