Grain Structure Statistics in As-Patterned and Annealed Interconnects

We have simulated the development of grain structures in polycrystalline films with lognormally distributed grain sizes, and carried out extensive characterization of grain cluster and bamboo cluster statistics. These statistics were characterized as a function of line-width to initial-grain-size ratios in as-patterned strips, and in strips in which we have simulated post-patterning grain structure evolution resulting from annealing. Among the important findings is that polygranular and bamboo cluster-length distributions for as-patterned lines are best fit by Weibull distribution functions instead of lognormal or exponential functions, as is often assumed. We report analytic formulae describing grain structure statistics that can be used in reliability calculations and simulations. We have carried out structure-sensitive electromigration simulations to show that the predicted failure statistics are essentially the same for grain structures generated using grain growth simulations and grain structures generated using analytic models. Analytic models provide computationally efficient means of determining the lifetime variations associated with grain-structure variations.