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The Discretization of the Basic Semiconductor Equations

  • Siegfried Selberherr

Abstract

The system of partial differential equations which forms the basic semiconductor equations together with appropriate boundary conditions has been investigated and characterized analytically in the previous chapter. This system cannot be solved explicitly in general. Therefore, the solution must be calculated by means of numerical approaches. We shall consider in this chapter such Solution procedures for the sealed equations which read:
$$\lambda ^2 \cdot \text{div}\,\text{grad}\,\psi - (n - p - C) = 0$$
(6-1)
$$\text{div}(D_n \cdot \text{grad}\,n - \mu _\text{n} \cdot n \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial n}} {{\partial t}}$$
(6-2)
$$\text{div}(D_p \cdot \text{grad}\,p - \mu _\text{p} \cdot p \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial p}} {{\partial t}}$$
(6-3)
Any numerical approach for the Solution of such a system consists essentially of three tasks. First, the domain, i.e. the Simulation geometry of the device, has to be partitioned into a finite number of subdomains, in which the Solution can be approximated easily with a desired accuracy. Secondly, the differential equations have to be approximated in each of the subdomains by algebraic equations which involve only values of the continuous dependent variables at discrete points in the domain and knowledge of the structure of the chosen funetions which approximate the dependent variables within each of the subdomains. In that way one obtains a fairly large system of, in general nonlinear, algebraic equations with unknowns comprised of approximations of the continuous dependent variables at discrete points. The Solution of this system is the final third task to be carried out. As this problem can be viewed rather independently it will be treated separately in Chapter 7.

Keywords

Finite Element Method Finite Difference Carrier Concentration Shape Function Finite Difference Method 
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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References

  1. [6.1]
    Adler, M. S.: A Method for Achieving and Choosing Variable Density Grids in Finite Difference Formulations and the Importance of Degeneracy and Band Gap Narrowing in Device Modeling. Proc. NASECODE I Conf., pp. 3–30. Dublin: Boole Press 1979.Google Scholar
  2. [6.2]
    Adler, M. S.: A Method for Terminating Mesh Lines in Finite Difference Formulations of the Semiconductor Device Equations. Solid-State Electron. 23, 845–853 (1980).CrossRefGoogle Scholar
  3. [6.3]
    Agier, W.: Die numerische Lösung der transienten Halbleitergleichungen. Diplomarbeit, Technische Universität Wien, 1983.Google Scholar
  4. [6.4]
    Babuska, I., Rheinboldt, W. C.: A Posteriori Error Analysis of Finite Element Solutions for One-Dimensional Problems. SIAM J. Numer. Anal 18, No. 3, 565–589 (1981).CrossRefGoogle Scholar
  5. [6.5]
    Barnes, J. J.: A Two-Dimensional Simulation of MESFET’s. Dissertation, University of Michigan, 1976.Google Scholar
  6. [6.6]
    Buturla, E. M., Cottrell, P. E., Grossman, B. M., Salsburg, K. A.: Finite-Element Analysis of Semiconductor Devices: The FIELDAY Program. IBM J. Res. Dev. 25, 218–231 (1981).CrossRefGoogle Scholar
  7. [6.7]
    Clough, R. W.: The Finite Element in Plane Stress Analysis. Proc. Conf. on Electronic Computation, pp. 345–378 (1960).Google Scholar
  8. [6.8]
    Cottrell, P. E., Buturla, E. M.: Two-Dimensional Static and Transient Simulation of Mobile Carrier Transport in a Semiconductor. Proc. NASECODE I Conf., pp. 31–64. Dublin: Boole Press 1979.Google Scholar
  9. [6.9]
    Davies, A. J.: The Finite Element Method: A First Approach. Oxford: Clarendon Press 1980.Google Scholar
  10. [6.10]
    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
  11. [6.11]
    Engl, W. L., Dirks, H.: Numerical Device Simulation Guided by Physical Approaches. Proc. NASECODE I Conf., pp. 65–93. Dublin: Boole Press 1979.Google Scholar
  12. [6.12]
    Engl, W. L., Dirks, H. K., Meinerzhagen, B.: Device Modeling. Proc. IEEE 71, No. 1, 10–33 (1983).CrossRefGoogle Scholar
  13. [6.13]
    Forsythe, G. E., Wasow, W. R.: Finite Difference Methods for Partial Differential Equations. New York: Wiley 1960.Google Scholar
  14. [6.14]
    Fox, L.: Finite-Difference Methods in Elliptic Boundary-Value Problems. In: The State of the Art in Numerical Analysis, pp. 799–881. London: Academic Press 1977.Google Scholar
  15. [6.15]
    Franz, A. F., Franz, G. A., Selberherr, S., Ringhofer, C., Markovich, P.: Finite Boxes — A Generalization of the Finite Difference Method Suitable for Semiconductor Device Simulation IEEE Trans. Electron Devices ED-30, No. 9, 1070–1082 (1983).CrossRefGoogle Scholar
  16. [6.16]
    Gummel, H. K.: A Self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations. IEEE Trans. Electron Devices ED-11, 455–465 (1964).CrossRefGoogle Scholar
  17. [6.17]
    Hachtel, G. D., Mack, M. H., O’Brien, R. R., Speelpennig, B.: Semiconductor Analysis Using Finite Elements — Part 1: Computational Aspects. IBM J. Res. Dev. 25, 232–245 (1981).CrossRefGoogle Scholar
  18. [6.18]
    Hachtel, G. D., Mack, M. H., O’Brien, R. R.: Semiconductor Analysis Using Finite Elements — Part 2: IGFET and BJT Case Studies. IBM J. Res. Dev. 25, 246–260 (1981).CrossRefGoogle Scholar
  19. [6.19]
    Hart, J. F., Cheney, E. W., Lawson, C. L., Maehly, H. J.: Computer Approximations. New York: Wiley 1968.Google Scholar
  20. [6.20]
    Hockney, R. W., Eastwood, J. W.: Computer Simulation Using Particles. New York: McGraw-Hill 1981.Google Scholar
  21. [6.21]
    Hrenikoff, A.: Solution of Problems in Elasticity by the Framework Method. J. Appl. Mech. A8, 169–175 (1941).Google Scholar
  22. [6.22]
    Kellogg, R. B.: Analysis of a Difference Approximation for a Singular Perturbation Problem in Two Dimensions. Proc. BAIL I Conf, pp. 113–117. Dublin: Boole Press 1980.Google Scholar
  23. [6.23]
    Kellogg, R. B., Shubin, G. R., Stephens, A. B.: Uniqueness and the Cell Reynolds Number. SIAM J. Numer. Anal. 17, No. 6, 733–739 (1980).CrossRefGoogle Scholar
  24. [6.24]
    Kellogg, R. B., Han, H.: The Finite Element Method for a Singular Perturbation Problem Using Enriched Subspaces. Report BN-978, University of Maryland, 1981.Google Scholar
  25. [6.25]
    Kraut, E. A., Murphy, W. D.: Application of Parabolic Partial Differential Equations to Semiconductor Device Modeling. Proc. NASECODE III Conf, pp. 150–154. Dublin: Boole Press 1983.Google Scholar
  26. [6.26]
    Kreskowsky, J. P., Grubin, H. L.: Numerical Solution of the Transient, Multidimensional Semiconductor Equations Using the LBI Techniques. Proc. NASECODE III Conf., pp. 155–160. Dublin: Boole Press 1983.Google Scholar
  27. [6.27]
    Kumar, R., Chamberlain, S. G., Roulston, D. J.: An Algorithm for Two-Dimensional Simulation of Reverse-Biased Beveled p-n Junctions. Solid-State Electron. 24, 309–311 (1981).CrossRefGoogle Scholar
  28. [6.28]
    Kumar, R., Roulston, D. J., Chamberlain, S. G.: Accurate Two-Dimensional Simulation of Double-Beveled p-n Junctions. Solid-State Electron. 24, 377–379 (1981).CrossRefGoogle Scholar
  29. [6.29]
    Laux, S. E.: Two-Dimensional Simulation of Gallium-Arsenide MESFET’s Using the Finite-Element Method. Dissertation, University of Michigan, 1981.Google Scholar
  30. [6.30]
    Laux, S. E., Lomax, R. J.: Numerical Investigation of Mesh Size Convergence Rate of the Finite Element Method in MESFET Simulation. Solid-State Electron. 24, 485–493 (1981).CrossRefGoogle Scholar
  31. [6.31]
    Machek, J., Selberherr, S.: A Novel Finite Element Approach to Device Modelling. IEEE Trans. Electron Devices ED-30, No. 9, 1083–1092 (1983).CrossRefGoogle Scholar
  32. [6.32]
    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
  33. [6.33]
    Markowich, P. A., Ringhofer, C. A., Selberherr, S.: A Singular Perturbation Approach for the Analysis of the Fundamental Semiconductor Equations. Report 2482, MRC, University of Wisconsin, 1983.Google Scholar
  34. [6.34]
    Marsal, D.: Die Numerische Lösung partieller Differentialgleichungen. Mannheim: Bibliographisches Institut 1976.Google Scholar
  35. [6.35]
    McHenry, D.: A Lattice Analogy for the Solution of Plane Stress Problems. J. Inst. Civ. Eng. 21, 59–82 (1943).CrossRefGoogle Scholar
  36. [6.36]
    Meis, T., Marcowitz, U.: Numerische Behandlung partieller Differentialgleichungen. Berlin-Heidelberg-New York: Springer 1978.Google Scholar
  37. [6.37]
    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
  38. [6.38]
    Mock, M. S.: An Initial Value Problem from Semiconductor Device Theory. SIAM J. Math. Anal. 5, No.4, 597–612 (1974).CrossRefGoogle Scholar
  39. [6.39]
    Mock, M. S.: Time Discretization of a Nonlinear Initial Value Problem. J. Comp. Phys. 21, 20–37 (1976).CrossRefGoogle Scholar
  40. [6.40]
    Mock, M. S.: The Charge-Neutral Approximation and Time Dependent Simulation. Proc. NASECODE I Conf., pp. 120–135. Dublin: Boole Press 1979.Google Scholar
  41. [6.41]
    Mock, M. S.: A Time-Dependent Numerical Model of the Insulated-Gate Field-Effect Transistor. Solid-State Electron. 24, 959–966 (1981).CrossRefGoogle Scholar
  42. [6.42]
    Mock, M. S.: The Stability Problem for Time-Dependent Models. In: An Introduction to the Numerical Analysis of Semiconductor Devices and Integrated Circuits, pp. 63–67. Dublin: Boole Press 1981.Google Scholar
  43. [6.43]
    Mock, M. S.: Analysis of Mathematical Models of Semiconductor Devices. Dublin: Boole Press 1983.Google Scholar
  44. [6.44]
    Mock, M. S.: Convergence and Accuracy in Stationary Numerical Models. In: An Introduction to the Numerical Analysis of Semiconductor Devices and Integrated Circuits, pp. 58–62. Dublin: Boole Press 1981.Google Scholar
  45. [6.45]
    O’Riordan, E.: Finite Element Methods for Singularly Perturbed Problems. Proc. BAIL II Conf., pp. 52–57. Dublin: Boole Press 1982.Google Scholar
  46. [6.46]
    Ortega, J. M., Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press 1970.Google Scholar
  47. [6.47]
    Parter, S. V.: Numerical Methods for Partial Differential Equations. New York: Academic Press 1979.Google Scholar
  48. [6.48]
    Scharfetter, D. L., Gummel, H. K.: Large-Signal Analysis of a Silicon Read Diode Oscillator. IEEE Trans. Electron Devices ED-16, 64–77 (1969).CrossRefGoogle Scholar
  49. [6.49]
    Schwarz, H. R.: Methode der finiten Elemente. Stuttgart: Teubner 1980.Google Scholar
  50. [6.50]
    Smith, G. D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford: Clarendon Press 1978.Google Scholar
  51. [6.51]
    Strang, G., Fix, G. J.: An Analysis of the Finite Element Method. Englewood Cliffs, N.J.: Prentice-Hall 1973.Google Scholar
  52. [6.52]
    Sutherland, A. D.: An Algorithm for Treating Interface Surface Charge in the Two-Dimensional Discretization of Poisson’s Equation for the Numerical Analysis of Semiconductor Devices such as MOSFET’s. Solid-State Electron. 23, 1085–1087 (1980).CrossRefGoogle Scholar
  53. [6.53]
    Szuhar, M.: Accurate Interface Handling for Mathematical Simulation of MOS Devices. Solid-State Electron. 25, No. 9, 963–965 (1982).CrossRefGoogle Scholar
  54. [6.54]
    Zarantonello, E. H.: Solving Functional Equations by Contractive Averaging. Report 160, MRC, University of Wisconsin, 1960.Google Scholar
  55. [6.55]
    Zienkiewicz, O. C.: The Finite Element Method. London: McGraw-Hill 1977.Google Scholar

Copyright information

© Springer-Verlag/Wien 1984

Authors and Affiliations

  • Siegfried Selberherr
    • 1
  1. 1.Institut für Allgemeine Elektrotechnik und ElektronikTechnische Universität WienAustria

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