2.5.1 Kirchheim Back Stress

In the work of Blech, presented in Section 2.1.2, it was found that electromigration gives rise to back stresses which may delay device failure. The nature and the origin of the stress were not clear. Kirchheim [91] was the first who incorporated the effect of the transient back stress build-up in a single model. Considering the movement of an atom, with an atomic volume Ω_a, from the grain boundary to the surface, the relaxation of the neighboring atoms within the grain boundary leads to a volume contraction of fΩ_a around the formed vacancy. The total volume change due to lattice relaxation is (1-f)Ω_a, where f is the relaxation factor which represents the ratio between the vacancy volume and the atomic volume. From the local strain field, induced by the volumetric change at the lattice site, a stress gradient is produced. The gradient of the mechanical stress acts as an additional driving force in the total vacancy flux equation, as follows

\[ \begin{equation} \vec{J}_\text{v}=\vec{J}^\text{d}_\text{v}+\vec{J}^\text{em}_\text{v}+\vec{J}^\text{$\sigma$}_\text{v}=-D_\text{v}\nabla C_\text{v}+\cfrac{C_\text{v}D_\text{v}}{k_\text{b}T}Z^*|e|\vec{E}-\cfrac{C_\text{v}D_\text{v}}{k_\text{b}T}f\Omega_\text{a} \nabla \sigma, \end{equation}\] (2.24)

where σ is the spherical part of the mechanical stress tensor. Considering a vacancy sink/source similar to that presented in equation (2.21), the continuity equation can be rewritten as

\[\begin{equation} \frac{\partial{C_\text{v}}}{\partial t}=- \nabla \cdot \left(-D_\text{v}\nabla C_\text{v}+\cfrac{C_\text{v}D_\text{v}}{k_\text{b}T}Z^*|e|\vec{E}-\cfrac{C_\text{v}D_\text{v}}{k_\text{b}T}f\Omega_\text{a} \nabla \sigma \right)+\cfrac{C_{\text{v}\text{,eq}}-C_\text{v}}{\tau_\text{v}}, \end{equation}\] (2.25)

where the expression for the equilibrium vacancy concentration in the grain boundary is related to the mechanical stress as follows [7]

\[ \begin{equation} C_{\text{v}\text{,eq}}=C_\text{v,0}\ \text{exp}\left(\cfrac{(1-f)\Omega_\text{a}\sigma}{k_\text{b}T}\right), \end{equation}\] (2.26)

and C_v,0 is the initial vacancy concentration in the absence of stress. The rate of the volumetric change ΔV/V, produced by the generation of a vacancy within a grain of diameter d, is related to the rate of vacancies accumulation within the grain boundary of thickness δ by

\[ \begin{equation} \cfrac{1}{V} \frac{\partial{V}}{\partial t}=(1-f) \Omega_\text{a} \cfrac{\delta}{d} \cfrac{C_{\text{v}\text{,eq}}-C_\text{v}}{\tau_\text{v}}. \end{equation}\] (2.27)

Using Hooke's law [151]

\[\begin{equation} \text{d}\sigma=B\cfrac{\text{d} V}{V}, \end{equation}\] (2.28)

the time evolution of the stress build-up due to the deviation of the vacancy concentration from its equilibrium is given by

\[ \begin{equation} \frac{\partial{\sigma}}{\partial t}=B(1-f)\Omega_\text{a} \cfrac{\delta}{d} \cfrac{C_{\text{v}\text{,eq}}-C_\text{v}}{\tau_\text{v}}, \end{equation}\] (2.29)

where B is the bulk modulus of the metal line.

By coupling the stress development (equation (2.29)) with the vacancy concentration dynamics (equation (2.25)) along the metal line, analytical solutions relating the mechanical stress to the production/annihilation of vacancies in the grain boundary can be derived for a few limiting cases. Furthermore, Kirchheim identified the three phases for vacancy concentration and stress evolution which significantly contribute to the understanding of the electromigration phenomenon in 3D interconnects.