Since the characteristic length scale for the roughness of the pad surface (e.g., 3) is
much larger than the feature sizes on the wafer, it was assumed that the wafer surface is
smooth and flat compared to the pad surface. This assumption makes it possible to use
the Greenwood-Williamson model [117] that describes the contact area, Areal, between a
flat surface and a randomly rough surface under a normal force (or load), L:
A ,real = TNA z h) (z)dz (2.16)
L = NAE12(z h)3/2 ()dz (2.17)
where N is the number density of the pad asperities (i.e., number of asperities per unit
area), Ao is the total area of the flat wafer surface, and h is the average gap between the
smooth wafer and the rough pad under the given conditions. Then, the down-pressure, P,
applied to the wafer with an area Ao is L/Ao whereas the actual mean contact pressure, Pr,
is L/Areal if the normal force applied to the wafer is entirely supported by the pad
asperities in contact with the wafer. That is
L =PA,= PrAreal (2.18)
E* is the composite modulus between the two surfaces in contact, that is defined as
1 1-v2 _V2
1 ~ (2.19)
E* Ep E
where E and v are the Young's modulus and the Poisson's ratio, and the subscripts p and
w represent the pad and the wafer, respectively.
Equations 2.16, 2.17 and 2.18 provide an explicit relationship between the contact
area (Ar) and the down pressure (P) that is needed in modeling the CMP process. From
Eqs. 2.16 and 2.17,