D2. Simplified analytical iterations for electron wavefunction using self-consistent solution for nm MOS gate stacks

In this paper, self consistent numerical solution of Poisson-Schrodinger equations is conducted using the shooting method, Numerov's integration, and damped Newton-Raphson iterative method, in order to obtain a reference electron distribution at high gate voltages for NMOS Metal gate/high-k stacks. The technology parameters are selected according to the ITRS roadmap for 22nm technologies, where a mid-gap metal gate is investigated. The results are compared to the using the analytical form of the Airy function envelope wavefunction, proposed in the literature as a solution for the first ground state. Instead of iterating over a very large number of points throughout the depletion region and inversion layer, only one analytical expression is solved using Newton-Raphson method, as a function of one fitting parameter. This procedure bypasses the numerical solution for discrete energy levels, and eliminates numerical integration needed to calculate charge and potential distributions in Poisson's equation. Analytical expressions for the charge, electric field and potential distribution as a function of distance are developed as a function of just one parameter.