Assuming that the hard particle maintains its spherical shape,
A, (r,z) = -a(r,z)(r o(r,z))+ r2 arcsin a(r,z) (3.11)
r
where co(r,z) is the indentation depth and a(r,z) is the radius of the circular indentation on
the wafer surface caused by the particle of radius r trapped under asperity of height z.
Because co(r,z) is very small compared to r, a(r,z)/r is also very small and Equation 3.11
can be linearized about a(r,z)/r as already discussed:
Az (r, a(r, z)(r, z) )+ ( (3.12)
3 r
Furthermore, since a2(r,z)=(2rco(r,z)-co2(r,z)) and co(r,z) is much smaller than r,
Equation 3.12 can be rewritten as
A4 (r, z) 2a3(r,z) (3.13)
3r
Once the indentation depth, co(r,z) (or a(r,z)) is known, the volumetric removal rate
by a single particle can be calculated from Eq. 3.13. It should be noted that co and a are
dependant on particle radius and height of the trapping asperity. Removal rate per particle
is given by:
RR, (r, z) = 2a(rz) V (3.14)
3r
Depending on the relative magnitude of the indentation depth (co(r,z)) as compared
to the surface layer thickness (t), the material removal may be classified into two
different regimes as depicted in Fig. 3-18: (a) o(r,z) < t and (b) o(r,z) > t and can be
calculated using force balance as follows.