3.2.1 Fast Diffusivity Paths

In Chapter 2 the role of diffusivity paths for material transport was discussed. It is clear that diffusion occurs along the various available paths and must be taken into account. Using the simple effective diffusion model is rather inconvenient for TCAD analysis, since one is not able to completely separate and understand the effect of each path on the distribution of material in the interconnect line. Consequently, a realistic picture of the mechanical stress distribution is not possible either. Moreover, each path has also a different effective valence [21,140], since the wind force depends on the electronic configuration surrounding an atom [58].

Thus, for generality of the model, the diffusion coefficient, $\DV$, and the effective valence, $\Z$, in (3.9) must be independently set for each region of the interconnet. In this way, combining (3.9) and (3.10) one obtains

$$\ensuremath{\ensuremath{\frac{\partial \CV^{bulk}}{\partial t}}}=... ...symAtomVol}{\kB\T}\CV^{bulk}\ensuremath{\nabla{\symHydStress}}\biggr)\biggr]}},$$ (3.11)

for the bulk and

$$\ensuremath{\ensuremath{\frac{\partial \CV^{int}}{\partial t}}}= ... ...}{\kB\T}\CV^{int}\ensuremath{\nabla{\symHydStress}}\biggr)\biggr]}} + \G_{int},$$ (3.12)

for interfaces which for copper dual-damascene interconnects can be either grain boundaries, the copper/capping layer interface or the copper/barrier layer interface.