Advertisement

Kinetic Transport Models for Semiconductors

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

Abstract

In this Chapter we shall derive and discuss transport equations, which model the flow of charge carriers in semiconductors. The common feature of these equations is that they describe the evolution of the phase space (position-momentum space) density function of the ensemble of negatively charged conduction electrons or, resp., positively charged holes, which are responsible for the current flow in semiconductor crystals.

Keywords

Boltzmann Equation Wigner Function Pseudodifferential Operator Liouville Equation Schrodinger Equation 
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]
    A. Arnold, P. Degond, P. A. Markowich, Η. Steinrück: The Wigner-Poisson Equation in a Crystal. Applied Mathematics Letters 2, 187–191 (1989).MathSciNetMATHCrossRefGoogle Scholar
  2. [1.2]
    A. Arnold, P. A. Markowich: The Periodic Quantum Liouville-Poisson Problem. Boll. U.M.I. (1989) (to appear).Google Scholar
  3. [1.3]
    A. Arnold, H. Steinrück: The Electromagnetic Wigner Equation for an Electron with Spin. ZAMP (1989) (to appear).Google Scholar
  4. [1.4]
    N. C. Ashcroft, Ν. D. Mermin: Solid State Physics. Holt-Sounders, New York (1976).Google Scholar
  5. [1.5]
    C. Bardos, P. Degond: Global Existence for the Vlasov-Poisson Equation in Three Space Variables with Small Initial Data. Ann. Inst. Henri Poincare, Analyse Non-linéaire 2, 101–118 (1985).MathSciNetMATHGoogle Scholar
  6. [1.6]
    C. Bender, S. Orszag: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1977).Google Scholar
  7. [1.7]
    G. F. Bertsch: Heavy Ion Dynamics at Intermediate Energy. Report, Cyclotron Laboratory and Physics Department, Michigan State University, East Lansing, MI 48824, USA (1978).Google Scholar
  8. [1.8]
    J. S. Blakemore: Semiconductor Statistics. Pergamon Press, Oxford (1962).MATHGoogle Scholar
  9. [1.9]
    Ν. N. Bogoliubov: Problems of a Dynamical Theory in Statistical Physics. In: Studies in Statistical Mechanics, Vol. I (J. de Boer, G. E. Uhlenbeck, eds.). North-Holland, Amsterdam (1962), p. 5.Google Scholar
  10. [1.10]
    M. Born, H. S. Green: A General Kinetic Theory of Fluids. Cambridge University Press, Cambridge (1949).Google Scholar
  11. [1.11]
    F. Brezzi, P. A. Markowich: The Three-Dimensional Wigner-Poisson Problem: Existence, Uniqueness and Approximation. Report, Centre de Mathématiques Appliquées, Ecole Polytechnique, F-91128 Palaiseau, France (1989).Google Scholar
  12. [1.12]
    P. Carruthers, F. Zachariasen: Quantum Collision Theory with Phase-Space Distributions. Reviews of Modem Physics 55, 245–285 (1983).MathSciNetCrossRefGoogle Scholar
  13. [1.13]
    C. Cercignani: The Boltzmann Equation and Its Applications (Applied Mathematical Sciences, Vol. 67). Springer-Verlag, Berlin (1988).Google Scholar
  14. [1.14]
    J. Cooper: Galerkin Approximations for the One-Dimensional Vlasov-Poisson Equation. Math. Method, in Appl. Sci. 5, 516–529 (1983).MATHCrossRefGoogle Scholar
  15. [1.15]
    R. Dautrey, J. L. Lions: Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques: Tome 3’sson, Paris (1985).Google Scholar
  16. [1.16]
    P. Degond, P. A. Marko wich: A Quantum Transport Model for Semiconductors: The Wigner-Poisson Problem on a Bounded Brillouin Zone. M 2 AN (to appear).Google Scholar
  17. [1.17]
    P. Degond, P. A. Markowich: A Mathematical Analysis of Quantum Transport in Three-Dimensional Crystals. Report, Centre de Math. Appl., Ecole Polytechnique, F-91128 Palaiseau, France (1989).Google Scholar
  18. [1.18]
    F. Nier: Etudes des Solutions Stationnaires de l’Équation de Wigner. Manuscript, Centre de Mathématiques Appliquées, Ecole Polytechnique, F-91128 Palaiseau, France (1989).Google Scholar
  19. [1.19]
    F. Nier: Existence d’une Solution Pour un Système Mono-dimensionelle d’Equations de Schrödinger et des Poisson Couplées. Manuscript, Centre de Mathématiques Appliquées, Ecole Polytechnique, F-91128 Palaiseau, France (1989).Google Scholar
  20. [1.20]
    S. de Groot: La Transformation de Weyl et la Fonction de Wigner: Use Forme Alternative de la Méchanique Quantique. Lex Presses de l’Université de Montréal, Montreal (1974).Google Scholar
  21. [1.21]
    R. Di Perna, P. L. Lions: Global Solutions of Vlasov-Poisson Type Equations. Report 8824, CEREMADE, Université Paris-Dauphine, F-15115 Paris, France (1989).Google Scholar
  22. [1.22]
    R. Di Perna, P. L. Lions: Global Weak Solutions of Vlasov-Maxwell Systems. Report 8817, CEREMADE, Université Paris-Dauphine, F-15115 Paris, France (1989).Google Scholar
  23. [1.23]
    R. Di Perna, P. L. Lions: On the Cauchy Problem for Boltzmann Equations: Global Existence and Weak Stability. Report CEREMADE, Université Paris-Dauphine, F-75775 Paris, France (1988).Google Scholar
  24. [1.24]
    D. K. Ferry, N. C. Kluksdahl, C. Ringhofer: Absorbing Boundary Conditions for the Simulation of Quantum Tunneling Phenomena. Transport Equ. and Statist. Phys. (to appear).Google Scholar
  25. [1.25]
    R. P. Feynman: The Feynman Lectures on Physics 3: Quantum Mechanics. Addison-Wesley, Reading (1965).Google Scholar
  26. [1.26]
    H. Hofmann: Das Elektromagnetische Feld. Springer-Verlag, Wien-New York (1982).Google Scholar
  27. [1.27]
    R. L. Hudson: When Is the Wigner Quasiprobability Nonnegative?. Reports on Math. Physics 6, 249–252 (1974).MathSciNetMATHCrossRefGoogle Scholar
  28. [1.28]
    L. P. Kadanoff, G. Baym: Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Non-Equilibrium Problems. Benjamin, New York (1962).Google Scholar
  29. [1.29]
    T. Kato: Perturbation Theory for Linear Operators. Springer-Verlag, New York (1966).MATHGoogle Scholar
  30. [1.30]
    J. G. Kirkwood: J. Chem. Phys. 14, 180 (1946).CrossRefGoogle Scholar
  31. [1.31]
    C. Kittel: Introduction to Solid State Physics. J. Wiley & Sons, New York (1968). [1.32] N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, C. Ringhofer: Self-Consistent Study of the Resonant Tunneling Diode. Phys. Rev. Β (to appear).Google Scholar
  32. [1.33]
    N. A. Krall, A. W. Trivelpiece: Principles of Plasma Physics. McGraw-Hill, New York (1973).Google Scholar
  33. [1.34]
    L. D. Landau, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, 3: Quantenmechanik. Akademie-Verlag, Berlin (1960).Google Scholar
  34. [1.35]
    L. D. Landau, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, 1: Mechanik, 2nd edn. Akademie-Verlag, Berlin (1963).Google Scholar
  35. [1.36]
    I. B. Levinson: Translational Invariance in Uniform Fields and the Equation for the Density Matrix in the Wigner Representation. Soviet Physics JETP 30, 362–367 (1970).MathSciNetGoogle Scholar
  36. [1.37]
    P. A. Markowich: The Stationary Semiconductor Device Equations. Springer-Verlag, Wien-New York (1986).Google Scholar
  37. [1.38]
    P. Α. Markowich: On the Equivalence of the Schrödinger and the Quantum Liouville Equations. Math. Meth. in the Appl. Sci. 11, 459–469 (1989).MathSciNetMATHCrossRefGoogle Scholar
  38. [1.39]
    P. A. Markowich, C. A. Ringhofer: An Analysis of the Quantum Liouville Equations. ZAMM 69, 121–127 (1989).MathSciNetMATHCrossRefGoogle Scholar
  39. [1.40]
    A. Messiah: Quantum Mechanics, North-Holland, Amsterdam (1965).Google Scholar
  40. [1.41]
    H. Neunzert: The Nuclear Vlasov Equation—Methods and Results that Cann(ot) be Taken Over from the ‘Classical’ Case. Proc. Workshop on Fluid Dynamical Approaches to the Many-Body Problem: Fundamental and Mathematical Aspects. Societa Italiana di Fisica, (1984).Google Scholar
  41. [1.42]
    H. Neunzert: An Introduction to the Nonlinear Boltzmann-Vlasov Equation. In: Lecture Notes in Math., Vol. 1048. Springer-Verlag, Berlin (1984).Google Scholar
  42. [1.43]
    B. Niclot, P. Degond, F. Poupaud: Deterministic Particle Simulations of the Boltzmann Transport Equation of Semiconductors. J. Comp. Phys. 78, 313–350 (1988).MATHCrossRefGoogle Scholar
  43. [1.44]
    F. Poupaud: On a System of Nonlinear Boltzmann Equations of Semiconductor Physics. SIAM J. Math. Anal, (to appear).Google Scholar
  44. [1.45]
    M. Reed, B. Simon: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1972).MATHGoogle Scholar
  45. [1.46]
    M. Reed, B. Simon: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, New York (1975).MATHGoogle Scholar
  46. [1.47]
    M. Reed, B. Simon: Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, New York (1978).MATHGoogle Scholar
  47. [1.48]
    L. Reggiani (ed.): Hot Electron Transport in Semiconductors. Springer-Verlag, Berlin (1985).Google Scholar
  48. [1.49]
    C. Ringhofer: A Spectral Method for Numerical Simulation of Quantum Tunneling Phenomena. SIAM J. Num. Anal, (to appear).Google Scholar
  49. [1.50]
    W. Rudin: Real and Complex Analysis, 2nd ed. McGraw-Hill, New York (1974).MATHGoogle Scholar
  50. [1.51]
    S. Selberherr: Analysis and Simulation of Semiconductor Devices. Springer-Verlag, Wien-New York (1984).Google Scholar
  51. [1.52]
    A. Shubin: Pseudodifferential Operators and Spectral Theory. Springer-Verlag, New York (1986).Google Scholar
  52. [1.53]
    H. Steinrück: Asymptotic Analysis of the Quantum Liouville Equation. Report, Inst. f. Ang. u. Num. Mathematik, TU-Wien, Austria (1988).Google Scholar
  53. [1.54]
    Η. Steinrück: The Wigner-Poisson Problem in a Crystal: Existence, Uniqueness, Semiclassical Limit in the One-Dimensional Case. Report, Inst. f. Ang. u. Num. Mathematik, TU-Wien, Austria (1989).Google Scholar
  54. [1.55]
    H. Steinrück: The One-Dimensional Wigner Poisson Problem and Its Relation to the Schrödinger Poisson Problem. Report, Inst. f. Ang. u. Num. Math., TU-Wien, Austria (1989).Google Scholar
  55. [1.56]
    H. Steinrück: Private communication (1989).Google Scholar
  56. [1.57]
    S. M. Sze: Physics of Semiconductor Devices. J. Wiley & Sons, New York (1981).Google Scholar
  57. [1.58]
    V. I. Tatarskii: The Wigner Representation of Quantum Mechanics, Sov. Phys. Usp. 26, 311–327 (1983).MathSciNetCrossRefGoogle Scholar
  58. [1.59]
    M. E. Taylor: Pseudodifferential Operators. Princeton University Press, Princeton (1981).MATHGoogle Scholar
  59. [1.60]
    F. Treves: Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1: Pseudodifferential Operators. Plenum Press, New York (1980).MATHGoogle Scholar
  60. [1.61]
    F. Treves: Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 2: Fourier Integral Operators. Plenum Press, New York (1980).MATHGoogle Scholar
  61. [1.62]
    S. Ukai, T. Okabe: On the Classical Solution in the Large in Time of the Two-Dimensional Vlasov Equation. Osaka J. of Math. 15, 245–261 (1978).MathSciNetMATHGoogle Scholar
  62. [1.63]
    E. Wigner: On the Quantum Correction for Thermodynamic Equilibrium. Physical Review 40, 749–759 (1932).MATHCrossRefGoogle Scholar
  63. [1.64]
    J. Yvon: La Théorie Statistique des Fluides (Actualités Scientifiques et Industrielles, No. 203). Hermann, Paris (1935).Google 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

Personalised recommendations