6. Conclusion and Outlook

Electromigration induced failure has been one of the major reliability issues for the microelectronic industry since its detection as a potential damage mechanism in metallization of integrated circuits. Since then, efforts have been made in order to design more resistant interconnects against electromigration damage. This has encompassed the development of new fabrication processes, the research of materials which yield adequate properties, and the design of specific geometrical features.

The continuous scaling of devices demands a continuous reduction of the metal line dimensions, which is accompanied by an increase in the density of interconnections in modern integrated circuits. Therefore, the interconnect lines have to operate at high temperatures and conduct high current densities, accentuating the electromigration transport. As a consequence, electromigration continues to be a challenge for the development of the new technological nodes, and the lifetime of interconnects have decreased from generation to generation, despite all the knowledge gained and efforts performed in the last 40 years.

The investigation of the physical phenomena behind the electromigration damage has become more and more important, because it can provide a stronger knowledge basis to counter the electromigration effect. In this context, mathematical modeling becomes an important tool which can significantly help to understand the electromigration failure. In the last decades, several continuum models have been proposed, and they have been able to partially explain several features of the electromigration damage. At the same time, the development of computational methods and resources has allowed to model complex systems and carry out numerical simulations in an efficient way.

In the scope of this work, the focus was put on developing a mathematical model suitable for implementation in a TCAD tool for numerical simulations. Electromigration modeling constitutes a complex problem due to the wide variety of physical effects which must be considered. A detailed study of the previous available models was carried out, identifying their main strengths and, at the same time, their main problems. Based on this analysis, these models were extended and further developed by taking into account the most relevant effects for electromigration simulation. As a result, a fully three-dimensional model is proposed, which connects the electromigration material transport problem with the electro-themal and mechanical problem in a general framework. The model equations were numerically described using the finite element method. The corresponding system of algebraic equations was derived and implemented in a TCAD tool.

In order to verify the developed model and numerical implementation, simulations in a simple interconnect line were carried out and compared with the available analytical solutions. The agreement between the numerical simulation results and the analytical solutions is remarkably good. Also, the importance of taking into account the effects of mechanical stress for electromigration modeling was demonstrated. The mechanical stress significantly affects the diffusivity of vacancies and, consequently, the electromigration induced transport. The non-uniform stress distribution which appears in the interconnect structure due to thermal processes leads to an anisotropic diffusivity. This effect is not readily visible under accelerated test conditions, however, it becomes much more important at real use conditions. Therefore, the activation energies for diffusion obtained from the accelerated tests are likely to be incorrect, posing a problem for the extrapolation of lifetimes to normal operating conditions.

A key feature of the proposed model is the inclusion of material interfaces and grain boundaries as paths of higher vacancy diffusion. Instead of considering a simple effective diffusion as valid for the entire interconnect, the correct diffusion coefficient can be independently set for each path, namely, the copper/capping layer interface, the copper/barrier interface, grain boundaries and bulk. In this way, it was shown that the most common experimental observations of electromigration induced voiding can be explained. Since the copper/capping layer interface is known to be the fastest path, it was shown that void nucleation typically occurs at this interface, at the cathode end of the line.

The model was further developed by considering grain boundaries as sites capable of trapping and releasing vacancies. This allowed the connection of the local dynamics of the grain boundary with the corresponding line deformation, and consequent mechanical stress build-up. Grain boundaries are introduced by a simple microstructure generator. In this way, it was shown that the combination of material interfaces as fast diffusivity paths together with the grain boundary model can explain void nucleation away from the cathode end. Voids nucleate at triple points formed by the intersection between the copper/capping layer interface with grain boundaries.

The microstructure has a major impact on the electromigration lifetime distribution. The simulation results indicate that the lognormal distribution of copper grain sizes is a primary cause for the lognormal distribution of the electromigration lifetimes. The increase of the standard deviation of the copper grain size distribution leads not only to longer mean times to failure, but also to larger variations of lifetimes. The first is naturally a beneficial effect, the latter, however, can be harmful for the interconnect reliability assessment. The increase of the standard deviation of lifetimes has a significant impact on correct lifetime estimation, since the increase of the standard deviation of lifetimes may indicate that a given interconnect technology fails earlier than expected, even if it shows a longer median lifetime.

To sum up, a complex and robust TCAD electromigration model which takes into account a wide diversity of physical phenomena was developed. Several numerical simulations of realistic three-dimensional interconnect structures were carried out, and several features of the electromigration failure were explained.

Nevertheless, there are still several points which should be improved, and they are suggestions for future work. From the physical modeling point of view a natural extension of this work is to develop a void evolution model also suitable for TCAD implementation and for three-dimensional simulations. This is a challenging task, since the void evolution problem requires the consideration of additional physical phenomena and, moreover, represents a moving boundary problem. This demands special numerical techniques for tracking the void surface as it evolves. Also, further investigations regarding the microstructure impact on the electromigration distribution are required. For that purpose, the use of a more realistic microstructure is needed. This will provide a more complete understanding of the underlying statistics of the electromigration lifetime distribution, which, in turn, can significantly contribute to the practical lifetime estimation and interconnect design.

R. L. de Orio: Electromigration Modeling and Simulation