NUMERICAL METHOD OF MIXED FINITE VOLUME-MODIFIED UPWIND FRACTIONAL STEP DIFFERENCE FOR THREE-DIMENSIONAL SEMICONDUCTOR DEVICE TRANSIENT BEHAVIOR PROBLEMS

doi:10.1016/S0252-9602(16)30129-1

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (1): 259-279.doi: 10.1016/S0252-9602(16)30129-1

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NUMERICAL METHOD OF MIXED FINITE VOLUME-MODIFIED UPWIND FRACTIONAL STEP DIFFERENCE FOR THREE-DIMENSIONAL SEMICONDUCTOR DEVICE TRANSIENT BEHAVIOR PROBLEMS

Yirang YUAN¹, Qing YANG², Changfeng LI³, Tongjun SUN³

1. Institute of Mathematics, Shandong University, Jinan 250100, China;
2. School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China;
3. School of Economics, Shandong University, Jinan 250100, China

Received:2014-08-31 Revised:2016-02-03 Online:2017-02-25 Published:2017-02-25
About author:Yirang YUAN,E-mail:yryuan@sdu.edu.cn;Qing YANG,E-mail:sd-yangq@163.com;Changfeng LI,E-mail:cfli@sdu.edu.cn;Tongjun SUN,E-mail:tjsun@sdu.edu.cn
Supported by:
This work is supported by National Natural Science Foundation of China (11101244, 11271231), National Tackling Key Problems Program (20050200069), and Doctorate Foundation of the Ministry of Education of China (20030422047).

Abstract

Abstract:

Transient behavior of three-dimensional semiconductor device with heat conduction is described by a coupled mathematical system of four quasi-linear partial differential equations with initial-boundary value conditions. The electric potential is defined by an elliptic equation and it appears in the following three equations via the electric field intensity. The electron concentration and the hole concentration are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite volume element approximation, keeping physical conservation law, is used to get numerical values of the electric potential and the accuracy is improved one order. Two concentrations and the heat conduction are computed by a fractional step method combined with second-order upwind differences. This method can overcome numerical oscillation, dispersion and decreases computational complexity. Then a three-dimensional problem is solved by computing three successive one-dimensional problems 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 using prior estimate theory and other special techniques of partial differential equations. This type of mass-conservative parallel method is important and is most valuable in numerical analysis and application of semiconductor device.

Key words: three dimensional transient behavior of heat conduction problem, mixed finite volume element, modified upwind fractional step difference, second-order error

Yirang YUAN, Qing YANG, Changfeng LI, Tongjun SUN. NUMERICAL METHOD OF MIXED FINITE VOLUME-MODIFIED UPWIND FRACTIONAL STEP DIFFERENCE FOR THREE-DIMENSIONAL SEMICONDUCTOR DEVICE TRANSIENT BEHAVIOR PROBLEMS[J].Acta mathematica scientia,Series B, 2017, 37(1): 259-279.

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References

[1] He Y, Wei T F. Computer Simulation Method for Semiconductor Device. Beijing:Scicence Press, 1989
[2] Shi M. Physics of modern semiconductor device. Beijing:Science Press, 2002
[3] Yuan Y R. Theory and Application of Reservoir Numerical Simulation, Chapter 7, Numercial Method of Transient Behavior of Semiconductor Device. Beijing:Science Press, 2013
[4] Yuan Y R. Recent progress in numerical methods for semiconductor devices. Chin J Comput Phy, 2009, 26(3):317-324
[5] Bank R E, Fichtner, W M, Rose, D J, et al. Transient simulation of sillcon devices and circuits. IEEE Computer-Aided Design, 1985, 6:436-451
[6] Jerome J W. Mathematical Theory and Approximation of Semiconductor Models. Philadelphia:SIAM, 1994
[7] Douglas Jr J, Yuan Y R. Finite difference methods for transient behavior of a semiconductor device. Mat Apli Comp, 1987, 6(1):25-38
[8] Yuan Y R. Finite element method and analysis of numerical simulation of semiconductor device. Acta Math Sci, 1993, 13(3):241-251
[9] Yuan Y R. Finite difference method and analysis for three-dimensional semiconductor device of heat conduction. Sci China Math, 1996, 39(11):1140-1151
[10] Gummel H K. A self-consistent iterative scheme for one-dimensional steady-state transistor calculation. IEEE Trans:Electron Device, 1964, 11:455-465
[11] Yuan Y R, Ding L Y, Yang H. A new method and theoretical analysis of numerical analog of semiconductor. Chin Sci Bull, 1982, 27(7):790-795
[12] Yuan Y R. Characteristics method with mixed finite element for transient behavior of semiconductor device. Chin Sci Bull, 1991, 36(17):1356-1357
[13] Yuan Y R. The approximation of the electronic potential by a mixed method in the simulation of semiconductor. J Systems Sci Math Sci, 1991, 11(2):117-120
[14] Lou Y. On basic semiconductor equation with heat conduction. J Partial Diff Eqs, 1995, 1:43-54
[15] Sun C W, Lu Q S, Fan Z X. Laser Irradiation Effect. Beijing:National Defence Industry Press, 2002
[16] Yuan Y R. Characteristic finite difference fractional step methods for three-dimensional semiconductor device of heat conduction. Chin Sci Bull, 2000, 45(2):123-131
[17] Yuan Y R. Finite difference fractional step method for transient behavior of a semiconductor device. Acta Math Sci, 2005, 25B(3):427-438
[18] Yuan Y R. Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device. Numer Meth Part D E, 2008, 24(2):400-417
[19] 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 methd. J Comp Appl Math, 2009, 228:391-411
[20] Yang Q, Yuan Y R. An approximation of semiconductor device by mixed finite element method and characteristics-mixed finite element method. Appl Math Comput, 2013, 225:407-424
[21] 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
[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, 2013, 70:42-57
[23] 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, 1994, 31(6):1605-1622
[24] Lazarov R D, Mischev I D, Vassilevski P S. Finite volume methods for convection-diffusion problems. SIAM J Numer Anal, 1996, 33(1):31-55
[25] Ewing R E. The Mathematics of Reservoir Simulation. Philadelphia:SIAM, 1983
[26] Douglas Jr J, Gunn J E. Two order correct difference analogues for the equation of multidimensional heat flow. Math Comp, 1963, 17(81):71-80
[27] Douglas Jr J, Gunn J E. A general formulation of alternating direction methods, part I, parabolic and hyperbolic problems. Numer Math, 1964, 6(5):428-453
[28] Douglas Jr J, Ewing R E, Wheeler M F. Approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Anal Numer, 1983, 17(1):17-33
[29] Douglas Jr J, 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. New York:Springer-Verlag, 1986:119-132
[30] Yang Q. Mixed covolume-upwind finite volume methods for semiconductor device. J Systems Sci Math Sci, 2011, 31(1):65-79
[31] Russll T F. Rigorous Block-Centered Discritizations on Irregular Grids:Improved Simulation of Complex Reservoir Systems. Project Report, Tulsa:Research Corporation, 1995
[32] Weiser A, Wheeler M F. On convergence of block-centered finite difference for elliptic problems. SIAM J Numer Anal, 1988, 25(2):351-375
[33] Jones J E. A Mixed Finite Volume Method for Accurate Computation of Fluid Velocities in Porous Media[D]. University of Colorado, 1995
[34] Nitsche J. Linear splint-funktioncn and dic methoden von Ritz for elliptishce randwert probleme. Arch Rational Mech Anal, 1968, 36:348-355
[35] Jiang L S, Pang Z Y. Finite Element Method and Its Theory. Beijing:People's Education Press, 1979
[36] Rui H X, Pan H A. A block-centered finite difference method for the Darcy-Forchheimer model. SIAM J Numer Anal, 2012, 50(5):2612-2631
[37] Yuan Y R. Time stepping along characteristics for the finite element approximation of compressible miscible displacement in porous media. Math Numer Sinica, 1992, 14(4):385-400
[38] Yuan Y R. Finite difference methods for a compressible miscible displacement problem in porous media. Math Numer Sinica, 1993, 15(1):16-28
[39] Douglas Jr J. Simulation of miscible displacement in porous media by a modified method of characteristics procedure//Numerical Analysis, Dundee, 1981, Lecture Note in Mathematics 912. Berlin:Springer-Verlag, 1982

NUMERICAL METHOD OF MIXED FINITE VOLUME-MODIFIED UPWIND FRACTIONAL STEP DIFFERENCE FOR THREE-DIMENSIONAL SEMICONDUCTOR DEVICE TRANSIENT BEHAVIOR PROBLEMS

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