2.2 Diffusivity Paths

There are several possible diffusivity paths in an interconnect line, so the total material transport is determined by the sum of the mass transport taking place along each of these paths. Typically, the material flux through these paths is taken into account by setting an effective diffusion coefficient of the form [4,21]

$$\Deff = \Dl + f_{gb}\Dgb + f_i\Di + f_c\Dc,$$ (2.8)

where $\Deff$ is the effective diffusivity, $\Dl$, $\Dgb$, $\Di$, $\Dc$ are the diffusion coefficients for diffusion through the lattice, grain boundary, material interfaces and dislocation cores (``pipe diffusion''), respectively, and $f_{gb}$, $f_i$, $f_c$ denote the corresponding fractions of atoms diffusing along these paths.

For a typical dual-damascene interconnect the effective diffusivity is given by [68,69]

$$\Deff = \Dl + \symGBthickness\frac{(w-\symGBd)}{w\symGBd}\Dgb + \symIntThick\frac{2(w+h)}{wh}\Di + \symDislDens\symDislArea\Dc,$$ (2.9)

where $\symGBthickness$, $\symIntThick$ are the grain boundary and interfaces thicknesses, respectively, $\symGBd$ is the average grain diameter, $\symDislDens$ is the dislocation density, $\symDislArea$ is the cross sectional area of a dislocation core, $w$ is the line width, and $h$ is the line height. The diffusion coefficients are expressed by the Arrhenius law

$$D=D_0\exp\left(-\frac{\Ea}{\kB\T}\right),$$ (2.10)

where $D_0$ is the pre-exponential factor and $\Ea$ is the activation energy. From these equations one can see that the effective diffusivity is determined by the dominant diffusion mechanism, i.e. by the fastest diffusivity path. The fastest diffusivity path depends on several factors, like the temperature, the microstructure, and the quality of the interface between the metal and adjacent layers. Typically, lattice diffusion has the highest activation energy, being the slowest path for mass transport, while the activation energy for diffusion along grain boundaries and interfaces is somewhat lower. In general, surfaces have the lowest values, being the fastest diffusivity paths.

Equation (2.9) allows to examine the relative influence of each path on the material transport due to electromigration. For example, when the linewidth is larger than the grain size, $w > d$, it is expected that grain boundaries form a continuous path in such a way that grain boundary diffusion might significantly contribute to the total mass transport along the line. On the other hand, interfacial diffusion becomes more and more important when the linewidth is less than the average grain size, $w < d$ (``bamboo-like structures''), since there is no continuous path for atomic transport along grain boundaries.

For aluminum based interconnects, the activation energy for grain boundary diffusion is significantly lower than that for interfacial diffusion. This is attributed to the formation of a stable native oxide on the aluminum surface, which reduces the interfacial diffusivity [3]. Thus, for polycrystalline lines diffusion along grain boundaries is expected to be the dominant transport mechanism. For a bamboo-like structure the interfacial diffusion becomes the dominant path, as mentioned above.

In turn, the activation energy for interfacial diffusion in copper seems to be lower than for diffusion along the grain boundary [13,70]. Consequently, the interface between copper and surrounding layers is the main diffusivity path, for both polycrystalline and bamboo lines. Nevertheless, it has been suggested that there should be a significant contribution of grain boundary diffusion to the total electromigration induced mass transport in copper polycrystalline lines [18,71]. This may become a key issue for the new technological nodes (32 nm and below), since the copper lines are expected to have polycrystalline structures at such small dimensions [18].