Wafer Die Yield

What is modelled

Dies are laid out on a round wafer with an unusable edge ring and thin scribe lanes between them. Random defects fall on the wafer; a die is good only if no killer defect lands on it. The share of good dies depends on the product of die area A and defect density D, and on whether defects are spread evenly or in clusters.

Equations

• Dies per wafer N ≈ π d² / (4A) − π d / √(2A) (compared with exact counting).
• Poisson Y = e−AD, Murphy Y = ((1 − e−AD) / AD)², Seeds Y = 1 / (1 + AD).
• Negative binomial Y = (1 + AD / α)−α, which tends to Poisson as α grows.
• Cost per good die wafer cost / (N Y).

Checks run in the page

• Exact die count against the standard formula.
• A 200,000-die gamma-Poisson simulation against the negative-binomial formula.
• Murphy yield by numerical integration.
• The Poisson limit and the ordering of the models.

Honest simplifications

Square dies of one size, all defects are killer defects, defect clusters are drawn independently for each die (no spatial correlation between neighbours), no edge-yield or systematic patterns, no repair or redundancy, no parametric yield or packaging and test losses, and an arbitrary wafer cost. The 3D page shows a packaging and test line; this page is the front-end yield arithmetic.

• Press Big GPU-size die and watch the red dies take over the wafer.
• Press Strong clustering: the same defect count kills fewer dies because they pile onto the same ones.
• Drag die area up and read how the cost per good die rises faster than the area.