Some nonlinear analytic aspects of VLSI semiconductor device modeling

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Abstract
This paper discusses some special features of VLSI semiconductor devices due to the nonlinearity of the partial differential equations defining the electric potential and the density of holes and electrons flowing within the device. Currently, solutions for these nonlinear partial differential equations are computed with great speed on very large modern computers. However, a theoretical analytical treatment of these systems has not been systematically carried through. It is hoped that this analytical study may shed some light on some aspects of VLSI device operation that, until now, have required ingenious physical intuition and arguments for their understanding. In particular, the analytic approach has the potential to consider the following problems of importance in VLSI semiconductor devices: 1) arbitrarily shaped devices, 2) high space dimensional effects (especially those with curved boundaries), 3) nonuniqueness of solutions, 4) bifurcation phenomena, 5) instabilities and parameter dependence, and 6) latch up phenomena and high bias effects. I personally, believe there is much to be gained from an interplay between the impressive numerical simulation already attained and an analytic consideration of the equations themselves. Typical questions are (for example) (i) Electronic Instabilities. How do these arise from the boundary value problems themselves? What are the mathematical mechanisms involved? (ii) Structural Problems. Do the equations themselves have special structures that may aid the numerical simulation of their solutions? Another key point here is that the partial differential equation systems for semiconductors are another nonlinear generalization of Maxwell's equations of electromagnetism. As semiconductor devices become smaller, there is, of course, a greater need to rely on theoretical analysis. The immense success of Maxwell's mathematical electromagnetic theory and his own statistical mechanics ideas (here fused together) gives one confidence of the ultimate validity of the partial differential equation approach. In this article we shall limit ourselves to a discussion of the standard Van Roosbroeck equations (assuming the validity of the Einstein relations and Boltzmann statistics) and to questions of steady-state processes (i.e., time-independent problems). We hope to address other aspects in latter publications.

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