Electromigration model for the prediction of lifetime based on the failure unit statistics in aluminum metallization

Abstract

A failure model for electromigration based on the ''failure unit model'' was presented for the prediction of lifetime in metal lines.The failure unit model, which consists of failure units in parallel and series, can predict both the median time to failure ͑MTTF͒ and the deviation in the time to failure ͑DTTF͒ in Al metal lines. The model can describe them only qualitatively. In our model, both the probability function of the failure unit in single grain segments and polygrain segments are considered instead of in polygrain segments alone. Based on our model, we calculated MTTF, DTTF, and activation energy for different median grain sizes, grain size distributions, linewidths, line lengths, current densities, and temperatures. Comparisons between our results and published experimental data showed good agreements and our model could explain the previously unexplained phenomena. Our advanced failure unit model might be further applied to other electromigration characteristics of metal lines.

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

“…The failure-unit, or structural defect, model has received quite general usage in electromigration theory, and versions of it have been considered as descriptions for electromigration failure on many occasions (e.g. [9,[15][16][17][18][19][20][21][22]). The model was originally developed by Agarwala et al [17] to describe the behaviour of the failure distribution as a function of the line length, and it was later developed by Cho and Thompson [19] to include the important dependence on linewidth.…”

“…For near-bamboo aluminium interconnects, there is some suggestion that the failure-units should be the grains, as any grain can be a potential failure site (e.g. [16] and [9] and references therein). However this ignores the complex microstructural effects such as those required for either a type-1 or a type-2 failure and which have always been known to be important [5,18,23,24].…”

“…As a result different authors have used widely differing values of N and left the exact nature of the failure-unit somewhat undefined. For example, in [19] N is chosen to be 40; in [5] calculations are performed for values of N = 1, 5, and 20; in [9] values of N are taken between 10 and 100 000; and in [16] values of N between 200 and 1000 are chosen. Where a failure-unit has been defined it has generally been taken, as here, to be a long cluster.…”

“…where • indicates that expectation values are to be taken over the distribution for N. In addition, if there are two different types of failure-unit in the stripe, with a number N 1 of type-1 failure-units and N 2 of type-2 failure-units, then for independent failure types, equation (8) has been extended to (e.g. [16])…”

“…Also shown are the best lognormal fits to the data. Note that while G approx (t) is a rigorous overestimate of the true failure rate, G(t), any analysis that assumes the approximate value, G approx (t) [15,16], but forces a lognormal fit, through an estimation of MTTF and DTTF, overestimates the failure time rate at low failure percentages and thus overestimates the reliability of the metallization.…”