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Journal of Computational Electronics

, Volume 14, Issue 3, pp 773–787 | Cite as

Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi–Dirac statistic functions

  • K. Gärtner
Article

Abstract

If in the classic van Roosbroeck system (Bell Syst Tech J 29:560–607, 1950) the statistic function is modified, the equations can be derived by a variational formulation or just using a generalized Einstein relation. In both cases a dissipative generalization of the Scharfetter–Gummel scheme (IEEE Trans Electr Dev 16, 64–77, 1969), understood as a one-dimensional constant current approximation, is derived for strictly monotone coefficient functions in the elliptic operator \(\nabla \cdot { {f}(v)} \nabla \), v chemical potential, while the hole density is defined by \(p={\mathcal {F}}(v)\le e^v.\) A closed form integration of the governing equation would simplify the practical use, but mean value theorem based results are sufficient to prove existence of bounded discrete steady state solutions on any boundary conforming Delaunay grid. These results hold for any piecewise, continuous, and monotone approximation of \({ {f}(v)}\) and \({\mathcal {F}}(v)\). Hence an implementation based on this discretization will inherit the same stability properties as the Boltzmann case based on the Scharfetter–Gummel scheme. Large chemical potentials and and related degeneracy effects in semiconductors can be approximated. A proven, stability focused blueprint for the discretization of a fairly general, steady state Fermi–Dirac like drift–diffusion setting for semiconductors using mainly new results to extend classic ideas is the main goal.

Keywords

Generalized Scharfetter–Gummel scheme Fermi–Dirac statistics Generalized Einstein relation  Dissipativity  Bounded discrete steady state solutions Unique thermodynamic equilibrium Degenerate semiconductors 

Mathematics Subject Classification

65N08 65N12 35J55 

Notes

Acknowledgments

The author thanks A. Glitzky and J. A. Griepentrog for very helpful discussions, H. Doan for the close to rounding error Polylog data used for the approximations, and T. Koprucki for the pointer to [19]. The reviewer comments are acknowledged, too. Their more outside point of view was very helpful to improve the readability.

References

  1. 1.
    Van Roosbroeck, W.: Theory of flow of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29, 560–607 (1950)CrossRefGoogle Scholar
  2. 2.
    Scharfetter, D.L.: Large-signal analysis of a silicon read diode oscillator. IEEE Trans. Electr. Dev 16, 64–77 (1969)CrossRefGoogle Scholar
  3. 3.
    Becker, Julian, Gärtner, Klaus, Klanner, Robert, Richter, Rainer: Simulation and experimental study of plasma effects in planar silicon sensors. Nucl. Instrum. Methods Phys. Res. Sect. A 624(3), 716–727 (2010)CrossRefGoogle Scholar
  4. 4.
    Pasveer, W.F., et al.: Unified description of charge-carrier mobilities in disordered semiconducting polymers. Phys. Rev. Lett. 94, 206601 (2005)CrossRefGoogle Scholar
  5. 5.
    Gajewski, H., Gärtner, K.: A dissipative discretization scheme for a nonlocal phase segregation model. Z. Angew. Math. Mech. 85, 815–822 (2005)CrossRefMATHGoogle Scholar
  6. 6.
    Gajewski, H., Gröger, K.: Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi–Dirac statistics. Math. Nachr. 140, 7–36 (1989)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gajewski, H., Gärtner, K.: On the discretization of van Roosbroeck’s equations with magnetic field. Z. Angew. Math. Mech. 76, 247–264 (1996)CrossRefMATHGoogle Scholar
  8. 8.
    Gajewski, H., Gröger, K.: Reaction–diffusion processes of electrically charged species. Math. Nachr. 177, 109–130 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gajewski, H., Albinus, G., Hünlich, R.: Thermodynamic design of energy models of semiconductor devices. Nonlinearity 15, 367–383 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Glitzky, A., Gärtner, K.: Existence of bounded steady state solutions to spin-polarized drift–diffusion systems. SIAM J. Math. Anal. 41, 2489–2513 (2010)CrossRefMATHGoogle Scholar
  11. 11.
    Glitzky, A., Gärtner, K.: Energy estimates for continuous and discretized electro-reaction–diffusion systems. Nonlinear Anal. 70, 788–805 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gärtner, K.: Existence of bounded discrete steady-state solutions of the van Roosbroeck system on boundary conforming delaunay grids. SIAM J. Sci. Comput. 31(2), 1347–1362 (2009)CrossRefMATHGoogle Scholar
  13. 13.
    Griepentrog, J.A.: On regularity, positivity and long-time behavior of solutions to an evolution system of nonlocally interacting particles. Report, WIAS, Preprint No. 1932 (2014)Google Scholar
  14. 14.
    Liero, M., Mielke, A.: Gradient structures and geodesic convexity for reaction-diffusion systems. Phil. Trans. R. Soc. A 371, 20120346 (2013). (28 pages) Google Scholar
  15. 15.
    Bonč-Bruevič, V.L., Kalashnikov, S.G.: Halbleiterphysik. Deutscher Verlag der Wissenschaften, Berlin (1982)CrossRefGoogle Scholar
  16. 16.
    Delaunay, B.: Sur La Sphére Vide. Izvestia Akademii Nauk SSSR. Otd. Matem. i Estestv. Nauk 7, 793–800 (1934)Google Scholar
  17. 17.
    Si, H.: Three dimensional boundary conforming delaunay mesh generation. PhD thesis, TU Berlin (2008)Google Scholar
  18. 18.
    Zeidler, E.: Vorlesung über nichtlineare Funktionalanalysis I—Fixpunktsätze. Teubner, Leipzig (1976). in GermanGoogle Scholar
  19. 19.
    Blakemore, J.S.: Approximations for Fermi–Dirac integrals, especially the function \({\cal F}_{1/2}(\eta )\) used to describe electron density in a semiconductor. Solid State Electron. 25, 1067–1076 (1982)CrossRefGoogle Scholar
  20. 20.
    Gradstein, I.S., Ryzhik, I.M.: Tables of Integrals, Sums, Series, and Products. Fizmatgiz, Moskow (1962). in RussianGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.ICSUniversitá della Svizzera italianaLuganoSwitzerland

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