the process especially regarding other process variables. By extending Brown's model
[76] of the metal polishing to the silicon polishing, Cook [24] developed a physical
model to address this limitation. The interactions between the abrasive particles and the
wafer surface is proposed as a Hertzian elastic penetration [77] of a spherical particle
under uniform pressure P into the wafer surface, sliding along the surface with a velocity
V and removing material volume proportional to the penetration. The MRR formulation
was proposed was
MRR= PV/(2E) (2.4)
where E is the Young's modulus of the wafer material. This model can be taken as a
theoretical equivalent to the Preston's equation since it supports the linear dependency of
MRR on pressure and velocity. The relationship between the wafer surface roughness Ra
the down pressure P, and abrasive size can also be obtained based on this model
Ra=3/4 b (P/2klE)2/3 (2 5)
where kl is the particle concentration, which is unity for a filled close hexagonal packing
[24] and b the diameter of the slurry particles. A similar model was developed by Liu et
al [78] based on the statistical method and Hertzian elastic penetration. Besides the wafer
material properties including wafer hardness Hw and wafer Young's modulus Ew, this
model included pad hardness Hp and abrasive Young's modulus Es resulting in
MRR = C, H,- E, + E PV (2.6)
H +H, P Es Ew
where Ce is a coefficient to account for the effects of slurry chemicals and other
consumable parameters. This model, similar to Cook's model, suggests that the material
removal is proportional to the applied pressure and relative speed