Electron-hole pair generation — Hinglish notes

1.3.6 · Hardware › Materials & Atomic Structure

WHAT is happening

WHY does a hole move & carry positive charge

HOW to derive the generation rate (first principles)

Hum thermal generation ko bina formula dump kiye derive karte hain.

Step 1 — Occupation probability. Ek electron ko gap cross karne ke liye $\geq E_{g}$ energy chahiye. Thermal equilibrium mein, reference se $E$ energy ke upar kisi state ke occupied hone ki probability Boltzmann statistics se governed hoti hai jab $E ≫ k T$ ho: $P (jump) \propto e^{- E_{g} / (2 k T)}$

Step 2 — Intrinsic carrier concentration. Kyunki generation $- E_{g} /2 k T$ mein exponential hai, thermally generated pairs ki equilibrium number hai: $n_{i} = N_{c} N_{v} e^{- E_{g} / (2 k T)}$ jahan $N_{c}, N_{v}$ effective densities of states hain. kyun? Pure material mein har freed electron ( $n$ ) ek hole ( $p$ ) banata hai, toh $n = p = n_{i}$ aur $n p = n_{i}^{2}$ ; $N_{c} N_{v} e^{- E_{g} / k T}$ ke product ka root lene se $n_{i}$ milta hai.

Step 3 — Mass-action law (born as pairs). $n p = n_{i}^{2}$ Yeh doping ke baad bhi hold karta hai: agar aap electrons add karo, toh holes ko decrease karna hoga product fixed rakhne ke liye — kyunki generation hamesha unhe pairs mein banata hai aur recombination unhe pairs mein destroy karta hai.

Ways to generate EHPs

Worked example Three energy sources

Source	Condition	Result
Thermal	$k T$ vibrations	$n_{i} \propto e^{- E_{g} /2 k T}$ , $T$ ke saath tezi se badhta hai
Optical	photon energy $h ν \geq E_{g}$	1 photon → 1 EHP (photodiode, solar cell)
Impact/field	strong E-field	avalanche multiplication

Yeh step kyun? Har ek sirf wahi same $E_{g}$ supply karta hai — mechanism alag hai, result (ek electron + ek hole) identical hai.

Common mistakes

Recall

Recall Active recall (hide & answer)

Generation mein saath mein kya do cheezein produce hoti hain? → ek free electron aur ek hole.
EHP generate karne ke liye minimum energy? → band gap $E_{g}$ .
$n_{i}$ mein exponent $E_{g} /2 k T$ kyun? → Fermi level mid-gap mein hai; aadha gap climb karo.
Cutoff wavelength condition? → $h ν \geq E_{g}$ , toh $λ \leq h c / E_{g}$ .
Mass-action law? → $n p = n_{i}^{2}$ .

Recall Feynman: explain to a 12-year-old

Ek bhara hua theatre socho jahan sab log baithe hain (electrons bonds mein stuck hain). Koi move nahi kar sakta — yahi insulator hai. Ab room ko kuch energy do (heat/light): ek banda upar balcony mein jump karta hai (conduction band) aur ab wahan walk kar sakta hai. Lekin uski purani seat ab khaali hai. Jab log us khaali seat ko fill karne ke liye shuffle karte hain, toh khaali seat row mein slide karti hui lagti hai — woh sliding khaali seat hi hole hai. Toh ek jump se ek walker (electron) + ek khaali seat (hole) milti hai — hamesha ek pair!

Flashcards

What is an electron-hole pair?

Valence band se conduction band mein freed ek electron aur us khali state (hole) jo woh peeche chhod jaata hai — saath mein create hote hain.

Minimum energy to generate an EHP?

Band gap energy

E_{g}

(

\approx 1.12

eV for Si).

Why does a hole carry positive charge?

Yeh ek missing electron hai; aas-paas ke valence electrons isko fill karne ke liye shift hote hain, jiski wajah se vacancy ek

+ q

carrier ki tarah move karti dikhti hai.

Formula for intrinsic carrier concentration?

n_{i} = N_{c} N_{v} e^{- E_{g} /2 k T}

Why

E_{g} /2 k T

and not

E_{g} / k T

n_{i}

Intrinsic Fermi level mid-gap mein hota hai, toh carrier sirf aadha gap hi climb karta hai.

Mass-action law?

n p = n_{i}^{2}

(equilibrium mein doping ke baath bhi hold karta hai).

Cutoff wavelength for optical generation?

λ_{ma x} = h c / E_{g}

;

h ν \geq E_{g}

chahiye.

Longest wavelength that frees an electron in Si?

\approx 1107

nm (near-infrared).

Three ways to generate EHPs?

Thermal (heat), optical (photons

h ν \geq E_{g}

), high-field impact/avalanche.

Effect of raising temperature on

n_{i}

Exponentially badhta hai, kyunki

n_{i} \propto e^{- E_{g} /2 k T}

What is generation-recombination balance in steady state?

G = R

; equilibrium mein

r n_{i}^{2} = G_{t h}

Connections

Energy Bands & Band Gap
Intrinsic vs Extrinsic Semiconductors
Doping — Donors & Acceptors
Recombination & Carrier Lifetime
Fermi-Dirac Distribution & Fermi Level
Photodiodes & Solar Cells
Drift & Diffusion Currents