On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case

Alabau, Fatiha

doi:10.1090/S0002-9947-98-02334-4

On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case
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Trans. Amer. Math. Soc. 350 (1998), 4709-4756 Request permission

Abstract:

We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.

References

Mauro de Oliveira Cesar and Ângelo Barone Netto, The existence of Liapunov functions for some nonconservative positional mechanical systems, J. Differential Equations 91 (1991), no. 2, 235–244. MR 1111175, DOI 10.1016/0022-0396(91)90140-5
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 10.1016/0362-546X(92)90227-6
Fatiha Alabau, New uniqueness theorems for the one-dimensional drift-diffusion semiconductor device equations, SIAM J. Math. Anal. 26 (1995), no. 3, 715–737. MR 1325911, DOI 10.1137/S003614109018823X
Fatiha Alabau, Structural properties of the one-dimensional drift-diffusion models for semiconductors, Trans. Amer. Math. Soc. 348 (1996), no. 3, 823–871. MR 1329526, DOI 10.1090/S0002-9947-96-01519-X
—, Uniqueness results for the steady-state electrodiffusion equations in case of monotonic potentials and multiple junctions, Nonlinear Anal. 29 (1997), 849–887.
F. Alabau and M. Moussaoui, Asymptotic estimates for the multi-dimensional electro-diffusion equations, M3AS, To appear.
Paul C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal. 52 (1973), 205–232. MR 374665, DOI 10.1007/BF00247733
C. Bandle, J. Bemelmans, M. Chipot, M. Grüter, and J. Saint Jean Paulin (eds.), Progress in partial differential equations: elliptic and parabolic problems, Pitman Research Notes in Mathematics Series, vol. 266, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1194209
Robert E. O’Malley Jr., Introduction to singular perturbations, Applied Mathematics and Mechanics, Vol. 14, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0402217
Peter A. Markowich, The stationary semiconductor device equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. MR 821965, DOI 10.1007/978-3-7091-3678-2
M. S. Mock, On equations describing steady-state carrier distributions in a semiconductor device, Comm. Pure Appl. Math. 25 (1972), 781–792. MR 323233, DOI 10.1002/cpa.3160250606
—, An example of nonuniqueness of stationary solutions in semiconductor device models, Compel 1 (1982), 165–174.
Michael S. Mock, Analysis of mathematical models of semiconductor devices, Advances in Numerical Computation Series, vol. 3, Boole Press, Dún Laoghaire, 1983. MR 697094
W.V. Van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell Syst. Techn. J. 29 (1950), 560–607.
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, DOI 10.1137/1.9781611970814
Thomas I. Seidman, Steady-state solutions of diffusion-reaction systems with electrostatic convection, Nonlinear Anal. 4 (1980), no. 3, 623–633. MR 574378, DOI 10.1016/0362-546X(80)90097-8
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 10.1137/0149066
S.M. Sze, Physics of semiconductor devices, John Wiley & Sons, New York, 1969.