Discretisation of the Stationary Device Problem

  • Peter A. Markowich
Part of the Computational Microelectronics book series (COMPUTATIONAL)


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:
  1. (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.

  2. (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.

  3. (iii)

    In each iteration step a usually large and sparse system of linear equations has to be solved.



Finite Difference Scheme Schottky Contact Discrete Solution Finite Element Solution Finite Element Discretisations 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [5.1]
    Axelsson, O., Barker, A. V.: Finite Element Solution of Boundary Value Problems, Theory and Computation. Orlando, Florida: Academic Press 1984.Google Scholar
  2. [5.2]
    Bank, R. E., Jerome, J. W., Rose, D. J.: Analytical and Numerical Aspects of Semiconductor Modelling. Report 82–11274–2, Bell Laboratories, 1982.Google Scholar
  3. [5.3]
    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).CrossRefGoogle Scholar
  4. [5.4]
    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).Google Scholar
  5. [5.5]
    Choo, S. C., Seidmann, T. I.: Iterative Scheme for Computer Simulations of Semiconductor Devices. Solid State Electronics 15, 1229–1235 (1972).CrossRefGoogle Scholar
  6. [5.6]
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. Amsterdam—New York—Oxford: North-Holland 1978.Google Scholar
  7. [5.7]
    Collatz, L.: Numerical Treatment of Differential Equations, 3rd ed. Berlin—HeidelbergNew York: Springer 1960.CrossRefGoogle Scholar
  8. [5.8]
    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.Google Scholar
  9. [5.9]
    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.Google Scholar
  10. [5.10]
    Forsythe, G. E., Wasow, W. R.: Finite Difference Methods for Partial Differential Equations. New York: Wiley 1960.Google Scholar
  11. [5.11]
    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).CrossRefGoogle Scholar
  12. [5.12]
    Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order, 2nd ed. Berlin—Heidelberg—New York: Springer 1983.CrossRefGoogle Scholar
  13. [5.13]
    Gummel, H. K.: A Self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations. IEEE Trans. Electron Devices. ED-11, 455–465 (1964).Google Scholar
  14. [5.14]
    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).CrossRefGoogle Scholar
  15. [5.15]
    Keller, H. B.: Approximation Methods for Nonlinear Problems with Application to Two-Point Boundary Value Problems. Math. Comp. 29, 464–474 (1974).CrossRefGoogle Scholar
  16. [5.16]
    Markowich, P. A., Ringhofer, C. A.: Collocation Methods for Boundary Value Problems on `Long’ Intervals. Math. Comp. 40, 123–150 (1983).Google Scholar
  17. [5.17]
    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).CrossRefGoogle Scholar
  18. [5.18]
    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).Google Scholar
  19. [5.19]
    Meis, T., Marcowitz, U.: Numerische Behandlung Partieller Differentialgleichungen. Berlin—Heidelberg—New York: Springer 1978.CrossRefGoogle Scholar
  20. [5.20]
    Mock, M. S.: On the Convergence of Gummel’s Numerical Algorithm. Solid State Electronics 15, 781–793 (1972).CrossRefGoogle Scholar
  21. [5.21]
    Mock, M. S.: Analysis of Mathematical Models of Semiconductor Devices. Dublin: Boole Press 1983.Google Scholar
  22. [5.22]
    Mock, M. S.: On the Computation of Semiconductor Device Current Characteristics by Finite Difference Methods. J. Engineering Math. 7, No. 3, 193–205 (1973).CrossRefGoogle Scholar
  23. [5.23]
    Mock, M. S.: Analysis of a Discretisation Algorithm for Stationary Continuity Equations in Semiconductor Device Models I. COMPEL 2, No. 3, 117–139 (1983).CrossRefGoogle Scholar
  24. [5.24]
    Mock, M. S.: Analysis of a Discretisation Algorithm for Stationary Continuity Equations in Semiconductor Device Models II. COMPEL 3, No. 3, 137–149 (1984).CrossRefGoogle Scholar
  25. [5.25]
    Oden, J. T.: Finite Elements of Nonlinear Continua. New York: McGraw-Hill 1972.Google Scholar
  26. [5.26]
    Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York—London: Academic Press 1970.Google Scholar
  27. [5.27]
    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).CrossRefGoogle Scholar
  28. [5.28]
    Scharfetter, D. L., Gummel, H. K.: Large Signal Analysis of a Silicon Read Diode Oscillator. IEEE Trans. Electron Devices. ED-16, 64–77 (1969).Google Scholar
  29. [5.29]
    Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Wien—New York: Springer 1984.CrossRefGoogle Scholar
  30. [5.30]
    Strang, G., Fix, G. J.: An Analysis of the Finite Element Method. Englewood Cliffs, N. J.: Prentice-Hall 1973.Google Scholar
  31. [5.31]
    Thompson, J. F. (ed.): Numerical Grid Generation. Amsterdam—New York—Oxford: North-Holland 1982.Google Scholar
  32. [5.32]
    Varga, R. S.: Matrix Iterative Analysis. Englewood Cliffs, N. J.: Prentice-Hall 1962.Google Scholar
  33. [5.33]
    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.Google Scholar
  34. [5.34]
    Zienkiewicz, O. C.: The Finite Element Method. London: McGraw-Hill 1977.Google Scholar
  35. [5.35]
    Zlamal, M.: Finite Element Solution of the Fundamental Equations of Semiconductor Devices I. Report, Department of Math., Technical University Brünn, CSSR, 1984.Google Scholar
  36. [5.36]
    Zlamal, M.: A Finite Element Solution of the Nonlinear Heat Equation. RAIRO Anal. Num. 14, 203–216 (1980).Google Scholar

Copyright information

© Springer-Verlag Wien 1986

Authors and Affiliations

  • Peter A. Markowich
    • 1
  1. 1.Institut für Angewandte und Numerische MathematikTechnische Universität WienViennaAustria

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