2.6 Void Initiation Condition

It has been shown in Section 2.3 that the condition for void nucleation was attributed to reaching a certain critical vacancy supersaturation concentration during the process of vacancy accumulation at sites of flux divergence, leading to vacancy condensation [138,37,129]. The void nucleation criterion requires an unrealistically high concentration of vacancies to reach the maximum saturation level for condensation and consequent spontaneous formation of a void [129]. Therefore, the condition of electromigration void nucleation based on the vacancy condensation mechanism was inappropriate since it cannot be justified from a thermodynamic point of view.

Several authors have recognized the importance of the electromigration stress build-up in an interconnect as the major cause for void nucleation [81,124,77,91,93]. The formation of a void occurs after a tensile stress threshold is reached. Different approaches for stress-driven void nucleation by vacancy condensation were discussed in literature [66,74]. All of those works were derived from the nucleation theory based on classical thermodynamics [50]. By employing this theory, Gleixner [63,64] analyzed the nucleation rates at various locations within passivated aluminum interconnects. For this case, the free energy change ΔF upon the creation of an embryo in the aluminum line is given by

\[\begin{equation} \Delta F= \Delta F_\text{$v$}V_\text{e}+\gamma_\text{$Al$}A_\text{$Al$}+(\gamma_\text{$Al_2O_3$}-\gamma_\text{$Al-Al_2O_3$})A_\text{$i$}-\gamma_\text{$gb$}A_\text{$gb$}, \end{equation}\] (2.52)

where ΔF_v is the Helmholtz free energy change per unit volume of the embryo, V_e is the volume of the embryo, γ_i refer to the interfacial free energies of the metal, passivation layer, metal/passivation interface, and grain boundary, respectively, and A_i are the areas of interfaces created/destroyed upon embryo formation. In the case of vacancy coalescence in a stressed material, ΔF_v is related to the hydrostatic stress in the line by

\[\begin{equation} \Delta F_\text{$v$}= -\sigma. \end{equation}\] (2.53)

This expression represents the energy released by the dissipation of the elastic strain energy in the metal. For positive σ, the free energy in equation (2.52) increases until a critical embryo volume value is reached. The maximum value referred to as the critical embryo is determined by the energy barrier for void nucleation ΔF^*, which is given by the condition

\[\begin{equation} \Delta F^*= \Delta F|_\text{$\partial(\Delta F)=0$}. \end{equation}\] (2.54)

Once the energy barrier and the site geometry are known, the void nucleation rate per unit volume at various sites within an interconnect can be calculated.

Small rates for nucleation are observed at metal/passivation interfaces, at grain boundaries, and at triple points, i.e. intersections between grain boundaries and metal/passivation layer interfaces. This means that void nucleation by vacancy condensation at these locations is not possible.

The last case is particularly interesting, since voids are typically observed to nucleate at the triple points [35]. An analysis carried out by Flinn [59] was useful to solve this apparent discrepancy by introducing a new concept for the understanding of void nucleation. The author suggested the idea that a flaw could form on the interface between the metal and the passivation layer with no adhesion. The so-called pre-existing adhesion-free patch can be the result of contamination or surface defect during the interconnect fabrication, which leads to weak adhesion between the metal and surrounding layer. By considering a circular patch of radius R_p, Flinn [59] derived an expression for the threshold stress σ_thr for void nucleation as follows

\[\begin{equation} \sigma_\text{thr}= \cfrac{2\gamma_\text{$Al$}}{R_\text{p}}. \end{equation}\] (2.55)

The formula is valid as long as the void grows in the region of weak adhesion. An extended version of the model proposed by Clemens [36] considered the void growth beyond the free surface of the contaminate region once the equilibrium contact angle ϑ_c is reached (2.4). ϑ_c is in the range between 0 and 90 leading to a decreased threshold stress value given by

\[\begin{equation} \sigma_\text{thr}= \cfrac{2\gamma_\text{$Al$}\sin\theta_\text{c}}{R_\text{p}}. \end{equation}\] (2.56)

If the stress is above the threshold value, the energy barrier between the embryo and a stable-growing void vanishes, leading to the formation of a void at different sites within an interconnect.

Figure 2.4: Schematic of the embryo at the line/passivation interface.