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3.5  Void Nucleation

In early days void nucleation was understood as a vacancy condensation in the structure.
For the vacancy condensation an unrealistic high concentration of the vacancies is needed to reach the supersaturation for condensation. Therefore, the process of condensation under classical thermodynamics can not be responsible for the void nucleation due to EM [49].
In contrast to the process of void nucleation mentioned above, various authors have determined the cause for void nucleation by the excess of the mechanical stress over a critical threshold [34]. Gleixner et al. [49] investigated nucleation rates within an interconnect line and gave a formula for the free energy \( F \) change due to the void formation in aluminium.

(3.44) \{begin}{gather} \triangle F=-\sigma V_{\mathrm {v}\mathrm {o}\mathrm {i}\mathrm {d}}+\gamma _{\mathrm {Al}}A_{\mathrm {Al}} \mathref {(3.44)}

\( V_{\mathrm {v}\mathrm {o}\mathrm {i}\mathrm {d}} \) is the volume of the void, \( A_{\mathrm {Al}} \) the area, \( \sigma   \) the hydraulic pressure, and \( \gamma _{\mathrm {Al}} \) the surface energies of aluminium. The first part stands for the energy freed by the dissipation of the elastic strain energy and the second denotes the energy consumed by the creation of new surfaces. The calculations of Gleixner et al. [49] revealed small rates for nucleation at grain boundaries, at interfaces of metals and surrounding isolation layers, and even at grain boundary intersecting metal/isolation interfaces. Therefore, this mechanism can be responsible for void nucleation.
Nevertheless, the nucleation of voids at interfaces between a isolation layer and a metal are frequently observed at positions where they intersect with grain boundaries. This phenomenon was resolved by Flinn [43], who assumed that the void forms at preexisting free surfaces. Free surfaces are formed due to contamination, while the line is fabricated. This contamination lowers the adhesion between the metal and the isolation layer. Based on this assumption, he derived a threshold stress \( \sigma _{\mathrm {t}\mathrm {h}} \) for a flaw of the radius \( R_{\mathrm {p}}.   \)

(3.45) \{begin}{gather} \sigma _{\mathrm {t}\mathrm {h}}=\frac {2\gamma _{\mathrm {s}}}{R_{\mathrm {p}}} \mathref {(3.45)} \{end}{gather}

Clemens et al. [25] showed that the formula of Flinn is only valid as long as the void grows on the free surface of the contaminated region. As the void grows further beyond the flaw area, the contact angle \( \theta _{\mathrm {c}} \) is in the range \( 0 < \theta _{\mathrm {c}} < 90^{\circ } \) leading to a reduced threshold stress given by

(3.46) \{begin}{gather} \displaystyle \sigma _{\mathrm {t}\mathrm {h}}=\ \frac {2\gamma _{\mathrm {s}}\sin \theta _{\mathrm {c}}}{R_{\mathrm {p}}}.
\mathref {(3.46)} \{end}{gather}

Both equations show an inverse dependence of the threshold stress on the flaw radius.
The stress required to form a void is in the order of \( 350\mathrm {M}\mathrm {N}/\mathrm {m}^{2} \) and can therefore be reached by the mechanical stress build-up by thermal stress [21]. For bigger flaw radii in the range of \( 100\mathrm {n}\mathrm {m} \) the threshold stress is in the range of \( 70\mathrm {M}\mathrm {N}/\mathrm {m}^{2} \) and can therefore also be built up by EM [34, 50].

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