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.