ISSN 1088-6842(online) ISSN 0025-5718(print)

Convergence of a finite element method for the drift-diffusion semiconductor device equations: the zero diffusion case

Authors: Bernardo Cockburn and Ioana Triandaf
Journal: Math. Comp. 59 (1992), 383-401, S29
MSC: Primary 65N30; Secondary 35Q60, 65N12, 78A55, 82D99
DOI: https://doi.org/10.1090/S0025-5718-1992-1145661-0
MathSciNet review: 1145661
Full-text PDF Free Access

Abstract: In this paper a new explicit finite element method for numerically solving the drift-diffusion semiconductor device equations is introduced and analyzed. The method uses a mixed finite element method for the approximation of the electric field. A finite element method using discontinuous finite elements is used to approximate the concentrations, which may display strong gradients. The use of discontinuous finite elements renders the scheme for the concentrations trivially conservative and fully parallelizable. In this paper we carry out the analysis of the model method (which employs a continuous piecewise-linear approximation to the electric field and a piecewise-constant approximation to the electron concentration) in a model problem, namely, the so-called unipolar case with the diffusion terms neglected. The resulting system of equations is equivalent to a conservation law whose flux, the electric field, depends globally on the solution, the concentration of electrons. By exploiting the similarities of this system with classical scalar conservation laws, the techniques to analyze the monotone schemes for conservation laws are adapted to the analysis of the new scheme. The scheme, considered as a scheme for the electron concentration, is shown to satisfy a maximum principle and to be total variation bounded. Its convergence to the unique weak solution is proven. Numerical experiments displaying the performance of the scheme are shown.

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• [1] R. E. Bank, D. J. Rose and W. Fichtner, Numerical methods for the semiconductor device simulation, IEEE Trans. Electron Devices ED-30 (1983), 1031-1041. MR 723107 (86c:65113b)
• [2] R. E. Bank, W. M. Coughran, Jr., W. Fichtner, E. H. Grosse, D. J. Rose, and R. K. Smith, Transient simulation of silicon devices and circuits, IEEE Trans. Computer-Aided Design CAD-4 (1985), 436-451.
• [3] F. Brezzi, Theoretical and numerical problems in reverse biased semiconductor devices, Computing methods in Applied Sciences and Engineering, VII (R. Glowinski and J. L. Lions, eds.), Elsevier, Amsterdam, 1986, pp. 45-58. MR 905286 (88h:65229)
• [4] F. Brezzi, A. Capelo, and L. D. Marini, A singular perturbation analysis for semiconductor devices, SIAM J. Math. Anal. 20 (1989), 372-387. MR 982665 (90e:82116)
• [5] F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217-235. MR 799685 (87g:65133)
• [6] F. Brezzi and L. Gastaldi, Mathematical properties of one-dimensional semiconductors, Mat. Apl. Comput. 5 (1986), 123-138. MR 884997 (88d:81131)
• [7] F. Brezzi, L. D. Marini, and P. Pietra, Two-dimensional exponential fitting and applications to drift-diffusion models, SIAM J. Numer. Anal. 26 (1989), 1324-1355. MR 1025092 (90m:65194)
• [8] F. Brezzi and P. A. Markowich, A convection-diffusion problem with small diffusion coefficient arising in semiconductor physics, Boll. Un. Mat. Ital. 2B (1988), 903-930. MR 977596 (90g:35071)
• [9] B. Cockburn and C. W. Shu, The Runge-Kutta local projection -discontinuous Galerkin method for scalar conservation laws, MAN 25 (1991), 337-361. MR 1103092 (92e:65128)
• [10] -, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework, Math. Comp. 52 (1989), 411-435. MR 983311 (90k:65160)
• [11] B. Cockburn, S. Y. Lin, and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems, J. Comput. Phys. 84 (1989), 90-113. MR 1015355 (90k:65161)
• [12] B. Cockburn, S. Hou, and C. W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comp. 54 (1990), 545-581. MR 1010597 (90k:65162)
• [13] B. Cockburn, The quasi-monotone schemes for scalar conservation laws. Part I, SIAM J. Numer. Anal. 26 (1989), 1325-1341. MR 1025091 (91b:65106)
• [14] W. M. Coughran, Jr. and J. W. Jerome, Modular algorithms for transient semiconductor device simulation, Part I: Analylsis of the outer iteration, Lectures in Appl. Math., vol. 25, Amer. Math. Soc., Providence, R.I., 1990, pp. 107-149.
• [15] M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 1-21. MR 551288 (81b:65079)
• [16] J. Douglas, Jr., I. Martinez-Gamba, and M. C. J. Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device, Mat. Apl. Comput. 5 (1986), 103-122. MR 884996 (88d:81132)
• [17] W. Fichtner, D. J. Rose, and R. E. Bank, Semiconductor device simulation, IEEE Trans. Electron Devices ED-30 (1983), 1018-1030. MR 723106 (86c:65113a)
• [18] A. Harten, On a class of high-resolution total-variation-stable finite-difference schemes, SIAM J. Numer. Anal. 21 (1984), 1-23. MR 731210 (85f:65085)
• [19] J. W. Jerome, Consistency of semiconductor modeling: An existence/stability analysis for the stationary Van Roosbroeck system, SIAM J. Appl. Math. 45 (1985), 565-590. MR 796097 (87j:82014)
• [20] J. W. Jerome and T. Kerkhoven, A finite element approximation theory for the drift diffusion semiconductor model, SIAM J. Numer. Anal. 28 (1991), 403-422. MR 1087512 (92k:65012)
• [21] J. W. Jerome, Evolution systems in semiconductor device modeling: A cyclic uncoupled line analysis for the Gummel map, Math. Methods Appl. Sci. 9 (1987), 455-492. MR 1200362 (93i:82087)
• [22] T. Kerkhoven, A proof of convergence of Gummel's algorithm for realistic device problems, SIAM J. Numer. Anal. 23 (1986), 1121-1137. MR 865946 (88a:35093)
• [23] S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243.
• [24] N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comput. Math. and Math. Phys. 16 (1976), 105-119.
• [25] A. Y. LeRoux, Etude du problème mixte pour une équation quasilinéaire du premier ordre, C. R. Acad. Sci. Paris, Sér. A 285 (1977), 351-354.
• [26] B. J. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), 1074-1081. MR 811184 (88a:65104)
• [27] -, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), 59-69. MR 815831 (87m:65141)
• [28] -, On nonlocal monotone difference schemes for scalar conservation laws, Math. Comp. 47 (1986), 19-36. MR 842121 (87j:65110)
• [29] P. A. Markovich, The stationary semiconductor equations, Springer-Verlag, New York, 1986.
• [30] -, Spatial-temporal structure of solutions of the semiconductor problem, computational aspects of VLSI Design with an emphasis on semiconductor device simulation, Lectures in Appl. Math., vol. 25, Amer. Math. Soc., Providence, R.I., 1990, pp. 1-26. MR 1046465 (90k:00025)
• [31] P. A. Markowich and P. Szmolyan, A system of convection-diffusion equations with small diffusion coefficient arising in semiconductor physics, preprint.
• [32] I. Martinez-Gamba and M. C. J. Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device, II, SIAM J. Numer. Anal. 26 (1989), 539-552. MR 997655 (90d:65219)
• [33] P. A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, Lecture Notes in Math., Springer-Verlag, 1977. MR 0483555 (58:3547)
• [34] C. Ringhofer, An asymptotic analysis of a transient pn-junction model, SIAM J. Appl. Math. 47 (1987), 624-642. MR 889643 (88e:35176)
• [35] C. Ringhofer and C. Schmeiser, An approximate Newton method for the solution of the basic semiconductor device equations, SIAM J. Numer. Anal. 26 (1989), 507-516. MR 997653 (90d:65220)
• [36] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91-106. MR 679435 (84a:65075)
• [37] P. Szmolyan, Initial transients of solutions of the semiconductor equations, preprint.
• [38] -, Asymptotic methods for transient semiconductor device equations, COMPEL 8 (1989), 113-122. MR 1020120
• [39] V. W. Van Roosbroeck, Theory of flow of electrons and holes in germanium and other semiconductors, Bell Syst. Tech. J. 29 (1950), 560-607.
• [40] M. Zlámal, Finite element solution of the fundamental equations of semiconductor devices, Math. Comp. 46 (1986), 27-43. MR 815829 (87d:65139)

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