Abstract
The cost-effective design of electronic microstructures requires an advanced modeling and coupled simulation of various physical effects. The classical isothermal approach leads to the basic drift-diffusion model for semiconductor device simulation. In the stationary case, it represents a coupled nonlinear system consisting of a Poisson equation for the electric potential and two continuity equations for the electron and hole flow. We discuss various discretization schemes with special emphasis on mixed finite element methods and we further address efficient numerical solution techniques including adaptive multilevel methods. Finally, to allow for ambient conditions such as external magnetic fields we consider consistent extensions of the classical model and discuss perspectives for their numerical treatment.
The first and third author were supported by FORTWIHR, Bavarian Consortium for High Performance Scientific Computing.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, post-processing and error estimates. M 2 AN Math. Modelling Numer. Anal. 19, 7–35 (1985).
B. Bachmann, Adaptive Mehrgitterverfahren zur Lösung der stationären Halbleitergleichungen. Dissertation, Universität Zürich (1993).
B.R. Baliga and S.V. Patankar, A new finite-element formulation for convection-diffusion problems. Numer. Heat Transfer 3, 393–409 (1980).
R. Bank, J.F. Bürgler, W. Fichtner and R.K. Smith, Some upwinding techniques for finite element approximations of convection-diffusion equations. Numer. Math. 58, 185–202 (1990).
F. Bornemann, B. Erdmann and Kornhuber, Adaptive multilevel methods in three space dimensions. Int. J. Numer. Methods Eng. 36, 3187–3203 (1993).
D. Braess and R. Verfürth, Multigrid methods for nonconforming finite element methods. SIAM J. Numer. Anal. 27, No. 4, 979–986 (1990).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, BerlinHeidelberg-New York, 1991.
F. Brezzi, L.D. Marini and P. Pietra, Two dimensional exponential fitting and application to drift-diffusion models. SIAM J. Numer. Anal. 26, 1347–1355 (1989).
F. Brezzi, L.D. Marini and P. Pietra, Numerical simulation of semiconductor devices. Comp. Math. Appl. Mech. Eng. 75, 493–514 (1989).
J.F. Bürgler, Discretization and Grid Adaption in Semiconductor Device Modeling. Hartung-Gorre, Konstanz, 1990.
B. Fraeijs de Veubeke, Displacement and equilibrium models in the finite element method. In: Stress Analysis, C. Zienkiewicz and G. Holister (eds.), John Wiley and Sons, New York, 1965.
H. Gajewski and K. Gröger, Semiconductor equations for mobilities based on Boltzmann statistics or Fermi-Dirac statistics. Math. Nachr. 140, 7–36 (1989).
H. Gajewski and K. Gärtner, On the discretization of van Roosbroeck’s equations with magnetic field. ETH Zürich, Integrated Systems Laboratory, Techn. Rep. No. 94 /14 (1994).
P.W.H. Hemker and H. Molenaar, A multigrid approach for the solution of the 2D semiconductor equations. IMPACT Comput. Sci. Engrg. 2, 219–243 (1990).
R. Hiptmair, Gemischte Finite-Elemente-Diskretisierung der Kontinuitätsgleichungen aus der Bauelementsimulation. Master Thesis, Technische Universität München, 1992.
R. Hiptmair and R. H. W. Hoppe, Mixed finite element discretization of continuity equations arising in semiconductor device simulation. In: Proc. Conf. Math. Modelling and Simulation of Electrical Circuits and Devices, R.E. Bank et al. (eds.), p. 197–217 Birkhäuser, Basel, 1994.
R.H.W. Hoppe and B. Wohlmuth, Adaptive multilevel techniques for mixed finite element discretizations of elliptic boundary value problems. Submitted to SIAM J. Numer. Anal..
R.H.W. Hoppe and B. Wohlmuth, Efficient Numerical Solution of Mixed Finite Element Discretizations by Adaptive Multilevel Methods. to appear in Apl. Mat..
W. Kampowsky, P. Rentrop and W. Schmidt, Classification and numerical simulation of electric circuits. Surv. Math. Ind. 2, 23–65 (1992).
P.A. Markowich, Chr.A. Ringhofer and Chr. Schmeiser, Semiconductor Equations. Springer, Berlin-Heidelberg-New York, 1990.
F. Montrone, The method of Baliga-Patankar in 3D device simulation. In: Proc. Conf. Math. Modelling and Simulation of Electrical Circuits and Devices, R.E. Bank et al. (eds.), p. 251–265 Birkhäuser, Basel, 1994
R.R.P. van Nooyen, Some aspects of mixed finite element methods for semiconductor simulation. Ph.D. Thesis, University of Amsterdam, 1992.
M. Paffrath, W. Jacobs, W. Klein, E. Rank, K. Steger, U. Weinert and U. Weyer, Concepts and algorithms in process simulation. Surv. Math. Ind. 3, 149–183 (1993).
A. Reusken, Multigrid applied to mixed finite elements schemes for current continuity equations. Preprint, Technical University Eindhoven (1990).
C. Riccobene, G. Wachutka, J.F. Bürgler and H. Baltes, Two-dimensional numerical modeling of dual-collector magnetotransistors: evidence for emitter efficiency modulation. Sensors and Actuators A, 31, 210–214 (1992).
C. Riccobene, G. Wachutka, J.F. Bürgler and H. Baltes, Operating principle of dual collector magnetotransistors studied by two-dimensional simulation. IEEE Trans. Electron Devices ED-41, 32–43 (1994).
D.L. Scharfetter and H.K. Gummel, Large-signal analysis of a silicon read diode oscillator. IEEE Trans Electron Devices ED-16, 64–77 (1969).
S. Selberherr, Analysis and Simulation of Semiconductor Devices. Springer, BerlinHeidelberg-New York, 1984.
N. Shigyo, T. Wada and S. Yasuda, Discretization problem for multidimensional current flow. IEEE Trans. CAD, CAD-8, 1046–1050 (1989).
G. Wachutka, Unified framework fir thermal, electrical, magnetic and optical semiconductor device modeling. COMPEL 10, 311–321 (1991).
G. Wachutka, Problem-oriented modeling of microtransducers: state of the art and future challenges. Sensors and Actuators A, 41–42, 279–283 (1994).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden
About this chapter
Cite this chapter
Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B. (1995). Coupling Problems in Microelectronic Device Simulation. In: Hackbusch, W., Wittum, G. (eds) Numerical Treatment of Coupled Systems. Notes on Numerical Fluid Mechanics (NNFM), vol 51. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-86859-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-322-86859-6_8
Publisher Name: Vieweg+Teubner Verlag
Print ISBN: 978-3-322-86861-9
Online ISBN: 978-3-322-86859-6
eBook Packages: Springer Book Archive