Finite element approximation to initial-boundary value problems of the semiconductor device equations with magnetic influence
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Abstract:
We shall consider Zlámal’s approach to the nonstationary equations of the semiconductor device theory under magnetic fields, with mixed boundary conditions. Owing to the reduced smoothness of the electric potential $\psi$ and carrier densities n and p caused by considering the mixed boundary conditions, we must use a nonstandard analysis for this procedure. Existence as well as uniqueness of the approximate solution is proved. The convergence rates obtained in this paper are slower than those previously obtained for pure Dirichlet or Neumann boundary conditions.References
-
W. Allegretto, Y. S. Mun, A. Nathan, and H. P. Baltes, Optimization of semiconductor magnetic field sensors using finite element analysis, Proc. NASECODE IV Conf., Boole Press, Dublin, 1985, pp. 129-133.
- J. Banasiak and G. F. Roach, On mixed boundary value problems of Dirichlet oblique-derivative type in plane domains with piecewise differentiable boundary, J. Differential Equations 79 (1989), no. 1, 111–131. MR 997612, DOI 10.1016/0022-0396(89)90116-2 R. E. Bank, W. M. Fichtner, Jr., D. J. Rose, and R. K. Smith, Transient simulation of silicon devices and circuits, IEEE Trans. Computer-Aided Design 4 (1985), 436-451.
- Randolph E. Bank, Joseph W. Jerome, and Donald J. Rose, Analytical and numerical aspects of semiconductor device modeling, Computing methods in applied sciences and engineering, V (Versailles, 1981) North-Holland, Amsterdam, 1982, pp. 593–597. MR 784655
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Jim Douglas Jr., Richard E. Ewing, and Mary Fanett Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numér. 17 (1983), no. 1, 17–33 (English, with French summary). MR 695450
- Jim Douglas Jr., Irene Martínez-Gamba, and M. Cristina J. Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device, Mat. Apl. Comput. 5 (1986), no. 2, 103–122 (English, with Portuguese summary). MR 884996
- Jim Douglas Jr. and Yuan Yirang, Finite difference methods for the transient behavior of a semiconductor device, Mat. Apl. Comput. 6 (1987), no. 1, 25–37 (English, with Portuguese summary). MR 903000 J. Douglas, Jr., Yirang Yuan, and Gang Li, A modified method of characteristic procedure for the transient behavior of a semiconductor device (preprint). —, A mixed method for the transient behavior of a semiconductor device (preprint).
- Richard E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978), no. 6, 1125–1150. MR 512687, DOI 10.1137/0715075 R. E. Ewing and M. F. Wheeler, Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case, Lectures on the Numerical Solution of Partial Differential Equations, University of Maryland, 1981, pp. 151-174.
- H. Gajewski, On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech. 65 (1985), no. 2, 101–108 (English, with German and Russian summaries). MR 841263, DOI 10.1002/zamm.19850650210
- H. Gajewski and K. Gröger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 (1986), no. 1, 12–35. MR 826656, DOI 10.1016/0022-247X(86)90330-6
- Irene Martínez-Gamba and Maria Cristina J. Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device. II, SIAM J. Numer. Anal. 26 (1989), no. 3, 539–552. MR 997655, DOI 10.1137/0726032
- Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
- Joseph W. Jerome, Consistency of semiconductor modeling: an existence/stability analysis for the stationary Van Roosbroeck system, SIAM J. Appl. Math. 45 (1985), no. 4, 565–590. MR 796097, DOI 10.1137/0145034 —, Evolution systems in semiconductor modeling: A cyclic uncoupled analysis for the Gummel map (to appear).
- Peter A. Markowich, A singular perturbation analysis of the fundamental semiconductor device equations, SIAM J. Appl. Math. 44 (1984), no. 5, 896–928. MR 759704, DOI 10.1137/0144064 —, The stationary semiconductor device equations, Springer-Verlag, Wien-New York, 1985.
- Peter A. Markowich and C. A. Ringhofer, A singularly perturbed boundary value problem modelling a semiconductor device, SIAM J. Appl. Math. 44 (1984), no. 2, 231–256. MR 739302, DOI 10.1137/0144018
- Peter A. Markowich and Miloš A. Zlámal, Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems, Math. Comp. 51 (1988), no. 184, 431–449. MR 930223, DOI 10.1090/S0025-5718-1988-0930223-7
- 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
- M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal. 5 (1974), 597–612. MR 417573, DOI 10.1137/0505061
- Michael S. Mock, Analysis of mathematical models of semiconductor devices, Advances in Numerical Computation Series, vol. 3, Boole Press, Dún Laoghaire, 1983. MR 697094
- C. Ringhofer and C. Schmeiser, An approximate Newton method for the solution of the basic semiconductor device equations, SIAM J. Numer. Anal. 26 (1989), no. 3, 507–516. MR 997653, DOI 10.1137/0726030 S. Selberherr and C. Ringhofer, Discretization methods for the semiconductor equations, Proc. NASECODE III Conf., Boole Press, Dublin, 1983, pp. 31-45.
- E. Stephan and J. R. Whiteman, Singularities of the Laplacian at corners and edges of three-dimensional domains and their treatment with finite element methods, Math. Methods Appl. Sci. 10 (1988), no. 3, 339–350. MR 949661, DOI 10.1002/mma.1670100309
- Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045 Yuanming Wang, Mathematical model and its analysis for the carrier transport in semiconductor devices, Appl. Math. J. Chinese Univ. 2 (1987), 228-240.
- Jiang Zhu, Finite element methods for nonlinear Sobolev equations, Northeast. Math. J. 5 (1989), no. 2, 179–196 (Chinese, with English summary). MR 1034130 Jiang Zhu, Finite difference methods for the semiconductor device equations with magnetic influence (to appear).
- Miloš Zlámal, Finite element solution of the fundamental equations of semiconductor devices. I, Math. Comp. 46 (1986), no. 173, 27–43. MR 815829, DOI 10.1090/S0025-5718-1986-0815829-6 —, Finite element solution of the fundamental equations of semiconductor devices. II (to appear).
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp. 59 (1992), 39-62
- MSC: Primary 65N30; Secondary 65N12
- DOI: https://doi.org/10.1090/S0025-5718-1992-1134742-3
- MathSciNet review: 1134742