1 Theoretical Formulation

We assumed unpassivated monocrystal isotropic interconnects where stress phenomena can be neglected. An interconnect is idealized as two-dimensional electrically conducting via which contains initially a circular void. For simplicity we also neglect the effects of grain boundaries and lattice diffusion. In this case there are two main forces which influence the shape of the evolving void interface: the chemical potential gradient and electron wind. The first force causes self-diffusion of metal atoms on the void interface and tends to minimize energy which results in circular void shapes. The electron wind force produces asymmetry in the void shape depending on the electrical field gradients.
Including contributions from both, electromigration and chemical potential-driven surface diffusion, gives the total surface atomic flux, $\bf {J_{A}}$ = $J_{A}\bf {t}$, where $\bf {t}$ is the unit vector tangent to the void surface [60,87]

$$J_A = D_s (-\vert e\vert Z^{*}E_s+\gamma_{s}\Omega \bf {t}\cdot\nabla_{s}\kappa)$$ (253)

$Z^{*}$ is the effective valence (Section 4.2), $e$ is the charge of an electron, $E_s\equiv \bf {E_s}\cdot{t}$ is the local component of the electric field tangential to the void surface, $\kappa$ is the local surface curvature, and $\nabla_s$ is the surface gradient operator; $\kappa \equiv \nabla \cdot \bf {n}$, where $\bf {n}$ is the local unit vector normal to the void surface. Further, $\gamma_{s}$ is the surface energy, $\Omega$ is the volume of an atom, and $D_s$ is given through an Arrhenius law:

$$D_s=\frac{D_0\delta_s}{k T} \Bigl(\frac{-Q_s}{k T}\Bigr)$$ (254)

Here, $\delta_s$ is the thickness of the diffusion layer, $T$ is the temperature, $Q_s$ is the activation energy for the surface diffusion, and $D_0$ is the pre-exponential factor for mass diffusion. Equation (4.43) is the Nernst-Einstein equation, where the sum in the parentheses on the right side expresses the driving force. Mass conservation gives the void propagation velocity normal to the void surface, $v_n$, through the continuity equation [61,63,87],

$$v_n = -\Omega \cdot \nabla_s \bf {J_A}.$$ (255)

The electric field $\bf {E}$ can be written as the gradient of the electric potential, i.e.,

$$\bf {E} \equiv -\nabla \varphi.$$ (256)

The field is also solenoidal, i.e.,

$$\nabla \cdot \bf {E} = 0.$$ (257)

Equations (4.46) and (4.47) imply that the potential $\varphi$ obeys Laplace's equation,

$$\Delta \varphi = 0.$$ (258)

The void surface and the interconnects edges are modeled as insulating boundaries, i.e.

$$\nabla \varphi \cdot \bf {n}=0,$$ (259)

at these boundaries.