Next: 2.5.3 Mechanical Constitutive Equations Up: 2.5 Stress Build-up due to Electromigration Previous: 2.5.1 Kirchheim Back Stress
The equations in Kirchheim's work were only capable to describe 1D finite lines blocked at both ends. The values of vacancy concentration and stress build-up due to electromigration were observed to be identical for every section on the line, because of the assumption of spherical stress. Further, the stress evolution equation succesfully incorporates the impact of the sink/source reactions, but neglects the contribution of the vacancy flow inside the line.
Korhonen [93] proposed a slightly different model than Kirchheim. He investigated the action of the stress gradient caused by electromigration in a narrow interconnect line deposited onto an oxidized silicon substrate, and covered by a rigid passivation layer. The formulation of the electromigration driving forces was based on the atomic flux instead of the vacancy flux. The major difference to Kirchheim's work was to consider the change of lattice sites per unit volume as source of deformation instead of the vacancy change. The material transport due to the passage of electric current along the line was assumed to be affected by grain boundary diffusion alone, as
where Da and Dgb are the atomic bulk diffusivity and the atomic grain boundary diffusivity, respectively. In this way, the atoms are deposited predominantly at the grain boundaries. The flux of atoms increases due to the differences in the chemical and electrical potentials between diverse locations of the interconnect line. Assuming thermal equilibrium of the vacancies (μv=0), the chemical potential function μ is given by [76]
| \[\begin{equation} \mu=\mu_\text{a}-\mu_\text{v}=\mu_\text{0}-\Omega_\text{a} \sigma, \end{equation}\] | (2.31) |
where μa is the atomic chemical potential, μ0 is the chemical potential at a stress free state, and σ is the tensile stress acting across the grain boundary. Including the impact of electric potentials due to electromigration, the atomic flux Ja is given by
| \[\begin{equation} \vec{J}_\text{a}=-\cfrac{C_\text{a}D_\text{a}}{k_\text{b}T}\left(\nabla\mu +Z^*|e|\vec{E}\right), \end{equation}\] | (2.32) |
where Ca is the atomic concentration and Da is the atomic diffusion coefficient.
In thermal equilibrium, the chemical potential is constant inside all grain boundaries. This implies that the deposition of atoms at the grain boundary is independent of its orientation. Furthermore, by including the effect of the rigid dielectric layer on the metallization line, the generation/annihilation of atoms in grain boundaries creates changes in the lattice sites concentration Cl per unit volume, resulting in the development of a uniform mechanical stress according to Hooke's law as follows
| \[\begin{equation} \cfrac{\text{d}C_\text{l}}{C_\text{l}}=-\cfrac{\text{d}\sigma}{B}. \end{equation}\] | (2.33) |
Since the lattice site occupied by an atom or a vacancy is assumed to have the same volume [102], the constitutive equation between stress and lattice site concentration leads to the reformulation of the vacancy continuity equation (equation (2.20)) as follows
Assuming that vacancy concentration is in thermal equilibrium with the stress inside the grain, then
| \[\begin{equation} C_\text{v}=C_{\text{v}\text{,eq}}=C_\text{v,0}\ \text{exp}\left(\cfrac{\Omega_\text{a}\sigma}{k_\text{b}T}\right). \end{equation}\] | (2.35) |
Substituting Cv (equation (2.35)) and μ (equation (2.31)), equation (2.34) becomes
Including the further assumption that just a very small number of vacancies is necessary to restore the vacancy equilibrium and to create stress (CvΩa/kbT≤Cl/B), the first term in the brackets on the left hand side of equation (2.36) is negligible. Taking into account that Cl=Ca=1/Ωa, the expression of the stress evolution along a metal line induced by electromigration is given by
| \[\begin{equation} \frac{\partial{\sigma}}{\partial t}= \nabla \cdot \left(\cfrac {D_\text{a}B}{k_\text{b}T}\left(\Omega_\text{a}\nabla\sigma -Z^*|e|\vec{E}\right)\right), \end{equation}\] | (2.37) |
where Da=DvCv/Cl.
Korhonen provided analytical solutions of equation (2.37) for several cases in order to estimate the stress build-up during electromigration. An interesting result is given with the solution for a semi-infinite line at x=0. (2.3) shows the stress build-up with time according to Korhonen's model. The inclusion of the stress dependence in the sink/source term of the continuity equation leads to a time-scaling change of the stress build-up. The lifetime predicted from Korhonen is calculated in hours, rather than in minutes obtained from the models presented in Section 2.3. The stress exhibits linear growth with time as predicted by Blech [16]. After a certain time, the stress starts to increase with the square root of time. Furthermore, reaching certain stress levels in the interconnect line represents a usual requirement for electromigration void nucleation.
The model well describes the origin of the stress development in a metal line. A closed equation was obtain for the stress distribution during electromigration, and the interconnect failure was associated with the build-up of a threshold stress value. The model does not consider the evolution of the components of the stress tensor which is influenced by external constraints imposed by passivation layers. An extended formulation of the problem which include contributions from different components of the stress tensor is therefore required.
