AL 4E* 3 h h)/2(z Jz (2.20)
L 4E (z -h)3/( Z
Following Yu et al.[63], the normal distribution function may be used for the
distribution of the asperity height, ). When )(z) = exp -z-- is substituted into
Cy [27j 2CY2 2
Equation 2.20, the integral cannot be obtained in an explicit form. Instead, a numerical
integration should be preformed. Johnson showed that the ratio of the two integrals in Eq.
2.20 is nearly a constant for a broad range of h/o [77]. Thus,
rea =-C 1 =C-1, PA, 22
A-C C (2.21)
where C is a constant whose value is between 0.3 and 0.4 when h/l is in the range of 0.5
and 3.0. Recalling that o is the standard deviation of the asperity height distribution,
h/o=3.0 represents a situation where only 0.13% of the pad asperities are in contact with
the wafer whereas about 30.85% is in contact when h/o=0.5. Practical situation in a
typical CMP process is well within this range ofh/l value. From Eqs. 2.18 and 2.21,
Pr = C E (2.22)
It should be noted that the contact area (Ar) increases linearly with the down-
pressure whereas the contact pressure (Pr) remains constant due to the linear increase of
Ar with P.
The underlying assumption for Eq. 2.18 is that the applied normal force on the
wafer is supported entirely by the pad asperities. As is mentioned previously,
experimental measurement on the friction coefficient indicates that the hydrodynamic