Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T12:17:44.643Z Has data issue: false hasContentIssue false
  • English
  • Français

An Approximation of Three-Dimensional Semiconductor Devices by Mixed Finite Element Method and Characteristics-Mixed Finite Element Method

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  05 August 2015

Qing Yang  and
Yirang Yuan
Qing Yang*
Affiliation:
School of Mathematical Science, Shandong Normal University, Jinan, 250014, China
Yirang Yuan
Affiliation:
Institute of Mathematics, Shandong University, Jinan, 250100, China
*
*Email address: sd_yangq@163.com (Q. Yang); yryuan@sdu.edu.cn (Y. Yuan)
Get access

Abstract

The mathematical model for semiconductor devices in three space dimensions are numerically discretized. The system consists of three quasi-linear partial differential equations about three physical variables: the electrostatic potential, the electron concentration and the hole concentration. We use standard mixed finite element method to approximate the elliptic electrostatic potential equation. For the two convection-dominated concentration equations, a characteristics-mixed finite element method is presented. The scheme is locally conservative. The optimal L2-norm error estimates are derived by the aid of a post-processing step. Finally, numerical experiments are presented to validate the theoretical analysis.

Keywords

Three-dimensional semiconductor devicescharacteristics-mixed finite element methodmixed finite element methodpost-processing steperror bound

MSC classification

Secondary: 65M15: Error bounds 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 3 , August 2015 , pp. 356 - 382
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arbogast, T. and Wheeler, M. F., A characteristics-mixed finite element method for ad-vection dominated transport problems, Siam J. Numer. Anal., vol. 32, no. 2 (1995), pp. 404–424.CrossRefGoogle Scholar
[2]Bank, R. E., Fjchtner, W. M., and Rose, D. J., Transient simulation of a silicon devices and circuits, IEEE Trans. on Cad of Integr. Syst., vol. 4, no. 4 (1985), pp. 436–451.CrossRefGoogle Scholar
[3]Chen, H. Z. and Jiang, Z. W., L∞ convergence of mixed finite element method for Laplacian operator, Korean J. Comput. Appl. Math., vol. 7, no. 1 (2000), pp. 61–82.CrossRefGoogle Scholar
[4]Douglas, J. Jr. and Yuan, Y. R., Finite difference methods for the transient behavior of a semiconductor device, Math. Appl. Comput., vol. 6, no. 1 (1987), pp. 25–38.Google Scholar
[5]Ewing, R. E., Russell, T. F. and Wheeler, M. F., Convergence analysis of an approximation of miscible displacement in porous meidia by mixed finite elements and a modified method of characteristics, Comput. Methods Appl. Mech. Eng., vol. 47, no. 1–2 (1984), pp. 73–92.CrossRefGoogle Scholar
[6]Jerome, J. W., Mathematical theory and approximation of semiconductor models, Siam, Philadelphia, 1994.Google Scholar
[7]Nédélec, J. C., Mixed finite elements in R3, Numer. Math., vol. 35 (1980), pp. 315–341.CrossRefGoogle Scholar
[8]Raviart, P. A. and Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, vol. 606, Springer-Verlag, Berlin, 1977, pp. 292–315.Google Scholar
[9]Russell, T. F., Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, Siam J. Numer. Anal., vol. 22, no. 5 (1985), pp. 970–1013.CrossRefGoogle Scholar
[10]Schatz, A. H., Thomée, V. and Wahlbin, L. B., Maximum norm stability and error estimates in parabolic finite element equations, Commu. on Pure and Appl. Math., vol. 33, no. 3 (1980), pp. 265–304.Google Scholar
[11]Scholz, R., Optimal L∞-estimates for a mixed finite element method for elliptic and parabolic problems, Calcolo, 20 (1983), pp. 355–377.Google Scholar
[12]Seidmann, T. I., Time dependent solution of a nonlinear system arising in semiconductor theory II, boundaries and periodicity, Nonlinear Anal., vol. 10 (1986), pp. 491–502.Google Scholar
[13]Sun, T. J. and Yuan, Y. R., An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method, J. Comput. Appl. Math., vol. 228, no. 1 (2009), pp. 391–411.CrossRefGoogle Scholar
[14]Yang, Q., Modified upwind finite volume scheme for semiconductor device, J. Syst. Sci. Math. Sci., vol. 28, no. 6 (2008), pp. 725–738.Google Scholar
[15]Yang, Q., Mixed covolume-upwind finite volume methods for semiconductor device, J. Syst. Sci. Math. Sci., vol. 31, no. 1 (2011), pp. 65–79.CrossRefGoogle Scholar
[16]Yang, Q., Upwind finite volume scheme for semiconductor device in three dimension, Acta Math. Sci., vol. 26A, no. 1 (2006), pp. 150–160.Google Scholar
[17]Yuan, Y. R., Recent progress in numerical methods for semiconductor device, Chin. J. Comput. Phys., vol. 26, no. 3 (2009), pp. 317–324.Google Scholar
[18]Yuan, Y. R., The characteristic finite element and analysis for the transient behavier of a semiconductor device, Acta Math. Sci., vol. 13, no. 3 (1993), pp. 241–251.Google Scholar
[19]Yuan, Y. R., Finite difference method and analysis for three-dimensional semiconductor device of heat conduction, Sci. China: Math., vol. 39, no. 11 (1996), pp. 1140–1151.Google Scholar