Yu's model yields a linear dependency of the material removal rate on the down pressure,
which agrees with Preston's equation.
Zhao and Shi [90] also proposed a model based on the wafer-asperity contact.
Unlike Yu's model [63], this model does not consider the Gaussian distribution of the
asperity heights. The contact area between an asperity and the wafer is given by Aa Ua P2/3
based on Hertz elastic contact theory. By combining Steigerwald's argument, the material
removal rate formulation can be obtained as
MRR= K(V)P(2/3) (2.13)
where K(V) is a function of the relative velocity V and other CMP parameters. It is
further considered by Zhao and Shi [90] that when the particles are rolling against the
wafer surface, their contribution to material removal will be negligible. They argued that
whether the particle is rolling or not is determined by the surface friction between the
particles and the wafer, and only when the down pressure P is larger than a threshold
down pressure Pth, the pure rolling can be avoided. The material removal rate is given as:
MRR = K(V)(P2/3 Pth2/3) for P > Pth (2.14a)
and
MRR = 0 for P< Pth (2.14b)
The fundamental difference between the above pad-based models and the particle-
based models by Cook and others is that pad-based models attribute the material removal
rate to the number of abrasive particles captured by the polishing pad while the later
attributes the material removal rate to the interaction between a single abrasive and the
wafer.