Chiplets and multi-die integration: Foundations — Hinglish deep dive

6.5.1 · D1 · Hardware › Advanced & Emerging Architectures › Chiplets and multi-die integration

Is page par kuch bhi assume nahi kiya gaya. Parent note padhne se pehle, tum neeche diye gaye har ek symbol ko point karke yeh batane mein capable hono chahiye ki woh kya hai, kaisa dikhta hai, aur topic ko uski zaroorat kyun hai. Chalte hain, ek ek karke samjhate hain.

1. Wafer, die, aur area $A$

Figure — Chiplets and multi-die integration

Figure dekho. Bada circle wafer hai. Squares ka grid dies hain. Jab hum die area $A$ kehte hain, toh hum ek us square ki size ki baat karte hain, jo square millimetres ( $mm^{2}$ ) mein measure hoti hai.

Symbol:

$A$ ::: die area, $mm^{2}$ mein. Picture: grid mein ek rectangle ki size.

2. Defects aur defect density $D_{0}$

Defects wafer par roughly randomly aur roughly evenly scatter hote hain — jaise pavement par rain ki boondein padti hain. Kuch areas mein zyada, kuch mein kam, lekin average par sprinkle uniform hai.

Hum sirf count ki jagah density kyun use karte hain? Kyunki count is baat par depend karta hai ki tum kitna bada patch dekh rahe ho. Density se hum pooch sakte hain: "area $A$ ki ek die par mujhe kitne defects expect karne chahiye?" — aur jawab seedha sprinkle rate times area hai.

Symbols:

$D_{0}$ ::: defect density, average defects per $mm^{2}$ . Picture: rain ki boondein kitni dense hain.
$D_{0} \cdot A$ ::: area $A$ ki ek die par expected number of defects. Picture: ek square par boondein = rate × square-size.

3. Probability, aur "zero defects" kyun goal hai

Ek die tab hi kaam karti hai jab uspar zero defects land hon. Toh hum chance ki baat karte hain.

Yahan "something" jis par hum dhyan dete hain woh hai ek die par kitne defects land hote hain. Hum likhenge $P (k)$ = probability ki exactly $k$ defects ek die par land hon, jahan $k = 0, 1, 2, \dots$

$k$ ::: ek die par defects ka ek whole-number count ( $0, 1, 2, \dots$ ).
$P (k)$ ::: exactly utni taadad milne ka chance.

Zero specifically kyun? Kyunki ek defect (usually) die ko khatam karne ke liye kaafi hota hai. Toh die tab hi achhi hoti hai jab $k = 0$ ho. Iska chance woh number hai jo hum dhoondh rahe hain — aur hum use abhi yield ka naam dene waale hain.

4. Yield formula ko step by step banana

Hum ek number chahte hain: probability ki ek die par zero defects hon. Hum us number ko yield $Y$ kehenge.

$Y$ ::: yield — un dies ka fraction jo achhi niklen; equivalently probability ki ek die par zero defects hon ( $0$ se $1$ ke beech ka number).

$Y$ paane ke liye hum pehle kisi bhi count $k$ ke liye poori probability $P (k)$ chahte hain, phir $k = 0$ set karte hain. Woh poora expression — Poisson formula — do cheezein use karta hai jo ek newcomer nahi pahchaanenge: factorial $k!$ aur exponential $e^{x}$ . Chaliye dono ko samjhate hain, phir unhe assemble karke $Y$ read karte hain.

4a. Factorial $k!$

Yeh kyun appear hota hai? Jab tum $k$ identical defects ek die par sprinkle karte ho, toh order jisme tumne unhe drop kiya woh matter nahi karta — "defect A phir B" wahi outcome hai jaise "B phir A". $k$ cheezein arrange karne ke $k!$ tarike hote hain, toh agar hum ordered sprinklings count karte toh har real outcome ko exactly $k!$ baar over-count kiya hota. $k!$ se divide karna woh over-count cancel karta hai. (Hamare purpose ke liye tumhe sirf ek fact chahiye: $0! = 1$ , woh value jo hum ek achhi die ke liye plug in karte hain.)

4b. Exponential $e^{x}$

Yahi function kyun aur, maano, $1/ x$ ya straight line kyun nahi? Kyunki hum pooch rahe hain: "agar defects independently land hote hain, toh chance kya hai ki unme se koi bhi meri die par nahi hit karta?" Independent survival multiply hoti hai. Die ko kai tiny patches mein tod do; poori die ke bachne ka chance har patch ke bachne ka product hai. Jab tum kai independent tiny survival-chances ko saath multiply karte ho, toh maths force karta hai ki ek exponential appear ho — exactly yahi $e^{- D_{0} A}$ hai. Koi aur function "independent, multiplying survival" ke saath fit nahi hota.

Red curve dekho: jaise $D_{0} A$ (expected defects) horizontal axis par badhta hai, survival $e^{- D_{0} A}$ ek cliff se gir jaati hai. Woh cliff hi wajah hai ki bade dies doomed hote hain.

4c. Poisson formula assemble karna — har factor kahan se aata hai

Ek die par exactly $k$ defects ki poori probability hai:

$P (k) = \frac{( D _{0} A ) ^{k} e ^{- D_{0} A}}{k !}$

Ise teen factors ki tarah padho, har ek ka ek kaam hai:

$(D_{0} A)^{k}$ — "har defect ka yahan land hone ka chance tha." Humne establish kiya ki $D_{0} A$ is die par defects ka expected count hai. Unme se $k$ ko land karne ka matlab hai is single-defect tendency ko khud se $k$ baar multiply karna → power $(D_{0} A)^{k}$ . Zyada expected defects, ya unme se zyada chahiye ( $k$ ), is factor ko upar push karta hai.
$\frac{1}{k !}$ — "orderings double-count mat karo." §4a se: $k$ defects identical hain, isliye hum $k!$ equivalent orderings ko divide out karte hain jo otherwise over-count ho jaate.
$e^{- D_{0} A}$ — "aur baaki saara area khali raha." Yeh §4b ka survival factor hai: probability ki koi extra defects nahi land kiye. Yeh woh exponential hai jiske baare mein humne argue kiya ki jab bhi independent survival-chances multiply hote hain toh zaroor appear hoti hai.

Teeno ko multiply karo aur tumhare paas standard Poisson model hai — rare, independent, uniformly-sprinkled events ka textbook description.

Ab payoff. Ek achhi die ke liye $k = 0$ set karo. Toh $(D_{0} A)^{0} = 1$ (factor 1 gayab — koi defects place nahi karne), $0! = 1$ (factor 2 trivial hai), sirf survival factor bachta hai:

$Y = P (0) = e^{- D_{0} A}$

Common mistake "Poisson model exact hai — real yield sach mein

$e^{- D_{0} A}$ hai." Kyun sahi lagta hai: derivation clean hai aur formula har jagah hai. Catch: Poisson assume karta hai ki defects uniform hain (har jagah same density) aur independent hain (ek defect doosre ko nearby nahi banata). Real wafers dono todte hain: defects cluster karte hain (ek scratch ya particle paas waale neighbours ka ek patch spawn karta hai) aur edges/centre ki non-uniform density hoti hai. Clustering ka matlab hai ki kuch dies mein bahut zyada defects hote hain jabki doosre saaf rehte hain, isliye true yield usually plain Poisson se zyada hoti hai. Fix: industry clustered models use karti hai (jaise negative-binomial / "Murphy" ya Seeds models) jo ek clustering knob add karte hain. Poisson sahi pehli picture hai — bas uski limits jaano. Dekho Wafer yield and defect density.

5. Ek die ko split karna: symbol $N$ (aur catch)

Agar bada die $800 mm^{2}$ tha aur $N = 4$ hai, toh har chiplet $800/4 = 200 mm^{2}$ ka hai. Chhota area yield formula mein plug karo:

$Y_{chiplet} = e^{- D_{0} (A / N)}$

Chhota exponent matlab exponential 1 ke kareeb hai — har individual chiplet achha hone ka far zyada chance hai. Abhi tak sab theek hai.

Common mistake "N chiplets mein split karna automatically finished product ki yield badhata hai."

Kyun sahi lagta hai: har chhota piece ki yield zyada hai, toh poori cheez ki bhi hogi. Catch: ek package tabhi kaam karta hai jab usme saare $N$ chiplets kaam karein. Agar woh independent hain, toh saare ke bachne ka chance har chiplet ki yield ko $N$ baar multiply karna hai: $Y_{package} = (e^{- D_{0} A / N})^{N} = e^{- D_{0} A}$ — exactly wahi monolithic yield phir se. Sirf kaatne se risk move hota hai, khatam nahi hota. Fix — real win kahan se aata hai: ① Known-good die (KGD): assembly se pehle har chiplet ko test karo aur kharab waalon ko discard karo, taaki tum kabhi doomed silicon ko package karne ki cost na chukao (tum cost bachate ho, aur sirf survivors assemble karte ho). ② Redundancy / spares / binning: kuch extra chiplets ya extra cores banao aur faulty waalon ko disable karo, taaki ek defect product ko khatam karne ki jagah degrade kare. Exponential wajah hai ki splitting zaroorat kyun hai; KGD aur redundancy wajah hain ki yeh actually pay kyun karta hai.

Toh honest summary: splitting har piece ka risk shrink karta hai aur — crucially — tumhe glue karne se pehle filter karne deta hai. Woh filtering, split khud nahi, economic engine hai.

6. Energy symbols: $E_{bit}$ , $C$ , $V$ , $L$

Parent ka doosra formula dies ke beech data move karne ki cost ke baare mein baat karta hai. Char naye symbols:

Picture: socho ek wire ek bucket ki tarah hai. "1" signal karne ke liye bucket ko charge se bharte ho; "0" signal karne ke liye use khaali karte ho. Ek lambi wire ek bada bucket hai — $C \propto L$ (capacitance length ke saath badhti hai). Har bit jo tum bheejte ho woh bucket bharne ki energy cost hai:

$E_{bit} = \frac{1}{2} C V^{2}$

Chiplets ke liye yeh kyun matter karta hai? Dies jo kareeb glued hain (micrometres apart, package par) ke chhote buckets hain → cheap bits. Data jo package se bahar circuit board par jaata hai (centimetres) ke bade buckets hain → expensive bits. Closeness = cheap communication. Isliye hum dies ko physically ek doosre ke kareeb laate hain.

Symbol $\propto$ ::: "ke proportional hai" — saath mein badhta hai, constants ignore karke. $C \propto L$ ka matlab hai "length double, capacitance double."

Aage kahan jaana hai

Wafer yield and defect density — $Y = e^{- D_{0} A}$ , defect clustering, aur $D_{0}$ kaise measure hoti hai ka deep dive.
Moore's Law and its slowdown — kyun hum sirf ek bade chip ko shrink karte nahi reh sakte.
Advanced packaging (2.5D 3D interposers) — achhe dies kaise glue hote hain.
Memory bandwidth and HBM — jahan $E_{bit} \propto L$ argument bahut zyada pay off karta hai.
Interconnect energy and dark silicon — energy side, expanded.
Heterogeneous integration — ek package mein alag alag dies mix karna.
Parent par wapas: Chiplets and multi-die integration.

Equipment checklist

Khud ko test karo — tum ready ho jab har jawab instantly aaye.

Symbol

A

kya stand karta hai, aur uski units kya hain?

Die area,

mm^{2}

mein — ek chip rectangle ki size.

D_{0}

kya measure karta hai?

Defect density: silicon ke

mm^{2}

par average number of defects.

Pure number

D_{0} A

kya hai?

Area

A

ki ek die par land hone waale defects ki expected (average) number.

P (k)

ka kya matlab hai?

Probability ki exactly

k

defects ek die par land hon.

Poisson formula mein

(D_{0} A)^{k}

aur

1/ k!

kya kaam karte hain?

(D_{0} A)^{k}

k

defects ke land hone ki tendency;

1/ k!

identical defects ki over-counted orderings cancel karta hai.

0!

aur

e^{0}

compute karo.

0! = 1

aur

e^{0} = 1

Kya

e^{- 4}

bada hai ya chhota, roughly?

Chhota — lagbhag

0.018

;

e^{- x}

zero ki taraf shrink hota hai jaise

x

badhta hai.

Yield

e^{- D_{0} A}

ka exponential kyun hona zaroori hai, straight line kyun nahi?

Kai patches ki independent survival multiply hoti hai, aur independent survival chances ko multiply karna ek exponential force karta hai.

k

ki kaun si value achhi die deti hai, aur formula kya reduce hota hai?

k = 0

; yeh

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

par reduce hota hai.

Poisson yield model ki do assumptions batao, aur ek tarika jisme reality unhe todti hai.

Uniform density aur independent defects; reality dono ko clustering ke zariye todti hai (defects neighbours spawn karte hain) aur wafer non-uniformity se.

Agar area

A

N

equal chiplets mein split karo, toh saare ke bachne ki yield kya hai, aur sirf split karna kyun akela help nahi karta?

(e^{- D_{0} A / N})^{N} = e^{- D_{0} A}

— monolithic jaisi hi; real gain kharab dies ko test/discard karne (KGD) aur redundancy se aati hai.

Energy-per-bit formula batao aur

C \propto L

ka kya matlab hai.

E_{bit} = \frac{1}{2} C V^{2}

; capacitance wire length ke saath badhti hai, toh lambi wires mein har bit mein zyada energy lagti hai.

Symbol

\propto

ka kya matlab hai?

"Ke proportional hai" — constant factors ignore karke, saath mein badhta hai.