Yield, defect density, and binning — Hinglish notes

4.3.21 · Hardware › Semiconductor Fabrication

Intuition Bada picture

Ek silicon wafer bahut mehenga hota hai. Uspe hum same chip ki hundreds of copies print karte hain (jise dies kehte hain). Random dust ke particles, crystal defects, aur process ki variation kuch dies ko kharab kar deti hai. Yield batata hai: maine jo saare dies banaye, unme se kitne fraction sach mein kaam karte hain? Defect density batati hai ki mera process kitna ganda hai (defects per unit area). Binning woh hai jo hum bachhne waalon ke saath karte hain: saare kaam karne wale chips equal nahi hote, toh hum unhe speed/quality grades mein sort karke alag-alag prices pe sell karte hain.

YEH kyun important hai? Yield directly cost per good chip set karta hai. Agar aadhe dies kharab ho jaayein, toh har bachne waale ko dono ki cost pay karni padti hai. Yeh chip economics ka sabse bada lever hai.

1. Dies per wafer

YEH $\frac{π r ^{2}}{A}$ kyun nahi hota: wafer ek circle hai lekin dies rectangles hain. Curved edge ke paas wale dies bahar gir jaate hain. Toh rim pe area lose hota hai. Ek common engineering estimate:

$DPW \approx \frac{π r ^{2}}{A} - \frac{π \cdot 2 r}{2 A}$

2. Defect density se Yield — Poisson yield derive karna

Hum $Y$ chahte hain = un dies ka fraction jo zero fatal defects ke saath hain.

KAISE derive karein. Maano defect density $D_{0}$ = average fatal defects per cm². Ek die ka area $A$ (cm²) hai. Ek die pe expected defects ki sankhya hai $λ = D_{0} A .$ Exactly $k$ defects dekhne ki Poisson probability hai $P (k) = \frac{λ ^{k} e ^{- λ}}{k !} .$ Ek die tabhi kaam karta hai jab $k = 0$ ho (maante hue ki koi bhi fatal defect use kharab kar deta hai): $P (0) = \frac{λ ^{0} e ^{- λ}}{0 !} = e^{- λ} .$

Real fabs: defects cluster karte hain

Real defects perfectly uniform nahi hote — dust clumps mein aata hai. Clustering ka matlab hai kuch dies pe bahut saare defects aa jaate hain jabki baaki clean rehte hain, toh good fraction pure Poisson ke prediction se zyada hoti hai. Negative-binomial model yeh capture karta hai:

3. Cost per good die

YEH kyun: aap poori wafer ki cost pay karte ho chahe kitne bhi dies survive karein. Cost ko good dies ki sankhya (gross dies × yield) se divide karo.

Figure — Yield, defect density, and binning

4. Binning

Do flavours:

Speed binning: same design, achievable frequency ke hisaab se sorted.
Functional binning (harvesting): ek die jisme defective core/cache block hai use us block ko disabled karke lower-tier SKU ke roop mein becha jaata hai (jaise, ek 8-core die jisme 2 bad cores hain use 6-core part ke roop mein becha jaata hai).

Worked examples

Common mistakes

Flashcards

Die defects model karne ke liye Poisson kyun use kiya jaata hai?

Fatal defects randomly, independently, area pe ek fixed average rate se girate hain — yeh exactly Poisson process ke conditions hain.

Poisson yield formula aur uska derivation origin?

Y = e^{- D_{0} A}

, mean

λ = D_{0} A

ke saath Poisson ke

P (0)

se.

D_{0}

kya represent karta hai aur kya yeh constant hai?

Per unit area average fatal defects; constant NAHI — yeh process mature hone ke saath girti hai (yield ramp).

Bade dies itni poorly yield kyun karte hain?

Yield area mein exponential hai (

e^{- D_{0} A}

); bada

A

yield ko super-linearly crush karta hai.

Clustering parameter

α

kya hai aur uski limit kya hai?

Negative-binomial defect-clustering factor; jab

α \to \infty

yield → Poisson

e^{- D_{0} A}

Cost per good die ka formula?

Cost_{wafer} / (DPW \times Y)

Binning kya hai?

Tested working dies ko measured performance ke basis pe quality/speed grades (SKUs) mein sort karna.

Speed binning vs functional binning?

Speed: max stable clock ke hisaab se sort karo. Functional/harvesting: defective cores/cache disable karo, lower tier ke roop mein becho.

Dies-per-wafer mein edge term subtract kyun karte hain?

Round wafer, rectangular dies — curved rim pe partial dies kho jaate hain.

Negative-binomial yield formula?

Y = (1 + D_{0} A / α)^{- α}

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek giant round cookie sheet pe chote square cookies bake kar rahe ho, sab identical hain. Kabhi kabhi ek keeda dough mein aa jaata hai aur ek cookie kharab kar deta hai. Yield = kitni cookies abhi bhi theek hain. Agar cookie badi hai, toh zyada chance hai ki keeda uspe aaya, toh badi cookies zyada kharab hoti hain — BAHUT zyada. Keede clumps mein aana chahte hain, toh agar tumhara luck achha ho toh clumps sirf thodi cookies kharab karte hain aur baaki spare ho jaati hain. Baking ke baad, tum achhi cookies taste karte ho aur jo sabse crispy hain unhe "premium" box mein daalo aur jo thodi soft hain unhe "cheap" box mein — wahi binning hai!

Connections

Photolithography — mask defects aur misalignment $D_{0}$ mein contribute karte hain.
Wafer testing and probe — binning se pehle good/bad dies kaise measure kiye jaate hain.
Process node scaling — chote nodes higher $D_{0}$ se start karte hain, longer yield ramp hota hai.
Poisson distribution — yield model ki statistical backbone.
Chiplets and MCM — ek bade die ko chote chiplets mein split karna taaki exponential yield loss se bacha ja sake.
Chip economics and cost per transistor — jahan cost-per-good-die strategy drive karta hai.