2.6 On the Void Evolution

The development of fatal voids, i.e. voids that trigger the line failure, is the ultimate cause for the electromigration induced interconnect failure [22,81]. The failure criterion is typically set as a maximum resistance increase tolerated for the corresponding interconnect line. Once a void is nucleated it can evolve, until it causes a significant resistance increase or even completely severs the line.

The void evolution phase can encompass several processes: a void can migrate along the interconnect [10,114], interact with the local microstructure [10,38,71] and grow, or even heal [10,11], undergo morphologic changes, assuming wedge-like shape or slit-like shape [27], before it definitely triggers interconnect failure. Furthermore, multiple voids can form in a line, so that their migration and agglomeration at a specific critical site can be the mechanism responsible for the interconnect failure [31,32,114].

The void surface acts as an additional path for atomic migration. The chemical potential of an atom on the void surface is given by [115,116,117]

$$\symSurfChemPot = \RefChemPot + \symAtomVol\left(\symElasticEnergy - \symSurfEnergy\symCurv\right),$$ (2.65)

where $\RefChemPot$ is a reference chemical potential, $\symElasticEnergy = \left(\boldsymbol\symHydStress : \boldsymbol\symStrain\right)/2$ is the elastic energy density of the material adjacent to the void, $\symSurfEnergy$ is the surface free energy, and $\symCurv$ is the curvature of the void surface. Thus, the atomic flux along the void surface due to gradients in chemical potential plus electromigration has the form

$$\vec\Js = -\frac{\Ds\symSurfThick}{\kB\T}\left(\ensuremath{\nabla_s{\symSurfChemPot}} + \ee\vert\Z\vert\vec\symSurfElecField\right),$$ (2.66)

where $\Ds$ is the surface diffusivity, $\symSurfThick$ is the surface thickness, $\vec\symSurfElecField$ is the electric field tangential to the void surface, and $\ensuremath{\nabla_s{}}$ denotes the gradient along the surface. By mass conservation the normal velocity at any point on the surface is given by [116,117]

$$\vn = -\ensuremath{\nabla_s\cdot{\vec\Js}}.$$ (2.67)

Void evolution due to electromigration is a complex dynamic process, for which modeling is a challenging task and, moreover, represents a moving boundary problem. Analytical solutions can only describe the asymptotic behavior of the moving boundary [105,118,119,120,121,122,123], since, in general, the shape changes which the void experiences cannot be analytically resolved. Therefore, a more general treatment demands the application of numerical methods and special techniques for tracking the void.

The most commonly used numerical method is based on sharp interface models [124,125,126,127,128], which requires an explicit tracking of the void surface and, consequently, a continuous remeshing procedure. As the void migrates, grows, and changes shape this explicit tracking becomes very demanding. Therefore, it can be satisfactorily applied only for simple two-dimensional cases and cannot be further extended. This shortcoming can be overcome with the introduction of the so-called diffuse interface model (or phase field model) [116,117,129,130,131,132,133] or the level set method [134,135,136,137,138]. The main advantage of these approaches is that the void is implicitly represented by a field parameter or level set function, so that void evolution is implicitly determined by the calculation of these functions. Thus, the demanding explicit void surface tracking can be avoided.