An analysis of the weakest-link model for early electromigration failure

The application of the weakest-link or failure-unit model to electromigration failure is discussed in relation to a simple model that should describe the early failures in fine-line, as-patterned aluminium reasonably well. The earliest failures are expected to be due to the presence of at least one polygranular cluster longer than some critical (Blech) length, while the next earliest are assumed to arise from linked shorter clusters. Using a simple model of the microstructure, the probability density function for the number of each of these types of failure-unit is determined as a function of the linewidth. For this simple model the overall failure distribution may be calculated essentially exactly using the Theory of Runs; consequently we can compare these exact results with some common approximations. It is demonstrated through Jensen's inequality that a common approximate method is in fact a rigorous upper bound for the failure distribution. The model exhibits the expected increase in median-time-to-failure and deviation-in-time-to-failure values with decreasing linewidth when forcing a lognormal distribution. Although this would suggest an eventual decrease in reliability with scale reduction, the exact results show that this is not the case