Anand L. Pardhanani

Advanced numerical methods and software approaches for semiconductor device simulation

In this article the authors concisely present several modern strategies that are applicable to drift-dominated carrier transport in higher-order deterministic models such as the drift-diffusion, hydrodynamic, and quantum hydrodynamic systems. The approaches include extensions of upwind and artificial dissipation schemes, generalization of the traditional Scharfetter-Gummel approach, Petrov-Galerkin and streamline-upwind Petrov Galerkin (SUPG), entropy variables, transformations, least-squares mixed methods and other stabilized Galerkin schemes such as Galerkin least squares and discontinuous Galerkin schemes. The treatment is representative rather than an exhaustive review and several schemes are mentioned only briefly with appropriate reference to the literature. Some of the methods have been applied to the semiconductor device problem while others are still in the early stages of development for this class of applications. They have included numerical examples from the recent research tests with some of the methods. A second aspect of the work deals with algorithms that employ unstructured grids in conjunction with adaptive refinement strategies. The full benefits of such approaches have not yet been developed in this application area and they emphasize the need for further work on analysis, data structures and software to support adaptivity. Finally, they briefly consider some aspects of software frameworks. These include dial-an-operator approaches such as that used in the industrial simulator PROPHET, and object-oriented software support such as those in the SANDIA National Laboratory framework SIERRA.

Mapped discretization strategies for curvilinear adaptively redistributed grids in semiconductor device modeling

Abstract We develop numerical solution schemes for semiconductor device models based on mapped discretization strategies and curvilinear, nonuniform grids. Such grids typically arise in adaptive redistribution schemes, and they offer several advantages when gridding irregular domain shapes or resolving complex solution profiles. We consider the hydrodynamic class of equations as a representative model for carrier transport. A mathematical transformation of variables is used to map the device equations from the physical coordinate system to a reference system in which the numerical discretization is performed. We develop a Scharfetter–Gummel type of discretization, formulated in the mapped reference domain, for the current density and energy flux terms in the transport model. The solution of the mapped discrete system is carried out in a fully-coupled, implicit form, and nonsymmetric gradient-type iterative strategies are used for solving the algebraic systems. Numerical results demonstrating the performance of the scheme with and without mesh adaptation are presented for test problems.

A mapped Scharfetter-Gummel formulation for the efficient simulation of semiconductor device models

An efficient numerical solution scheme based on a new mapped finite difference discretization and iterative strategies is developed for submicron semiconductor devices. As a representative model we consider a nonparabolic hydrodynamic system. The discretization is formulated in a mapped reference domain, and incorporates a transformed Scharfetter-Gummel treatment for the current density and energy flux. This permits the use of graded, nonuniform curvilinear grids in the physical domain of interest, which has advantages when gridding irregular domain shapes or grading meshes for steep solution profiles. The solution of the discrete system is carried out in a fully coupled, implicit form, and nonsymmetric gradient-type iterative strategies are investigated. Numerical results demonstrating the performance and reliability of the scheme are presented for test problems.

Efficient simulation of complex patterns in reaction-diffusion systems

Abstract Efficient numerical simulation techniques based on iterative and multigrid concepts are developed for solving coupled, nonlinear, reaction-diffusion systems. The solution approach is developed in a general setting and then applied to specific reaction-diffusion systems that give rise to complex dynamical patterns. The numerical strategy is based on semidiscretizing the coupled equations using a finite-difference formulation, with time integration of the resulting system of ordinary differential equations. Iterative and multigrid strategies are used to improve integration efficiency and to accelerate convergence. Numerical experiments are carried out to demonstrate the performance of the methods.

Simulation of macroscopic superconductivity for microelectronics

Abstract A mathematical framework has been developed for numerical analysis and simulation of applications in superconducting microelectronics. The approach is similar to those used successfully in semiconductor modeling. Here we investigate semidiscrete simulation of the time dependent Ginzburg-Landau equations. Several interesting numerical and modeling issues regarding the structure of the solutions and their sensitivity to the data and mesh resolution are described using results from a representative problem.

Time-integration and iterative techniques for semiconductor diffusion modeling

We investigate numerical integration, preconditioning, iterative solution and multigrid strategies for a class of réaction-diffusion systems used for modeling nonequilibrium phosphorus diffusion in silicon. These problems typically yield stiff systems of equations, and their efficient numerical simulation requires the use of stable integration strategies along with fast, robust algebraic system solvers. We compare the numerical performance of semi-implicit Runge-Kutta methods in conjunction with several standard nonsymmetric iterative solvers and multigrid methods. Our results demonstrate that block-diagonal preconditioning with node-based assembly of the discrete system dramatically improves the performance of iterative solvers. Numerical studies also reveal some interesting new aspects regarding the choice of integration schemes when using iterative methods to solve the linear systems. Unlike the case of direct solvers, where higher-order integration methods typically yield higher computational efficiency, the use of iterative solvers can significantly change or even reverse this trend.

Investigation of time-integration and iterative techniques for nonequilibrium phosphorus diffusion modeling

We investigate numerical integration and iterative solution strategies for a class of reaction-diffusion systems used for modeling nonequilibrium phosphorus diffusion in silicon. These problems typically yield stiff systems of equations, and their efficient numerical simulation requires the use of stable integration strategies along with fast, robust algebraic system solvers. We compare the numerical performance of semi-implicit Runge-Kutta methods in conjunction with several standard nonsymmetric iterative solvers and multigrid methods.