surface and the pad asperity average plane under the given conditions. E* is the
composite modulus between the two surfaces in contact, defined as
1 1-v2 1v (34)
-- + W (3.4)
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 (3.1) and (3.2) are derived
from Hertz theory as in Greenwood-Williamson model [77, 117], yielding contact area
and load at a given asperity of height 'z' to be:
Areal(Z)=7 P (z-h) (3.5)
L(z)=(4/3)E* 1/ 2(z-h)3/2 (3.6)
where 'h' is calculated from Eq. 3.2, since applied load and pad properties are known.
The underlying assumption for Eq. 3.2 is that the applied normal force on the wafer is
supported entirely by the pad asperities. As was mentioned previously, experimental
measurements indicate that the hydrodynamic pressure is negligible under typical CMP
operational conditions [69].
From equations 3.5 and 3.6, contact pressure and contact area at a given asperity
height can be calculated assuming that the pad asperity distribution is known. The pad
asperity distribution can be determined from routine surface roughness measurement
techniques like optical and force profilometers or atomic force microscopy (AFM).
Contact pressure according to Hertz theory, on an asperity of height z is given by:
P(z) =4E* (z- h)2 (3.7)
3z-f1i
35c/72