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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Vydáno v:SIAM journal on scientific computing Ročník 17; číslo 1; s. 118 - 132
Hlavní autoři: Meza, Juan C., Tuminaro, Ray S.
Médium: Journal Article Konferenční příspěvek
Jazyk:angličtina
Vydáno: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.1996
Témata:
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
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Abstract 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.
AbstractList 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.
Author Meza, Juan C.
Tuminaro, Ray S.
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Cites_doi 10.1109/T-ED.1964.15364
10.1137/0902035
10.1007/978-3-7091-8752-4
10.1090/S0025-5718-1977-0431719-X
10.1007/978-3-662-02427-0
10.1002/nme.1620240408
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Issue 1
Keywords 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
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References Wesseling Pieter (R14) 1992
R3
Golub Gene H. (R4) 1983
R12
R5
R11
R6
Mock Michael S. (R9) 1983
Sze S. M. (R13) 1981
R1
References_xml – volume-title: Matrix computations
  year: 1983
  ident: R4
– volume-title: An introduction to multigrid methods
  year: 1992
  ident: R14
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  doi: 10.1109/T-ED.1964.15364
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  doi: 10.1137/0902035
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  doi: 10.1007/978-3-7091-8752-4
– ident: R3
  doi: 10.1090/S0025-5718-1977-0431719-X
– volume-title: Physics of Semiconductor Devices
  year: 1981
  ident: R13
– volume-title: Analysis of mathematical models of semiconductor devices
  year: 1983
  ident: R9
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  doi: 10.1007/978-3-662-02427-0
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  doi: 10.1002/nme.1620240408
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Snippet A multigrid preconditioned conjugate gradient algorithm is introduced into a semiconductor device modeling code DANCIR. This code simulates a wide variety of...
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StartPage 118
SubjectTerms 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
Title A Multigrid Preconditioner for the Semiconductor Equations
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