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The vacancy source/sink term given in equation (3.27) models vacancy generation/annihilation processes, which occur only inside the
fast diffusivity paths in the interconnect.
Typical diffusion paths, which act as sources and sinks for vacancies in metals, are grain boundaries, interfaces, and dislocations [7].
The properties of microstructure and diffusion paths have a strong impact on vacancy transport in the interconnect line.
Vacancy dynamics at these locations are composed of three different mechanisms:
As discussed in Section 2.4, each diffusion path has a different effective vacancy diffusion coefficient Dvn and an effective charge number Zn*. It is therefore possible to describe the distribution of vacancy concentration Cvn independently for each region of the interconnect line in the following way
where n represents the available diffusion path in the interconnect. In copper-based interconnects, typical diffusion paths are bulk (b), grain boundaries (gb), and material interfaces (i). The diffusion of vacancies along the various available paths must be taken into account in order to understand the impact of each region on the distribution of vacancies in the interconnect line. The fastest diffusivity path will dominate the development of electromigration failure in the interconnect.
For this purpose, a detailed model is presented in the following section which describes the effect of two diffusivity paths, grain boundary and interface. These paths affect the vacancy dynamics and the consequent build-up of mechanical stress inside the interconnect line.
The network of grain boundaries influences the electromigration-induced transport of vacancies during electrical operation [41]. The vacancy diffusion inside the grain boundary is faster than the bulk diffusion, because the barrier energies for the formation and migration of vacancies inside grain boundaries are lower than those for the bulk material. This is due to the larger diversity of vacancy exchange mechanisms into the grain boundary [143]. The combination of the works of Herring [76], Fisher [58], and Ceric [25] enables to understand the vacancy dynamics in the presence of grain boundaries. The grain boundary model is developed by considering diffusion in a semi-infinite bulk line containing a single grain boundary of width δ normal to the surface, as illustrated in (3.2).
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Herring [76] obtained a relationship for the chemical potential of a vacancy on the grain boundary μvgb by investigating the equilibrium during the process of material exchange between the grain boundary and grain bulk. In the case that vacancy concentration varies along the grain boundary, the relationship can be expressed as follows
| \[\begin{equation} \mu^\text{gb}_\text{v}=\mu_\text{0}+\Omega_\text{a}\sigma_\text{nn}+k_\text{b}T\ln\left(\cfrac{C^\text{gb}_\text{v}}{C_\text{v,0}}\right), \end{equation}\] | (3.31) |
where μ0 is the reference chemical potential and Cvgb is the vacancy concentration in the grain boundary. The expression for the chemical potential of a vacancy in the bulk μv [100] is given by
| \[\begin{equation} \mu_\text{v}=\mu_\text{0}+\cfrac{1}{3}f\Omega_\text{a}\mathrm{Tr}(\overline{\overline\sigma})+k_\text{b}T\ln\left(\cfrac{C_\text{v}}{C_\text{v,0}}\right), \end{equation}\] | (3.32) |
When μvgb=μv, the local equilibrium between surface stress and vacancy concentration [7] permits to obtain the equilibrium concentration of vacancies inside the grain boundary Cv,eqgb as follows
| \[\begin{equation} C^\text{gb}_{\text{v}\text{,eq}} = C_\text{v,0} \ \text{exp}\left(-\cfrac{\Omega_\text{a}\sigma_\text{nn}}{k_\text{b}T}\right). \end{equation}\] | (3.33) |
By following equation (3.18), the general expression for the vacancy flux along the grain boundary Jvgb is given by
| \[\begin{equation} J^\text{gb}_\text{v} = \cfrac{D^\text{gb}_\text{v}C^\text{gb}_\text{v}}{k_\text{b}T}\frac{\partial{\mu^\text{gb}_\text{v}}}{\partial x}, \end{equation}\] | (3.34) |
where Dvgb is the vacancy diffusion coefficient for the grain boundary.
The grain boundary model of Fisher [58] is able to describe the vacancy transport inside the grain boundary by considering the contribution of the vacancy exchange mechanisms between grain boundary and grain bulk, and vice versa. The material balance equation that drives the vacancy concentration distribution inside the grain boundary is given by
| \[\begin{equation} \frac{\partial {C^\text{gb}_\text{v}}}{\partial t}=-\frac{\partial{J^\text{gb}_\text{v}}}{\partial x}-\cfrac{J^{2}_\text{v}-J^{1}_\text{v}}{\delta}, \end{equation}\] | (3.35) |
where Jv2 and Jv1 are the normal components of the flux from both sides of the grain boundary( (3.2)). The second term on the right hand side of equation (3.35) represents the recombination rate of vacancies at the grain boundary region due to the diffusing fluxes from/to the bulk. In the continuum modeling approach, the chemical potential on the grain boundary (equation (3.31)) is constant throughout the width δ, and is equal to the chemical potential in the bulk (equation (3.32)) at the interfaces between bulk material and grain boundary such that
| \[\begin{equation} \mu^{1}_\text{v}(-\delta/2)=\mu^{2}_\text{v}(\delta/2)=\mu^\text{gb}_\text{v}. \end{equation}\] | (3.36) |
The difference of the vacancy fluxes on both sides of the grain boundary in equation (3.35) corresponds to the gain/loss of vacancies localized at the grain boundary and approximates the vacancy generation/annihilation rate G in equation (3.27) with
| \[\begin{equation} G=-\cfrac{J^{2}_\text{v}- J^{1}_\text{v}}{\delta}=-\cfrac{D_\text{v}C_\text{v}}{\delta k_\text{b}T}(\nabla \mu^{2}_\text{v}-\nabla \mu^{1}_\text{v}). \end{equation}\] | (3.37) |
Ceric [25] provided a different interpretation of the last term by introducing the concept of trapped vacancies inside the grain boundary, with concentration Cvt, in order to enable a more convenient numerical implementation of the grain boundary model. The model is based on the theory of segregation at the interfaces developed by Lau [101]. The grain boundary is assumed to be able to trap vacancies from both sides of the grain boundary with trapping rate ωt and release them to the grains with release rate ωr. The fluxes in equation (3.37) can therefore be expressed as follows
| \[\begin{equation} J^{1}_\text{v}=\omega_\text{t}(C^\text{gb}_{\text{v}\text{,eq}}-C^\text{t}_\text{v})C^{1}_\text{v}-\omega_\text{r}C^\text{t}_\text{v}, \end{equation}\] | (3.38) |
| \[\begin{equation} J^{2}_\text{v}=-\omega_\text{t}(C^\text{gb}_{\text{v}\text{,eq}}-C^\text{t}_\text{v})C^{2}_\text{v}+\omega_\text{r}C^\text{t}_\text{v}, \end{equation}\] | (3.39) |
where Cv1 and Cv2 are the vacancy concentrations in each grain. By substituting equation (3.38) and (3.39) into equation (3.37), it is possible to obtain the vacancy source/sink term as follows
Since the grain boundary represents the interface between grains, Cv1=Cv2=Cv, and equation (3.40) simplifies to
where
| \[\begin{equation} \tau^\text{gb}_\text{v}=\cfrac{\delta}{2\omega_\text{t}C_\text{v}} \end{equation}\] | (3.42) |
is the characteristic time of vacancy generation/annihilation for the grain boundary. Small values of τvgb indicates that the grain boundary acts as an efficient source/sink of vacancies.
The vacancy dynamics in the presence of material interfaces can be described by using the same approach employed above for the grain boundary model. Typically, interfaces separate materials with different properties. Interfaces have higher diffusion coefficients than the bulk material, and act as sources/sinks for vacancies. If the interface separates a conducting material from a non-conducting one, the interface acts as a blocking boundary for vacancy transport from the metal line to the interface ( (3.3)). Furthermore, since there is no flux of vacancies due to electromigration in the non-conducting material, the flux term Jv2 in equation (3.37) vanishes. In this situation, equation (3.41) reduces to
where
| \[\begin{equation} \tau^\text{i}_\text{v}=\cfrac{\delta}{\omega_\text{t}C_\text{v}} \end{equation}\] | (3.44) |
is the characteristic generation/annihilation time for the material interface.
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It should be noted that under the condition
equations (3.41) and (3.43) reduce to the Rosenberg-Ohring term presented in equations (2.21). The grain boundary and material interface models presented above include the effect of fast diffusivity paths as vacancy sinks/sources into the bulk vacancy transport model. Furthermore, this behavior is important for the mechanical stress calculations, described in the following section.