Comparative study in work-function variation: Gaussian vs. Rayleigh distribution for grain size

Because of the significantly-increasing work-function variation (WFV) in high-k/metal-gate technology in sub-30-nm nodes, a simple but reasonable model for quantitatively estimating the WFV is currently required. In this study, a Monte Carlo simulation for statistically generating the grain sizes following two different probability distributions (ie Gaussian and Rayleigh distributions) is suggested and performed. The shapes of the grains created by following the Rayleigh distribution (vs the Gaussian distribution) are significantly closer to the real shapes of the grains in the metal gate of TiN. Thus, the WFV estimated by using the Rayleigh distribution is well matched to the previous results.