A Multigrid Preconditioner for the Semiconductor Equations

A multigrid preconditioned conjugate gradient algorithm is introduced into a semiconductor device modeling code DANCIR. This code simulates a wide variety of semiconductor devices by numerically solving the drift-diffusion equations. The most time-consuming aspect of the simulation is the solution o...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 17; no. 1; pp. 118 - 132
Main Authors: Meza, Juan C., Tuminaro, Ray S.
Format: Journal Article Conference Proceeding
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.1996
Subjects:
Algorithms
Applied sciences
Computer-aided design of microcircuits; layout and modeling
Dependent variables
Design. Technologies. Operation analysis. Testing
Electronics
Exact sciences and technology
Integrated circuits
Laboratories
Mathematical methods in physics
Numerical approximation and analysis
Numerical linear algebra
Physics
Semiconductor electronics. Microelectronics. Optoelectronics. Solid state devices
Simulation
Conjugate gradient method
Electronic properties
Semiconductor devices
Junction field effect transistor
Non linear equation
Discretization
Linear equation
Multigrid
MOS transistors
Sparse matrix
Relaxation method
Numerical simulation
Drift diffusion equation
Iterative methods
Preconditioning
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:A multigrid preconditioned conjugate gradient algorithm is introduced into a semiconductor device modeling code DANCIR. This code simulates a wide variety of semiconductor devices by numerically solving the drift-diffusion equations. The most time-consuming aspect of the simulation is the solution of three linear systems within each iteration of the Gummel method. The original version of DANCIR uses a conjugate gradient iteration preconditioned by an incomplete Cholesky factorization. In this paper, we consider the replacement of the Cholesky preconditioner by a multigrid preconditioner. To adapt the multigrid method to the drift-diffusion equations, interpolation, projection, and coarse grid discretization operators need to be developed. These operators must take into account a number of physical aspects that are present in typical devices: wide-scale variation in the partial differential equation (PDE) coefficients, small-scale phenomena such as contact points, and an oxide layer. Additionally, suitable relaxation procedures must be designed that give good smoothing numbers in the presence of anisotropic behavior. The resulting method is compared with the Cholesky preconditioner on a variety of devices in terms of iterations, storage, and run time.
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ISSN:1064-8275
1095-7197
DOI:10.1137/0917010