Channel length and short-channel effects — Hinglish notes

2.4.15 · Hardware › Transistors: BJT & FET

Channel length KYA hoti hai?

Yeh definition KYU? Gate sirf wohi mobile charge "own" kar sakta hai jiska depletion charge seedha iske neeche baitha ho. Source aur drain junctions apne khud ke depletion regions (triangular wedges) banate hain jo channel mein $x_{d S}$ aur $x_{d D}$ tak laterally extend karte hain. Jab $L$ bada hota hai toh yeh wedges channel ka ek negligible fraction hain; jab $L$ shrink hoke $x_{d S} + x_{d D}$ ke paas pahunchta hai toh woh poora channel consume kar lete hain, isliye gate simple model ke assumption se kaafi kam charge govern karta hai.

Effect 1 — Threshold voltage roll-off ( $V_{T}$ lowering)

Long-channel threshold (bulk-charge term): $V_{T 0} = V_{F B} + 2 ϕ_{F} + \frac{Q _{B}}{C _{o x}}, Q_{B} = 2 q ε_{s i} N_{A} (2 ϕ_{F})$

Gate ke neeche depletion region ko rectangle ke bajaye trapezoid ki tarah model karo. Gate trapezoidal charge "own" karta hai; source/drain junctions dono triangular corners own karte hain. Maano $x_{j}$ = junction depth aur $x_{d m}$ = vertical (bulk ke andar) gate depletion depth. Trapezoid ki geometry gate ke paas bacha hua bulk charge ka fraction deti hai:

$\frac{Q _{B}^{'}}{Q _{B}} = 1 - \frac{x _{j}}{L} (1 + \frac{2 x _{d m}}{x _{j}} - 1)$

Yeh step KYU? Rectangle se remove kiye gaye dono triangles ka total area $x_{j}$ (junctions kitni dur pahunchti hain) ke proportional hai aur $L$ ke inversely (ek fixed corner ek short channel ka zyada bada fraction hota hai). Reduced charge substitute karne par:

Effect 2 — Drain-Induced Barrier Lowering (DIBL)

Effect 3 — Velocity saturation & mobility degradation

Short device mein do alag mobility-related effects aate hain — ek lateral field se, ek vertical field se.

3a — Velocity saturation (lateral field)

$v = \frac{μ E}{1 + E / E _{cr i t}} E ≫ E_{cr i t} v_{s a t} = μ E_{cr i t}$

Consequence: saturation current overdrive mein linear ho jaata hai, quadratic nahi: $I_{D, s a t} \approx W C_{o x} v_{s a t} (V_{GS} - V_{T}) (vs. long-channel \propto (V_{GS} - V_{T})^{2}) .$

3b — Vertical-field mobility degradation (surface scattering)

$μ_{e f f} = \frac{μ _{0}}{1 + θ ( V _{GS} - V _{T} )}$

Yeh step KYU? $θ$ (units $V^{- 1}$ ) capture karta hai ki vertical field $E_{⊥} \propto (V_{GS} - V_{T}) / t_{o x}$ kitni strongly carriers ko interface mein crush karta hai. Zyada overdrive → chhota $μ_{e f f}$ → drive current $V_{GS}$ ke saath sublinearly badhti hai, velocity- saturation ceiling ke upar se bhi. Dono effects milke ideal $\propto (V_{GS} - V_{T})^{2}$ law ko severely blunt kar dete hain.

Effect 4 — Channel Length Modulation (CLM)

Saturation mein drain-side depletion channel ko $L - Δ L$ length par pinch off kar deta hai. Jab $V_{D S}$ badhta hai, $Δ L$ barhta hai → effective channel chhota hota hai → $I_{D}$ badhta rehta hai (finite output resistance): $I_{D} = I_{D, s a t} (1 + λ V_{D S}), λ \propto \frac{1}{L} .$

Figure — Channel length and short-channel effects

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko explain karo

Ek garden hose (source→drain) imagine karo jisme tumhara haath (gate) flow control karne ke liye use dabaata hai. Agar hose lamba hai, tumhara haath clearly in-charge hai. Ab hose ko super short karo — tumhara dabaav aur dono end-fittings almost touch kar rahe hain. Fittings khud hose ko squeeze karti hain, toh paani tab bhi leak hota hai jab tum band karne ki koshish karo, aur door waale end (drain voltage) ka pressure tumhare haath se past push kar sakta hai. Woh "ends ab flow ko boss kar rahe hain tumhare bajaye" exactly short-channel effects hai.

Flashcards

MOSFET ko "short-channel" kab maana jaata hai?

Jab

L

, source aur drain ke lateral depletion widths ke sum ke comparable ho, roughly

L ≲ x_{d S} + x_{d D}

(woh point jahan dono depletion regions milti hain).

V_{T}

chhote

L

ke saath kyun roll off (decrease) hota hai?

Source/drain junction depletion regions bulk charge ka kuch hissa support karti hain, isliye gate kam charge

Q_{B}^{'}

support karta hai aur kam voltage chahiye;

Δ V_{T} \propto 1/ L

DIBL kya hai?

Drain-Induced Barrier Lowering: drain field source-channel barrier ko neeche karta hai, isliye zyada

V_{D S}

effectively

V_{T}

lower karta hai aur leakage badhata hai.

V_{T} = V_{T} (0) - η V_{D S}

Velocity saturation saturation current law ke saath kya karta hai?

I_{D, s a t}

\propto (V_{GS} - V_{T})^{2}

\propto (V_{GS} - V_{T})

mein change kar deta hai, kyunki

v \to v_{s a t} = μ E_{cr i t}

Vertical-field mobility degradation kis wajah se hoti hai?

High

V_{GS}

carriers ko rough Si–SiO₂ interface ke against press karta hai, surface-roughness/remote-coulomb scattering add karta hai;

μ_{e f f} = μ_{0} / (1 + θ (V_{GS} - V_{T}))

overdrive badhne par girta hai.

Channel-length modulation kya hai aur yeh output resistance ko kaise affect karta hai?

Saturation pinch-off effective

L

Δ L

se shorten karta hai jab

V_{D S}

badhta hai;

I_{D} = I_{D, s a t} (1 + λ V_{D S})

, finite output resistance deta hai,

λ \propto 1/ L

Punch-through kya hota hai?

Jab drain aur source depletion regions merge ho jaayein, ek ungated current path banta hai; gate control kho deta hai aur switch fail ho jaata hai.

DIBL numerically off-state leakage ko kaise affect karta hai?

Δ

V_{T}

drop

I_{o f f}

1 0^{Δ/ S}

se badhata hai, jahaan

S

subthreshold slope hai.

Charge sharing se threshold roll-off ka formula kya hai?

Δ V_{T} = - \frac{Q _{B}}{C _{o x}} \frac{x _{j}}{L} (1 + 2 x_{d m} / x_{j} - 1)

, jahaan

x_{d m}

vertical gate depletion depth hai.

Dono alag mobility effects ke liye field ki kaunsi do directions zimmedaar hain?

Lateral field (

V_{D S} / L

) → velocity saturation; vertical field (

\propto (V_{GS} - V_{T}) / t_{o x}

) → surface-scattering mobility degradation.

Connections

MOSFET operation and regions
Threshold voltage and body effect
Subthreshold conduction and leakage
Velocity saturation and carrier transport
Surface scattering and effective mobility
Scaling theory (Dennard scaling)
Drain-Induced Barrier Lowering (DIBL)
Depletion region physics of pn junctions