Finite element solution of the fundamental equations of semiconductor devices. I
Author:
Miloš Zlámal
Journal:
Math. Comp. 46 (1986), 2743
MSC:
Primary 65N30
MathSciNet review:
815829
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Abstract: We investigate the nonstationary equations of the semiconductor device theory consisting of a Poisson equation for the electric potential and of two highly nonlinear continuity equations for carrier densities n and p. We use simplicial elements with linear polynomials and fournode twodimensional and eightnode threedimensional isoparametric elements. There are constructed finite element solutions such that the current densities , and the electric field strength are constant on each element. Two schemes are proposed: one is nonlinear, the other is partly linear. The schemes preserve the property of the exact solution (corresponding to the physical meaning) that the carrier densities n and p are positive. Existence of the solution is proved in both cases, unicity in the second case. A subsequent paper II will be devoted to problems of stability and convergence.
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 [2]
 H. Gajewski, "On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors," Z. Angew. Math. Mech., v. 65, 1985, pp. 101108. MR 841263 (87k:35216)
 [3]
 V. Girault & P. A. Raviart, Finite Element Approximation of the NavierStokes Equations, SpringerVerlag, Berlin and New York, 1979. MR 548867 (83b:65122)
 [4]
 P. A. Markowich, "A singular perturbation analysis of the fundamental semiconductor device equations," SIAM J. Appl. Math., v. 44, 1984, pp. 896928. MR 759704 (86e:78024)
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 M. S. Mock, "An initial value problem from semiconductor device theory," SIAM J. Math. Anal., v. 5, 1974, pp. 597612. MR 0417573 (54:5623)
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 M. S. Mock, Analysis of Mathematical Models of Semiconductor Devices, Boole Press, Dublin, 1983. MR 697094 (84m:78002)
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 J. M. Ortega & W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
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 D. L. Scharfetter & H. K. Gummel, "Large signal analysis of a silicon Read diode oscillator," IEEE Trans. Electron. Devices, v. ED16, 1969, pp. 6477.
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 O. C. Zienkiewicz, The Finite Element Method, McGrawHill, London, 1977.
 [10]
 M. Zlámal, "A finite element solution of the nonlinear heat equation," RAIRO Anal. Numér., v. 14, 1980, pp. 203216. MR 571315 (81f:65089)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608158296
PII:
S 00255718(1986)08158296
Article copyright:
© Copyright 1986
American Mathematical Society
