Question bank — Yield, defect density, and binning

True — $D_0$ ek property hai process/wafer ki, die ki nahi. Die size ke saath jo badalta hai woh hai $\lambda =D_{0}A$, yani expected count per die, density khud nahi.

False — yield hai $Y=e^{-D_{0}A}$, toh $A$ double karne se survival factor square ho jaata hai: $e^{-2\lambda }=(e^{-\lambda }{)}^2$. 60% yield 36% ban jaati hai, 30% nahi.

False in general — ek chip ek redundant/harvestable block mein defect carry kar sakta hai aur phir bhi lower SKU ke roop mein kaam kar sakta hai. "Fatal defect" (jo die ko kill kare) yield-relevant category hai, "koi bhi defect" nahi.

False — ${\lim }_{\alpha \to \infty }(1+D_{0}A/\alpha {)}^{-\alpha }=e^{-D_{0}A}$, toh yeh wapas Poisson par collapse ho jaata hai. Infinite $\alpha$ ka matlab hai "no clustering," yaani bilkul uniform defects — exactly wahi Poisson assumption hai.

True finite $\alpha$ ke liye — clumping defects ko kam dies par dher kar deta hai, baaki zyada ko bachata hai, toh good fraction badhta hai. Equality sirf $\alpha \to \infty$ limit mein milti hai.

False — yeh round rim ko ignore karta hai. Curved edge par straddling karne wale rectangular dies kho jaate hain, toh hum ek edge-loss term subtract karte hain; $\pi r^2/A$ sirf ek upper estimate hai.

False — speed aur harvest binning same die aur mask use karte hain. Sorting test ke baad hoti hai; SKUs ka ladder variation aur fusing se aata hai, naye designs se nahi.

True — wafer ka paisa tum dete hi ho chahe survivors kitne bhi hon. Yield cost per good die ko badalta hai denominator $\text{DPW}\times Y$ ko change karke, wafer price ko nahi.

Formula sahi hai lekin conclusion absurd hai — ek real die ki irreducible minimum area hoti hai, aur shrinking $D_0$ bhi badhata hai (finer features print karna mushkil hota hai, dekho Process node scaling). Yeh limit mathematical hai, physical nahi.

Galat — tuneable input $D_0$ hai, ek process-maturity metric jo naye node par high shuru hoti hai aur yield ramp ke dauran fab ke seekhne ke saath girti hai. Yield product ki life bhar badlti rehti hai.

Galat — yeh usually same die hota hai jismein do defective cores fuse off hote hain (harvesting). Ek mask, kaafi SKUs.

Random ka matlab uniform nahi hota. Real dust clumps karta hai, spatial clustering create karta hai; yahi reason hai ki pure Poisson real yield under-predict karta hai aur negative-binomial correction exist karta hai.

Yeh yield omit karta hai. Tumhe good dies se divide karna chahiye $=\text{DPW}\times Y$, kyunki dead dies wafer area toh cost karti hain lekin kuch earn nahi karti.

Yeh ek estimate hai. $\sqrt{2A}$ orientation ke upar ek average die edge length approximate karta hai aur $2\pi r$ ko loosely $\pi \cdot 2r$ likha gaya hai — poora correction engineering-grade hai, koi derivation nahi.

Ek trap — clustering sirf ek given defect population ki statistical distribution badalta hai; defects hata nahi deta. Asli goal $D_0$ kum karna hai. Clustering ek modelling reality hai, koi lever nahi jise khaincha jaaye.

Poisson rare, independent events ke counts describe karta hai jo ek region par fixed average rate pe hote hain, jo exactly hai "is die area par kitne random defects padte hain." Dekho Poisson distribution.

Ek die tabhi kaam karta hai jab usmein zero fatal defects hon; koi bhi $k\ge 1$ use kill kar deta hai. Toh working fraction precisely $P(k=0)=e^{-\lambda }$ hai.

Yield area mein exponential hai, toh bada $A$ survivor fraction ko super-linearly crush karta hai, aur har survivor ko kaafi dead neighbours ke wafer share ka bhi kharch uthana padta hai. Yeh Chiplets and MCM ki ek key motivation hai.

Process variation ki wajah se "working" chips genuinely speed/power/functional units mein differ karte hain; binning us continuous spread ko ek product ladder mein convert karta hai, toh ek chip jo top grade se thodi miss ho jaaye bechne ki bajaye scrap nahi hoti.

Har chote chiplet ka $A$ chhota hota hai, toh $e^{-D_{0}A}$ high rehta hai; tum sirf good chiplets assemble karte ho, ek giant monolithic die ki exponential penalty avoid karke. (Details: Chiplets and MCM.)

Cost per good die $={\text{Cost}}_{\text{wafer}}/(\text{DPW}\times Y)$ hai, aur $Y=e^{-D_{0}A}$, $D_0$ par exponentially respond karta hai — small density improvements good dies ki sankhya multiply kar sakti hai, linear wafer savings se zyada. Dekho Chip economics and cost per transistor.

Poisson ek fixed rate everywhere assume karta hai; real defects wafer par density mein vary karte hain, aur $\alpha$ encode karta hai ki woh local rate kitna fluctuate karta hai (clustering). Ek number ($\lambda$) local rate ke mean aur spread dono capture nahi kar sakta.

$Y=e^0=1$, yaani 100% — koi defects nahi matlab har die survive karti hai, jaise formula demand karta hai.

$Y=e^{-D_{0}A}\to 0$ — infinitely bada die almost certainly kam se kam ek fatal defect pakad leta hai. Yeh "big-die punishment" ka limiting form hai.

$Y=(1+D_{0}A/\alpha {)}^{-\alpha }\to 1$ jab $\alpha \to 0^{+}$ — saare defects dies ke ek vanishing fraction par pile ho jaate hain, toh almost har die bach jaati hai. Yeh Poisson $\alpha \to \infty$ limit ka bilkul opposite extreme hai.

Nahi — jab kam dies fit hoti hain toh edge loss dominate karta hai, toh correction term total ka ek bada fraction hota hai aur DPW bahut uncertain hoti hai; yeh estimate un wafers ke liye hai jo kaafi dies hold karti hain.

Yield ka 1 hona kehta hai ki har die functional hai, lekin binning phir bhi speed/power variation se sort kar sakti hai — functional yield aur speed grade independent axes hain. Dekho Wafer testing and probe.

Zaroori nahi — equal $Y$ ka matlab hai equal $\lambda =D_{0}A$, toh jo fab bade die ke saath hai uska $D_0$ compensate karne ke liye lower hona chahiye. Sirf yield se process cleanliness rank nahi kar sakte $A$ jaane bina.

Haan — sirf fatal defects (jo actually die kill karein) $Y$ mein count hote hain; kaafi measured particles benign hote hain ya non-critical areas mein padte hain, toh yield-relevant $D_0$ typically raw particle count se lower hoti hai.