Skip to main content

Drift-Diffusion Equations

  • Chapter
  • First Online:
Transport Equations for Semiconductors

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

This and the following chapters are concerned with the formal derivation of semi-classical macroscopic transport models from the semiconductor Boltzmann equation. We start in this chapter with the derivation of drift-diffusion equations, which are the simplest semiconductor model in the hierarchy. It was first derived by van Roosbroeck in 1950 [1]. We derive the model using the moment method introduced in Chap. 2. A derivation using a simple collision operator was presented in Sect. 2.4. In this chapter, we will employ the low-density operator (4.22). The derivation was made rigorous by Poupaud [2].

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. van Roosbroeck. Theory of flow of electron and holes in germanium and other semiconductors. Bell Syst. Techn. J. 29 (1950), 560–607.

    Google Scholar 

  2. F. Poupaud. Diffusion approximation of the linear semiconductor Boltzmann equation: analysis of boundary layers. Asympt. Anal. 4 (1991), 293–317.

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. F. Golse and F. Poupaud. Limite fluide des \’equations de Boltzmann des semiconducteurs pour une statistique de Fermi–Dirac. Asympt. Anal. 6 (1992), 135–160.

    MATH  MathSciNet  Google Scholar 

  5. E. Zeidler. Nonlinear Functional Analysis and Its Applications, Vol. II. Springer, New York, 1990.

    Google Scholar 

  6. N. Ben Abdallah and M. Tayeb. Diffusion approximation for the one dimensional Boltzmann–Poisson system. Discrete Contin. Dyn. Sys. B 4 (2004), 1129–1142.

    Article  MATH  MathSciNet  Google Scholar 

  7. N. Masmoudi and M. Tayeb. Diffusion limit of a semiconductor Boltzmann–Poisson system. SIAM J. Math. Anal. 38 (2007), 1788–1807.

    Article  MATH  MathSciNet  Google Scholar 

  8. F. Poupaud and C. Schmeiser. Charge transport in semiconductors with degeneracy effects. Math. Meth. Appl. Sci. 14 (1991), 301–318.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Markowich and C. Schmeiser. Relaxation time approximation for electron–phonon interaction in semiconductors. Math. Models Meth. Appl. Sci. 5 (1995), 519–527.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Markowich, F. Poupaud, and C. Schmeiser. Diffusion approximation of nonlinear electron phonon collision mechanisms. RAIRO Modél. Math. Anal. Numér. 29 (1995), 857–869.

    MATH  MathSciNet  Google Scholar 

  11. N. Ben Abdallah and M. Tayeb. Diffusion approximation and homogenization of the semiconductor Boltzmann equation. SIAM Multiscale Model. Simul. 4 (2005), 896–914.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. Poupaud and J. Soler. Parabolic limit and stability of the Vlasov-Fokker–Planck system. Math. Models Meth. Appl. Sci. 10 (2000), 1027–1045.

    MATH  MathSciNet  Google Scholar 

  13. H. Gummel. A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE Trans. Electr. Devices ED-11 (1964), 455–465.

    Article  Google Scholar 

  14. D. Scharfetter and H. Gummel. Large signal analysis of a silicon Read diode oscillator. IEEE Trans. Electr. Devices ED-16 (1969), 64–77.

    Article  Google Scholar 

  15. F. Brezzi, L. Marini, and P. Pietra. Méthodes d’éléments finis mixtes et schéma de Scharfetter–Gummel. C. R. Acad. Sci. Paris, Sér. I 305 (1987), 599–604.

    MATH  MathSciNet  Google Scholar 

  16. F. Brezzi, L. Marini, and P. Pietra. Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26 (1989), 1342–1355.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. C. Chainais-Hillairet, J.-G. Liu, and Y.-J. Peng. Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis. Math. Model. Numer. Anal. 37 (2003), 319–338.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Sacco and F. Saleri. Mixed finite volume methods for semiconductor device simulation. Numer. Meth. Part. Diff. Eqs. 13 (1997), 215–236.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Mock. On equations describing steady-state carrier distributions in a semiconductor device. Commun. Pure Appl. Math. 25 (1972), 781–792.

    Article  MathSciNet  Google Scholar 

  20. M. Mock. An initial value problem from semiconductor device theory. SIAM J. Math. Anal. 5 (1974), 597–612.

    Article  MATH  MathSciNet  Google Scholar 

  21. H. Gajewski and K. Gröger. On the basic equations for carrier transport in semiconductors. J. Math. Anal. Appl. 113 (1986), 12–35.

    Article  MATH  MathSciNet  Google Scholar 

  22. H. Gajewski and K. Gröger. Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi–Dirac statistics. Math. Nachr. 140 (1989), 7–36.

    Article  MATH  MathSciNet  Google Scholar 

  23. K. Gröger and J. Rehberg. Uniqueness for the two-dimensional semiconductor equations in case of high carrier densities. Math. Z. 213 (1993), 523–530.

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Jerome. Analysis of Charge Transport. A Mathematical Study of Semiconductor Devices. Springer, Berlin, 1996.

    Google Scholar 

  25. P. Markowich. The Stationary Semiconductor Device Equations. Springer, Vienna, 1986.

    Google Scholar 

  26. F. Brezzi, L. Marini, P. Markowich, and P. Pietra. On some numerical problems in semiconductor device simulation. In: G. Toscani, V. Boffi, and S. Rionero (eds.), Mathematical Aspects of Fluid and Plasma Dynamics (Salice Terme, 1988), Lecture Notes in Math. 1460, 31–42. Springer, Berlin, 1991.

    Chapter  Google Scholar 

  27. F. Brezzi, L. Marini, S. Micheletti, P. Pietra, R. Sacco, and S. Wang. Discretization of semiconductor device problems. In: W. Schilders and E. ter Maten, Handbook of Numerical Analysis, Vol. 13: Numerical Methods in Electromagnetics, 317–441. North-Holland, Amsterdam, 2005.

    Google Scholar 

  28. J. Jerome. The approximation problem for drift-diffusion systems. SIAM Rev. 37 (1995), 552–572.

    Article  MATH  MathSciNet  Google Scholar 

  29. W. Hänsch. The Drift-Diffusion Equation and Its Applications in MOSFET Modeling. Springer, Vienna, 1991.

    Google Scholar 

  30. S. Selberherr. Analysis and Simulation of Semiconductor Devices. Springer, Vienna, 1984.

    Google Scholar 

  31. A. Yamnahakki. Second order boundary conditions for the drift-diffusion equations of semiconductors. Math. Models Meth. Appl. Sci. 5 (1995), 429–455.

    Article  MATH  MathSciNet  Google Scholar 

  32. S. Taguchi and A. Jüngel. Kinetic theory of a two-surface problem of electron flow in a semiconductor. J. Stat. Phys. 130 (2007), 313–342.

    Article  ADS  Google Scholar 

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

    Google Scholar 

  34. M. Grundmann. The Physics of Semiconductors. Springer, Berlin, 2006.

    Google Scholar 

  35. T. Goudon, V. Miljanovic, and C. Schmeiser. On the Shockley–Read–Hall model: generation–recombination in semiconductors. SIAM J. Appl. Math. 67 (2007), 1183–1201.

    Article  MATH  MathSciNet  Google Scholar 

  36. F. Poupaud. Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory. Z. Angew. Math. Mech. 72 (1992), 359–372.

    Article  MATH  MathSciNet  Google Scholar 

  37. L. Arlotti and G. Frosali. Runaway particles for a Boltzmann-like transport equation. Math. Models Meth. Appl. Sci. 2 (1992), 203–221.

    Article  MATH  MathSciNet  Google Scholar 

  38. G. Frosali and C. Van der Mee. Scattering theory in the linear transport theory of particle swarms. J. Stat. Phys. 56 (1989), 139–148.

    Article  ADS  MathSciNet  Google Scholar 

  39. W. Hänsch and M. Miura-Mattausch. The hot-electron problem in small semiconductor devices. J. Appl. Phys. 60 (1986), 650–656.

    Article  ADS  Google Scholar 

  40. P. Degond and A. Jüngel. High-field approximations of the energy-transport model for semiconductors with non-parabolic band structure. Z. Angew. Math. Phys. 52 (2001), 1053–1070.

    Article  MATH  MathSciNet  Google Scholar 

  41. E. Kan, U. Ravaioli, and T. Kerkhoven. Calculation of velocity overshoot in submicron devices using an augmented drift-diffusion model. Solid State Electr. 34 (1991), 995–999.

    Article  ADS  Google Scholar 

  42. K. Thornber. Current equations for velocity overshoot. IEEE Electr. Device Letters 3 (1983), 6971.

    Google Scholar 

  43. N. Zakhleniuk. Nonequilibrium drift-diffusion transport in semiconductors in presence of strong inhomogeneous electric fields. Appl. Phys. Letters 89 (2006), 252112.

    Article  ADS  Google Scholar 

  44. C. Cercignani, I. Gamba, and C. Levermore. High-field approximations to a Boltzmann–Poisson system and boundary conditions in a semiconductor. Appl. Math. Letters 10 (1997), no 4, 111–117.

    Article  MATH  MathSciNet  Google Scholar 

  45. C. Cercignani, I. Gamba, and C. Levermore. A drift-collision balance for a Boltzmann–Poisson system in bounded domains. SIAM J. Appl. Math. 61 (2001), 1932–1958.

    Article  MATH  MathSciNet  Google Scholar 

  46. T. Goudon, J. Nieto, F. Poupaud, and J. Soler. Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker–Planck system. J. Diff. Eqs. 213 (2005), 418–442.

    Article  MATH  MathSciNet  Google Scholar 

  47. J. Nieto, F. Poupaud, and J. Soler. High-field limit for the Vlasov-Poisson-Fokker–Planck system. Arch. Rat. Mech. Anal. 158 (2001), 29–59.

    Article  MATH  MathSciNet  Google Scholar 

  48. N. Ben Abdallah and H. Chaker. The high field asymptotics for degenerate semiconductors. Math. Models Meth. Appl. Sci. 11 (2001), 1253–1272.

    Article  MATH  MathSciNet  Google Scholar 

  49. N. Ben Abdallah, H. Chaker, and C. Schmeiser. The high field asymptotics for a fermionic Boltzmann equation: entropy solutions and kinetic shock profiles. J. Hyperb. Diff. Eqs. 4 (2007), 679–704.

    Article  MATH  MathSciNet  Google Scholar 

  50. V. Bonch-Bruevich and S. Kalashnikov. Halbleiterphysik. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982.

    Google Scholar 

  51. J. Viallet and S. Mottet. Transient simulation of heterostructure. In: J. Miller et al. (eds.), NASECODE IV Conference Proceedings. Dublin, Boole Press, 1985.

    Google Scholar 

  52. A. Jüngel. Asymptotic analysis of a semiconductor model based on Fermi–Dirac statistics. Math. Meth. Appl. Sci. 19 (1996), 401–424.

    Article  MATH  Google Scholar 

  53. J. I. Díaz, G. Galiano, and A. Jüngel. On a quasilinear degenerate system arising in semiconductor theory. Part I: existence and uniqueness of solutions. Nonlin. Anal.: Real-World Appl. 2 (2001), 305–336.

    Article  MATH  Google Scholar 

  54. A. Jüngel. On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Models Meth. Appl. Sci. 4 (1994), 677–703.

    Article  MATH  Google Scholar 

  55. A. Jüngel. A nonlinear drift-diffusion system with electric convection arising in semiconductor and electrophoretic modeling. Math. Nachr. 185 (1997), 85–110.

    Article  MATH  MathSciNet  Google Scholar 

  56. A. Jüngel. Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. Z. Angew. Math. Mech. 75 (1995), 783–799.

    Article  MATH  MathSciNet  Google Scholar 

  57. A. Jüngel and P. Pietra. A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Models Meth. Appl. Sci. 7 (1997), 935–955.

    Article  MATH  Google Scholar 

  58. C. Chainais-Hillairet and F. Filbet. Asymptotic behavior of a finite volume scheme for the transient drift-diffusion model. IMA J. Numer. Anal. 27 (2007), 689–716.

    Article  MATH  MathSciNet  Google Scholar 

  59. C. Chainais-Hillairet and Y.-J. Peng. Finite volume approximation for degenerate drift-diffusion system in several space dimensions. Math. Models Meth. Appl. Sci. 14 (2004), 461–481.

    Article  MATH  MathSciNet  Google Scholar 

  60. P. Guan and B. Wu. Existence of weak solutions to a degenerate time-dependent semiconductor equations with temperature effects. J. Math. Anal. Appl. 332 (2007), 367–380.

    Article  MATH  MathSciNet  Google Scholar 

  61. R. Courant and K. Friedrichs. Supersonic Flow and Shock Waves. Interscience, New York, 1967.

    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). Drift-Diffusion Equations. In: Transport Equations for Semiconductors. Lecture Notes in Physics, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89526-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-89526-8_5

  • 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)

Publish with us

Policies and ethics