On the Physically Based Compact Gate $C$ –$V$ Model for Ultrathin Gate Dielectric MOS Devices Using the Modified Airy Function Approximation

The exponent $(\lambda)$ of the modified Airy function solution of the quantized energy levels in the MOS potential well, which is used in the physically based quantum-mechanical compact gate $C$– $V$ model of Li , has been found to be dependent on the barrier height at the Si–dielectric interface and the substrate doping density. The physical origins of this dependence are discussed. An empirical equation that considers these effects is proposed for $\lambda$. Comparison with the experimental $C$–$V$ data of MOS devices with high-$k$ gate dielectrics shows that inclusion of these effects in the compact $C$– $V$ model of Li is necessary for the accurate simulation of MOS field-effect transistors with high-$k$ gate dielectrics.