Subthreshold leakage current — Hinglish notes

2.4.17 · Hardware › Transistors: BJT & FET

WHAT hai yeh?

Threshold se neeche current kyun flow karta hai? Threshold voltage $V_{t h}$ ek convention hai, koi physical cliff nahi. Iske neeche bhi surface potential bands ko bend kar raha hota hai; electrons ka ek Boltzmann-small fraction itna thermal energy rakhta hai ki surface ko populate kar sake. Kyunki woh fraction $\propto e^{q ψ_{s} / k T}$ hota hai, aur $ψ_{s}$ $V_{GS}$ ke saath move karta hai, toh current $V_{GS}$ mein exponential hota hai.

HOW: scratch se derive karte hain

Threshold ke neeche channel BJT ke base ki tarah behave karta hai: carriers source se drain ki taraf diffuse karte hain. Electrons ke diffusion current se start karo:

$I_{s u b} = q A D_{n} \frac{d n}{d x} \approx q A D_{n} \frac{n ( 0 ) - n ( L )}{L}$

Yeh step kyun? Weak inversion → channel ke along negligible drift field → transport diffusion hai, bilkul BJT base mein minority carriers ki tarah.

Surface ke source end par carrier concentration surface potential $ψ_{s}$ se Boltzmann statistics ke zariye set hoti hai:

$n (0) \propto e^{q ψ_{s} / k T}$

Yeh step kyun? Thermal equilibrium mein band bending $ψ_{s}$ electron population ko ek Boltzmann factor se badhata hai — yahi exponential ka source hai.

Ab, $ψ_{s}$ gate par kaise depend karta hai? Gate voltage oxide capacitance $C_{o x}$ aur depletion capacitance $C_{d e p}$ ke beech divide hoti hai (ek capacitive divider):

$\frac{d ψ _{s}}{d V _{GS}} = \frac{C _{o x}}{C _{o x} + C _{d e p}} = \frac{1}{n}, n \equiv 1 + \frac{C _{d e p}}{C _{o x}}$

Yeh step kyun? $V_{GS}$ ka poora hissa surface tak nahi pahunchta — kuch depletion layer ke across drop hota hai. $n$ body-effect / subthreshold slope factor hai, typically $1.1$ – $1.5$ .

Combine karo: $n (0) \propto e^{q V_{GS} / (nk T)}$ . Sab milake aur drain-voltage dependence add karke (drain end carriers deplete karta hai, $n (L) \propto e^{- q V_{D S} / k T}$ ):

Subthreshold swing $S$

WHAT: kitne millivolts ka $V_{GS}$ hatana padta hai taaki $I_{s u b}$ 10 ke factor se cut ho jaye.

$I_{s u b}$ ka $lo g_{10}$ lo aur slope invert karo:

\;\Rightarrow\; S = \frac{dV_{GS}}{d(\log_{10}I_{sub})}$$ > [!formula] Subthreshold swing > $$\boxed{\,S = n\,V_T\ln 10 = \left(1+\frac{C_{dep}}{C_{ox}}\right)\frac{kT}{q}\ln 10\,}$$ > 300 K par, **ideal** ($n=1$) limit hai $S_{min}= V_T\ln 10 \approx 60\ \text{mV/decade}$. > [!intuition] 60 mV/dec ek wall kyun hai > $\ln 10$ aur $kT/q$ physics se fixed hain. Aap room temperature par ordinary MOSFET se 60 mV/dec ko beat *nahi* kar sakte — yahi "Boltzmann tyranny" hai jo limit karti hai ki $V_{DD}$ kitna neeche ja sakta hai. (Tunnel-FETs jaise exotic devices ise todne ki koshish karte hain.) ![[2.4.17-Subthreshold-leakage-current.png]] --- ## Worked examples > [!example] 1 — Swing compute karo > Ek transistor mein $C_{ox}=2C_{dep}$ hai, yaani $C_{dep}/C_{ox}=0.5$, at 300 K. > - $n = 1 + 0.5 = 1.5$. *Kyun?* Slope factor ki definition. > - $V_T = 25.9$ mV. *Kyun?* 300 K par $kT/q$. > - $S = 1.5 \times 25.9 \times 2.303 \approx 89.5$ mV/decade. > **Interpretation:** $V_{GS}$ ko ~90 mV drop karne se leakage 10× cut ho jaati hai. > [!example] 2 — Do bias points ke beech leakage ratio > Same device, $n=1.5$, $V_T=25.9$ mV. Leakage kitni badi hai jab gate $V_{th}$ se 120 mV *neeche* ho vs 300 mV neeche? > $$\frac{I(-120)}{I(-300)} = \exp\!\left(\frac{-0.120-(-0.300)}{1.5\times0.0259}\right)=\exp\!\left(\frac{0.180}{0.0389}\right)=e^{4.63}\approx 102.$$ > **Yeh step kyun?** Sirf $V_{GS}$ ka *difference* exponent mein jaata hai; $V_{th}$, $I_0$ cancel ho jaate hain. 180 mV deeper OFF-bias se leakage ~100× cut ho jaati hai. > [!example] 3 — Temperature leakage ko blow up karti hai > $S \propto T$: 300 K se 360 K jaane par $S$ factor $360/300=1.2$ se badhta hai, toh $S: 89.5\to107$ mV/dec. Aur bura yeh hai ki $I_0\propto V_T^2$ *aur* exponent mein $V_T$ dono badhte hain, toh leakage $T$ ke saath **super-linearly** badhti hai. Yahi reason hai ki hot chips zyada leak karti hain, aur zyada heat karti hain → thermal runaway ka risk. --- ## Common mistakes > [!mistake] "Threshold ke neeche current exactly zero hota hai." > **Kyun sahi lagta hai:** hume sikhaya jaata hai $I_D = \tfrac12\mu C_{ox}\tfrac{W}{L}(V_{GS}-V_{th})^2$, jo $V_{th}$ ke neeche 0 (ya imaginary) deta hai. > **Fix:** woh square-law model sirf *strong inversion* ke liye hai. $V_{th}$ ke neeche ek **alag, exponential** law govern karta hai. Transistor ek leaky switch hai, perfect nahi. Square-law wahan simply apply nahi hota. > [!mistake] "Subthreshold current $V_{DS}$ par strongly depend karta hai." > **Kyun sahi lagta hai:** saturation mein hum expect karte hain ki $I_D$ channel-length modulation ke zariye $V_{DS}$ se tied ho. > **Fix:** $I_{sub}\propto(1-e^{-V_{DS}/V_T})$ **saturate** ho jaata hai jab $V_{DS}\gtrsim 3V_T\approx 78$ mV. Dominant control knob $V_{GS}$ hai (exponentially), $V_{DS}$ *nahi*. > [!mistake] "Better fabrication se $S$ ko 60 mV/dec se neeche push kar sakte hain." > **Kyun sahi lagta hai:** better oxides $C_{ox}$ ko bada banate hain, $C_{dep}/C_{ox}\to 0$, $n\to 1$. > **Fix:** $n=1$ par bhi aap $S=V_T\ln10=60$ mV/dec tak pahunchte ho — jo $kT/q$ se set hota hai. Fabrication aapko 60 *tak* le jaata hai, neeche kabhi nahi (room T par). Ise todne ke liye ek naya transport mechanism chahiye (TFETs mein band-to-band tunneling). --- ## #flashcards/hardware Subthreshold conduction kis region mein hota hai? ::: Weak inversion mein, $V_{GS}<V_{th}$, channel sirf weakly inverted hoti hai. Subthreshold current kis transport mechanism se flow karta hai? ::: Diffusion se (BJT base ki tarah), drift se nahi. Subthreshold current formula likhо. ::: $I_{sub}=I_0\,e^{(V_{GS}-V_{th})/nV_T}\,(1-e^{-V_{DS}/V_T})$. Subthreshold swing $S$ define karo. ::: $S=nV_T\ln 10$; mV of $V_{GS}$ jo $I_{sub}$ ko 10× change kare. $S$ ka ideal room-temperature limit kya hai aur kyun? ::: 60 mV/decade, kyunki $n\ge1$ aur $S_{min}=(kT/q)\ln10$ (Boltzmann limit). Slope factor $n$ kya hai? ::: $n=1+C_{dep}/C_{ox}$, capacitive-divider factor; typically 1.1–1.5. Kis $V_{DS}$ ke upar $I_{sub}$ saturate hota hai? ::: 300 K par approximately $3V_T\approx78$ mV. Temperature ke saath leakage kyun badh jaati hai? ::: $S\propto T$ aur $V_T$ exponent mein + $I_0\propto V_T^2$ → super-linear rise, thermal runaway ka risk. 300 K par $V_T=kT/q$ kya hai? ::: ≈ 25.9 mV. Subthreshold current mein $V_{GS}$ ya $V_{DS}$ dominate karta hai? ::: $V_{GS}$, exponentially; $V_{DS}$ sirf ~78 mV tak, phir saturate ho jaata hai. --- > [!recall]- Feynman: ek 12-saal ke bacche ko samjhao > Socho ek paani ka tap jo tumne "off" kar diya, lekin phir bhi paani ki ek patli dhaar trickle karti rehti hai. Jitna zyada handle band ki taraf ghuma'o, *das-guna-chhoti* dhaar ho jaati hai har thodi extra ghoomahat ke liye — lekin poori band kabhi nahi hoti. Ek transistor ka "off" us leaky tap jaisa hi hai: usme hamesha ek tiny current trickle karta hai. Billions of taps ek chip par — woh saari tiny dhaarein milke ek real puddle of wasted battery ban jaati hain. Aur ek fundamental rule (is baat se ki heat electrons ko kaise jiggles karti hai) kehta hai ki har "10× chhoti dhaar" ke liye kam se kam 60 millivolts of handle-turning chahiye — room temperature par aap isse better nahi kar sakte. > [!mnemonic] Yaad rakho > **"Sixty is the floor, Volt-Thermal times ln-ten no more."** > $S=nV_T\ln10 \ge 60$ mV/dec. Aur **SUB = Slow, Under, Boltzmann**: **S**low exponential tail, **U**nder threshold, driven by **B**oltzmann statistics. ## Connections - [[MOSFET operating regions]] — subthreshold woh region hai jo square-law regime se *neeche* hai. - [[Threshold voltage Vth]] — exponential ka reference point. - [[BJT diffusion current]] — same diffusion physics; MOSFET-in-weak-inversion ≈ BJT. - [[Thermal voltage kT-q]] — 60 mV/dec Boltzmann limit set karta hai. - [[Static vs dynamic power dissipation]] — leakage main static-power culprit hai. - [[Body effect]] — same capacitive divider $n$ aur $V_{th}$ shift dono deta hai. - [[Short-channel effects / DIBL]] — effective $V_{th}$ ko lower karta hai, leakage badhata hai. ## 🖼️ Concept Map ```mermaid flowchart TD OFF[MOSFET OFF, VGS below Vth] -->|leaks| ISUB[Subthreshold leakage Isub] WEAK[Weak inversion] -->|carriers move by| DIFF[Diffusion transport] DIFF -->|analogous to| BJT[BJT base diffusion] DIFF -->|gives| DIFFEQ[Isub = qADn dn/dx] BOLTZ[Boltzmann surface carriers] -->|n0 ~ exp qPsi/kT| EXP[Exponential in VGS] CDIV[Capacitive divider Cox and Cdep] -->|defines| NFACTOR[Slope factor n] NFACTOR -->|scales exponent| FORMULA[Isub = I0 exp of VGS-Vth over nVT] EXP -->|leads to| FORMULA DIFFEQ -->|combined with Boltzmann| FORMULA FORMULA -->|VDS above 3VT| SAT[Saturates in VDS] FORMULA -->|yields| SWING[Subthreshold swing S] ISUB -->|wastes| POWER[Static power in billions of transistors] ```