Direct vs indirect band gap materials — Hinglish notes

2.1.5 · Hardware › Band Theory & Carrier Physics

WHAT is a band gap in $E$ – $k$ space?

Figure — Direct vs indirect band gap materials

WHY does momentum matter? (Derive the selection rule)

Hum interband transition par constraint do conservation laws se derive karte hain.

Step 1 — Energy conservation. Electron conduction energy $E_{c} (k_{c})$ se valence energy $E_{v} (k_{v})$ mein girta hai: $E_{c} (k_{c}) - E_{v} (k_{v}) = ℏ ω_{photon} (\pm E_{phonon})$

Yeh step kyun? Emitted photon (aur koi bhi phonon) ko exactly woh energy carry karni hogi jo electron khota hai.

Step 2 — Crystal-momentum conservation. $ℏ k_{c} = ℏ k_{v} + q_{photon} (\pm q_{phonon})$

Yeh step kyun? Total crystal momentum transition mein conserve hota hai, bilkul linear momentum ki tarah.

Step 3 — Photon momentum estimate karo. Near-gap energy ke photon ki wavelength $λ \sim 1 μ m$ hoti hai: $q_{photon} = \frac{2 π}{λ} \sim \frac{2 π}{1 0 ^{- 6} m} \approx 6 \times 1 0^{6} m^{- 1}$

Brillouin zone ki width se compare karo: $k_{BZ} \sim \frac{π}{a} \sim \frac{π}{5 \times 1 0 ^{- 10} m} \approx 6 \times 1 0^{9} m^{- 1}$

Yeh step kyun? Humein dekhna hai ki $q_{photon}$ kitna bada hai zone ke across momentum change $∣ k_{c} - k_{v} ∣$ ke comparison mein.

Step 4 — The ratio. $\frac{q _{photon}}{k _{BZ}} \approx \frac{6 \times 1 0 ^{6}}{6 \times 1 0 ^{9}} \approx 1 0^{- 3} \approx 0$

Yeh step kyun? Kyunki photon ka momentum zone ko bridge karne ke liye ~1000× zyada chhota hai, hum conclude karte hain:

WHY the absorption edges differ

Kyunki indirect absorption weak hoti hai, silicon ko sunlight absorb karne ke liye hundreds of microns material chahiye, jabki direct-gap GaAs cell wahi light ~1 μm mein absorb kar leta hai.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

E

–

k

space mein direct band gap kya define karta hai?

Conduction band minimum aur valence band maximum

k

ki same value par occur karte hain.

Photon akela direct transition kyun cause kar sakta hai?

Near-gap photon ka momentum Brillouin-zone width ka ~

1 0^{- 3}

hota hai, isliye vertical (

k_{c} \approx k_{v}

) transition automatically crystal momentum conserve karta hai.

Indirect transition mein konsa extra particle involved hona chahiye, aur kyun?

Ek phonon, crystal-momentum difference

ℏΔ k

supply karne ke liye jo photon provide nahi kar sakta.

Do direct-gap aur do indirect-gap semiconductors batao.

Direct: GaAs, GaN (also InP, CdTe). Indirect: Si, Ge (also GaP).

Silicon ek poor light emitter kyun hai?

Indirect gap → radiative recombination ko phonon chahiye (three-body process), isliye yeh slow hoti hai aur zyada tar non-radiative (heat) hoti hai.

Direct gap ke liye absorption edge?

α \propto (h ν - E_{g})^{1/2}

Indirect gap ke liye absorption edge?

α \propto (h ν - E_{g} \pm E_{p h})^{2}

, phonon-absorption/emission branches mein split hota hai.

Band-gap magnitude aur type related hain?

Nahi — yeh independent hain. Wide gaps direct (GaN) ya indirect (AlAs) dono ho sakte hain.

GaAs ko light absorb karne ke liye sirf ~1 μm chahiye lekin Si ko hundreds of μm kyun?

Direct absorption strong hoti hai (large

α

); indirect (phonon-assisted) absorption weak hoti hai, isliye Si ko bahut lamba optical path chahiye.

Recall Feynman: explain to a 12-year-old

Socho ek electron ek aisa baccha hai jise ek top bunk (high energy) se bottom bunk (low energy) par jump karna hai. Lekin ek rule hai: bacche ko left-to-right same spot par land karna hai, sirf neeche nahi. Agar dono beds lined up hain (direct), baccha bas seedha girata hai aur ek note gaata hai (light emit karta hai) — easy! Agar bottom bed sideways shifted hai (indirect), baccha sirf gir nahi sakta; kisi ko use usi waqt sideways dhakka bhi dena padta hai. Woh dhakka ek "phonon" hai — poore bed frame ka hilna. Ek saath drop AND dhakka chahiye hona almost kabhi nahi hota, isliye indirect bacche heat ke roop mein silently girte hain instead of light gaane ke. Isliye hum LED "singers" direct materials jaise GaAs se banate hain, silicon se nahi.

Connections

Band Theory Basics — $E (k)$ aur Brillouin zones ki origin
Phonons and Lattice Vibrations — indirect transitions ke liye momentum source
Recombination Mechanisms — radiative vs. Shockley–Read–Hall
LEDs and Laser Diodes — direct-gap materials kyun use hote hain
Optical Absorption in Semiconductors — Tauc plots aur absorption edges
Silicon Solar Cells — indirect gap ke consequences