Transistor Modeling for the Semi-conductor Industry

 

Ellis Cumberbatch

Mathematics Department

Claremont Graduate University

 

Software packages, under the generic name SPICE, simulate C-MOS integrated circuits. Since there may be up to 106 transistors on a chip, each transistor must have a simple representation in SPICE.

 

The current/voltage characteristics of a single transistor are derived from the flow of electrons and holes, governed by non-linear partial differential equations, the so-called drift-diffusion equations (in the classical limit). Quasi-one-dimensional approximations were found suitable in early models, and as device size reduced these were adapted empirically to fit data so much so that now there can be as many as 200 fitting parameters needed.

 

After a brief introduction to the physics of semi-conductors, I shall describe an approach to the basic equations using asymptotic analysis that, for some transistor characteristics including the current/voltage relation, yields accurate formulae needing much fewer parameters. Also I shall describe enhanced equations that have been introduced to model quantum effects that are present in the ultra-small devices. We have obtained solutions to these equations using asymptotic techniques.