Effect of Slurry Flow Rate on Tribological, Thermal, and Removal ...

Journal <strong>of</strong> The Electrochemical Society, 151 7 G482-G487 2004
G485
Figure 10. Simulation and experimental
removal rate as a function <strong>of</strong> p V.
f s /(2h 0 s r), where r is the radius on the pad, f s is the slurry flow
rate, and h 0
s is the slurry thickness between grooves. Under the
wafer, it is assumed that the slurry is entrained by the pad with no
outward convective motion. The slurry flow model is very rudimentary,
capturing only the general outward flow. It can be seen that all
other factors being equal, increasing slurry flow rate f s increases the
rate at which heat is transported <strong>of</strong>f the pad, resulting in cooling <strong>of</strong>
the wafer due to lowering <strong>of</strong> T s . In Eq. 2, the convective heattransfer
coefficient h ws between the wafer and slurry varies as the
square root <strong>of</strong> the sliding speed. In Ref. 6, h ws was estimated directly
from correlation theory for pad and wafer rotation rates <strong>of</strong> 120
rpm 0.93 m/s. At the same separation between the pad and wafer
centers, we can relate the heat-transfer coefficient at sliding speed V
to the speed at 120 rpm
h ws V h ws V 120 V
V 120
The ratio <strong>of</strong> the sliding speeds in Eq. 3 is also the same as the ratio
<strong>of</strong> the rotation rates.
Reaction rate model.—In this report, copper removal rate can be
described well by a subset <strong>of</strong> the Langmuir-Hinshelwood model in
Ref. 7, summarized below. In this model, n moles <strong>of</strong> an unspecified
reactant R in the slurry react at a rate k 1 with the copper film on the
wafer to form a product layer L on the surface
k 1
Cu nR → L 4
The reacted layer is then removed by mechanical abrasion with a
rate k 2
3
L → L 5
k 2
The abraded material L is carried away by the slurry and is not
redeposited. The local removal rate in this sequential mechanism is
RR M w
k 1 C
1 k 1C
k 2
where M w is the molecular weight <strong>of</strong> copper, is the density, and C
is the local molar concentration <strong>of</strong> reactant. It is assumed that there
is little reactant depletion so that C remains constant. This allows C
to be absorbed into k 1 and be set to unity. In Eq. 6, the rate <strong>of</strong> the
chemical reaction is expressed as k 1 A exp( E/kT w ) with C
6
1. Assuming that the mechanical removal rate is proportional to
p V, then k 2 c p pV, where c p is an assumed proportionality
constant. In the mechanically limited extreme, the polish rate is
RR (M w c p /)pV and in the opposite limit it is RR
(M w /)k 1 .
It is important to note the physical dependence <strong>of</strong> the variables in
the removal rate law Eq. 5 with this choice <strong>of</strong> k 1 and k 2 . While the
experimental value <strong>of</strong> p V can be chosen at will, the local wafer
temperature T w cannot because it is a function <strong>of</strong> p V through the
boundary condition expressed by Eq. 2. T w is also independently
influenced by COF, w , the slurry temperature, and the wafer-slurry
heat-transfer coefficient, so that p V does not completely determine
the wafer temperature.
Comparison <strong>of</strong> Theory and Experiments
The calibration <strong>of</strong> the thermal model for copper polishing and
comparisons <strong>of</strong> pad temperature histories predicted by the model
with measured pad temperatures, including thermal effects from in
situ conditioning, are described in Ref. 8. This model in turn is a
slight reparameterization <strong>of</strong> the same model for oxide polish. 6 In the
present analysis, the important parameters involve the heat partition
fraction in Eq. 1 for which the value <strong>of</strong> w 217 W/m K was chosen
vs. the value <strong>of</strong> 124 for silicon, and 385 for pure copper in Ref.
8, and the wafer-slurry heat-transfer coefficient in Eq. 3, which was
estimated as 762 W/m 2 °C at 120 rpm for the application in Ref. 6.
As mentioned in Eq. 1, the effective thermal conductivity <strong>of</strong> wafer
was associated with the layers presented in the wafer surface. During
the polishing, the wafer surface was fully or partly covered with
oxide copper rather other just pure copper; thus, the effective thermal
conductivity was lower than that <strong>of</strong> pure copper. The value <strong>of</strong>
w 217 W/m K was an optimized one by fitting the model to the
experimental measured data.
Based on this calibration, the model was run for all pressures,
relative pad-wafer velocities, and slurry flow rates. The measured
COF from each condition was smoothed with a linear regression line
as described in Ref. 8. The ambient temperature that was employed
as an initial condition in each case for the pad, wafer, and slurry and
for calculating the convective heat loss to the air was estimated by
averaging the five initial temperatures measured on the surface <strong>of</strong>
the pad near the leading edge <strong>of</strong> the wafer. The ambient temperature
was found to vary by as much as 1.5°C about a mean <strong>of</strong> 25°C, as
shown in Fig. 7a. Figure 7b compares the calculated mean pad temperature
rise at sample point locations with the measured mean temperatures.
These are in good agreement over all conditions, thus
providing added confidence in the calibration <strong>of</strong> the model. The
model then provided the wafer temperature field needed for comput-