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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]

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

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

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

next up previous contents
Next: 3. A General TCAD Up: 2. Physics of Electromigration Previous: 2.5 Void Nucleation

R. L. de Orio: Electromigration Modeling and Simulation