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

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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.

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  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)

  2. 2.

    Scharfetter, D.L.: Large-signal analysis of a silicon read diode oscillator. IEEE Trans. Electr. Dev 16, 64–77 (1969)

  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)

  4. 4.

    Pasveer, W.F., et al.: Unified description of charge-carrier mobilities in disordered semiconducting polymers. Phys. Rev. Lett. 94, 206601 (2005)

  5. 5.

    Gajewski, H., Gärtner, K.: A dissipative discretization scheme for a nonlocal phase segregation model. Z. Angew. Math. Mech. 85, 815–822 (2005)

  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)

  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)

  8. 8.

    Gajewski, H., Gröger, K.: Reaction–diffusion processes of electrically charged species. Math. Nachr. 177, 109–130 (1996)

  9. 9.

    Gajewski, H., Albinus, G., Hünlich, R.: Thermodynamic design of energy models of semiconductor devices. Nonlinearity 15, 367–383 (2002)

  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)

  11. 11.

    Glitzky, A., Gärtner, K.: Energy estimates for continuous and discretized electro-reaction–diffusion systems. Nonlinear Anal. 70, 788–805 (2009)

  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)

  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)

  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)

  15. 15.

    Bonč-Bruevič, V.L., Kalashnikov, S.G.: Halbleiterphysik. Deutscher Verlag der Wissenschaften, Berlin (1982)

  16. 16.

    Delaunay, B.: Sur La Sphére Vide. Izvestia Akademii Nauk SSSR. Otd. Matem. i Estestv. Nauk 7, 793–800 (1934)

  17. 17.

    Si, H.: Three dimensional boundary conforming delaunay mesh generation. PhD thesis, TU Berlin (2008)

  18. 18.

    Zeidler, E.: Vorlesung über nichtlineare Funktionalanalysis I—Fixpunktsätze. Teubner, Leipzig (1976). in German

  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)

  20. 20.

    Gradstein, I.S., Ryzhik, I.M.: Tables of Integrals, Sums, Series, and Products. Fizmatgiz, Moskow (1962). in Russian

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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.

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Correspondence to K. Gärtner.

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Gärtner, K. Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi–Dirac statistic functions. J Comput Electron 14, 773–787 (2015).

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  • 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