Silicon Manufacturing and Yield

The GDSII file left the building at tape-out. What happens next is arguably the most precise manufacturing process humanity has ever run — and it ends not with "the chip" but with a statistical population of chips, some perfect, some dead, some merely wounded. This lesson is about that population: how silicon is actually printed, why some dice are born broken, and the single exponential formula — yield — that quietly dictates the economics of the entire industry, from why chips cost what they cost to why the biggest designs are now being cut into pieces.

Masks, light, and layers

Each layer of the layout becomes a mask — a quartz plate patterned with the layer's shapes. In photolithography, the wafer is coated with light-sensitive photoresist, light is projected through the mask (shrunk by the projection optics onto the wafer), and the exposed resist is developed away — leaving a stencil. Etching, ion implantation or material deposition then acts only where the stencil permits. Strip the resist, coat again, align the next mask to nanometre precision, repeat: transistor layers first, then via and metal layers, dozens of masks in all, over roughly three months of around-the-clock processing.

The finest layers are smaller than the wavelength of the light printing them — decades of heroic optical tricks made that possible, and at the bleeding edge the light itself changed: EUV (extreme ultraviolet, 13.5 nm) machines the size of a bus, whose light is made by vaporising tin droplets with a laser 50,000 times a second and steered by mirrors, because no lens is transparent at that wavelength. Each one costs a few hundred million dollars. This is the machinery your polygons ride through.

Wafers, defects, and the yield exponential

Chips are printed hundreds at a time on a 300 mm wafer — a single crystal of absurdly pure silicon, sliced into a disc. Simple geometry says how many dice fit: roughly the wafer area divided by the die area, minus the partial dice wasted around the circular edge (a correction that bites big dice hardest).

Now the villain. Despite cleanrooms thousands of times cleaner than an operating theatre, stray particles and process imperfections land on wafers at some defect density D — defects per unit area. One killer defect in a die's area, and that die is scrap. If defects sprinkle randomly (a Poisson process), the chance a die of area A escapes untouched is

Y \approx e^{-A \cdot D},

the Poisson yield model. The exponent is the expected number of defects landing in one die. And because A sits in an exponent, the model punishes big dice without mercy. Watch what doubling the area does. Since e^{-2AD} = \left(e^{-AD}\right)^2,

Y(2A) = Y(A)^2 .

Doubling the area squares the yield fraction. A die yielding 90% doubles into one yielding 81% — mildly annoying. But a die yielding 50% doubles into one yielding 25%, and doubling again gives 6.25%. Worse, it is a double penalty: the doubled die also fits only half as many times on the wafer, so good dice per wafer fall by the halving and by the squaring together. This one fact explains why giant dice are reserved for products that can charge thousands of dollars — and, as the next lesson shows, why the industry started cutting big designs into small chiplets.

See the exponential

Drag the defect density — a mature process sits near the low end, a brand-new node near the high end — and watch what each defect density does to a 100 mm² phone chip versus an 800 mm² giant:

Run the fab's spreadsheet

Put the pieces together: dice per wafer (with the standard edge-loss correction), the yield exponential, and the cost model the whole business runs on — \text{cost per good die} = \frac{\text{wafer cost}}{\text{good dice per wafer}}.

// From a $12,000 wafer to a price tag, at D = 0.1 defects/cm². const waferDiameter = 300; // mm const waferCost = 12000; // dollars const D = 0.1; // defects per cm² const dicePerWafer = (a: number): number => { const r = waferDiameter / 2; // area term minus the partial dice lost around the circular edge const gross = (Math.PI * r * r) / a - (Math.PI * waferDiameter) / Math.sqrt(2 * a); return Math.floor(gross); }; const dieYield = (a: number): number => Math.exp(-(a / 100) * D); // a in mm² → cm² console.log("area mm² | dice/wafer | yield | good dice | $/good die"); for (const a of [25, 100, 200, 400, 800]) { const n = dicePerWafer(a); const y = dieYield(a); const good = Math.floor(n * y); console.log( String(a).padStart(8) + " | " + String(n).padStart(10) + " | " + (100 * y).toFixed(0).padStart(4) + "% | " + String(good).padStart(9) + " | $" + (waferCost / good).toFixed(2), ); } console.log(""); console.log("32× the area → far more than 32× the cost: the exponential's bill.");

The 800 mm² monster costs on the order of a hundred times more per good die than the 25 mm² chip, though it holds only 32× the silicon. Every extra square millimetre pays twice: fewer candidate dice, and a worse survival rate for each.

Binning: selling the wounded

"Good" versus "dead" is not the whole story. Testing happens twice — wafer sort, where a probe card touches each die's pads while still on the wafer (no point paying to package a corpse), and final test after packaging, which catches assembly damage and verifies full speed and temperature behaviour. And between perfect and dead lies a profitable middle: binning. A die whose logic all works but only at a lower clock becomes the slower, cheaper SKU. A die with a defect in one core of eight has that core fused off and ships as the six- or seven-core model. GPUs are the masters of this — a flagship die with a few dead shader clusters is the mid-range product; whole product lines are one die sorted into bins. Salvage turns yield loss back into revenue, which is why architects deliberately design in redundancy — spare cores, spare memory rows — to give defects somewhere harmless to land.

chips are printed layer by layer through dozens of masks (photolithography; EUV at the bleeding edge), hundreds of dice per 300 mm wafer;
random defects at density D kill dice: the Poisson model gives Y \approx e^{-AD};
doubling die area squares the yield fraction — and halves dice per wafer: a double penalty that makes big dice disproportionately expensive;
cost per good die = wafer cost ÷ good dice per wafer;
wafer sort screens dice before packaging; final test screens packaged parts; binning sells partly-defective or slow dice as cheaper SKUs, with designed-in redundancy making salvage possible.

Wafers grew for decades — 100 mm, 150, 200, then 300 mm around 2001 — because a bigger disc means more dice per pass through every machine, and area grows with the square of the diameter. The obvious next step, 450 mm, was seriously attempted: consortia were formed, prototype tools built, billions spent. It never happened. Every machine in a fab — steppers, etchers, furnaces, robots — would have needed reinventing at once, for tens of billions of dollars, and the equipment makers looked at who would capture the savings (their customers, not them) and quietly walked away. Twenty-plus years later, 300 mm is still the standard — a rare example of an exponential industry hitting a limit set not by physics but by economics and negotiation.

Notice what yield is: multiply it by wafer cost and dice per wafer and you have a competitor's entire cost structure — their margins, their pricing floor, how much room they have to undercut you. That is why real yield numbers are among the most closely guarded secrets in the industry, wrapped in NDAs, known precisely only to the foundry and its customer. So treat every public yield claim — a keynote's "healthy yields", a leaker's "80%", an analyst's estimate — as marketing, rumour, or both. Definitions slide too: yield of what? Perfect dice, or anything salvageable into some bin? At what clock? Measured on which month's wafers, on a node that improves weekly? A number with none of that context is a press release, not data. The model in this lesson is real and load-bearing; any specific public number plugged into it deserves a raised eyebrow.