1/2
a=R (2.31)
This equation along with equations 2.22 and 2.28 gives the following expression for the
volumetric removal rate by a single particle of radius R.
(2 "-3/4 E 3/2
(RR) C3/2R2 ~/2 V (2.32)
S3 P Be
Indentation depth greater than the surface layer thickness (i.e., o > t)
When the bulk wafer material beyond the surface layer is relatively soft (e.g.,
copper) compared to the hardness of the particles, it is possible for the particle to cause
deformation into the wafer beyond the surface layer (Fig. 2-3b). As it was described in
the previous section, the deformation is expected to be plastic for both the surface layer
and the bulk wafer, and the force causing the deformation may be given as
F, = [2tRB, + (a2 2tR)B j (2.33)
where Bw and Be are the Brinell hardness of the bulk wafer material and the chemically
modified surface layer, respectively. Because this force is equivalent to the one imposed
on the particle by the down-pressure, Eq. 2.33 is equated with Eq. 2.29 with b=R to give
R2P, + 2(B, Bj)Rt 12
a R + 2(B (2.34)
Then, Eq. 2.28 for the volumetric removal rate by a single particle is reduced to
(RR) =2C(j R2 +21 Rt (2.35)
3 B BR