3.4.1 Sharp Interface Model

Most of the existing models for electromigration void evolution are based on sharp interface approaches [96,156,71,135,62]. They model the void surface as a distinct sharp boundary between the material of the metal line and the empty space in the void. The surface is described by specifying a large number of points on it. Thus, sharp interface methods require an explicit tracking of void surfaces during their evolution and those surfaces define moving boundary problems. As nucleated voids can grow, migrate, and change shape during the course of the evolution, the void surface tracking can be demanding and needs automatic adaptive mesh generation to trace the large shape modifications which normally occur [148].

Before describing the physical principles behind the sharp interface model, an overview of the assumptions which underlie the analysis should be provided [11]. The interconnect line is assumed to function as a 2D conducting deformable solid, with a void surface located inside the line, as illustrated in (3.5).

Figure 3.5: Idealization of the void surface in a 2D conducting interconnect line.

The interconnect line is subjected to a voltage ΔV_e applied across its ends, inducing a local electrical field E in the line. The metal line is idealized as an isotropic, linear elastic solid, and diffusion in the bulk of the line is neglected. Under the influence of the induced electrical field, the void evolution mechanism due to electromigration is mainly related to surface diffusion. Surface diffusion is also influenced by the mechanical stress gradient and the void curvature. The diffusion of atoms along the void surface causes it to change its shape, leading to void growth. Another small contribution governing the void growth arises from feeding the void with vacancies from the bulk. Additional features can also contribute to the evolution of the void, such as grain boundary diffusion and anisotropic surface diffusivity.

In the absence of electric currents, the transport of atoms lying on the void surface is driven by the gradient of the chemical potential. The chemical potential μ_s of an atom on the surface is given by [62]

\[\begin{equation} \mu_\text{s}=\mu_\text{0}- \Omega_\text{a}\gamma_\text{s}(\theta)\kappa_\text{s}+\cfrac{\Omega_\text{a}(\overline{\overline\sigma} : \overline{\overline\epsilon})}{2}, \end{equation}\] (3.73)

where μ₀ is the reference chemical potential, γ_s(θ) is the orientation angle θ dependent constant related to the surface energy, κ_s is the local curvature of the metal/void interface, and the double-dot product is defined for the tensors σ and ε as follows

\[\begin{equation} \overline{\overline\sigma} : \overline{\overline\epsilon}=\sum_{i,j} \sigma_\text{ij}\epsilon_\text{ij}. \end{equation}\] (3.74)

The second term on the right hand side of equation (3.73) refers to the free energy of the surface on which κ_s assigns negative curvature to convex surfaces and positive curvature to concave surfaces. The third term is related to the local elastic strain energy. Electric current provides an additional driving force for surface transport, namely the electron wind force. The surface chemical potential and the electromigration driving force relate to the total atomic flux J_s along the metal/void interface as follows

\[\begin{equation} J_\text{s}=\cfrac{D_\text{s}(\theta)\delta_\text{s}}{k_\text{b}T}\left(-\nabla_\text{s}\mu_\text{s}+Z^*e\nabla_\text{s} V_\text{e}\right). \end{equation}\] (3.75)

Here, ∇_s is the surface Laplacian operator, δ_s is the thickness of the diffusion layer, and D_s(θ) is the temperature and orientation dependent constant related to the surface diffusion coefficient through the Arrhenius law

\[\begin{equation} D_\text{s}(\theta) = D_\text{s,0}\left[1+m\left(1-\cos(n\theta - \theta_{0})\right)\right] \ \text{exp}\left(-\cfrac{Q_\text{s}}{k_\text{b}T}\right), \end{equation}\] (3.76)

where D_s,0 is the pre-exponential coefficient for surface diffusion, m is the degree of anisotropy, n is the crystallographic symmetry, θ₀ is the orientation of the interconnect line with respect to the face-centered-cubic crystal planes, and Q_s is the activation energy for surface diffusion [62]. The isotropic medium is a special case in which γ_s is a constant and m=0. The assumption of a dense network of grain boundaries, where all the grains are equally distributed in every direction, permits to neglect anisotropic effects and grain boundary diffusion in the line. As a consequence, the diffusion along the void surface leads to void growth and changes of the void shape due to anisotropic and grain boundary diffusivities should be neglected.

The resulting atomic flux along the void surface is therefore proportional to the component of the electrical field E_s in the tangential surface direction [25]. The normal velocity v_n at each point of the void surface can be obtained from equation (3.75) as follows

\[\begin{equation} v_\text{n}=-\nabla_\text{s}J_\text{s}=-\cfrac{D_\text{s}\delta_\text{s}}{k_\text{b}T}\left(-\nabla^2_\text{s}\mu_\text{s} +Z^*e\nabla^2_\text{s} V_\text{e}\right), \end{equation}\] (3.77)

where D_s is the isotropic surface diffusivity (m=0 in equation (3.76)). This equation implies conservation of the void size during important morphological void changes [11]. The void surface moves as a result of the contributions of the unbalanced fluxes due to gradients in the curvature, elastic strain energy, and electrical potential. It should be pointed out that elastic strain energy effects have been shown to dominate electromigration in passivated copper lines during the void nucleation period, while for void evolution, electron wind becomes the major driving force [118,152]. The effect of mechanical stress on the void evolution mechanism can therefore be neglected, when performing an analysis of a copper interconnect line.

As mentioned before, sharp interface modeling appears convenient to describe the void shape changes during void evolution, but special techniques to track the metal/void interface for every simulation time step are required. The front tracking problem requires re-meshing at every time step, which gives rise to converging issues and is generally difficult to handle. It should be mentioned, however, that the computational demands of surface tracking through a sharp interface model have been largely addressed by combining the front tracking method with more efficient methods of solving the corresponding boundary-value problems with which the surface evolution problem is coupled. Such an example is the state-of-the-art boundary-integral methods with numerous applications to the void dynamics problem in the literature [68,69,4]. The boundary-integral method has found an advantage over the finite-element method (FEM) when it comes to tracking the interface corresponding to the void surface.