Finite element solution of the fundamental equations of semiconductor devices. I

Author:
Miloš Zlámal

Journal:
Math. Comp. **46** (1986), 27-43

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815829-6

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 four-node two-dimensional and eight-node three-dimensional 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.

**[1]**E. M. Buturla, P. E. Cottrell, B. M. Grossman & K. A. Salsburg, "Finite-element analysis of semiconductor devices: The FIELDAY PROGRAM,"*IBM J. Res. Develop.*, v. 25, 1981, pp. 218-231.**[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. 101-108. MR**841263 (87k:35216)****[3]**V. Girault & P. A. Raviart,*Finite Element Approximation of the Navier-Stokes Equations*, Springer-Verlag, 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. 896-928. MR**759704 (86e:78024)****[5]**M. S. Mock, "An initial value problem from semiconductor device theory,"*SIAM J. Math. Anal.*, v. 5, 1974, pp. 597-612. MR**0417573 (54:5623)****[6]**M. S. Mock,*Analysis of Mathematical Models of Semiconductor Devices*, Boole Press, Dublin, 1983. MR**697094 (84m:78002)****[7]**J. M. Ortega & W. C. Rheinboldt,*Iterative Solution of Nonlinear Equations in Several Variables*, Academic Press, New York, 1970. MR**0273810 (42:8686)****[8]**D. L. Scharfetter & H. K. Gummel, "Large signal analysis of a silicon Read diode oscillator,"*IEEE Trans. Electron. Devices*, v. ED-16, 1969, pp. 64-77.**[9]**O. C. Zienkiewicz,*The Finite Element Method*, McGraw-Hill, London, 1977.**[10]**M. Zlámal, "A finite element solution of the nonlinear heat equation,"*RAIRO Anal. Numér.*, v. 14, 1980, pp. 203-216. MR**571315 (81f:65089)**

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0815829-6

Article copyright:
© Copyright 1986
American Mathematical Society