Threshold voltage from numerical solution of the two-dimensional MOS transistor

The MOST model of Pao and Sah is extended to take into account the two-dimensional nature of the electrostatic potential. By keepig the current one dimensional, the basic equations can be cast into a form suitable for Gummel's iterative scheme. The numerical model is based on a finite-difference approximation to Poisson's equation and a closed-form expression for the current flow. The model is verified by comparing its results with experimental data. Good agreement is obtained. Deviations of the threshold voltage from the conventional expression for short-channel structures outside the range of the gradual channel approximation are investigated. In particular, the dependence of threshold voltage on channel length, drain-source, and substrate-source bias are illustrated with numerical and experimental results. Practical results from these investigations are summarized in graphical form.