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A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems Applications to specific types of physical systems

Published online by Cambridge University Press:  20 June 2017

Yirang Yuan ,
Qing Yang ,
Changfeng Li  and
Tongjun Sun
Yirang Yuan*
Affiliation:
Institute of Mathematics, Shandong University, Jinan, 250100, China
Qing Yang
Affiliation:
School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China
Changfeng Li
Affiliation:
School of Economics, Shandong University, Jinan, 250100, China
Tongjun Sun
Affiliation:
Institute of Mathematics, Shandong University, Jinan, 250100, China
*
*Corresponding author. Email address:yryuan@sdu.edu.cn (Y. R. Yuan)
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Abstract

A coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

Keywords

Three dimensional transient behavior of heat conduction problemnumerical simulation computationmixed finite elementmodified upwind fractional step differencesecond-order error estimates in L2 norm

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 82D37: Semiconductors
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 541 - 561
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bank, R. E., Fichtner, W. M. and Rose, D. J., Transient simulation of sillcon devices and circuits, IEEE Computer-Aided Design, 6 (1985), pp. 436–451.CrossRefGoogle Scholar
[2] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RATRO., Anal. Numer., 2 (1974), pp. 129–151.Google Scholar
[3] Ciariet, P. G., The finite element methods for elliptic problem, North-Holland, Amsterdam, 1978.Google Scholar
[4] Douglas, J. Jr, Gunn, J. E., Two order correct difference analogues for the equation of multidimensional heat flow, Math. Comp., 17 (1963), pp. 71–80.CrossRefGoogle Scholar
[5] Douglas, J. Jr and Gunn, J. E., A general formulation of alternating direction methods, Part I. Parabolic and hyperbolic problems, Numer. Math., 6(5) (1964), pp. 428–453.Google Scholar
[6] Douglas, J. Jr, Simulation of miscible displacement in porous media by a modified method of characteristics procedure, In Numerical Analysis, Dundee, 1981, Lecture Note in Mathematics 912, Springer-Verlag, Berlin, 1982.Google Scholar
[7] Douglas, J. Jr, Ewing, R. E., Wheeler, M. F., Approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numer., 17(1) (1983), pp. 17–33.Google Scholar
[8] Douglas, J. Jr, Yuan, Y. R., Numerical simulation of immiscible flow in porous media based on combining the method of characteristics with mixed finite element procedure, Numerical Simulation in Oil Recovery, Springer-Verlag, New York, (1986), pp. 119–132.Google Scholar
[9] Douglas, J. Jr, Yuan, Y. R., Finite difference methods for transient behavior of a semiconductor device, Mat. Apli. Comp., 6(1) (1987), pp. 25–38.Google Scholar
[10] Ewing, R. E., The Mathematics of reservoir simulation, SIAM, Philadelphia, 1983.Google Scholar
[11] Ewing, R. E., Lazarov, R. D., Vassilev, A. T., Finite difference scheme for parabolic problems on a composite grids with refinement in time and space, SIAM J. Numer. Anal., 31(6) (1994), pp. 1605–1622.CrossRefGoogle Scholar
[12] Gummel, H. K., A self-consistent iterative scheme for one-dimensional steady-state transistor calculation, IEEE Trans: Electron Device, 11(10) (1964), pp. 455–465.Google Scholar
[13] He, Y., Wei, T. F., Computer simulation method for semiconductor device, Scicence Press, Beijing, 1989 (in Chinese).Google Scholar
[14] Jerome, J. W., Mathematical Theory and Approximation of Semiconductor Models, SIAM, Philadelphia 1994.Google Scholar
[15] Lazarov, R. D., Mischev, I. D., Vassilevski, P. S., Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33(1) (1996), pp. 31–55.Google Scholar
[16] Lou, Y., On basic semiconductor equation with heat conduction, J. Partial. Diff. Eqs., 8(1) (1995), pp. 43ĺC-54.Google Scholar
[17] Raviart, R. A., Thomas, J. M., A mixed finite element method for second order elliptic problems, Lecture Notes in Math., 606, Springer-Verlag, Berlin, 1977, pp. 292–315.Google Scholar
[18] Shi, M., Physics of modern semiconductor device, Science Press, Beijing, 2002 (in Chinese).Google Scholar
[19] Sun, C. W., LU, Q. S., Fan, Z. X., Laser irradiation effect, National Defence Industry Press, Beijing, 2002 (in Chinese).Google Scholar
[20] Sun, T. J., Yuan, Y. R., An approximation of incompressible miscible displacment in porous media by mixed finite element method and characteristics-mixed finite element method, J. Comp. Appl. Math., 228(2009), pp. 391–411.Google Scholar
[21] Thomas, J. M., SurL’analyse numerique des methods d’elements finis hybrides et mixtes, These. Universite Pierre et Marie Curie, 1977.Google Scholar
[22] Yang, Q., Yuan, Y. R., An approximation of semiconductor device of heat conduction by mixed finite element method characteristics-mixed finite element method, Appl. Numer. Math., 70(2013), pp. 42–57.Google Scholar
[23] Yang, Q., Yuan, Y. R., An approximation of semiconductor device by mixed finite element method and characteristics-mixed finite element method, Appl. Math. Comput., 225(2013), pp. 407–424.Google Scholar
[24] Yang, Q., Yuan, Y. R., An approximation of three-dimensional semiconductor device by mixed finite element method and characteristics-mixed finite element method, Numer. Math-Theory. ME., to appear.Google Scholar
[25] Yuan, Y. R., Ding, L. Y., Yang, H. A., A new method and theoretical analysis of numerical analog of semiconductor, Chin. Sci. Bull., 27(7) (1982), pp. 790–795.Google Scholar
[26] Yuan, Y. R., Characteristics method with mixed finite element for transient behavior of semiconductor device, Chin. Sci. Bull., 36(17) (1991), pp. 1356–1357.Google Scholar
[27] Yuan, Y. R., The approximation of the electronic potential by a mixed method in the simulation of semiconductor, J. Systems Sci. Math. Sci., 11(2) (1991), pp. 117–120.Google Scholar
[28] Yuan, Y. R., Time stepping along characteristics for the finite element approximation of compressible miscible displacement in porous media, Math. Numer. Sinica, 14(4) (1992), pp. 386–406.Google Scholar
[29] Yuan, Y. R., Finite element method and analysis of numerical simulation of semiconductor device, Acta Math. Sci., 13(3) (1993), pp. 241–251.Google Scholar
[30] Yuan, Y. R., Finite difference methods for a compressible miscible displacement problem in porous media, Math. Numer. Sinica, 15(1) (1993), pp. 16–28.Google Scholar
[31] Yuan, Y. R., Finite difference method and analysis for three-dimensional semiconductor device of heat conduction, Sci. China Math., 39(11) (1996), pp. 1140–1151.Google Scholar
[32] Yuan, Y. R., Characteristic finite difference fractional step methods for three-dimensional semiconductor device of heat conduction, Chin. Sci. Bull., 45(2) (2000), pp. 123–131.CrossRefGoogle Scholar
[33] Yuan, Y. R., Finite difference fractional step method for transient behavior of a semiconductor device, Acta. Math. Sci. (Ser. B), 25(3) (2005), pp. 427–438.Google Scholar
[34] Yuan, Y. R., Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device, Numer. Meth. Part. D. E., 24(2) (2008), pp. 400–417.Google Scholar
[35] Yuan, Y. R., Recent progress in numerical methods for semiconductor devices, Chinese J. Computational Physics, 26(3) (2009), pp. 317–324.Google Scholar
[36] Yuan, Y. R., Theory and application of reservoir numerical simulation, Chapter 7, numercial method of transient behavior of semiconductor device, Science Press, Beijing, 2013 (in Chinese).Google Scholar