Theory of electromigration failure in polycrystalline metal Slms

We introduce a dynamic fuse model for the damage done to a current-carrying polycrystalline metal thin film by electromigration. We determine the exact scaling behavior of the crack tip velocity for a single crack oriented perpendicularly to the direction of the ambient current. A Lifshitz-type theory works well only when the initial density of defects (p} is small. For all p, the mean failure time is to an excellent approximation proportional to the average length of the shortest path across the film in a certain metric. This conclusion is supported by our simulations and by analytical work based on a variational formulation of our problem. Using the shortest-path approximation, we show that the mean failure time tends to zero as (p, — p) ' as the percolation threshold p =p, is approached. We also use the shortest-path approximation to account for experimental results on the lifetime of long polycrystalline metal wires of varying lengths and widths.