Analysis of multistate models for electromigration failure

Abstract

Additional Information:• This article was published in the Journal of Applied Physics [ c Amer- The application of a multistate Markov chain is considered as a model of electromigration interconnect degradation and eventual failure. Such a model has already been used ͓Tan et al., J. Appl. Phys. 102, 103703 ͑2007͔͒, maintaining that, in general, it leads to a failure distribution described by a gamma mixture, and that as a result, this type of distribution ͑rather than a lognormal͒ should be used as a prior in any Bayesian mode fitting and subsequent reliability budgeting. Although it appears that the model is able to produce reasonably realistic resistance curves R͑t͒, we are unable to find any evidence that the failure distribution is a simple gamma mixture except under contrived conditions. The distributions generated are largely sums of exponentials ͑phase-type distributions͒, convolutions of gamma distributions with different scales, or roughly normal. We note also some inconsistencies in the derivation of the gamma mixture in the work cited above and conclude that, as it stands, the Markov chain model is probably unsuitable for electromigration modeling and a change from lognormal to gamma mixture distribution generally cannot be justified in this way. A hidden Markov model, which describes the interconnect behavior at time t rather than its resistance, in terms of generally observed physical processes such as void nucleating, slitlike growth ͑where the growth is slow and steady͒, transverse growth, current shunting ͑where the resistance jumps in value͒, etc., seems a more likely prospect, but treating failure in such a manner would still require significant justification.