4.2 The Physics of Electromigration

The essential ingredients of electromigration are diffusion and a driving force. The diffusion produces random motion of atoms. Grain boundary diffusion is the dominant diffusion mechanism at lower temperatures, but at higher temperatures the main contribution comes from bulk diffusion. The second ingredient of electromigration, the driving force, brings a bias into the random motion of atoms. The contributions of the driving force are the wind force $\mathbf{F}_{wind}$ and direct force $\mathbf{F}_{direct}$. The source of the wind force is the scattering of the atoms by the moving atom. A well-established quantum-mechanical expression for the wind force induced by the electrical field $\mathbf{E}$ is available [71],

$$\mathbf{F}_{wind}=\sum_k \delta f(k)\langle\Psi_k\vert-\nabla V(\mathbf{r})\Bigl\vert _{\mathbf{r}=\mathbf{r}_P}\vert\Psi_k\rangle,$$ (4.1)

with,

$$\delta f(k)=e\tau_k \mathbf{E}\cdot \mathbf{v}_k \frac{d f_{FD}(\epsilon_k)}{d\epsilon_k}.$$ (4.2)

Equation (4.1) expresses the electron wind force for an atom at the position given with the vector $\mathbf{r}_P$. $\Psi_k$ is the electron wave function and $\mathbf{v}_k$ is the electron velocity, both are labeled by crystal momentum $\mathbf{k}$ and band index $n$ ( $k = (\mathbf{k},n)$). The transport relaxation time $\tau_k$ is inversely proportional to the resistivity. $f_{FD}$ is Fermi-Dirac distribution and $V(\mathbf{r})$ is the potential of the atom.

An atom is subject to the direct force $\mathbf{F}_{direct}$ of the electrical field if it's nuclear charge is not completely screened. This force directly pushes the atom towards the cathode.

Both direct and wind force are proportional to the electric field [72,71] and the total electromigration driving force can be written as,

$$\mathbf{F}=\mathbf{F}_{wind}+\mathbf{F}_{direct}=(Z_{wind}+Z_{direct})e\mathbf{E}=Z^*e\mathbf{E}.$$ (4.3)

The effective valence $Z^*$, which is the sum of the wind ($Z_{wind}$) and direct valence ( $Z_{direct}$), has been measured for a lot of systems [73]. The wind valences $Z_{wind}$ calculated on the basis of (4.1) are in good agreement with measurement results [74], but in this work we make use of measurement results only.