Abstract
The numerical solution of boundary value problems for nonlinear systems of elliptic partial differential equations in general and the static simulation of semiconductor devices in particular usually proceeds in the following steps:
-
(i)
The ‘continuous’ problem is replaced by a suitable approximating ‘discrete’ nonlinear system of algebraic equations, whose solutions are intrinsically related to point-values of approximate solutions. This procedure is called discretisation of the boundary value problem.
-
(ii)
Since, usually, the nonlinear system of equations generated by the discretisation cannot be solved exactly, an iteration scheme based on (quasi-) linearisation is set up in order to obtain an approximate discrete solution.
-
(iii)
In each iteration step a usually large and sparse system of linear equations has to be solved.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Axelsson, O., Barker, A. V.: Finite Element Solution of Boundary Value Problems, Theory and Computation. Orlando, Florida: Academic Press 1984.
Bank, R. E., Jerome, J. W., Rose, D. J.: Analytical and Numerical Aspects of Semiconductor Modelling. Report 82–11274–2, Bell Laboratories, 1982.
Bramble, J. H., Hubbard, B. E.: On the Formulation of Finite Difference Analogues of the Dirichlet Problem for Poisson’s Equation. Num. Math. 4, 313–327 (1962).
Buturla, E. M., Cottrell, P. E.: Two-Dimensional Finite Element Analysis of Semiconductor Steady State Transport Equations. Proc. International Conference “Computer Methods in Nonlinear Mechanics”, Austin, Texas, pp. 512–530 (1974).
Choo, S. C., Seidmann, T. I.: Iterative Scheme for Computer Simulations of Semiconductor Devices. Solid State Electronics 15, 1229–1235 (1972).
Ciarlet, P.: The Finite Element Method for Elliptic Problems. Amsterdam—New York—Oxford: North-Holland 1978.
Collatz, L.: Numerical Treatment of Differential Equations, 3rd ed. Berlin—HeidelbergNew York: Springer 1960.
Doolan, E. P., Miller, J. J. H., Schilders, W. H. A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Dublin: Boole Press 1980.
Fichtner, W., Rose, D. J.: On the Numerical Solution of Nonlinear Elliptic PDEs Arising from Semiconductor Device Modelling. Report 80–2111–12, Bell Laboratories, 1980.
Forsythe, G. E., Wasow, W. R.: Finite Difference Methods for Partial Differential Equations. New York: Wiley 1960.
Franz, A. F., Franz, G. A., Selberherr, S., Ringhofer, C. A., Markowich, P. A.: Finite Boxes — A Generalisation of the Finite Difference Method Suitable for Semiconductor Device Simulation. IEEE Trans. Electron Devices. ED-30, No. 9, 1070–1082 (1983).
Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order, 2nd ed. Berlin—Heidelberg—New York: Springer 1983.
Gummel, H. K.: A Self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations. IEEE Trans. Electron Devices. ED-11, 455–465 (1964).
Jüngling, W., Pichler, P., Selberherr, S., Guerrero, E., Pötzl, H.: Simulation of Critical IC Fabrication Processes Using Advanced Physical and Numerical Methods. IEEE Trans. Electron Devices ED-32, No. 2, 156–167 (1985).
Keller, H. B.: Approximation Methods for Nonlinear Problems with Application to Two-Point Boundary Value Problems. Math. Comp. 29, 464–474 (1974).
Markowich, P. A., Ringhofer, C. A.: Collocation Methods for Boundary Value Problems on `Long’ Intervals. Math. Comp. 40, 123–150 (1983).
Markowich, P. A., Ringhofer, C. A., Selberherr, S., Lentini, M.: A Singular Perturbation Approach for the Analysis of the Fundamental Semiconductor Equations. IEEE Trans. Electron Devices. ED-30, No. 9, 1165–1180 (1983).
Markowich, P. A., Ringhofer, C. A., Steindl, A.: Arclength Continuation Methods for the Computation of Semiconductor Device Characteristics. IMA J. Num. Anal. 33, 175–187 (1984).
Meis, T., Marcowitz, U.: Numerische Behandlung Partieller Differentialgleichungen. Berlin—Heidelberg—New York: Springer 1978.
Mock, M. S.: On the Convergence of Gummel’s Numerical Algorithm. Solid State Electronics 15, 781–793 (1972).
Mock, M. S.: Analysis of Mathematical Models of Semiconductor Devices. Dublin: Boole Press 1983.
Mock, M. S.: On the Computation of Semiconductor Device Current Characteristics by Finite Difference Methods. J. Engineering Math. 7, No. 3, 193–205 (1973).
Mock, M. S.: Analysis of a Discretisation Algorithm for Stationary Continuity Equations in Semiconductor Device Models I. COMPEL 2, No. 3, 117–139 (1983).
Mock, M. S.: Analysis of a Discretisation Algorithm for Stationary Continuity Equations in Semiconductor Device Models II. COMPEL 3, No. 3, 137–149 (1984).
Oden, J. T.: Finite Elements of Nonlinear Continua. New York: McGraw-Hill 1972.
Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York—London: Academic Press 1970.
den Heijer, C., Rheinboldt, W. C.: On Steplength Algorithms for a Class of Continuation Methods. SIAM J. Num. Anal. 18, Nr. 5, 925–948 (1981).
Scharfetter, D. L., Gummel, H. K.: Large Signal Analysis of a Silicon Read Diode Oscillator. IEEE Trans. Electron Devices. ED-16, 64–77 (1969).
Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Wien—New York: Springer 1984.
Strang, G., Fix, G. J.: An Analysis of the Finite Element Method. Englewood Cliffs, N. J.: Prentice-Hall 1973.
Thompson, J. F. (ed.): Numerical Grid Generation. Amsterdam—New York—Oxford: North-Holland 1982.
Varga, R. S.: Matrix Iterative Analysis. Englewood Cliffs, N. J.: Prentice-Hall 1962.
Watanabe, D. S., Sheikh, Q. M., Slamed, S.: Convergence of Quasi-Newton Methods for Semiconductor Equations. Report, Department of Computer Science, University of Illinois—Urbana, U.S.A., 1984.
Zienkiewicz, O. C.: The Finite Element Method. London: McGraw-Hill 1977.
Zlamal, M.: Finite Element Solution of the Fundamental Equations of Semiconductor Devices I. Report, Department of Math., Technical University Brünn, CSSR, 1984.
Zlamal, M.: A Finite Element Solution of the Nonlinear Heat Equation. RAIRO Anal. Num. 14, 203–216 (1980).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag Wien
About this chapter
Cite this chapter
Markowich, P.A. (1986). Discretisation of the Stationary Device Problem. In: The Stationary Semiconductor Device Equations. Computational Microelectronics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-3678-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-7091-3678-2_5
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-99937-0
Online ISBN: 978-3-7091-3678-2
eBook Packages: Springer Book Archive