Advertisement

Introduction

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

Abstract

Semiconductor device modeling started in the early fifties just after Van Roosbroeck had formulated the so-called fundamental semiconductor device equations, a nonlinear system of partial differential equations, which describes potential distribution, carrier concentrations and current flow in arbitrary semiconductor devices (see [1.32]). In the early stages highly simplified one-dimensional models accessible to direct analytic treatment were used in order to understand device characteristics and to improve device design (see [1.28], [1.29]). The trend towards miniaturisation in VLSI and device design, mainly caused by the increasing demand for fast computers with large storage, rendered the simplified models and consequently the fully analytic approach obsolete. Instead, the emphasis shifted towards numerical simulation techniques, i.e. the computational solution of the semiconductor device equations based on numerical discretisation methods. This approach was suggested by Gummel [1.9] for the bipolar transistor. De Mari [1.3], [1.4] applied the fully computational approach to pn-junction diodes. It became clear very soon that standard methods and theories of discretisation techniques are inappropriate because they require an enormous amount of computer resources in order to give reasonably accurate results when modeling practically relevant devices. The main reason for this is that the equations are stiff and allow for solutions of locally different behaviour. The stiffness problem was — to a certain extent — overcome by the ingenuity of Scharfetter and Gummel, who developed a nonstandard, special purpose discretisation method, which — sometimes in a modified and extended way — is being used up to now (see [1.25]).

Keywords

Semiconductor Device Singular Perturbation Semiconductor Crystal Singularly Perturb Intrinsic Carrier Concentration 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1.1]
    Browne, B. T., Miller, J. J. H. (eds.): Numerical Analysis of Semiconductor Devices. Dublin: Boole Press 1979.Google Scholar
  2. [1.2]
    Browne, B. T., Miller, J. J. H. (eds.): Numerical Analysis of Semiconductor Devices and Integrated Circuits. Dublin: Boole Press 1981.Google Scholar
  3. [1.3]
    De Mari, A.: An Accurate Numerical Steady State One-Dimensional Solution of the P-N Junction. Solid State Electron. 11, 33–58 (1968).CrossRefGoogle Scholar
  4. [1.4]
    De Mari, A.: An Accurate Numerical One-Dimensional Solution of the P-N Junction under Arbitrary Transient Conditions. Solid State Electron. 11, 1021–2053 (1968).CrossRefGoogle Scholar
  5. [1.5]
    Eckhaus, W.: Asymptotic Analysis of Singular Perturbations. Amsterdam-New York-Oxford: North-Holland 1979.Google Scholar
  6. [1.6]
    Fichtner, W., Rose, D. (eds.): Special Issue on Numerical Simulation of VLSI Devices. IEEE Trans. Electron Devices. ED-30 (1983).Google Scholar
  7. [1.
    ] Fichtner, W., Rose, D. (eds.): Special Issue on Numerical Simulation of VLSI Devices. IEEE Trans. Electron Devices. (To appear.)Google Scholar
  8. [1.8]
    Franz, A. F., Franz, G. A., Selberherr, S., Ringhofer, C., Markowich, P.: Finite Boxes-A Generalisation of the Finite Difference Method Suitable for Semiconductor Device Simulation. IEEE Trans. Electron Devices. 30, No. 9, 1070–1082 (1983).CrossRefGoogle Scholar
  9. [1.9]
    Gummel, H. K.: A Self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations. IEEE Trans. Electron Devices. 11, 455–465 (1964).CrossRefGoogle Scholar
  10. [1.10]
    Heywang, W., Pötzl, H. W.: Bandstruktur and Stromtransport. Berlin-Heidelberg-New York: Springer 1976.CrossRefGoogle Scholar
  11. [1.11]
    Kurata, M.: Numerical Analysis for Semiconductor Devices. Lexington, Mass.: Lexington Press 1982.Google Scholar
  12. [1.12]
    O’Malley, R. E., jr.: Introduction to Singular Perturbations. New York: Academic Press 1974.Google Scholar
  13. [1.13]
    Markowich, P. A., Ringhofer, C.: A Singularly Perturbed Boundary Value Problem Modelling a Semiconductor Device. SIAM J. Appl. Math. 44, No. 2, 231–256 (1984).Google Scholar
  14. [1.14]
    Markowich, P. A.: A Singular Perturbation Analysis of the Fundamental Semiconductor Device Equations. SIAM J. Appl. Math. 44, No. 5, 896–928 (1984).Google Scholar
  15. [1.15]
    Markowich, P. A.: A Qualitative Analysis of the Fundamental Semiconductor Device Equations. COMPEL 2, No. 3, 97–115 (1983).CrossRefGoogle Scholar
  16. [1.16]
    Markowich, P. A., Ringhofer, C., Selberherr, S., Lentini, M.: A Singular Perturbation Approach for the Analysis of the Fundamental Semiconductor Equations. IEEE Trans. Electron Devices. 30, No. 9, 1165–1180 (1983).CrossRefGoogle Scholar
  17. [1.17]
    Miller, J. J. H. (ed.): Numerical Analysis of Semiconductor Devices and Integrated Circuits. Dublin: Boole Press 1983.Google Scholar
  18. [1.18]
    Miller, J. J. H. (ed.): Numerical Analysis of Semiconductor Devices and Integrated Circuits. Dublin: Boole Press 1985.Google Scholar
  19. [1.19]
    Mock, M. S.: On Equations Describing Steady State Carrier Distributions in a Semiconductor Device. Comm Pure and Appl. Math. 25, 781–792 (1972).Google Scholar
  20. [1.20]
    Mock, M. S.: An Initial Value Problem from Semiconductor Device Theory. SIAM J. Math. Anal. 5, No. 4, 597–612 (1974).Google Scholar
  21. [1.21]
    Mock, M. S.: Time Discretisation of a Nonlinear Initial Value Problem. J. Comp. Phys. 21, 20–37 (1976).CrossRefGoogle Scholar
  22. [1.22]
    Mock, M. S.: On the Computation of Semiconductor Device Current Characteristics by Finite Difference Methods. J. Eng. Math. 7, No. 3, 193–205 (1973).CrossRefGoogle Scholar
  23. [1.23]
    Mock, M. S.: Analysis of Mathematical Models of Semiconductor Devices. Dublin: Boole Press 1983.Google Scholar
  24. [1.24]
    Ringhofer, C., Selberherr, S.: Implications of Analytical Investigations about the Semiconductor Equations on Device Modelling Programs. MRC-TSR 2513, Math. Res. Center, University of Wisconsin-Madison, U.S.A., 1980.Google Scholar
  25. [1.25]
    Scharfetter, D. L., Gummel, H. K.: Large-Signal Analysis of a Silicon Read Diode Oscillator. IEEE Trans. Electron Devices. 16, 64–77 (1969).CrossRefGoogle Scholar
  26. [1.26]
    Seidman, T. I.: Steady State Solutions of Diffusion-Reaction Systems with Electrostatic Convection. Nonlinear Analysis. 4, No. 3, 623–637 (1980).CrossRefGoogle Scholar
  27. [1.27]
    Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Wien-New York: Springer 1984.CrossRefGoogle Scholar
  28. [1.28]
    Smith, R. A.: Semiconductors, 2nd ed. Cambridge: Cambridge Univ. Press 1978.Google Scholar
  29. [1.29]
    Sze, S. M.: Physics of Semiconductor Devices, 2nd ed. New York: J. Wiley 1981.Google Scholar
  30. [1.30]
    Vasileva, A. B., Stelmakh, V. F.: Singularly Disturbed Systems of the Theory of Semiconductor Devices. USSR Comput. Math. Phys. 17, 48–58 (1977).Google Scholar
  31. [1.31]
    Vasileva, A. B., Butuzow, V. F.: Singularly Perturbed Equations in the Critical Case. MRC-TSR 2039, Math. Res. Center, University of Wisconsin-Madison, U.S.A., 1980.Google Scholar
  32. [1.32]
    van Roosbroeck, W. V.: Theory of Flow of Electrons and Holes in Germanium and Other Semiconductors. Bell Syst. Techn. J. 29, 560–607 (1950).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

Personalised recommendations