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Structural properties of the one-dimensional drift-diffusion models for semiconductors

Author: Fatiha Alabau
Journal: Trans. Amer. Math. Soc. 348 (1996), 823-871
MSC (1991): Primary 35G30, 35J25, 35B50
MathSciNet review: 1329526
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Abstract: This paper is devoted to the analysis of the one-dimensional current and voltage drift-diffusion models for arbitrary types of semiconductor devices and under the assumption of vanishing generation recombination. We show in the course of this paper, that these models satisfy structural properties, which are due to the particular form of the coupling of the involved systems. These structural properties allow us to prove an existence and uniqueness result for the solutions of the current driven model together with monotonicity properties with respect to the total current $I$, of the electron and hole current densities and of the electric field at the contacts. We also prove analytic dependence of the solutions on $I$. These results allow us to establish several qualitative properties of the current voltage characteristic. In particular, we give the nature of the (possible) bifurcation points of this curve, we show that the voltage function is an analytic function of the total current and we characterize the asymptotic behavior of the currents for large voltages. As a consequence, we show that the currents never saturate as the voltage goes to $\pm \infty$, contrary to what was predicted by numerical simulations by M. S. Mock (Compel. 1 (1982), pp. 165–174). We also analyze the drift-diffusion models under the assumption of quasi-neutral approximation. We show, in particular, that the reduced current driven model has at most one solution, but that it does not always have a solution. Then, we compare the full and the reduced voltage driven models and we show that, in general, the quasi-neutral approximation is not accurate for large voltages, even if no saturation phenomenon occurs. Finally, we prove a local existence and uniqueness result for the current driven model in the case of small generation recombination terms.

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  • F. Alabau. New uniqueness theorems for the one-dimensional drift-diffusion semiconductor device equations. Siam J. Math. Anal., 26:715–737, 1995.
  • F. Alabau. A uniqueness theorem for reverse-biased diodes. To appear in Applicable Anal.
  • F. Alabau. Analyse asymptotique et simulation numérique des équations des semi-conducteurs. PhD thesis, Université Paris 6, 1987.
  • Fatiha Alabau, Uniform asymptotic error estimates for semiconductor device and electrochemistry equations, Nonlinear Anal. 14 (1990), no. 2, 123–139. MR 1036203, DOI
  • Fatiha Alabau, A method for proving uniqueness theorems for the stationary semiconductor device and electrochemistry equations, Nonlinear Anal. 18 (1992), no. 9, 861–872. MR 1162478, DOI
  • Fatiha Alabau, Étude des modèles de dérive-diffusion à courant donné et à potentiel donné dans le cas des semi-conducteurs mono-dimensionnels, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 9, 885–890 (French, with English and French summaries). MR 1218281
  • E.A. Coddington and N. Levinson. Theory of ordinary differential equations. McGraw-Hill, New York, 1955.
  • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • Jacques Henry and Bento Louro, Singular perturbation theory applied to the electrochemistry equations in the case of electroneutrality, Nonlinear Anal. 13 (1989), no. 7, 787–801. MR 999329, DOI
  • Thomas Kerkhoven, On the one-dimensional current driven semiconductor equations, SIAM J. Appl. Math. 51 (1991), no. 3, 748–774. MR 1094517, DOI
  • Peter A. Markowich, The stationary semiconductor device equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. MR 821965
  • Peter A. Markowich and Christian Schmeiser, Uniform asymptotic representation of solutions of the basic semiconductor-device equations, IMA J. Appl. Math. 36 (1986), no. 1, 43–57. MR 984458, DOI
  • M.S. Mock. An example of nonuniqueness of stationary solutions in semiconductor device models. Compel, 1:165–174, 1982.
  • W.V. Van Roosbroeck. Theory of flow of electrons and holes in germanium and other semiconductors. Bell Syst. Techn. J., 29:560–607, 1950.
  • Isaak Rubinstein, Electro-diffusion of ions, SIAM Studies in Applied Mathematics, vol. 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1075016
  • Herbert Steinrück, A bifurcation analysis of the one-dimensional steady-state semiconductor device equations, SIAM J. Appl. Math. 49 (1989), no. 4, 1102–1121. MR 1005499, DOI
  • M. J. Ward, L. G. Reyna, and F. M. Odeh, Multiple steady-state solutions in a multijunction semiconductor device, SIAM J. Appl. Math. 51 (1991), no. 1, 90–123. MR 1089133, DOI

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Additional Information

Fatiha Alabau
Affiliation: CeReMaB, Université Bordeaux I, 351, cours de la Libération 33405 Talence Cedex, France

Keywords: Semiconductor, electrochemistry, nonlinear system, elliptic, existence, uniqueness, monotonicity
Received by editor(s): October 23, 1993
Received by editor(s) in revised form: October 31, 1994
Article copyright: © Copyright 1996 American Mathematical Society