Intuition The one core ideaWe print many identical rectangular chips onto one round silicon disk; random specks ruin some, and yield is the fraction that survive. Everything on the parent page is built from three things — how many chips fit, how dirty the process is, and how much survival costs — so this page defines every letter and picture those ideas stand on.
This is the ground-floor page for Yield, defect density, and binning . If a symbol confused you there, it is defined here from zero. Read top to bottom — each item uses only things defined above it.
r d wafer is a thin circular slice of pure silicon. We describe its size two ways:
==r
==d d = 2 r
Picture a round pizza. The radius is centre-to-crust; the diameter is crust-to-crust straight through the middle. Fabs quote wafers by diameter ("a 300 mm wafer"), so you constantly convert: if d = 300 r = 150 = 15
r area question, and the area of a circle is written with the radius, not the diameter. So we carry r
Definition Area, and the symbolA Area is the amount of flat surface something covers, measured in square units. Two areas matter here:
Area of the whole round wafer: a circle of radius r π r 2
==A one single chip (one rectangle).
Why the square (the little 2 two directions — width and height — so its units are a length times a length, e.g. centimetres times centimetres, written cm 2
π π ≈ 3.14159
We will divide one by the other in a moment — that is the whole idea of "how many chips fit."
A die is one finished rectangular chip cut from the wafer. DPW (gross dies per wafer) is how many die-rectangles geometrically fit on the round disk , counting before any defects ruin any of them.
The very first estimate is just "big area ÷ small area":
ideal count = A π r 2 = room one chip needs room on the whole wafer .
Intuition Why this is only an estimate — the round-vs-square problemThe wafer is a circle but every die is a rectangle . Rectangles that hang over the curved edge fall off and are wasted. So the true count is a little less than π r 2 / A
π r 2 / A Why it feels right: it is clean and looks like a total.
Fix: it over counts, because partial dies at the round rim can't be sold. DPW is always somewhat smaller. (See Photolithography for how the die grid is actually printed.)
Definition Defect, and defect densityD 0 defect is anything that kills a chip: a dust speck, a crystal flaw, a printing wobble. ==D 0 defect density — the average number of fatal defects per unit area , e.g. "0.5 defects per cm²".
The picture: sprinkle salt over the whole wafer. D 0 where the grains land — it is how heavily you sprinkled, on average, per square centimetre.
Intuition Why "per area" and not "per wafer"Chips come in different sizes. A number per area lets us ask about any die size just by multiplying by that die's area A
D 0 Why it feels right: it looks like a material constant.
Fix: D 0 process , not the metal. A brand-new manufacturing node starts dirty (high D 0 Process node scaling .
λ λ expected number of fatal defects on one single die . You get it by multiplying "defects per area" by "area of one die":
λ = D 0 A .
Units cancel to give a pure count: cm 2 defects × cm 2 = defects 0.5 2 expect 1 on average .
Intuition Why we bother naming this productThe picture: a bigger die is a bigger landing pad, so more specks hit it. λ λ
e e − λ e ≈ 2.71828 π same fraction over each equal step — "compounding" decay. ==e − λ e minus λ 0 1 λ
e − λ every patch is clean is a product of many "almost 1" factors. When you make the patches infinitely small, that product collapses exactly onto e − λ e − λ
Read the shape from the figure: at λ = 0 e 0 = 1 = 100% λ never touches zero — there is always a slim chance a big die got lucky.
The full reason this exact curve (and not another) governs defects is the Poisson distribution — the D2 page derives it; here you only need to trust the shape .
Y Y 0 1 zero fatal defects, and that probability is e − λ Y = e − D 0 A .
Intuition What each letter is doing in one sentenceD 0 A λ e − ( … )
Everything downstream — good dies per wafer, cost — is just this Y Wafer testing and probe for how Y
α α clustering parameter . Real specks land in clumps , not evenly. α α α
helps yield
If ten specks all pile onto one already-dead die, they wasted themselves — nine other dies stayed clean that a "spread-out" sprinkle would have killed. So clumping saves extra dies, and the real yield is higher than the plain e − λ Y = ( 1 + D 0 A / α ) − α α → ∞ e − D 0 A
Definition Cost per good dieIf a whole wafer costs Cost wafer DPW × Y working dies, then each survivor must carry the whole bill:
Cost good die = DPW × Y Cost wafer .
The picture: you pay for the entire pizza whether or not every slice is edible, so you divide the price only among the good slices. This is the bridge to Chip economics and cost per transistor and to why giant dies get chopped into Chiplets and MCM .
Binning is sorting the dies that passed into quality grades — by top stable clock speed, by power use, or by how many cores/cache blocks still work — and selling each grade as a different product.
The picture: after tasting the good cookies, the crunchiest go in the "premium" box and the softer ones in the "budget" box. No new recipe — the same die at different prices.
survival e to the minus lambda
What does r d Radius = centre-to-edge distance; d = 2 r
Why is die area written with a little 2 cm 2 Area covers two directions, so its unit is length × length.
What is DPW and why is it less than π r 2 / A Dies that fit on the round wafer before defects; the round rim wastes partial rectangular dies.
What does D 0 Average fatal defects per unit area; not constant — it drops as a process matures.
How do you get λ λ = D 0 A
What is e − λ e
Write the Poisson yield and say what each letter does. Y = e − D 0 A D 0 A e − ( ⋅ )
What does α Clustering; clumps waste defects on already-dead dies, sparing clean ones.
Give the cost-per-good-die formula. Cost wafer / ( DPW × Y )
What is binning in one line? Sorting passing dies into speed/quality grades sold as different products.
