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Electromigration is a kinetic process which results in the net transport of metal atoms over macroscopic distances due to the correlation of two competing processes, namely diffusion and migration. Since metal atoms migrate via a vacancy exchange mechanism [20], the atomic flux can be expressed in terms of vacancy species. The vacancy flux is in the opposite direction to the atomic transport. In the following, an expression of the vacancy flux induced by the contributions of vacancy diffusion and electromigration will be provided.
Diffusion is a non-equilibrium process, and it ceases when the system reaches the full thermodynamic equilibrium. The laws of diffusion are mathematical relationships which relate the rate of diffusion to the concentration gradient responsible for the mass transfer [65]. Vacancies flow from regions of high concentration to regions of low concentration. According to the first Fick's law, the flux of vacancies Jvd, due to diffusion in any part of the system, is proportional to the gradient of the vacancy concentration Cv as follows
where Dv is the vacancy diffusion coefficient or vacancy diffusivity. Diffusion coefficients are related to temperature by the Arrhenius law
| \[\begin{equation} D=D_\text{0}\ \text{exp}\left(-\cfrac{E_\text{n}}{k_\text{b}T}\right), \end{equation}\] | (2.16) |
where D0 is the pre-exponential factor and En is the activation energy for diffusion of a given diffusive species N.
The action of the electromigration driving force Fem, presented in Section 2.2, on the vacancies gives rise to an extra velocity component vd for the affected diffusive species, in the direction of the force. Using the Nerst-Einstein relationship, the drift velocity is given by
| \[\begin{equation} \vec{v}_\text{d}=M \vec{F}_\text{em}=\cfrac{D_\text{v}}{k_\text{b}T}Z^*|e|\vec{E}, \end{equation}\] | (2.17) |
and the vacancy flux Jemv due to electromigration is calculated as follows
| \[\begin{equation} \vec{J}^\text{em}_\text{v}=C_\text{v}\vec{v}_\text{d}=\cfrac{C_\text{v}D_\text{v}}{k_\text{b}T}Z^*|e|\vec{E}. \end{equation}\] | (2.18) |
In a stress-assisted diffusion condition, the migration process is superimposed on diffusion. The tendency of vacancies to diffuse is opposite to their tendency to migrate under the influence of the electromigration force.
Therefore, the net vacancy flux Jv in any part of the system due to the competition of Fickian diffusion and electromigration is given by
| \[\begin{equation} \vec{J}_\text{v}=\vec{J}^\text{d}_\text{v}+\vec{J}^\text{em}_\text{v}=-D_\text{v}\nabla C_\text{v}+\cfrac{C_\text{v}D_\text{v}}{k_\text{b}T}Z^*|e|\vec{E}. \end{equation}\] | (2.19) |
Diffusion and electromigration fluxes cause a redistribution of the vacancies in the metal line. Since vacancy concentration is not a conserved quantity, the balance equation for vacancy conservation is given by Fick's second law as
| \[ \begin{equation} \frac{\partial{C_\text{v}}}{\partial{t}}=- \nabla \cdot \vec{J}_\text{v}+G, \end{equation}\] | (2.20) |
where G is the sink/source term which models the creation and annihilation of vacancies at particular sites inside the metal line [20]. These sites are grain boundaries, extended defects, and interfaces. Vacancies will accumulate or deplete depending on whether the divergence operator has a negative or positive sign, respectively.
The equations (2.19) and (2.20) are the fundamental continuum model equations which describe the vacancy concentration behavior resulting in accumulation or depletion of vacancies due to diffusion and electromigration, together with the contribution of vacancy generation/annihilation, along an interconnect line.
Shatzkes and Lloyd [138] investigated the transport of vacancies to the end of a semi-infinite line and derived the first solution of equation (2.20) with a zero sink/source term. As a consequence of their model, the expression of Black's equation with the current density exponent n=2 was obtained (equation (2.11)). Another interesting analytic solution of equation (2.20), for the case where G is null, was obtained by Clement for a finite line with blocking boundary conditions at both ends of the line [37]. The solutions of the two aforementioned models are calculated by imposing the steady-state condition of vacancy saturation, i.e. the vacancy concentration reaches a certain critical value significantly higher than the initial value. However, the models are inadequate because the time evaluated to reach the steady-state vacancy supersaturation is smaller than that observed experimentally in [129].
Rosenberg and Ohring [129] provided improvements on the previous models by introducing the source/sink term in the continuity equation as follows
| \[\begin{equation} \frac{\partial{C_\text{v}}}{\partial{t}}=- \nabla \cdot \vec{J}_\text{v}+\cfrac{C_{\text{v}\text{,eq}}-C_\text{v}}{\tau_\text{v}}, \end{equation}\] | (2.21) |
where Cv,eq is the equilibrium vacancy concentration, and τv the vacancy relaxation time. The second term of the right-hand side of equation (2.21) describes the process of vacancy annihilation or generation at the sites acting as sinks or sources of vacancies. In such a way, if the vacancy concentration is smaller than its equilibrium value, vacancies are generated near vacancy sources and reach the steady-state condition in a time regulated by τv. Typically, the magnitude of τv is milliseconds and the vacancy supersaturation is reached quickly [20]. Furthermore, the maximum saturation level is low and it cannot be used to determine the void nucleation condition for electromigration failure. Solutions for these shortcomings will be shown in the next sections, with a consideration of the stress build-up mechanism at the sites of vacancy generation/annihilation. Before, a general overview concerning the fast diffusivity paths inside the metal line, along which vacancies are mainly transported, is provided.