We derive thermal generation without dumping a formula.
Step 1 — Occupation probability.
An electron needs energy ≥Eg to cross the gap. In thermal equilibrium the probability of a state at energy E above the reference being occupied is governed by Boltzmann statistics for E≫kT:
P(jump)∝e−Eg/(2kT)
Step 2 — Intrinsic carrier concentration.
Because generation is exponential in −Eg/2kT, the equilibrium number of thermally generated pairs is:
ni=NcNve−Eg/(2kT)
where Nc,Nv are the effective densities of states. Why ? In a pure material every freed electron (n) makes exactly one hole (p), so n=p=ni and np=ni2; taking the root of the product NcNve−Eg/kT gives ni.
Step 3 — Mass-action law (born as pairs).np=ni2
This holds even after doping: if you add electrons, holes must decrease to keep the product fixed — because generation always makes them in pairs and recombination destroys them in pairs.
What two things are produced together in generation? → a free electron and a hole.
Minimum energy to generate an EHP? → the band gap Eg.
Why exponent Eg/2kT in ni? → Fermi level is mid-gap; climb half the gap.
Cutoff wavelength condition? → hν≥Eg, so λ≤hc/Eg.
Mass-action law? → np=ni2.
Recall Feynman: explain to a 12-year-old
Think of a full theatre where everyone is sitting (electrons stuck in bonds). Nobody can move — that's an insulator. Now give the room some energy (heat/light): a person jumps up to the balcony (conduction band) and can now walk around. But their old seat is now empty. As people shuffle to fill that empty seat, the empty seat seems to slide across the row — that sliding empty seat is the hole. So one jump makes one walker (electron) + one empty seat (hole) — always a pair!
Dekho, ek pure semiconductor (jaise silicon) mein sabhi electrons apne covalent bonds mein "fase" hote hain — matlab freeze, koi current nahi. Ab jab hum usko energy dete hain (heat se, ya light photon se, ya strong electric field se), toh ek electron itni energy pakad leta hai ki wo bond tod ke valence band se conduction band mein jump maar deta hai. Ab wo free hai aur ghoom sakta hai. Lekin jab wo bhaaga, toh peeche ek khaali jagah chhod gaya — usi ko hum hole kehte hain. Toh yaad rakho: energy do → ek free electron + ek hole, hamesha jodi (pair) mein paida hote hain.
Hole ko positive charge maana jaata hai — kyunki wo missing electron ki jagah hai. Socho ek bhari parking lot jismein ek hi khaali space hai; jaise-jaise gaadiyan us space mein aati hain, wo khaali jagah ulti direction mein "chalti" dikhti hai. Bas wahi khaali jagah hamara hole hai — koi real particle nahi, ek convenient soch hai.
Formula wise, thermal generation exponential hota hai: ni=NcNve−Eg/2kT. Yahan 2 isliye aata hai kyunki intrinsic Fermi level gap ke beech mein hota hai, toh electron ko sirf aadha gap chadhna padta hai. Aur ek magic rule: np=ni2 — chahe doping karo, electrons aur holes ka product constant rehta hai. Temperature badhao toh ni tezi se badhta hai — isiliye pure semiconductor temperature ke saath pagal ho jaata hai, aur usable banane ke liye doping karni padti hai. Optical case mein, photon ki energy hν band gap Eg se badi honi chahiye, warna EHP banega hi nahi — isi wajah se silicon ki cutoff wavelength ~1107 nm hoti hai.