On some numerical problems in semiconductor device simulation

  • F. Brezzi
  • L. D. Marini
  • P. Markowich
  • P. Pietra
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1460)


We recall in the introduction the main features of the drift-diffusion model for semiconductor devices, pointing out its physical meaning, its possible derivation, and its limits. Then, in Section 2, we present a mixed finite element method for the discretization of this model. Finally, using asymptotic analysis techniques, we compare the qualitative behaviour of the mixed method with other methods (classical conforming Galerking method and harmonic average methods). This asymptotic analysis provides some indication of the advantages of the mixed method.


Mixed Method Mixed Finite Element Method Order Elliptic Problem Harmonic Average Mixed Approximation 
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  1. [1]
    D.N. Arnold-F. Brezzi: Mixed and non-conforming finite element methods: implementation, post-processing and error estimates. M 2 AN 19, 7–32, 1985.MathSciNetzbMATHGoogle Scholar
  2. [2]
    R.E. Bank-D.J. Rose-W. Fichtner: Numerical methods for semiconductor device simulation. IEEE Trans. El. Dev. 30, 1031–1041, 1983.ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    F. Brezzi: On the existence uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. R.A.I.R.O. 8-R2, 129–151, 1974.MathSciNetzbMATHGoogle Scholar
  4. [4]
    F.Brezzi-L.D. Marini-P.Pietra: Two-dimensional exponential fitting and applications to drift-diffusion models. (To appear in SIAM J.Numer.Anal.).Google Scholar
  5. [5]
    F.Brezzi-L.D. Marini-P.Pietra: Numerical simulation of semiconductor devices. (To appear in Comp.Meths.Appl. Mech.and Engr.).Google Scholar
  6. [6]
    P.G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.zbMATHGoogle Scholar
  7. [7]
    B.X. Fraeijs de Veubeke: Displacement and equilibrium models in the finite element method. In: Stress Analysis, O.C. Zienkiewicz and G. Hollister eds., Wiley, New York, 1965.Google Scholar
  8. [8]
    L.D.Marini-P.Pietra: New mixed finite element schemes for current continuity equations. (Submitted to COMPEL).Google Scholar
  9. [9]
    P.A.Markowich: The Stationary Semiconductor Device Equations. Springer, 1986.Google Scholar
  10. [10]
    P.A.Markowich-C.Ringhofer-C.Schmeiser: Semiconductor equations. Springer, 1989. (To appear).Google Scholar
  11. [11]
    P.A. Markowich-M. Zlámal: Inverse-average-type finite element discretisations of self-adjoint second order elliptic problems, Math. of Comp. 51, 431–449, 1988.CrossRefGoogle Scholar
  12. [12]
    M.S. Mock: Analysis of a discretisation algorithm for stationary continuity equations in semiconductor device models II. COMPEL 3, 137–149, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Niclot-P. Degond-F. Poupaud: Deterministic particle simulations of the Boltzmann transport equation of semiconductors. J. Comp. Phys., 78, 313–350, 1988.ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    F.Poupaud: On a system of nonlinear Boltzmann equations of semiconductor physics. (To appear in SIAM J. Math. Anal.).Google Scholar
  15. [15]
    P.A.Raviart-J.M.Thomas: A mixed finite element method for second order elliptic problems. In Mathematical aspects of the finite element method, Lecture Notes in Math. 606, 292–315, Springer, 1977.Google Scholar
  16. [16]
    S.Selberherr: Analysis and simulation of semiconductor devices. Springer, 1984.Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • F. Brezzi
    • 1
    • 2
  • L. D. Marini
    • 2
  • P. Markowich
    • 3
  • P. Pietra
    • 2
  1. 1.Istituto di Meccanica StrutturaleUniversità di PaviaItaly
  2. 2.Istituto di Analisi Numerica del C.N.R. di PaviaItaly
  3. 3.Fachbereich Mathematik, TU-BerlinFRG

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