Skip to main content

Scattering Models

  • 2018 Accesses

Part of the Lecture Notes in Physics book series (LNP,volume 773)

The Vlasov (or Liouville) equation of the previous chapter does not take into account short-range particle interactions, like collisions of the particles with other particles or with the crystal lattice. In this chapter, we extend the Vlasov equation to include scattering mechanisms which leads to the Boltzmann equation. We present only a phenomenological derivation. For rigorous results, we refer to [1, Sect. 1.5.3] and [2, Chap. 4].

Keywords

  • Boltzmann Equation
  • Acoustic Phonon
  • Collision Operator
  • Vlasov Equation
  • Dirac Distribution

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-540-89526-8_4
  • Chapter length: 25 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-3-540-89526-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   89.99
Price excludes VAT (USA)
Hardcover Book
USD   89.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Babovsky. Die Boltzmann-Gleichung. Teubner, Stuttgart, 1998.

    MATH  Google Scholar 

  2. C. Cercignani, R. Illner, and M. Pulvirenti. The Mathematical Theory of Dilute Gases. Springer, New York, 1994.

    MATH  Google Scholar 

  3. L. Boltzmann. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte Akad. Wiss. Wien 66 (1872), 275–370. Translation: Further studies on the thermal equilibrium of gas molecules. In: S. Brush (ed.), Kinetic Theory, Vol. 2, 88–174. Pergamon Press, Oxford, 1966.

    Google Scholar 

  4. T. Carleman. Sur la théorie de l’équation intégro-différentielle de Boltzmann. Acta Mathematica 60 (1933), 91–146.

    CrossRef  MathSciNet  Google Scholar 

  5. L. Arkeryd. On the Boltzmann equation. Arch. Rat. Mech. Anal. 45 (1971), 1–34.

    MathSciNet  Google Scholar 

  6. R. DiPerna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130 (1989), 321–366.

    CrossRef  MathSciNet  Google Scholar 

  7. F. Golse, B. Perthame, P.-L. Lions, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988), 110–125.

    MATH  CrossRef  MathSciNet  Google Scholar 

  8. P. Gérard. Solutions globales du problème de Cauchy pour l’équation de Boltzmann (d’après R. Di Perna et P.-L. Lions). Séminaire Bourbaki, Vol. 1988–89, Astérisque 161–162, Exp. No. 699 (1989), 257–281.

    Google Scholar 

  9. P.-L. Lions. Global solutions of kinetic models and related problems. In: C. Cercignani and M. Pulvirenti (eds.), Nonequilibrium Problems in Many-Particle Systems, Lecture Notes in Math. 1551, 58–86. Springer, Berlin, 1992.

    Google Scholar 

  10. C. Villani. A review of mathematical topics in collisional kinetic theory. In: S. Friedlander and D. Serre (eds.), Handbook of Mathematical Fluid Dynamics, Vol. 1, 71–305. Elsevier, Amsterdam, 2002.

    CrossRef  Google Scholar 

  11. F. Poupaud. On a system of nonlinear Boltzmann equations of semiconductors physics. SIAM J. Appl. Math. 50 (1990), 1593–1606.

    MATH  CrossRef  MathSciNet  Google Scholar 

  12. F. Mustieles. Global existence of solutions for the nonlinear Boltzmann equation of semiconductor physics. Rev. Mat. Iberoamer. 6 (1990), 43–59.

    MATH  MathSciNet  Google Scholar 

  13. F. Mustieles. Global existence of weak solutions for a system of nonlinear Boltzmann equations in semiconductor physics. Math. Meth. Appl. Sci. 14 (1991), 139–153.

    MATH  CrossRef  MathSciNet  Google Scholar 

  14. H. Andréasson. Global existence of smooth solutions in three dimensions for the semiconductor Vlasov-Poisson-Boltzmann equation. Nonlin. Anal.: Theory Meth. Appl. 28 (1990), 1193–1211.

    CrossRef  Google Scholar 

  15. A. Majorana and S. Marano. Space homogeneous solutions to the Cauchy problem for semiconductor Boltzmann equations. SIAM J. Math. Anal. 28 (1997), 1294–1308.

    MATH  CrossRef  MathSciNet  Google Scholar 

  16. A. Majorana and S. Marano. On the Cauchy problem for spatially homogeneous semiconductor Boltzmann equations: existence and uniqueness. Annali Math. 184 (2005), 275–296.

    MATH  CrossRef  MathSciNet  Google Scholar 

  17. A. Majorana and C. Milazzo. Space homogeneous solutions of the linear semiconductor Boltzmann equation. J. Math. Anal. Appl. 259 (2001), 609–629.

    MATH  CrossRef  MathSciNet  Google Scholar 

  18. G. Bird. Molecular Gas Dynamics and Direct Simulation of Gas Flows. Clarendon Press, Oxford, 1994.

    Google Scholar 

  19. M. Fischetti and S. Laux. Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects. Phys. Rev. B 38 (1988), 9721–9745.

    CrossRef  ADS  Google Scholar 

  20. K. Nanbu. Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent gases. J. Phys. Soc. Japan 52 (1983), 2042–2049.

    Google Scholar 

  21. A. Gnudi, D. Ventura, and G. Baccarani. Modeling impact ionization in a BJT by means of a spherical harmonics expansion of the Boltzmann equation. IEEE Trans. Computer-Aided Design 12 (1993), 1706–1713.

    CrossRef  Google Scholar 

  22. N. Goldsman, L. Henrickson, and J. Frey. A physics-based analytical/numerical solution to the Boltzmann transport equation for use in device simulation. Solid State Electr. 34 (1991), 389–396.

    CrossRef  ADS  Google Scholar 

  23. C. Gray and H. Ralph. Solution of Boltzmann’s equation for semiconductors using a spherical harmonic expansion. J. Phys. C: Solid State Phys. 5 (1972), 55–62.

    CrossRef  ADS  Google Scholar 

  24. C. Buet. A discrete velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transp. Theory Stat. Phys. 25 (1996), 33–60.

    MATH  CrossRef  ADS  MathSciNet  Google Scholar 

  25. L. Pareschi and B. Perthame. A Fourier spectral method for homogeneous Boltzmann equations. Transp. Theory Stat. Phys. 25 (1996), 369–383.

    MATH  CrossRef  ADS  MathSciNet  Google Scholar 

  26. E. Gabetta, L. Pareschi, and G. Toscani. Relaxation schemes for nonlinear kinetic equations. SIAM J. Numer. Anal. 34 (1997), 2168–2194.

    MATH  CrossRef  MathSciNet  Google Scholar 

  27. C. Auer, A. Majorana, and F. Schürrer. Numerical schemes for solving the non-stationary Boltzmann–Poisson system for two-dimensional semiconductor devices. In: T. Goudon, E. Sonnendrücker, and D. Talay (eds.), ESAIM: Proceedings 15 (2005), 75–86.

    Google Scholar 

  28. M. Cáceres, J. A. Carrillo, and A. Majorana. Deterministic simulation of the Boltzmann–Poisson system in GaAs-based semiconductors. SIAM J. Sci. Comput. 27 (2006), 1981–2009.

    MATH  CrossRef  MathSciNet  Google Scholar 

  29. M. Galler and F. Schürrer. A direct multigroup-WENO solver for the 2D non-stationary Boltzmann–Poisson system for GaAs devices: GaAs-MESFET. J. Comput. Phys. 212 (2006), 778–797.

    MATH  CrossRef  ADS  Google Scholar 

  30. G. Ossig and F. Schürrer. Simulation of non-equilibrium electron transport in silicon quantum wires. I. Comput. Electr. 7 (2008), 367–370.

    Google Scholar 

  31. V. Aristov. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer, Dordrecht, 2001.

    MATH  Google Scholar 

  32. L. Pareschi. Computational methods and fast algorithms for Boltzmann equations. In: N. Bellomo (ed.), Lecture Notes on the Discretization of the Boltzmann Equation, Series Adv. Math. Appl. Sci. 63, Chapter 7. World Scientific, Singapore, 2003.

    Google Scholar 

  33. K. Brennan. The Physics of Semiconductors. Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  34. H. Grahn. Introduction to Semiconductor Physics. World Scientific, Singapore, 1999.

    MATH  Google Scholar 

  35. M. Lundstrom. Fundamentals of Carrier Transport. 2nd edition, Cambridge University Press, Cambridge, 2000.

    CrossRef  Google Scholar 

  36. K. Seeger. Semiconductor Physics. An Introduction. Springer, Berlin, 2004.

    Google Scholar 

  37. V. Gantmakher and Y. Levinson. Carrier Scattering in Metals and Semiconductors. North Holland, New York, 1987.

    Google Scholar 

  38. B. Ridley. Quantum Processes in Semiconductors. Clarendon Press, Oxford, 1982.

    Google Scholar 

  39. W. Zawadzki. Mechanics of electron scattering in semiconductors. In: T. Moss (ed.), Handbook of Semiconductors, Vol. 1, Chapter 12. North-Holland, New York, 1982.

    Google Scholar 

  40. W. Wenckebach. Essentials of Semiconductor Physics. John Wiley & Sons, Chichester, 1999.

    Google Scholar 

  41. N. Ben Abdallah and P. Degond. On a hierarchy of macroscopic models for semiconductors. J. Math. Phys. 37 (1996), 3308–3333.

    ADS  MathSciNet  Google Scholar 

  42. L. Reggiani (ed.). Hot Electron Transport in Semiconductors. Springer, Berlin, 1985.

    Google Scholar 

  43. C. Kittel. Introduction to Solid State Physcis. 7th edition, John Wiley & Sons, New York, 1996.

    Google Scholar 

  44. N. Ben Abdallah, P. Degond, and S. Génieys. An energy-transport model for semiconductors derived from the Boltzmann equation. J. Stat. Phys. 84 (1996), 205–231.

    MATH  CrossRef  ADS  Google Scholar 

  45. N. Ashcroft and N. Mermin. Solid State Physics. Sanners College, Philadelphia, 1976.

    Google Scholar 

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

    CrossRef  ADS  Google Scholar 

  47. F. Poupaud. Mathematical theory of kinetic equations for transport modelling in semiconductors. In: B. Perthame (ed.), Advances in Kinetic Theory and Computing: Selected Papers, Ser. Adv. Math. Appl. Sci. 22, 141–168. World Scientific, Singapore, 1994.

    Google Scholar 

  48. Y. Sone. Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston, 2002.

    MATH  Google Scholar 

  49. P. Markowich, C. Ringhofer, and C. Schmeiser. Semiconductor Equations. Springer, Vienna, 1990.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ansgar Jüngel .

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Jüngel, A. (2009). Scattering Models. In: Transport Equations for Semiconductors. Lecture Notes in Physics, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89526-8_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-89526-8_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-89525-1

  • Online ISBN: 978-3-540-89526-8

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)