3.2 Vacancy Dynamics Problem

Since atomic diffusion in copper metallization is a result of the vacancy exchange mechanism [20], the mass transport which occurs in the interconnect during electromigration can be modeled in terms of vacancy transport. The vacancy flux is caused by the contributions of vacancy diffusion and vacancy migration due to the action of different driving forces.

The electromigration driving force, presented in Section 2.2, is not sufficient to quantitatively describe the vacancy migration which opposes to the Fickian diffusional vacancy flux. Other contributions can be obtained from the general expression of the electrochemical potential μ' of a vacancy in the bulk [20], which is a thermodynamic measure of the chemical potential μ taking into account the contribution of electrostatic energy, as follows

\[\begin{equation} \mu' = \mu'_\text{0}+\mu'_\text{e}+\mu'_\text{t}+\mu'_\text{$\sigma$}, \end{equation}\] (3.16)

where μ'₀, μ'_e, μ'_t, and μ'_σ are the electrochemical potentials related to the reference state, electromigration, thermomigration, and stress-migration, respectively. Any electrochemical potential relates to the respective driving force F_m for vacancy migration [20] as follows

\[\begin{equation} \vec F_\text{m} = -\nabla\mu' \end{equation}\] (3.17)

and, by using the Nerst-Einstein relationship, to the vacancy flux along the line due to the different migration processes J_v^m as

\[\begin{equation} \vec J^\text{m}_\text{v} = \cfrac{D_\text{v}C_\text{v}}{k_\text{b}T}\vec F_\text{m}, \end{equation}\] (3.18)

where D_v is the diffusion coefficient, C_v is the vacancy concentration, and k_b is the Boltzmann constant. Adding the Fickian term J_v^d, the total vacancy flux J_v is obtained by

\[\begin{equation} \vec J_\text{v} = \vec J^\text{d}_\text{v}+\vec J^\text{m}_\text{v}=\vec J^\text{d}_\text{v}+\vec J^\text{em}_\text{v}+\vec J^\text{t}_\text{v}+\vec J^\text{$\sigma$}_\text{v}. \end{equation}\] (3.19)

The first flux term represents the typical flux of vacancies due to diffusion, and is expressed by the first Fick's law as follows

\[\begin{equation} \vec J^\text{d}_\text{v}=-D_\text{v}\nabla C_\text{v}. \end{equation}\] (3.20)

The second flux contribution is the flux induced by the electromigration driving force F_em in the form

\[\begin{equation} \vec J^\text{em}_\text{v}=\cfrac{D_\text{v}C_\text{v}}{k_\text{b}T}\vec{F}_\text{em}=-\cfrac{D_\text{v}C_\text{v}}{k_\text{b}T}|Z^*|e \nabla V_\text{e}, \end{equation}\] (3.21)

where Z^* is the effective valence and e is the elementary charge. The third and the fourth flux terms are related to the driving forces of thermomigration F_t [32] and stress-migration F_σ [112], respectively, as follows

\[\begin{equation} \vec J^\text{t}_\text{v}=\cfrac{D_\text{v}C_\text{v}}{k_\text{b}T}\vec F_\text{t}=-\cfrac{D_\text{v}C_\text{v}}{k_\text{b}T}\cfrac{Q^*\nabla T}{T} \end{equation}\] (3.22)

and

\[\begin{equation} \vec J^\text{$\sigma$}_\text{v}=\cfrac{D_\text{v}C_\text{v}}{k_\text{b}T}\vec F_\text{$\sigma$}=-\cfrac{D_\text{v}C_\text{v}}{k_\text{b}T}f \Omega_\text{a} \nabla \sigma, \end{equation}\] (3.23)

where Q^* is the heat transport, f is the vacancy relaxation factor, Ω_a is the volume of an atom, and σ is the hydrostatic mechanical stress. The heat transport Q^* is described as the isothermal heat generated during the movement of the atom in the process of jumping a lattice site [20]. Assuming that a vacancy is compared to a substitutional atom with smaller volume in the crystal lattice, the vacancy relaxation factor represents the relationship between the volume of a vacancy and the atomic volume. Typical values of f are in the range between 0 and 1. The total vacancy flux due to the combination of the different driving forces is given by

\[\begin{equation} \vec J_\text{v} = -D_\text{v}\left(\nabla C_\text{v}+\cfrac{C_\text{v}|Z^*|e}{k_\text{b}T}\nabla V_\text{e}+\cfrac{C_\text{v}Q^*}{k_\text{b}T^2}\nabla T+\cfrac{C_\text{v}f \Omega_\text{a}}{k_\text{b}T}\nabla \sigma\right). \end{equation}\] (3.24)

The vacancy diffusion coefficient D_v in equation (3.24) depends on the temperature and the hydrostatic stress build-up through the exponential dependence given by the Arrhenius law [40]

\[\begin{equation} D_\text{v} = D_\text{v,0} \ \text{exp}\left(\cfrac{(1-f)\Omega_\text{a}\sigma -E_\text{a}}{k_\text{b}T}\right), \end{equation}\] (3.25)

where D_v,0 is the pre-exponential factor, and E_a is the activation energy for the diffusion of a vacancy.

The driving forces cause a redistribution of the vacancy concentration in the interconnect line. The vacancy concentration distribution obeys the material balance equation as follows

\[\begin{equation} \frac{\partial{C_\text{v}}}{\partial t}=-\nabla \cdot\vec J_\text{v}+G. \end{equation}\] (3.26)

The local vacancy concentration changes due to the contribution of two mechanisms. The first term on the right hand side of equation (3.26) represents the vacancy accumulation/depletion due to the existence of flux divergence during vacancy transport. The second term, G, is the source/sink term (also known as the Rosenberg-Ohring term [129]) which models the vacancy generation/annihilation processes due to the change of lattice sites at particular locations in the interconnect. This term describes the production/annihilation of vacancies, when their concentration is larger/lower than the equilibrium value C_v,eq, respectively, and is given by

\[\begin{equation} G=\cfrac{C_{\text{v}\text{,eq}}-C_\text{v}}{\tau_\text{v}}, \end{equation}\] (3.27)

where τ_v is the characteristic generation/annihilation time. The equilibrium concentration of vacancies C_v,eq is given by the Arrhenius law

\[\begin{equation} C_{\text{v}\text{,eq}} = C_\text{v,0} \ \text{exp}\left(\cfrac{(1-f)\Omega_\text{a}\sigma -E_\text{a}}{k_\text{b}T}\right), \end{equation}\] (3.28)

where C_v,0 is the equilibrium vacancy concentration in the absence of stress.