Next: 2.6 Void Initiation Condition Up: 2.5 Stress Build-up due to Electromigration Previous: 2.5.2 Korhonen's Model
Sarychev [132,133] treated the stress build-up due to electromigration similar to the thermal expansion-induced stress, and extended the previous works within the scope of a standard 3D continuum model. The vacancy flux as well as the creation and annihilation of vacancies lead to local volume changes inside a metal line and induce its deformation. The total strain tensor εik has an inelastic component εikv due to vacancy migration and generation/annihilation processes and a purely elastic component εikσ due to the direct action of mechanical stress
The rate of the inelastic strain tensor εikv includes contributions from both the pile up and the generation/annihilation of vacancies, and is described by the following kinetic equation
where δik is the Kronecker delta function. The inelastic strain tensor is diagonal with equal diagonal entries. The equation (2.39) is therefore replaced by a single kinetic equation as follows
The inhomogeneities in the vacancy concentration distribution and the interaction of the metal with the surrounding medium produce volumetric elastic strains in the line. According to Hooke's law [151], metals respond to elastic strains by a build-up of stresses as follows
| \[\begin{equation} \epsilon^\text{$\sigma$}_\text{ik}= S_\text{iklm}\sigma_\text{lm}, \end{equation}\] | (2.41) |
where Siklm is the matrix of elastic compliances. The rate of the elastic strain tensor is given by
| \[\begin{equation} \frac{\partial{\epsilon^\text{$\sigma$}_\text{ik}}}{\partial t}= S_\text{iklm}\frac{\partial{\sigma_\text{lm}}}{\partial t}. \end{equation}\] | (2.42) |
Substituting equations (2.40) and (2.42) into equation (2.38), and performing the inversion of the tensor Siklm, the rate of mechanical stress becomes
where Ciklm is the matrix of elastic stiffness. The substitution of equation (2.41) into equation (2.38) results in the important stress-strain relationship given by
| \[\begin{equation} \sigma_\text{ij}= -B\epsilon^\text{v}_\text{kk}\delta_\text{ij}+\lambda\epsilon_\text{kk}\delta_\text{ij}+2\mu\epsilon_\text{ij}, \end{equation}\] | (2.44) |
where λ and μ are the Lame constants, and B is the bulk modulus. Since the accelerations are small during electromigration, the mechanical equilibrium equation [98] is obtained by
| \[\begin{equation} \sum_{j=1}^3\frac{\partial{\sigma_\text{ij}}}{\partial{x_\text{j}}}= 0. \end{equation}\] | (2.45) |
Substituing equation (2.44) into equation (2.45), and using the small displacement approximation
| \[\begin{equation} \epsilon_\text{ik}= \cfrac{1}{2} \left(\frac{\partial{u_\text{i}}}{{\partial x_\text{k}}}+\frac{\partial{u_\text{k}}}{{\partial x_\text{i}}}\right), \end{equation}\] | (2.46) |
the three equations which define the displacement vector u are obtained as follows
In this way, a set of 3D self-consistent equations (2.25), (2.40), (2.47), and (2.44) for the evolution of mechanical stress inside the passivated metal line during electromigration is obtained.
The main innovation of the model was to connect the evolution of all the components of the stress tensor with the vacancy transport, by including the impact of the geometry of the metallization and imposed displacement boundary conditions. Furthermore, Sarychev's model is the first general 3D model for describing the electromigration stress build-up, which can be applied for an arbitrary 3D geometry.
Similarly to Sarychev's work, Sukharev [144,145,146,147] independently developed a model for describing the electromigration stress build-up where different diffusion paths for bulk, interfaces, and grain boundaries were considered. Sukharev distinguished between different vacancy concentrations for bulk Cvb and the others fast diffusivity paths Cvn. Plated atoms were introduced to describe the atomic exchange between bulk and fast diffusivity paths. In this way, the vacancy generation/annihilation is represented by atom plating/removal from those locations where the activation energy for diffusion is lower. The rate of the plated atom exchange has the standard form of the G term presented in Section 2.3
| \[\begin{equation} G_\text{n}= -\cfrac{C^\text{n}_\text{v}-C_{\text{v}\text{,eq}}}{\tau_\text{v}}. \end{equation}\] | (2.48) |
Considering immobile plated atoms with a concentration Cpl, the plated atom continuity equations for bulk and diffusivity paths are given by
and
| \[\begin{equation} \frac{\partial{C^\text{n}_\text{pl}}}{\partial t}= \cfrac{C^\text{n}_\text{v}-C_{\text{v}\text{,eq}}}{\tau_\text{v}}, \end{equation}\] | (2.50) |
respectively. Following this approach, Sukharev obtained an equation which combines the electromigration-induced strain tensor with the vacancy concentration and the plated atom concentration as follows
| \[\begin{equation} \epsilon^\text{v}_\text{ik}= \Omega_\text{a} ((f-1)(C^\text{n}_\text{v}-C^\text{n}_\text{v,0})+(C_\text{pl}-C_\text{pl,0})\delta_\text{ik}, \end{equation}\] | (2.51) |
where Cv,0n and Cpl,0 are the vacancy concentration and the atom concentration in the plated layer at zero stress, respectively. He concluded that the evolution of the plated atom concentration is mainly responsible for the electromigration stress build-up.
The models independently developed by Sarychev and Sukharev allow for the study of the stress build-up due to electromigration in complex 3D interconnect structures. Further aspects concerning the coupling to the electro-thermal problem and extensions with the fast diffusivity paths approach will be presented in Chapter 3.