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A mixed-finite volume element coupled with the method of characteristic fractional step difference for simulating transient behavior of semiconductor device of heat conductor and its numerical analysis

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Abstract

The mathematical system is formulated by four partial differential equations combined with initialboundary value conditions to describe transient behavior of three-dimensional semiconductor device with heat conduction. The first equation of an elliptic type is defined with respect to the electric potential, the successive two equations of convection dominated diffusion type are given to define the electron concentration and the hole concentration, and the fourth equation of heat conductor is for the temperature. The electric potential appears in the equations of electron concentration, hole concentration and the temperature in the formation of the intensity. A mass conservative numerical approximation of the electric potential is presented by using the mixed finite volume element, and the accuracy of computation of the electric intensity is improved one order. The method of characteristic fractional step difference is applied to discretize the other three equations, where the hyperbolic terms are approximated by a difference quotient in the characteristics and the diffusion terms are discretized by the method of fractional step difference. The computation of three-dimensional problem works efficiently by dividing it into three one-dimensional subproblems and every subproblem is solved by the method of speedup in parallel. Using a pair of different grids (coarse partition and refined partition), piecewise threefold quadratic interpolation, variation theory, multiplicative commutation rule of differential operators, mathematical induction and priori estimates theory and special technique of differential equations, we derive an optimal second order estimate in L2-norm. This numerical method is valuable in the simulation of semiconductor device theoretically and actually, and gives a powerful tool to solve the international problem presented by J. Douglas, Jr.

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Acknowledgements

The authors express their deep appreciation to Prof. J. Douglas Jr, and Prof. Jiang Lishang for their helpful suggestions in the serial of research on numerical simulation of semiconductor device.

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Authors and Affiliations

  1. Institute of Mathematics, Shandong University, Jinan, 250100, China

    Yi-rang Yuan, Chang-feng Li & Tong-jun Sun

  2. School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China

    Qing Yang

  3. School of Economics, Shandong University, Jinan, 250100, China

    Chang-feng Li

Authors
  1. Yi-rang Yuan
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  2. Qing Yang
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  3. Chang-feng Li
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  4. Tong-jun Sun
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Corresponding author

Correspondence to Yi-rang Yuan.

Additional information

Project supported by the National Natural Science Foundation of China (Grant Nos. 11101124 and 11271231), the National Tackling Key Problems Program for Science and Technology (Grant No. 20050200069), and the Doctorate Foundation of the Ministry of Education of China (Grant No. 20030422047).

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Yuan, Yr., Yang, Q., Li, Cf. et al. A mixed-finite volume element coupled with the method of characteristic fractional step difference for simulating transient behavior of semiconductor device of heat conductor and its numerical analysis. Acta Math. Appl. Sin. Engl. Ser. 33, 1053–1072 (2017). https://doi.org/10.1007/s10255-017-0721-y

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  • DOI: https://doi.org/10.1007/s10255-017-0721-y

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