Hajdin Ceric Associate Prof. Dipl.-Ing. Dr.techn. Publications

The reliability of each technological improvement of copper-based and copper-replacement-metals-based interconnects is characterized by dozens of material parameters related to the metals' microstructure and interfaces. The most important of these parameters are the effective valences and the diffusion coefficients. As the proportionality factor between the intensity of the electromigration force and the external electric field, the product of the effective valence and the diffusivity coefficient plays a central role in the dynamics of the degradation and wear-out of interconnect metals. During the degradation process, vacancies are transported through the bulk of the interconnect metal, along the grain boundaries and along the interfaces. In Fig. 1, the major migration paths in a polycrystalline metallic structure are presented. Generally, values of effective valence and diffusivity along these three transport paths vary significantly.
While the numerical determination of the diffusion coefficient is possible using a wide spectrum of methods, the numerical methods for determining the effective valence have still not reached a level of maturity such that they can be routinely applied with the desired accuracy. This is because the diversity and complexity of practical applications would demand a general approach. Of the many different quantum mechanical approaches to matter available, only density functional theory (DFT) would be suitable.
In our research, we apply different methods to estimate the values of the effective valences at different types of copper grain boundaries and interfaces(and similarly for some of the most promising copper-replacement metals). Some of these estimation methods are analytical calculations based on a simple model of grain boundaries, in which they act as a repulsive barrier for the current-carrying electron. Others include DFT calculations of the equilibrium electron density in order to determine the atomic structure that defines a specific grain boundary, as well as the application of the Feynman-Hellman theorem to obtain the electromigration force. More advanced methods are based on the theory of non-equilibrium Green's functions. By applying and comparing different estimation methods, a standard method that presents a trade-off between accuracy and simplicity should be found, which can be routinely applied while assessing new materials and technologies. Prior to the application of the effective valence estimation method for a specific grain boundary, the atomic structure of the grain boundary needs to be constructed. The constructing of grain boundaries is performed in two steps. The first step is a purely geometrical operation, where the intended type of grain boundary is created by aligning two differently oriented crystallographic planes. In the second step, by applying a molecular dynamics simulation, the minimum-energy configuration is identified. The relaxed structure of the copper grain boundary is shown in Fig. 2. Blue and light-blue spheres mark atomic positions in alternating [002] planes perpendicular to the tilt axis [001]. On the left-hand side, the calculated effective valence (Z*) along the grain boundary is presented. As can been seen, the estimated value of the effective valence drops to 20% of its bulk value.

Fig. 1: Migration paths in a polycrystalline metallic interconnect line: Blue - bulk, red - interfaces and yellow - grain boundaries.

Fig. 2: Calculated values of the effective valence (Z*) for the case of a copper grain boundary with a reciprocal density of coincident sites equal to 5 and a tilt angle of 53.1 degrees (210).