Journal of Statistical Physics

, Volume 130, Issue 2, pp 313–342 | Cite as

A Two-Surface Problem of the Electron Flow in a Semiconductor on the Basis of Kinetic Theory



A steady flow of electrons in a semiconductor between two parallel plane Ohmic contacts is studied on the basis of the semiconductor Boltzmann equation, assuming a relaxation-time collision term, and the Poisson equation for the electrostatic potential. A systematic asymptotic analysis of the Boltzmann–Poisson system for small Knudsen numbers (scaled mean free paths) is carried out in the case where the Debye length is of the same order as the distance between the contacts and where the applied potential is of the same order as the thermal potential. A system of drift-diffusion-type equations and their boundary conditions is obtained up to second order in the Knudsen number. A numerical comparison is made between the obtained system and the original Boltzmann–Poisson system.


Semiconductor Boltzmann equation Hilbert expansion Drift-diffusion equations Second-order boundary conditions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Van Roosbroeck, W.: Theory of flow of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29, 560 (1950) Google Scholar
  2. 2.
    Bløtekjær, K.: Transport equations for electrons in two-valley semiconductors. IEEE Trans. Electron. Devices 17, 38 (1970) Google Scholar
  3. 3.
    Stratton, R.: Diffusion of hot and cold electrons in semiconductor barriers. Phys. Rev. 126, 2002 (1962) CrossRefADSGoogle Scholar
  4. 4.
    Ben Abdallah, N., Degond, P.: On a hierarchy of macroscopic models for semiconductors. J. Math. Phys. 37, 3306 (1996) MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Ben Abdallah, N., Degond, P., Génieys, S.: An energy-transport model for semiconductors derived from the Boltzmann equation. J. Stat. Phys. 84, 205 (1996) MATHCrossRefGoogle Scholar
  6. 6.
    Degond, P., Génieys, S., Jüngel, A.: A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects. J. Math. Pures Appl. 76, 991 (1997) MATHMathSciNetGoogle Scholar
  7. 7.
    Ben Abdallah, N., Desvillettes, L., Génieys, S.: On the convergence of the Boltzmann equation for semiconductors toward the energy transport model. J. Stat. Phys. 98, 835 (2000) MATHCrossRefGoogle Scholar
  8. 8.
    Degond, P., Jüngel, A., Pietra, P.: Numerical discretization of energy-transport models for semiconductors with non-parabolic band structure. SIAM J. Sci. Comput. 22, 986 (2000) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Anile, A., Muscato, O.: Improved hydrodynamic model for carrier transport in semiconductors. Phys. Rev. B 51, 16728 (1995) CrossRefADSGoogle Scholar
  10. 10.
    Grasser, T., Kosina, H., Gritsch, M.: Using six moments of Boltzmann’s transport equation for device simulation. J. Appl. Phys. 90, 2389 (2001) CrossRefADSGoogle Scholar
  11. 11.
    Yamnahakki, A.: Second order boundary conditions for the drift-diffusion equations for semiconductors. Math. Models Methods Appl. Sci. 5, 429 (1995) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cercignani, C., Gamba, I.M., Levermore, C.D.: A drift-collision balance for a Boltzmann–Poisson system in bounded domains. SIAM J. Appl. Math. 61, 1932 (2001) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ringhofer, C., Schmeiser, C., Zwirchmayr, A.: Moment methods for the semiconductor Boltzmann equation on bounded position domains. SIAM J. Numer. Anal. 39, 1078 (2001) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Baranger, H.U., Wilkins, J.W.: Ballistic structure in the electron distribution function of small semiconducting structures: General features and specific trends. Phys. Rev. B 36, 1487 (1987) CrossRefADSGoogle Scholar
  15. 15.
    Baranger, H.U., Wilkins, J.W.: Phys. Rev. B 30, 7349 (1984) CrossRefADSGoogle Scholar
  16. 16.
    Trugman, S.A., Taylor, A.J.: Analytic solution of the Boltzmann equation with applications to electrons transport in inhomogeneous semiconductors. Phys. Rev. B 33, 5575 (1986) CrossRefADSGoogle Scholar
  17. 17.
    Kuhn, T., Mahler, G.: Carrier kinetics in a surface-excited semiconductor slab: Influence of boundary conditions. Phys. Rev. B 35, 2827 (1987) CrossRefADSGoogle Scholar
  18. 18.
    Sano, N.: Kinetic study of velocity distributions in nanoscale semiconductor devices under room-temperature operation. Appl. Phys. Lett. 85, 4208 (2004) CrossRefADSGoogle Scholar
  19. 19.
    Csontos, D., Ulloa, S.E.: Quasiballistic, nonequilibrium electron distribution in inhomogeneous semiconductor structures. Appl. Phys. Lett. 86, 253103 (2005) CrossRefGoogle Scholar
  20. 20.
    Poupaud, F.: Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers. Asymptot. Anal. 4, 293 (1991) MATHMathSciNetGoogle Scholar
  21. 21.
    Golse, F., Poupaud, F.: Limite fluide des équations de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac. Asymptot. Anal. 6, 135 (1992) MATHMathSciNetGoogle Scholar
  22. 22.
    Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284, 617 (1984) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Degond, P., Schmeiser, C.: Kinetic boundary layers and fluid-kinetic coupling in semiconductors. Trans. Theory Stat. Phys. 28, 31 (1999) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sone, Y.: Asymptotic theory of flow of rarefied gas over a smooth boundary I. In: Trilling, L., Wachman, H.Y. (eds.) Rarefied Gas Dynamics, vol. 1, p. 243. Academic Press, New York (1969) Google Scholar
  25. 25.
    Sone, Y., Yamamoto, K.: Flow of rarefied gas over plane wall. J. Phys. Soc. Jpn. 29, 495 (1970) CrossRefADSGoogle Scholar
  26. 26.
    Sone, Y., Onishi, Y.: J. Phys. Soc. Jpn. 47, 672 (1979) CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Sone, Y.: Asymptotic theory of flow of rarefied gas over a smooth boundary II. In: Dini, D. (ed.) Rarefied Gas Dynamics, vol. 2, p. 737. Editrice Tecnico Scientfica, Pisa (1971) Google Scholar
  28. 28.
    Sone, Y.: Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers. In: Gatignol, R., Soubbaramayer. (eds.) Advances in Kinetic Theory and Continuum Mechanics, p. 19. Springer, Berlin (1991) Google Scholar
  29. 29.
    Sone, Y., Aoki, K., Takata, S., Sugimoto, H., Bobylev, A.V.: Inappropriateness of the heat-conduction equation for description of a temperature field of a stationary gas in the continuum limit: Examination by asymptotic analysis and numerical computation of the Boltzmann equation. Phys. Fluids 8, 628 (1996). Erratum 8, 841 (1996) MATHCrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Sone, Y., Bardos, C., Golse, F., Sugimoto, H.: Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number: General theory. Eur. J. Mech. B/Fluids 19, 325 (2000) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Sone, Y.: Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston (2002) MATHGoogle Scholar
  32. 32.
    Sone, Y.: Molecular Gas Dynamics: Theory, Techniques, and Applications. Birkhäuser, Boston (2006) Google Scholar
  33. 33.
    Aoki, K., Takata, S., Nakanishi, T.: Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force. Phys. Rev. E 65, 026315 (2002) CrossRefADSGoogle Scholar
  34. 34.
    Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511 (1954) MATHCrossRefADSGoogle Scholar
  35. 35.
    Welander, P.: On the temperature jump in a rarefied gas. Ark. Fys. 7, 507 (1954) MathSciNetGoogle Scholar
  36. 36.
    Kogan, M.N.: On the equations of motion of a rarefied gas. Appl. Math. Mech. 22, 597 (1958) MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Chu, C.K.: Kinetic-theoretic description of the formation of a shock wave. Phys. Fluids 8, 12 (1965) CrossRefGoogle Scholar
  38. 38.
    Aoki, K., Nishino, K., Sone, Y., Sugimoto, H.: Numerical analysis of steady flows of a gas condensing on or evaporating from its plane condensed phase on the basis of kinetic theory: Effect of gas motion along the condensed phase. Phys. Fluids A 3, 2260 (1991) MATHCrossRefADSGoogle Scholar
  39. 39.
    Ben Abdallah, N., Degond, P.: The Child–Langmuir law for the Boltzmann equation of semiconductors. SIAM J. Math. Anal. 26, 364 (1995) MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Poupaud, F.: Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory. Z. Angew. Math. Mech. 72, 359 (1992) MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Cercignani, C., Gamba, I., Levermore, C.: High field approximations to a Boltzmann–Poisson system and boundary conditions in a semiconductor. Appl. Math. Lett. 10, 111 (1997) MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Willis, D.R.: Comparison of kinetic theory analyses of linearized Couette flow. Phys. Fluids 5, 127 (1962) MATHCrossRefGoogle Scholar
  43. 43.
    Sone, Y.: Kinetic theory analysis of linearized Rayleigh problem. J. Phys. Soc. Jpn. 19, 1463 (1964) CrossRefADSGoogle Scholar
  44. 44.
    Sone, Y.: Some remarks on Knudsen layer. J. Phys. Soc. Jpn. 21, 1620 (1966) CrossRefADSGoogle Scholar
  45. 45.
    Tamada, K., Sone, Y.: Some studies on rarefied gas flows. J. Phys. Soc. Jpn. 21, 1439 (1966) CrossRefADSGoogle Scholar
  46. 46.
    Sone, Y.: Thermal creep in rarefied gas. J. Phys. Soc. Jpn. 21, 1836 (1966) CrossRefADSGoogle Scholar
  47. 47.
    Sone, Y., Onishi, Y.: Kinetic theory of evaporation and condensation—hydrodynamic equation and slip boundary condition. J. Phys. Soc. Jpn. 44, 1981 (1978) CrossRefADSGoogle Scholar
  48. 48.
    Sone, Y., Yamamoto, K.: Flow of rarefied gas through a circular pipe. Phys. Fluids 11, 1672 (1968). Erratum 13, 1651 (1970) MATHCrossRefGoogle Scholar
  49. 49.
    Sone, Y.: Effect of sudden change of wall temperature in rarefied gas. J. Phys. Soc. Jpn. 20, 222 (1965) CrossRefADSGoogle Scholar
  50. 50.
    Sone, Y., Onishi, Y.: Kinetic theory of evaporation and condensation. J. Phys. Soc. Jpn. 35, 1773 (1973) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Organization of Advanced Science and TechnologyKobe UniversityKobeJapan
  2. 2.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria

Personalised recommendations