2.4 Effect of Diffusion Paths

There are several transport paths along which vacancies may flow in metallic interconnects. These paths are considered as sources or sinks for vacancies in metal lines. Typical paths are dislocations, grain boundaries, and interfaces. In Section 1.4.5, it has been shown that the properties of microstructure, such as grain size distribution, crystal orientation in the grains, and structure of interfaces, have a strong impact on vacancy transport in the line.

The vacancy flow is directly proportional to the vacancy diffusion coefficient, which is related to several possible diffusion mechanisms. The dominant diffusion mechanism is determined by the fastest diffusivity path. Therefore, vacancy transport due to electromigration is a function of the available diffusivity paths which cause vacancy accumulation or depletion in the metallization. One or more vacancy diffusivity paths dominate the electromigration failure while others remain ineffective [149]. The different diffusivity paths depend on several factors:

• path type,
• path width,
• path thickness,
• activation energy,
• effective charge number.

In order to reduce the complexity of a system where different diffusion paths are available and active, the vacancy flux in equation (2.19) through these paths is expressed as follows

\[\begin{equation} \vec{J}_\text{v}=-\cfrac{D_\text{v}Z^*}{k_\text{b}T} \left(\cfrac{\nabla C_\text{v}k_\text{b}T}{Z^*} - C_\text{v}|e|\vec{E}\right), \end{equation}\] (2.22)

where

\[\begin{equation} D_\text{v}Z^*=\sum_{n} D^\text{n}_\text{v}Z_\text{n}^*f_\text{n}. \end{equation}\] (2.23)

The effective diffusion coefficient Dⁿ_v and the effective charge number Z^*_n are different for each diffusion path n, and are related to the fraction of vacancies f_n diffusing through a given pathway. The main diffusion paths n are bulk (b), grain boundaries (gb), material interfaces (i), surfaces (s), and dislocations (d).