## A multigrid preconditioner for the semiconductor equations (1996)

Venue: | SIAM J. Sci. Comput |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Meza96amultigrid,

author = {Juan C. Meza and Ray and S. Tuminaro},

title = {A multigrid preconditioner for the semiconductor equations},

journal = {SIAM J. Sci. Comput},

year = {1996},

pages = {17--1}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Amultigrid preconditioned conjugate gradient algorithm is introduced into a semiconductor device modeling code, DANCIR. This code simulates a wide variety of semiconductor devices bynumerically solving the drift-di usion 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 amultigrid preconditioner. To adapt the multigrid method to the drift-di usion equations, interpolation, projection, and coarse grid discretization operators need to be developed. These operators must take into account anumber of physical aspects that are present intypical devices: wide scale variation in the partial di erential equation (PDE) coe cients, small scale phenomena suchascontact 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.

### Citations

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Multi-Grid Methods and Applications
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Citation Context ...e systems of equations that arise from many PDE applications. We giveonly a brief sketch of one type of multigrid algorithm. Detailed descriptions of more general multigrid algorithms can be found in =-=[3, 5]-=-. One iteration of a simple multigrid `V' cycle consists of smoothing the error using a relaxation technique (such as Gauss-Seidel), `solving' an approximation to the smooth error equation on a coarse... |

358 |
Multi-level adaptive solutions to boundary-value problems
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Citation Context ...e systems of equations that arise from many PDE applications. We giveonly a brief sketch of one type of multigrid algorithm. Detailed descriptions of more general multigrid algorithms can be found in =-=[3, 5]-=-. One iteration of a simple multigrid `V' cycle consists of smoothing the error using a relaxation technique (such as Gauss-Seidel), `solving' an approximation to the smooth error equation on a coarse... |

341 |
An Introduction to Multigrid Methods
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Citation Context .... That is, interpolating such thatu is smooth (i.e. ux is constant) at the interpolation point is not the right criterion. Instead, we need to take into account the function w(x). One possibility (see=-=[14]-=- or [1]) is to require that the term w(x)ux be constant attheinterpolated points. This results in an interpolation formula of the form (3.13) u(x) = w(x ; h 2 )u(x ; h)+w(x + h 2 w(x + h 2 )+w(x ; h 2... |

307 |
Physics of Semiconductor Devices
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- 1981
(Show Context)
Citation Context ...al equation plus two continuity equations, one each for the electron and hole current densities. For a complete derivation of the equations the reader can consult a variety of references, for example =-=[9, 11, 13]-=-. The potential or Poisson equation is given by r E = ; r2 (2.1) = � where is the scalar permittivity of the semiconductor, is the electric potential, and E = ;r is the electric eld. The total electri... |

143 |
Analysis and Simulation of Semiconductor Devices
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(Show Context)
Citation Context ...al equation plus two continuity equations, one each for the electron and hole current densities. For a complete derivation of the equations the reader can consult a variety of references, for example =-=[9, 11, 13]-=-. The potential or Poisson equation is given by r E = ; r2 (2.1) = � where is the scalar permittivity of the semiconductor, is the electric potential, and E = ;r is the electric eld. The total electri... |

37 |
A self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations
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(Show Context)
Citation Context ...linear system of equations arising from the spatial discretization of the drift-di usion equations can be written more compactly as: F ( � u� v) =[f1�f2�f3] T =0: The DANCIR code uses Gummel's method =-=[6]-=- (also known as nonlinear block Gauss-Seidel) to solve this nonlinear equation. One Gummel iteration consists of solving f2 for u, f3 for v, and then f1 for . Gummel's method has the advantage of only... |

24 |
A multigrid method and a combined multigrid-conjugate gradient method for elliptic problems with strongly discontinuous coefficients in general domains
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- 1981
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Citation Context ... r b ; Au ~r Rr /* R is a projection operator */ v 0 v MG(~r� v� level +1) u u + Pv /* P is an interpolation operator */ u relax2(b� u� level) endif Fig. 1. MG Algorithm for A levelu = b in Figure 1) =-=[8]-=-. The Jacobi iteration is one smoother which has this property when it is used for prerelaxation and postrelaxation. Another smoother combination with this property consists of using red-black GaussSe... |

22 |
Analysis of mathematical models of semiconductor devices
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(Show Context)
Citation Context ...al equation plus two continuity equations, one each for the electron and hole current densities. For a complete derivation of the equations the reader can consult a variety of references, for example =-=[9, 11, 13]-=-. The potential or Poisson equation is given by r E = ; r2 (2.1) = � where is the scalar permittivity of the semiconductor, is the electric potential, and E = ;r is the electric eld. The total electri... |

12 |
The multi-grid method for the di usion equation with strongly discontinuous coe cients
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(Show Context)
Citation Context ...s, interpolating such thatu is smooth (i.e. ux is constant) at the interpolation point is not the right criterion. Instead, we need to take into account the function w(x). One possibility (see[14] or =-=[1]-=-) is to require that the term w(x)ux be constant attheinterpolated points. This results in an interpolation formula of the form (3.13) u(x) = w(x ; h 2 )u(x ; h)+w(x + h 2 w(x + h 2 )+w(x ; h 2 ) )u(x... |

11 |
Semiconductor device modelling from the numerical point of view., Int
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(Show Context)
Citation Context .... The di culties associated with the wide range in the magnitude of the dependent variables can be circumvented to a certain extent by employing di erent variables. However it has been noted by Polak =-=[12]-=- thatchanging variables amounts to trading high variability in the dependent variables for increased nonlinearity in the equations. In the DANCIR code, the carrier concentrations are scaled by using t... |

4 | Multigrid Methods for Semiconductor Device Simulation - Molenaar - 1993 |

2 | A multigrid solver for the semiconductor equations. Technical report, Institut fur Angewandte Mathematik der Universitat - Bachmann - 1993 |

1 |
A three-dimensional steady-state semiconductor device simulator
- DANCIR
- 1990
(Show Context)
Citation Context ...rformance computers are necessary. In this paper we discuss the use of multigrid preconditioning in conjunction with a conjugate gradient algorithm inside a semiconductor device modeling code, DANCIR =-=[7]-=-. DANCIR is a three-dimensional semiconductor device simulator capable of computing the solution of the steady-state drift-di usion equations. The solution of the drift-di usion equations involves the... |