Skip to main content
SpringerLink
Log in

Maximum entropy moment system of the semiconductor Boltzmann equation using Kane’s dispersion relation

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

It is known that the maximum entropy moment systems of the gas-dynamical Boltzmann equation suffer from severe disadvantages which are related to the non-solvability of an underlying maximum entropy moment problem unless restrictions on the choice of the macroscopic variables are made. In this article, we show that no such difficulties appear in the semiconductor case if Kane’s dispersion relation is used for the energy band of electrons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anile, A.M., Mascali, G., Romano, V.: Recent developments in hydrodynamical modeling of semiconductors (2003) 1, 54 in Mathematical Problems in Semiconductor Physics, Lecture Notes in Mathematics 1832. Springer (2003)

  2. Shannon, C.E.: Bell System Tech. J. 27, 379, 623, also reprinted in Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1949)

    Google Scholar 

  3. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)

    Article  Google Scholar 

  4. Dreyer, W.: Maximization of the entropy in non–equilibrium. J. Phys. A: Math. Gen. 20, 6505–6517 (1987)

    Article  Google Scholar 

  5. Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996)

    Google Scholar 

  6. Junk, M.: Domain of definition of Levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)

    Article  Google Scholar 

  7. Junk, M.: Maximum entropy for reduced moment problems. Math. Models Methods Appl. Sci. 10, 1001–1025 (2000)

    Google Scholar 

  8. Junk, M.: Moment Problems in Kinetic Theory. Universität Kaiserslautern, Habilitationsschrift (2001)

    Google Scholar 

  9. Markowich, P., Ringhofer, C., Schmeiser, C.: Semiconductor Equations. Springer, Wien (1990)

    Google Scholar 

  10. Ashcroft, N.C., Mermin, N.D.: Solid State Physics. Philadelphia, Holt-Sounders (1976)

    Google Scholar 

  11. Jacoboni, C., Reggiani, L.: The Monte Carlo method for the solution of charge transport in semiconductors with application to covalent materials. Review Modern Phys. 55, 645–705 (1983)

    Article  Google Scholar 

  12. Jacoboni, C., Lugli, P.: The Monte Carlo Method for Semiconductor Device Simulation. Vienna–New York, Springer-Berlin Heidelberg, New York (1989)

    Google Scholar 

  13. Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. The structure of extended thermodynamics. Cont. Mech. Thermodyn. 1, 3–20 (1989)

    Article  Google Scholar 

  14. Anile, A.M., Romano, V.: Non parabolic band transport in semiconductors: closure of the moment equations. Cont. Mech. Thermodyn. 11, 307–325 (1999)

    Article  Google Scholar 

  15. Eu, B.C.: A modified moment method and irreversible thermodynamics. J. Chem. Phys. 73, 2958–2969 (1980)

    Article  Google Scholar 

  16. Gorban, A.N., Karlin, I.V.: Method of invariant manifolds and the regularization of acoustic spectra. Transp. Theory Stat. Phys. 23, 559–632 (1994)

    Google Scholar 

  17. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer Berlin Heidelberg, New York (1998)

    Google Scholar 

  18. Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer-Verlag, Berlin (1993)

    Google Scholar 

  19. Anile, A.M., Romano, V., Russo, G.: Extended hydrodynamical model of carrier transport in semiconductors. SIAM J. Appl. Math. 61, 74–101 (2000)

    Article  Google Scholar 

  20. Romano, V.: Non parabolic band transport in semiconductors: closure of the production terms in the moment equations. Cont. Mech. Thermodyn. 12, 31–51 (2000)

    Article  Google Scholar 

  21. Romano, V.: Non parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices. Math. Meth. Appl. Sci. 24, 439–471 (2001)

    Article  Google Scholar 

  22. Mascali, G., Romano, V.: Hydrodynamical model of charge transport in GaAs based on the maximum entropy principle. Cont. Meh. Thermodyn. 14, 405–423 (2002)

    Article  Google Scholar 

  23. Majorana, A.: Conservation laws from the Boltzmann equation describing electron-phonon interactions in semiconductors. Transp. Theory Stat. Phys. 22, 849–859 (1993)

    Google Scholar 

  24. Majorana, A.: Equilibrium solutions of the non-linear Boltzmann equation for an electron gas in a semiconductors. Il Nuovo Cimento 108B, 871–877 (1993)

    Google Scholar 

  25. Romano, V.: Maximum entropy principle for electron transport in semiconductors. In: Proceedings InternationalConference WASCOM 99. Vulcano (Eolie Isles) Italy (1999)

  26. Dreyer, W., Junk, M., Kunik, M.: On the approximation of the Fokker-Planck equation by moment systems. Nonlinearity 14, 881–906 (2001)

    Article  Google Scholar 

  27. Junk, M., Unterreiter, A.: Maximum entropy moment systems and Galilean invariance. Continuum Mech. Thermodyn. 14, 563–576 (2002)

    Article  Google Scholar 

  28. Csiszár, I.: I-divergence geometry of probabaility distributions and minimization problems. Ann. of Prob. 3, 146–158 (1975)

    Google Scholar 

  29. Lewis, A.S.: Consistency of moment systems. Can. J. Math. 47, 995–1006 (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FB Mathematik, Universitäat Kaiserslautern, Erwin-Schrödingerstraße, 67663, Kaiserslautern, Germany

    Michael Junk

  2. Dipartimento di Matematica e Informatica, Università di Catania, viale A. Doria 6, 95125, Catania, Italy

    Vittorio Romano

Authors
  1. Michael Junk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vittorio Romano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vittorio Romano.

Additional information

Communicated by H. Struchtrup

PACS 73.50-h; 73.61.-r

Rights and permissions

Reprints and permissions

About this article

Cite this article

Junk, M., Romano, V. Maximum entropy moment system of the semiconductor Boltzmann equation using Kane’s dispersion relation. Continuum Mech. Thermodyn. 17, 247–267 (2005). https://doi.org/10.1007/s00161-004-0201-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-004-0201-5

Search

Navigation