2.1.3
Intuition Ek-line intuitionKoi material conduct karega ya nahi, yeh depend karta hai kitna mushkil hai ek electron ko ek full band se empty band mein uthana . Woh "uthane ki height" hi band gap E g
Valence band (VB): sabse upar wala energy band jo (T = 0 bhara hota hai. Yeh electrons atoms se bound hote hain.Conduction band (CB): usse agla band, T = 0 khali hota hai. Yahaan ke electrons move karne ke liye free hain → current.Band gap E g VB ke top (E v E c E g = E c − E v
EK full band current kyun nahi carry karta? Ek bilkul full band mein, har electron jo right move karta hai, uske liye ek aur left move karta hai → net current = 0 empty states chahiye jismein electrons accelerate ho sakein. Toh conduction ke liye ya toh ek partly-full band chahiye, ya electrons gap ke paar promote hone chahiye aur holes chhodne chahiye.
Definition Classification
Class
Band gap E g
Room T
Conductor (metal) E g ≈ 0 Hamesha bahut saare free carriers
Semiconductor chhota, ∼ 0.1 3 eV = 1.12 = 0.67 = 1.42
Kam carriers, thermally excited
Insulator bada, ≳ 4 eV ≈ 5.5
Almost koi carriers nahi
Jo electrons gap ke paar pahunchte hain unki sankhya Boltzmann/Fermi factor follow karti hai. Chaliye ise scratch se build karte hain.
Sabse important relation yeh hai:
n i ∝ e − E g /2 k T E g exponent mein hai, ek modest gap change carriers mein bahut bada change laata hai. Yeh akela exponential poori conductor/semiconductor/insulator split explain karta hai.
Worked example Example 1 — Carriers ka ratio: Si vs diamondSi (E g = 1.12 E g = 5.5 T = 300 k T = 0.02585 n i
n i d ia n i S i = e − ( E g S i − E g d ia ) /2 k T = e ( 5.5 − 1.12 ) / ( 2 ⋅ 0.02585 ) Yeh step kyun? Same formula, ratio lo toh prefactors roughly cancel ho jaate hain.
= e 4.38/0.0517 = e 84.7 ≈ 1 0 37 Yeh step kyun? e 84.7 ln 10 = 2.303 84.7/2.303 ≈ 36.8
Matlab: Si mein diamond se ~1 0 37 E g
Worked example Example 2 — Si ko 300 K se 600 K tak heat karne ka effectFactor change = exp [ 2 k E g ( 300 1 − 600 1 ) ] Yeh step kyun? n i ∝ e − E g /2 k T E g /2 k = 1.12/ ( 2 ⋅ 8.617 × 1 0 − 5 ) = 6499 exp [ 6499 ( 1/300 − 1/600 )] = exp [ 6499 ⋅ 0.001667 ] = e 10.8 ≈ 5 × 1 0 4 Matlab: T 50 , 000 × zyada conductive ho jaate hain — metals ke ulta.
Worked example Example 3 — Metals kyun alag hainEk metal mein E g = 0 e − E g /2 k T = e 0 = 1 T Yeh step kyun? E g = 0 Matlab: metal conductivity fixed electron count se set hoti hai; resistance aslmein T badhti hai kyunki zyada lattice vibrations un electrons ko scatter karte hain — yeh mobility effect hai, carrier-count effect nahi.
Common mistake "Metals garam hone par better conduct karte hain, jaise semiconductors."Kyun sahi lagta hai: semiconductors clearly heat se improve karte hain, toh aap generalize kar lete ho.
Fix: Metals mein carrier count fixed hai; heating sirf phonon scattering badhaata hai → conductivity girती hai. Semiconductors mein heating exponentially zyada carriers create karta hai, jo scattering par heavy padta hai → conductivity badhti hai. Different dominant mechanism.
Common mistake "Gap factore − E g / k T Kyun sahi lagta hai: plain Boltzmann factor jaisa lagta hai.
Fix: Intrinsic carriers ke liye Fermi level mid-gap ke paas hota hai, toh har carrier average par sirf E g /2 − E g /2 k T
Common mistake "Insulator = infinite gap, semiconductor = finite gap — yeh fundamentally alag cheezein hain."Kyun sahi lagta hai: hum unhe alag label karte hain.
Fix: Yeh ek continuum hai. Diamond (5.5 1.12 same equation follow karte hain. Line is based on whether n i T
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho electrons wo bachche hain jo building ki ground floor par phanse hain, aur carts push karne mein help karne ke liye (electricity banana) unhein ek khali top floor par jump karna padta hai. Band gap us jump ki height hai.
Metal: floors touch karte hain — bachche freely cross karte hain, hamesha ready → great conductor.Semiconductor: chhota jump — kuch energetic bachche kar lete hain, garam hone par zyada kar lete hain (unke paas zyada energy hoti hai) → kabhi kabhi conduct karta hai.Insulator: bahut bada jump — basically koi nahi kar paata → koi electricity nahi.
Woh "jump height" ek exponent mein hone ki wajah se thoda sa uncha jump matlab WAY kam bachche upar.
Mnemonic Order yaad rakho"Metals Merge, Semis Squeeze, Insulators Isolate."
M erge: bands overlap karte hain, E g ≈ 0 S queeze: chhota gap, thermally thode electrons squeeze hokar paar jaate hain.I solate: gap itna bada ki electrons neeche isolated hain.
Aur: "Half in the exponent" → yeh E g / 2 k T
Band gap E g Valence band ke top aur conduction band ke bottom ke beech forbidden energy range, E g = E c − E v
Ek bilkul full band net current kyun carry nahi karta? Har electron jo ek taraf move karta hai uske liye ek doosri taraf move karta hai; accelerate karne ke liye koi empty states nahi → net current zero.
Conductor/metal ka typical band gap? ≈ 0
Semiconductor vs insulator ke liye typical band gap ranges? Semiconductor ∼ 0.1 3 ≳ 4
Intrinsic carrier count ko kaun si single relation govern karti hai? n i ∝ e − E g /2 k T
Exponent E g /2 k T E g / k T Intrinsic Fermi level mid-gap ke paas hota hai, toh har carrier average par ~E g /2
Semiconductor conductivity temperature ke saath kyun badhti hai lekin metal conductivity kyun girti hai? Semiconductor: exponentially zyada carriers dominate karte hain. Metal: fixed carriers, sirf extra phonon scattering → resistance upar.
Room temperature (300 K) par k T Lagbhag 0.0259 ≈ 26
Exponential factor kaun si starting statistics se milta hai? Fermi–Dirac distribution f ( E ) = 1/ ( 1 + e ( E − E F ) / k T ) E − E F ≫ k T
Electron promoted across gap
