A new simplified two-dimensional model for the threshold voltage of MOSFET´s with nonuniformly doped substrate

A new simplified two-dimensional model for the threshold voltage of MOSFETs is derived in terms of simple characteristic functions. These characteristic functions are transformed from the exact series solution of the two-dimensional Poisson´s equation, in which the effects of a nonuniformly doped substrate and a finite graded source-drain junction depth are included. In this model, charge-screening effects are proposed to account for the weak dependence of the threshold voltage on the substrate bias for short-channel MOSFETs, and exact source and drain boundary potentials can be approximated by their equivalent power functions. The accuracy of the simplified 2-D model is verified by 2-D numerical analysis. Moreover, comparisons between the simplified 2-D model and the experimental results are made, and good agreement is obtained for wide ranges of channel lengths, applied substrate, and drain biases