Considering that the pad is sufficiently hard as compared to the modified layer, the
value of b in Fig. 3-18 can be close to the particle radius R,
( -1/2
P(z)
a(r,z) = r i when co(r,z) < t (3.15)
and
a(r, z) = r2p(z) + 2(BW Be)rt when co(r,z) > t (3.16)
BI
where, Be is the effective Brinell hardness of the chemically modified surface layer. In
the following section, the number of particles in contact with the wafer surface is
estimated to determine the overall removal rate.
Number of Active Abrasive Particles
When the wafer is pressed against the pad and is in sliding motion relative to the
pad, the slurry particles can be entrapped only in the region where the local distance
between the wafer and the pad asperity is smaller than the particle diameter, 2R. For a
slurry consisting of particles with Gaussian distribution, mean particle size (Rm) is used
to calculate the number of particles that can be trapped on the pad asperity of height z:
N(z) 2Rm -Areai(z) p (3.17)
where Rm is the mean abrasive particle radius and Areai(z) is the real contact area between
the wafer surface and the pad asperity as given by eq. 3.5. p, is the number density of the
particles in the slurry (i.e., number of particles per unit slurry volume), that can be
calculated from the solid loading Ca (wt%) of the slurry particles as
3C, p,
Pn ,= 3a 0p (3.18)
4;r pr3 (r (r)dr