Intuition The one core ideaTo understand CMP , you only need to believe one sentence: the amount of material you rub off a surface grows when you press harder and when you slide farther . Everything else on the parent page — Preston's equation, the velocity trick, dishing — is just this one idea dressed up in symbols, and this page builds every one of those symbols from nothing.
Before you can read a formula like R R = k p P v letter and each little sign (∝ × d / d t z ^
Definition Height / thicknessh h how tall the material is at a spot on the wafer, measured straight up from some baseline — like the depth of paint on a wall. Units: metres (m), though in chips it's usually nanometres : 1 nm = 1 0 − 9 m
Look at the figure. The wavy line is the wafer surface seen edge-on. The vertical red arrow is h bump is a place where h valley is where h h
Δ h Δ "change in" . So Δ h how much the height changed = (height before) − (height after). If we polish away 5 nm Δ h = 5 nm
Why the topic needs it: CMP is about removing height, so the whole subject is a story about Δ h
Intuition Why "pressure" and not just "force"Press a wafer against a pad with your whole hand vs. one fingertip using the same push . The fingertip digs in more. So what actually decides how deep the abrasives cut is not the total push but the push per unit of area . That "per area" quantity is pressure.
P P = Contact area Force pressing down = A F pascal (Pa). A kilopascal is 1 kPa = 1000 Pa ∼ 30 kPa
In the figure, the same downward force (black arrows) is spread over a big area on the left and a tiny area on the right. The red region is the actual contact patch. Same force, smaller patch ⇒ higher pressure ⇒ deeper cut. This is why CMP cares about P
Why the topic needs it: Preston says R R ∝ P
Definition Speed and velocityv Speed is distance covered per unit time (metres per second, m/s). Velocity v with a direction . When the parent writes v relative sliding speed between the wafer surface and the pad rubbing it — how fast one skids across the other.
distance controls removal
Each abrasive grain scratches a groove as it drags. The farther the surface slides, the more grooves get cut, so more material comes off. "Distance slid per second" is exactly velocity. That is why v
Why the topic needs it: R R ∝ v v same everywhere on the wafer, which needs the ideas below.
Intuition What "rate" means, in picturesA rate answers "how fast is something changing per second?" If height drops by 3 nm rate of removal is 3 nm/s steepness (slope) of the line.
Definition The derivatived t d h d t d h "the rate of change of h t The d d t d h the slope of the height-vs-time graph at an instant .
The figure shows height falling as we polish. The red line touching the curve is the slope at that moment — that slope is d t d h Removal Rate R R
Why the topic needs it: Preston's whole output is R R P v
∝ A ∝ B A B B A not yet say by how much — it just says they track each other. To turn a ∝ = constant that supplies the missing scale.
Definition Preston coefficientk p R R = k p P v k p bucket that holds everything we chose not to model in detail: the chemistry, the abrasive type, pad texture, temperature. A better oxidizer (see Slurry Chemistry and Colloids ) makes k p k p Pa − 1 k p ⋅ P ⋅ v
k p Why it feels right: it multiplies pressure and speed, like a wear constant.
The fix: k p chemistry . Change the slurry, not the mechanics, and k p chemical in CMP.
Why the topic needs it: It converts the intuition "R R ∝ P v Archard Wear Law ) into a usable number.
Intuition Why we need a new speed for spinning thingsThe pad and wafer don't move in a straight line — they spin . For spinning, we measure how fast the angle sweeps, not how far a point travels (points near the edge travel farther than points near the center per turn). "Angle swept per second" is called angular speed.
Definition Angle in radians, andω angle measured in radians is (arc length) ÷ (radius). A full circle is 2 π ≈ 6.28 Angular speed ω ("omega") is radians swept per second (rad/s). A common lab unit is rpm (revolutions per minute); convert with
ω ( rad/s ) = rpm × 60 2 π .
× 60 2 π 40 rpm 2 π 40 × 60 2 π = 4.19 rad/s .
Definition Straight-line speed from spin:v = ω r r speed = ω r . This is why the wafer's edge would move faster than its center if it spun about its own axis alone.
Why the topic needs it: The relative-velocity formula that guarantees uniform polishing is built entirely from ω r
Intuition Why arrows (vectors) enter hereA spinning point's velocity has both a size and a direction (it points along the circle). A plain number can't hold a direction; an arrow can. So we upgrade positions and velocities to vectors — arrows with length and heading.
Definition Position vectorr d r arrow from the wafer's own center to a point on it . d arrow from the pad's center to the wafer's center (how far off-center the wafer sits, magnitude d
Definition The unit vectorz ^ z ^ straight up out of the wafer , along the spin axis. The little hat "^
Definition The cross productz ^ × r × cross product . For our case there's one fact to remember: z ^ × r r rotates it 90° counter-clockwise , keeping its length. That rotated arrow points exactly the way a spinning point moves . So ω ( z ^ × r ) is the velocity of that point. Look at the figure: black is r z ^ × r
Why the topic needs it: It lets us write each point's velocity as one tidy arrow, then subtract the pad's arrow from the wafer's arrow to get the relative sliding velocity — the thing Preston's v
Height h and change delta h
Removal Rate = slope dh/dt
Pressure P = force over area
Velocity v = sliding speed
Angular speed omega and rpm
Vectors r, d, z-hat, cross product
Uniform velocity v equals omega d
Each box is one section above; the arrows show which idea you must own before the next makes sense. All roads lead to Preston's law, which is CMP in a single line.
Cover the right side and test yourself — you are ready for the parent note only if each reveals cleanly.
What does Δ h "Change in height" = removed minus starting height; measured in nanometres (1 nm = 1 0 − 9 m
Why use pressure P F Because cutting depth depends on force per unit area ; P = F / A
What does the slope of a height-vs-time graph represent? The removal rate R R = d t d h
Turn A ∝ B Multiply by a constant: A = k B
What lives inside the coefficient k p Everything unmodelled — chemistry, abrasive, pad, temperature; units Pa − 1
Convert 40 rpm 40 × 60 2 π ≈ 4.19 rad/s
Why does the wafer edge move faster than its center under self-spin? Straight-line speed v = ω r r
What does z ^ A length-1 arrow pointing up along the spin axis (direction only).
What does z ^ × r r Rotates it 90° counter-clockwise (same length) → points the way the spinning point moves.
Why does ω p = ω w The wafer-radius
r terms cancel, leaving
∣ v ∣ = ω d , independent of position.
Next: return to Chemical mechanical planarization (CMP) and read Preston's derivation — every symbol there is now one you built.
