The Drift Diffusion Equations

  • Peter A. Markowich
  • Christian A. Ringhofer
  • Christian Schmeiser


The drift diffusion equations are the most widely used model to describe semiconductor devices today. The bulk of the literature on mathematical models for device simulation is concerned with this nonlinear system of partial differential equations and numerical software for its solution is commonplace at practically every research facility in the field. From an engineering point of view, the interest in the drift diffusion model is to replace as much laboratory testing as possible by numerical simulation in order to minimize costs. To this end, it is important that computations can be performed in a reasonable amount of time. This implies that the involved mathematical models cannot be too complicated, such as, for instance, the higher dimensional transport equations described in Chapter 1. For the current state of technology the drift diffusion equations seem to represent a reasonable compromise between computational efficiency and an accurate description of the underlying device physics. Therefore transport equations are used mainly to compute data for the model parameters in the drift diffusion equations in the engineering environment. It should be pointed out, however, that, with the increased miniaturization of semiconductor devices, one comes closer and closer to the limits of validity of the drift diffusion equations, even in an industrial environment.


Singular Perturbation Reverse Bias Depletion Region Forward Bias Applied Bias 
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  1. [3.1]
    R. A. Adams: Sobolev Spaces. Academic Press, New York (1975).MATHGoogle Scholar
  2. [3.2]
    F. Alabeau: A Singular Perturbation Analysis of the Semiconductor Device and the Electrochemistry Equations. Report, Institut National de Recherche en Informatique et en Automatique, Paris (1984).Google Scholar
  3. [3.3]
    U. Ascher, P. A. Markowich, C. Schmeiser, H. Steinrück, R. Weiß: Conditioning of the Steady State Semiconductor Device Problem. SIAM J. Appl. Math. 49, 165–185 (1989).MathSciNetMATHCrossRefGoogle Scholar
  4. [3.4]
    R. E. Bank, et al: Analytical and Numerical Aspects of Semiconductor Device Modelling. Report 82–11274–2, Bell Laboratories, Murray Hill (1982).Google Scholar
  5. [3.5]
    R. E. Bank, D. J. Rose: Global Approximate Newton Methods. Numerische Mathematik 37, 279–295 (1981).MathSciNetMATHCrossRefGoogle Scholar
  6. [3.6]
    F. Brezzi: Theoretical and Numerical Problems in Reverse Biased Semiconductor Devices. Proc. 7th Intern. Conf. on Comput. Meth. in Appl. Sci. and Engng., INRIA, Paris (1985).Google Scholar
  7. [3.7]
    F. Brezzi, A. C. S. Capelo, L. Gastaldi: A Singular Perturbation Analysis of Reverse Biased Semiconductor Diodes. SIAM J. Math. Anal. 20, 372–387 (1989).MathSciNetMATHCrossRefGoogle Scholar
  8. [3.8]
    F. Brezzi, L. Gastaldi: Mathematical Properties of One-Dimensional Semiconductors. Mat. Apl. Comp. 5, 123–137 (1986).MathSciNetMATHGoogle Scholar
  9. [3.9]
    L. Caffarelli, A. Friedman: A Singular Perturbation Problem for Semiconductors. Bolletino U.M.I. 1-B (7), 409–421 (1987).MathSciNetGoogle Scholar
  10. [3.10]
    H. Gajewski: On the Existence of Steady-State Carrier Distributions in Semiconductors. ZAMM (to appear).Google Scholar
  11. [3.11]
    H. Gajewski: On Existence, Uniqueness and Asymptotic Behavior of Solutions of the Basic Equations for Carrier Transport in Semiconductors. ZAMM 65, 101–108 (1985).MathSciNetMATHCrossRefGoogle Scholar
  12. [3.12]
    D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1984).Google Scholar
  13. [3.13]
    E. Griepentrog, R. Maerz: Differential Algebraic Equations and Their Numerical Treatment. Teubner, Leipzig (1986).MATHGoogle Scholar
  14. [3.14]
    H. K. Gummel: A Self-Consistent Iterative Scheme for One-Dimensional Steady State Transistor Calculations. IEEE Trans. Electron. Devices 11, 455–465 (1964).CrossRefGoogle Scholar
  15. [3.15]
    J. Henri, B. Louro: Singular Perturbation Theory Applied to the Electrochemistry Equations in the Case of Electroneutrality. Nonlinear Analysis TMA 13, 787–801 (1989).CrossRefGoogle Scholar
  16. [3.16]
    J. W. Jerome: Consistency of Semiconductor Modelling: An Existence/Stability Analysis for the Stationary Van Roosbroeck System. SIAM J. Appl. Math. 45, 565–590 (1985).MathSciNetMATHCrossRefGoogle Scholar
  17. [3.17]
    T. Kerkhoven: A Proof of the Convergence of Gummel’s Method for Realistic Device Geometries. SIAM J. Num. Anal. 23, 1121–1137 (1986).MathSciNetMATHCrossRefGoogle Scholar
  18. [3.18]
    P. Markowich: A Singular Perturbation Analysis of the Fundamental Semiconductor Device Equations. SIAM J. Appl. Math. 44, 896–928 (1984).MathSciNetMATHCrossRefGoogle Scholar
  19. [3.19]
    P. Markowich: The Stationary Semiconductor Device Equations. Springer, Wien-New York (1986).Google Scholar
  20. [3.20]
    P. Markowich, P. Szmolyan: A System of Convection-Diffusion Equations with Small Diffusion Coefficient Arising in Semiconductor Physics. J. Diff. Equ. 81, 234–254 (1989).MathSciNetMATHCrossRefGoogle Scholar
  21. [3.21]
    P. Markowich, C. Ringhofer: A Singularly Perturbed Boundary Value Problem Modelling a Semiconductor Device. SIAM J. Appl. Math. 44, 231–256 (1984).MathSciNetMATHCrossRefGoogle Scholar
  22. [3.22]
    P. Markowich, C. Ringhofer: Stability of the Linearized Transient Semiconductor Device Equations. ZAMM 67, 319–332 (1987).MathSciNetMATHCrossRefGoogle Scholar
  23. [3.23]
    P. Markowich, C. Schmeiser: Uniform Asymptotic Representations of the Basic Semiconductor Device Equations. IMA J. Appl Math. 36, 43–57 (1986).MathSciNetMATHCrossRefGoogle Scholar
  24. [3.24]
    M. S. Mock: Analysis of Mathematical Models of Semiconductor Devices. Boole Press, Dublin (1983).MATHGoogle Scholar
  25. [3.25]
    R. E. O’Malley jr.: Introduction to Singular Perturbations. Academic Press, New York (1974).MATHGoogle Scholar
  26. [3.26]
    C. Ringhofer: An Asymptotic Analysis of a Transient P-N Junction Model. SIAM J. Appl. Math. 47, 624–642 (1987).MathSciNetMATHCrossRefGoogle Scholar
  27. [3.27]
    C. Ringhofer: A Singular Perturbation Analysis for the Transient Semiconductor Device Equations in One Space Dimension. IMA J. Appl. Math. 39, 17–32 (1987).MathSciNetMATHCrossRefGoogle Scholar
  28. [3.28]
    C. Ringhofer, C. Schmeiser: An Approximate Newton Method for the Solution of the Basic Semiconductor Device Equations. SIAM J. Num. Anal. 26, 507–516 (1989).MathSciNetMATHCrossRefGoogle Scholar
  29. [3.29]
    C. Ringhofer, C. Schmeiser: A Modified Gummel Method for the Basic Semiconductor Device Equations. IEEE Trans. CAD 7, 251–253 (1988).Google Scholar
  30. [3.30]
    C. Schmeiser: On Strongly Reverse Biased Semiconductor Diodes. SIAM J. Appl. Math. (1989) (to appear).Google Scholar
  31. [3.31]
    C. Schmeiser, R. Weiss: Asymptotic Analysis of Singular Singularly Perturbed Boundary Value Problems. SIAM J. Math. Anal. 17, 560–579 (1986).MathSciNetMATHCrossRefGoogle Scholar
  32. [3.32]
    T. Seidman, G. Troianiello: Time Dependent Solution of a Nonlinear System Arising in Semiconductor Theory. Nonlinear Analysis T.M.A. 9, 1137–1157 (1985).MathSciNetMATHCrossRefGoogle Scholar
  33. [3.33]
    T. Seidman, G. Troianiello: Time Dependent Solution of a Nonlinear System Arising in Semiconductor Theory, II: Boundedness and Periodicity. Nonlinear Analysis T.M.A. 10, 491–502 (1986).MATHCrossRefGoogle Scholar
  34. [3.34]
    S. Selberherr: Analysis and Simulation of Semiconductor Devices. Springer-Verlag, Wien New York (1984).Google Scholar
  35. [3.35]
    S. M. Sze: Physics of Semiconductor Devices, 2nd edn. John Wiley & Sons, New York (1981).Google Scholar
  36. [3.36]
    P. Szmolyan: Initial Transients of Solutions of the Semiconductor Device Equations. Nonlinear Analysis TMA (to appear).Google Scholar
  37. [3.37]
    H. Triebel: Interpolation Theory, Function Spaces, Differential Operators. Verlag der Wissenschaften, Berlin (1973).Google Scholar
  38. [3.38]
    J. H. Wilkinson: Rounding Errors in Algebraic Processes. Prentice Hall, New Jersey (1963).MATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1990

Authors and Affiliations

  • Peter A. Markowich
    • 1
  • Christian A. Ringhofer
    • 2
  • Christian Schmeiser
    • 3
  1. 1.Fachbereich MathematikTechnische Universität BerlinBerlin 12Germany
  2. 2.Department of MathematicsArizona State UniversityTempeUSA
  3. 3.Institut für Angewandte und Numerische MathematikTechnische Universität WienWienAustria

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