Electromigration reliability — Hinglish notes

6.4.8 · Hardware › Power, Thermal & Reliability

Electromigration KYA hai?

YEH kyun matter karta hai: Jaise-jaise transistors shrink hote hain, wires narrower hoti hain lekin aksar similar ya zyada current carry karti hain, isliye current density $J$ badhti hai. EM lifetime $J$ ke saath steeply girti hai, isliye EM ek hard design rule set karta hai: ek max allowed current per wire width.

Force KAISE arise hoti hai (first principles se derivation)

Ek biased wire mein metal ion par do forces act karti hain:

Direct field force $F_{f i e l d} = q E$ — electric field positive ion ko cathode ki taraf kheenchti hai.
Electron-wind force $F_{w in d}$ — moving electrons ions se collide karte hain, momentum anode ki taraf transfer karte hain (electron flow direction).

Acche conductors mein wind dominate karti hai. Hum dono ko ek effective charge $Z^{*}$ mein bundle karte hain:

$F = Z^{*} q E = Z^{*} q ρ J$

Yeh step kyun? Kyunki $E = ρ J$ (Ohm's law local form mein, $ρ$ = resistivity), hum driving force ko directly current density $J$ ke terms mein express karte hain — woh cheez jo designers control karte hain.

Yeh force ek atomic flux drive karti hai. Einstein relation use karke (mobility $= D / k_{B} T$ ):

$J_{a t o m} = \frac{N D}{k _{B} T} F = \frac{N D}{k _{B} T} Z^{*} q ρ J$

jahan $N$ = atom density, $D = D_{0} e^{- E_{a} / k_{B} T}$ diffusivity hai. Is flux mein divergence ( $\nabla \cdot J_{a t o m} \neq = 0$ ) hi mass accumulate karti hai → voids/hillocks.

Black's Equation (lifetime law)

Combine karo: failure tab hoti hai jab ek critical amount of mass move ho chuki ho. Rate $\propto J_{a t o m} \propto J e^{- E_{a} / k_{B} T}$ . Empirically Black ne paaya ki current dependence ek power $n$ hai, jo Median-Time-To-Failure deta hai:

$e^{+ E_{a} / k T}$ (positive) kyun? Zyada $T$ → faster diffusion → shorter life, toh MTTF ko $T$ badhne par girna chahiye. $e^{+ E_{a} / k T}$ $T$ ke saath decrease karta hai. ✔ Sanity check.

Blech effect (short-line immunity)

Worked examples

Common mistakes

Recall Ek 12-saal ke bacche ko explain karo (Feynman)

Ek wire ek aisa hallway hai jisme tiny metal marbles bhari hain. Jab tum usme electric "wind" ka strong stream push karte ho, toh hawa marbles ko hall mein gira deti hai. Dheere-dheere, ek jagah marbles khatam ho jaati hain aur ek hole ban jaata hai → wire snap ho jaati hai (open). Doosri jagah ek pile-up ho jaata hai → neighbor wire ko touch kar leta hai (short). Zyada push karo (more current) ya garam karo, marbles tez move karti hain aur jaldi toot jaati hai. Lekin agar hallway bahut short ho, toh marbles end mein pile ho jaati hain aur itna push back karti hain ki ruk jaati hain — woh wire kabhi nahi toot ti.

Recall — flashcards

Electromigration ko kaunsa physical mechanism drive karta hai?

Conducting electrons se momentum transfer (electron wind) jo metal atoms ko shove karta hai.

EM ke under metal atoms kis direction mein migrate karte hain?

Anode ki taraf — electron flow ki same direction mein (conventional current ke opposite).

Black's equation batao.

MTTF = A J^{- n} e^{E_{a} / k_{B} T}

Exponential

e^{+ E_{a} / k T}

(positive) kyun hai, negative kyun nahi?

Zyada T matlab faster diffusion → shorter life, toh MTTF ko T badhne par girna chahiye;

e^{+ E_{a} / k T}

T ke saath decrease karta hai.

Current exponent

n

physically kya represent karta hai?

n \approx 1

= void-nucleation limited;

n \approx 2

= void-growth limited.

Blech effect / Blech length kya hai?

Short wires immortal hoti hain kyunki back-stress EM force balance kar leta hai jab

J \cdot L

ek critical product se kam ho.

Voids aur hillocks kya hain?

Void = atom depletion → open circuit (cathode side); hillock = atom pile-up → short (anode side).

Modern nodes mein EM kyun zyada bura hai?

Wires narrow hoti hain lekin current high rehta hai → current density J badhta hai → MTTF

\propto J^{- n}

collapse ho jaata hai.

Copper aluminum se zyada EM-resistant kyun hai?

Higher activation energy; lekin Cu phir bhi surfaces/interfaces ke along migrate karta hai, isliye cap/barrier engineering zaroori hai.

Effective charge

Z^{*}

kya hai?

Ek lumped factor jo direct field force aur (dominant) electron-wind force combine karta hai:

F = Z^{*} q ρ J

Connections

Joule heating & self-heating — local $T$ badhata hai, EM accelerate karta hai.
Interconnect RC delay — same wires; narrow karna EM aur delay dono ko worse karta hai.
Stress migration — thermomechanical stress se driven sibling wear-out.
Arrhenius reliability model — $e^{E_{a} / k T}$ temperature-acceleration backbone.
Copper damascene process — barriers/caps jo Cu EM paths gate karte hain.
Design rules & current density limits — kaise EM ek layout constraint banta hai.
MTTF and FIT rates — per-wire life ko chip reliability budgets mein convert karna.