Die Yield from Defect Density (Poisson Model)

Defects land on a wafer more or less at random: a particle in the lithography, a pinhole in an oxide, a break in a metal line. If they are independent and uniformly distributed at density $D_0$, then the number landing on a die of area $A$ is Poisson-distributed with mean $D_0 A$, and the chance a given die catches none of them — the yield — is the Poisson probability of zero, $e^{-D_0 A}$.

The area is in an exponent, and that changes everything about chip economics. Doubling the die area does not halve the yield; it SQUARES it. A 90% yield becomes 81%, a 50% yield becomes 25%. Combine that with the fact that a bigger die also means fewer dies per wafer, and cost per good die rises far faster than area does. This is the single strongest economic argument against large monolithic chips, and it is why chiplets exist: cut a large design into several small dies, yield each of them well, and join them on a package substrate. The packaging is not free, but it is cheaper than the exponential.

Pure Poisson is the pessimistic member of a family, and it is pessimistic for an identifiable reason: real defects cluster. Particles arrive in bursts, a scratch takes out a swathe, an edge effect ruins a ring. Clustering means the defects that do occur concentrate on fewer dies, so more dies come through clean than independence predicts. Foundries therefore use the negative-binomial model, $Y = (1 + D_0 A/\alpha)^{-\alpha}$, where the clustering parameter $\alpha$ is fitted from production data, or Murphy's model, which assumes a distribution of defect densities across the wafer. Both give higher yields than Poisson at the same $D_0$ and $A$.

Two more honesties belong on this figure. Not every defect is fatal — redundancy is designed in deliberately, most obviously in memories, where spare rows and columns are blown in at test so a die with a handful of bad cells still ships. And the yield that matters commercially is not this one but the fraction of dies that meet full specification, including speed and power binning. A die that works only at a lower clock is not a loss; it is a cheaper product, and binning is a substantial part of how the economics actually close.