WHY it isn't just Aπr2: the wafer is a circle but dies are rectangles. Dies near the curved edge fall off. So we lose area at the rim. A common engineering estimate:
We want Y = fraction of dies with zero fatal defects.
HOW to derive. Let defect density D0 = average fatal defects per cm². A die has area A (cm²). The expected number of defects on one die is
λ=D0A.
The Poisson probability of seeing exactly k defects is
P(k)=k!λke−λ.
A die works only if k=0 (assuming any fatal defect kills it):
P(0)=0!λ0e−λ=e−λ.
Real defects aren't perfectly uniform — dust clumps. Clustering means a few dies eat many defects while others stay clean, so the good fraction is higher than pure Poisson predicts. The negative-binomial model captures this:
Speed binning: same design, sorted by achievable frequency.
Functional binning (harvesting): a die with a defective core/cache block is sold with that block disabled as a lower-tier SKU (e.g., an 8-core die with 2 bad cores sold as a 6-core part).
Fatal defects land randomly, independently, at a fixed average rate over area — the exact conditions for a Poisson process.
Poisson yield formula and its derivation origin?
Y=e−D0A, from P(0) of Poisson with mean λ=D0A.
What does D0 represent and is it constant?
Average fatal defects per unit area; NOT constant — it falls as a process matures (yield ramp).
Why do big dies yield so poorly?
Yield is exponential in area (e−D0A); larger A crushes yield super-linearly.
What is the clustering parameter α and its limit?
Negative-binomial defect-clustering factor; as α→∞ yield → Poisson e−D0A.
Formula for cost per good die?
Costwafer/(DPW×Y).
What is binning?
Sorting tested working dies into quality/speed grades (SKUs) based on measured performance.
Speed binning vs functional binning?
Speed: sort by max stable clock. Functional/harvesting: disable defective cores/cache, sell as lower tier.
Why subtract an edge term in dies-per-wafer?
Round wafer, rectangular dies — partial dies at the curved rim are lost.
Negative-binomial yield formula?
Y=(1+D0A/α)−α.
Recall Feynman: explain to a 12-year-old
Imagine baking a giant round cookie sheet covered in tiny square cookies, all identical. Sometimes a bug lands in the dough and ruins a cookie. Yield = how many cookies are still good. If a cookie is bigger, it's more likely a bug landed on it, so big cookies get ruined more — a LOT more. The bugs like to land in clumps, so if you're lucky the clumps ruin just a few cookies and spare the rest. After baking, you taste the good cookies and put the crunchiest ones in the "premium" box and the softer ones in the "cheap" box — that's binning!
Dekho, ek silicon wafer ek badi round plate hai jispe hum sैकड़ों same chips (dies) chhaapte hain. Har wafer mahenga hota hai, aur dust ke kan ya crystal defects kuch dies ko maar dete hain. Yield ka matlab hai — jitni dies banayi, unme se kitni actually kaam karti hain. Yeh economics ka sabse bada lever hai: agar aadhi dies mar gayi, to har working chip ki cost double ho gayi.
Yield nikalne ke liye hum Poisson distribution use karte hain, kyunki defects random aur independent tarike se girte hain. Ek die pe average defects λ=D0×A, jahan D0 defect density (defects per cm²) aur A die ka area. Zero defect ki probability e−λ hoti hai, isliye Y=e−D0A. Yaad rakho — yeh exponential hai, isliye badi die (jaise GPU) ka yield bahut tezi se girta hai. Real fabs me defects clump karte hain, isliye negative-binomial model (1+D0A/α)−α zyada accurate hai, aur α→∞ pe wapas Poisson ban jata hai.
Binning ka funda simple hai: jo chips test me pass hui, sab ek jaisi nahi hoti. Process variation ki wajah se koi 5 GHz chalti hai, koi sirf 3.5 GHz. To hum unhe grade ke hisaab se sort karke alag price pe bechte hain — yeh speed binning hai. Aur agar ek 8-core die me 2 core kharab hain, to unhe disable karke 6-core part bechte hain — yeh functional binning / harvesting hai. Same die, alag SKU, alag price. Is tarah company kharab-lekin-usable silicon se bhi paisa recover karti hai.
