The advantages of Cook's and Liu's models over Preston's equation are that they
provide more insights into the roles of the consumable parameters. The contributions of
the slurry abrasives and pad, for example, have been attributed to their size and hardness.
An additional benefit is that not only material removal rate, but also surface quality issues
such as roughness (Eq. 2.5) can be addressed using these models.
Slurry Erosion
In Cook's and Liu's models, the mechanical removal by abrasive particles was
assumed to be the dominant mechanism. However, some researchers instead assumed the
material removal to be by "slurry erosion" approach, i.e., due to mechanical-enhanced
erosion. Runnels et al. [79-81] developed an erosion-based model for CMP. They
assumed that a fluid film exists between the wafer and pad interface, which affects the
erosion/material removal rates at each single point through the fluid stress tensors.
Motivated by observations at feature scale, modification to Preston's equation was
proposed, with relative velocity V, replaced by the tangential stress on the wafer surface;
MRR=Ceoton (2.7)
where Ce is an all purpose coefficient, ot is the shear stress due to the slurry flow and on
the normal stress.
The model uses a steady state incompressible Navier-Stokes equation to model the
slurry flow at the wafer-polishing pad interface. The wafer surface is assumed to be
smooth and spherical with a large radius of curvature. The fluid layer thickness and the
angle of attack between the pad and the wafer are obtained through an iterative procedure
satisfying force and momentum balance. Runnels' model has been integrated into several
particle-scale models by researchers including Tseng and Wang [82] and Zhang and