Study of High-k/Metal-Gate Work-Function Variation Using Rayleigh Distribution

By using a Monte Carlo simulation for the stochastic distribution of grain sizes, the work-function variation (WFV) in high-k/metal-gate (HK/MG) is quantitatively and simply estimated with improved physical validity, with a Rayleigh distribution. Based on the Rayleigh distribution for the grain sizes, the WFV calculation for a TiN gate-stack is validated by previous experimental and simulation results. Additionally, a parameter for the WFV, i.e., ratio of the average grain size to the gate area (RGG), is suggested in this paper. This paves a new path to answer the following questions: 1) to what extent can the grain size of metal-gate materials be minimized to satisfy statistical targets? 2) to what extent can the physical gate area of metal oxide semiconductor field-effect transistors be scaled down whether the total variation is mainly limited by the WFV? Finally, it is concluded that a new HK/MG gate-stack should be developed to have the slope of <; 122 mV in the σ (WFV) versus RGG plot.