An analytical model of the microstructure in near-bamboo interconnects

We aim here to describe electromigration-induced interconnect failure using a one-dimensional microstructure model (already one-dimensional models are used for the stress evolution). One might expect such a model to be reasonably successful because either (i) line widths are well below grain-boundary diameters (e.g. in as-patterned aluminium) or (ii) sidewall or topside surfaces provides the dominant diffusion path (e.g. in post-pattern annealed aluminium or copper). Our simple model is based on the Theory of Runs from probability theory and consequently may be solved analytically. As a test, we demonstrate that the model is able to reproduce the cluster length statistics of the two-dimensional simulator MIT/EmSIM for narrow metal (near-bamboo) stripes to a very good approximation. The importance of developing such a model is that, because of its analytical nature, it may access important areas where statistics are difficult for the simulator, such as rare events—e.g. time-to-first-failure rather than average events such as the mean-time-to-failure—or impossible events such as σd ≠ 0.27. In addition, it should allow some light to be thrown on the validity of the failure-unit model used to obtain interconnect lifetimes.