2.1.3 1980s: Modification of Black's Equation

In the following ten years, the development of very large-scale integrated (VLSI) circuits led to the investigation of electromigration in small-dimension conductors assembled into a multilayered structure with insulation and barrier layers [79]. With the continuous shrinkage of the interconnect line width, researchers moved toward the improvement of the electromigration lifetime in the miniaturized electronic devices. For this purpose, the incorporation of copper or silicon in pure aluminum interconnects proved to be beneficial [154]. This advantage was mainly due to the significant reduction of the atom flow in fast diffusivity paths, when the grain size of the metal film is similar to the film width [149]. At this time, a drastic improvement in the electromigration performance was provided by the introduction of barrier materials, such as titanium and tungsten [139]. The adoption of refractory materials for metallization resulted in the protection from inter-diffusion between aluminum and silicon in the VLSI circuits, reducing electromigration failure.

With the ongoing miniaturization of ICs as well as the development of new interconnect materials, the validity and reliability of Black's equation, in predicting the interconnect lifetime, were becoming controversial [138]. The original formulation of TTF had already been modified in 1971 by Blair [15] as follows

\[\begin{equation} MTTF =Aj^{-n} \ \text{exp}\left(\cfrac{E_\text{a}}{k_\text{B}T}\right), \end{equation}\] (2.10)

where the current density exponent n is a parameter which can be experimentally determined. He observed that experimental data could be fitted to equation (2.10) by allowing a variable current density exponent in the range between 1 and 2. In the 1980s, the different interpretation of the value of this parameter led to the separation of electromigration lifetime prediction models in three classes. The categorization was based on the impact of the contribution of two electromigration failure phases on the MTTF estimation, namely void nucleation and void evolution. An exponent value close to 2, as in the original Black's equation, means that the contribution of the void nucleation phase, due to the electromigration stress build-up, represents the major portion of the lifetime. In turn, the time necessary to grow a void and trigger a failure strongly dominates the electromigration lifetime estimation after a void was already formed. The void evolution mechanism implies a failure time proportional to the inverse of the current density because the mass transport due to electromigration is linearly dependent on the current density [138]. A failure model for lifetime evaluation based on the combination of the nucleation and the growth mechanisms should provide a more sophisticated understanding of the electromigration behavior [40].

Furthermore, Shatzkes and Lloyd [138] argued the application of both original and modified Black's equations, concluding that significant errors may arise in the lifetime extrapolation from accelerated tests. The experimental determination of A, n, and E_a in equation (2.10) yielded incorrect parameter values and, consequently, an incorrect lifetime estimation. The major shortcoming of Black's equation was that temperature gradients, together with Joule heating effects, were not included in the lifetime expression. From the solution of the continuity equation for the vacancy concentration with a perfectly blocking boundary, and assuming that failure occurs when the critical vacancy concentration value is reached (i.e. the void nucleation condition), the lifetime is obtained as follows

\[\begin{equation} MTTF =BT^{2}j^{-2} \ \text{exp}\left(\cfrac{E_\text{a}}{k_\text{B}T}\right), \end{equation}\] (2.11)

where the T² proportionality term does not appear in the original Black's equation. It should be pointed out that the theoretical model proposed by Shatzkes and Lloyd was the first one which rigorously explained the inverse square dependence on current density for the lifetime. Values above 2 can be attributed to extensive Joule heating effects.