Next: 3. Modeling Electromigration Using TCAD Up: 2. Physical Description of Electromigration Previous: 2.6 Void Initiation Condition
Once the fatal void has nucleated, the void evolution mechanism is the ultimate cause for electromigration induced interconnect wear-out failure which results in open circuit failure. The electromigration void evolution mechanism is typically related to the development of the interconnect resistance as a function of time. As soon as the void grows, a rapid non-linear interconnect resistance increase begins. The interconnect fails after a maximum tolerable resistance level is reached.
The void evolution mechanism leads to the line failure as the result of a competition between growth, shape change, and motion of the void [3]. Given the different influences on the void evolution, modeling interconnect failures poses a challenging task. Two different modeling methodologies for electromigration void evolution have been developed in the 1990s.
The first approach is the sharp interface method which attempts to model the void surface Γint as a sharp discontinuity between the metal and the empty space in the void, as shown in (2.5(a)) [96,156,71,135,62]. It requires a direct tracking of the void surface during its evolution with a continuously re-adapted mesh as the structure changes its shape [62]. Applications of this method provided numerical simulations of the void shape change and well explained some experimental observations. Due to the complexity of the surface tracking as the void is subjected simultaneously to motion, shape change and growth, the sharp interface model tends to have poor numerical stability and hence is cumbersome.
The diffuse interface method (or phase field method) offers an attractive alternative to the sharp interface model for solving the electromigration void surface evolution problems. The idea of a sharp interface between the conductor material and the empty void is abandoned. The entire domain Ω is described by a continuous variation of an order parameter φ (or phase field variable) from +1 in the metal "phase" to -1 in the void "phase" over a narrow metal-void interfacial layer associated with the void surface, as shown in (2.5(b)) [107]. The value of the order parameter defines the material and the void at any point on a fixed grid and therefore the diffuse interface model does not require exact tracking of the surface elements and their geometry. In general, the diffuse interface model is an approximation to the sharp interface model. Furthermore, in the limit of vanishing interface thickness, the two models will show matching physical behavior. Diffuse interface models for the electromigration void evolution analysis successfully predicted void motion and growth in Al and Cu interconnects [107,108,10,11].
For the last 20 years, new methodologies [60,61,42,43,45,152], based on semi-empirical modeling, were employed for the simulation of void evolution and resistance increase in interconnects. Semi-empirical methods are analytical expressions which are partially derived from purely theoretical assumptions and partially obtained from empirical evidence as a result of experimental observations or by fitting to the experimental results. For those reasons, void evolution semi-empirical based methods are capable to numerically calculate the growth of the void and the consequent resistance increase efficiently.
In the next chapter, the different approaches for modeling the void surface evolution as well as a simple semi-empirical model for void growth will be presented in details.