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CMP removal rate calculator (Preston equation)

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KEY POINTS

CMP removal rate follows the Preston equation, RR = Kp × pressure × relative velocity. When platen and head rotate at the same speed, the relative velocity at every point on the wafer equals the platen angular velocity × center distance, eliminating velocity-driven nonuniformity. Kp is an empirical constant set by slurry, pad and film, so it is back-calculated from a measured rate.

Calibrating the Preston coefficient (reference conditions)

Removal rate measured at reference conditions (nm/min)Pressure at that time (psi)Relative velocity at that time (m/s)

Preston coefficient Kp66.7nm/min/(psi·m/s)

Kp is an empirical constant set by the combination of slurry, pad and film. Back-calculate it once from a removal rate measured on your tool, then use it to predict other conditions.

Polishing conditions

Polishing pressure (downforce) (psi)Platen speed (rpm)Head speed (rpm)Platen center to head center distance (mm)Thickness to remove (nm)Overpolish (%)

Average removal rate351nm/min

Polish time (including overpolish)1.71 min (103 s)

Average relative velocity1.75m/s

Within-wafer nonuniformity from velocity (1σ)0.02%

Pressure (SI units)20.7kPa

Removal rate across the wafer radius

This is the distribution caused by relative velocity alone. Real within-wafer profiles change substantially with head zone pressures, pressure concentration at the edge from the retaining ring, uneven slurry supply and pad temperature, and production tools correct for these with zone pressures.

Head speed and uniformity (fixed platen speed)

Platen 93 rpm, center distance 180 mm. The velocity profile is perfectly uniform when head and platen speeds are equal.
Head speed	Average relative velocity	Average removal rate	Nonuniformity (1σ)
74 rpm (×0.80)	1.76 m/s	352 nm/min	0.21 %
84 rpm (×0.90)	1.75 m/s	351 nm/min	0.05 %
87 rpm (×0.94)	1.75 m/s	351 nm/min	0.02 %
93 rpm (×1.00)	1.75 m/s	351 nm/min	0.00 %
99 rpm (×1.06)	1.75 m/s	351 nm/min	0.02 %
102 rpm (×1.10)	1.75 m/s	351 nm/min	0.05 %
112 rpm (×1.20)	1.76 m/s	352 nm/min	0.21 %

The Preston equation and CMP settings

The Preston equation RR = Kp·P·V was proposed in 1927 as an empirical rule for glass polishing; it states that removal rate is proportional to the product of pressure and relative velocity. It holds reasonably well for each film type in CMP, so raising pressure or rotation speed speeds up polishing. Too much pressure, however, increases scratches and dishing and risks delamination of mechanically weak films such as low-k, so around 2 to 4 psi is common in practice.

When platen and head rotate in the same direction at the same speed, the relative velocity at every point on the wafer equals platen speed × center distance. Real tools still run the head at a slightly different speed so that the same spots on wafer and pad do not line up periodically, averaging out pad wear and slurry distribution.

FAQ

Why run the head at a slightly different speed from the platen?

Equal speeds make relative velocity uniform, but slightly different speeds keep the same spots on wafer and pad from lining up periodically, which evens out pad wear and slurry distribution.

What pressure is typical for CMP?

Around 2 to 4 psi is common. Higher pressure polishes faster but increases scratches and dishing and risks delaminating weak films such as low-k.

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Last updated: October 23, 2026