Energy bands - valence band and conduction band

2.1.1 · Hardware › Band Theory & Carrier Physics

Bands bante kyun hain?

Figure — Energy bands - valence band and conduction band

Yeh metal vs insulator vs semiconductor kaise decide karta hai?

Formula Gap ke across thermal population (WHY the factor appears)

KYA: Ek electron ke paas itni thermal energy hone ki probability ki woh gap cross kar sake. KAISE (derivation sketch): Boltzmann statistics kehta hai ki reference se $Δ E$ upar ki state ek electron dwara thermal fluctuation se occupied hone ki probability $e^{- Δ E / k_{B} T}$ ke scale par hoti hai. Ek electron ko $E_{g}$ climb karna padta hai, isliye $n \propto e^{- E_{g} / (2 k_{B} T)}$ Factor of 2 KYUN? Ek free carrier banana matlab ek electron in CB AUR ek hole in VB — intrinsic carrier density $n_{i}$ gap ko energetically dono ke beech split kar deti hai. Poora result (density of states se derive kiya hua) yeh hai: $n_{i} = N_{c} N_{v} e^{- E_{g} / (2 k_{B} T)}$ jahan $N_{c}, N_{v}$ effective densities of states hain. Bada $E_{g}$ ⇒ exponentially kam carriers.

Worked examples

Common mistakes (Steel-man + fix)

Flashcards

What is the valence band?

Sabse upar ki energy band jo

T = 0

par (bonding) electrons se completely filled hoti hai.

What is the conduction band?

Sabse neeche ki band jo

T = 0

par khaali hoti hai; yahan ke electrons freely conduct kar sakte hain.

Define the band gap

E_{g}

E_{g} = E_{c} - E_{v}

, forbidden energy range jahan VB top aur CB bottom ke beech koi allowed electron states nahi hote.

Why do discrete atomic levels become bands in a solid?

Pauli exclusion

N

atoms ke overlapping identical orbitals ko

N

closely-spaced levels mein split hone par majboor karta hai ≈ ek continuous band.

Why can't a completely full band conduct?

Har rightward-moving electron ek leftward electron se cancel ho jaata hai; net current zero hai. Conduction ke liye empty adjacent states chahiye.

How does intrinsic carrier density depend on the gap and temperature?

n_{i} = N_{c} N_{v} e^{- E_{g} / (2 k_{B} T)}

; chhote

E_{g}

aur zyada

T

ke liye zyada carriers.

Why the factor of 2 in the exponent of

n_{i}

Ek carrier banana matlab CB mein ek electron AUR VB mein ek hole, gap energy dono species ke beech split ho jaati hai.

Metal vs insulator vs semiconductor by band picture?

Metal: overlapping/partly-filled bands (no effective gap). Insulator: bada gap (>~5 eV). Semiconductor: chhota gap (~0.5–2 eV).

E_{g}

of Si, Ge, GaAs at 300 K?

Si 1.12 eV, Ge 0.66 eV, GaAs 1.42 eV.

Photon absorption condition across the gap?

Tabhi absorb hota hai jab

E_{p h o t o n} = h c / λ \geq E_{g}

; usse neeche material transparent hota hai.

Recall Feynman: 12-saal ke bacche ko samjhao

Ek parking garage imagine karo. Neeche ka floor (valence band) gadiyon se bilkul bhara hua hai — koi hil nahi sakta, poora gridlock hai, koi traffic flow nahi. Upar ka floor (conduction band) bilkul khaali hai — gaadi chalane ki poori jagah hai. Dono ke beech ek aisa floor hai jahan na ramp hai na parking spots — wahi band gap hai; gaadiyaan wahan ruk nahi sakti. Traffic chalane ke liye ek gaadi ko poore upar khaali floor tak uthana padta hai. Agar jump chhota hai (semiconductor), thodi heat ya light gaadi ko upar uchhal sakti hai. Agar bahut bada hai (insulator), koi gaadi kabhi nahi pahunch sakti. Metal mein toh upar ka floor pehle se adha bhara aur khula hai — traffic hamesha freely chalta rehta hai.

Connections

Fermi level and Fermi-Dirac distribution — decide karta hai ki temperature $T$ par bands kaise filled hote hain.
Density of states — $n_{i}$ formula mein use hone wale $N_{c}$ , $N_{v}$ deta hai.
Intrinsic and extrinsic semiconductors — doping $E_{g}$ zyada change kiye bina carriers shift karta hai.
Holes as charge carriers — VB mein electrons ke peeche chhoot jaane wali empty states.
Direct vs indirect band gap — kyun GaAs light emit karta hai lekin Si nahi karta.
Kronig-Penney model — allowed/forbidden bands ka quantum-mechanical origin.