**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 10^{6 } 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.