Silicon (group IV) has 4 valence electrons, all locked into covalent bonds at T=0 K. A locked electron cannot move → insulator behaviour at absolute zero.
At T>0, thermal vibration can supply enough energy Eg (the band gap, ≈ 1.12 eV for Si) to break a bond. This does two things at once:
A free electron appears in the conduction band.
The vacated bond is a hole that behaves like a positive mobile charge.
Because breaking a bond always makes one of each, in a pure crystal:
n=p=ni
WHY derive: so you never memorise the formula blindly.
The number of electrons in the conduction band is (electrons available to promote) × (probability of promotion). Statistical mechanics gives the promotion probability of crossing an energy gap as a Boltzmann-like factor. Carrying through the full density-of-states integral (which we won't redo here) yields:
n=Nce−(Ec−EF)/kT,p=Nve−(EF−Ev)/kT
where Nc,Nv are the "effective density of states" in each band.
Multiply them — why multiply? because the product kills the unknown Fermi level EF:
np=NcNve−(Ec−Ev)/kT=NcNve−Eg/kT
For intrinsic material n=p, so:
ni=NcNve−Eg/2kT
Replace a Si atom with a group-V atom (P, As, Sb). It forms 4 bonds and has 1 electron left over, bound only very weakly (≈ 0.05 eV). At room temperature that electron is essentially free.
Each donor gives one electron without creating a hole → n≫p.
Imagine a classroom where every kid is holding hands (bonds) — nobody can walk around, so no "current." Warm the room up and a few kids let go and wander (free electrons), leaving empty hand-spots that also seem to move around (holes). That's a pure semiconductor: weak, few wanderers.
Now sneak in a kid with an extra hand who can't hold anyone with it — that spare hand's kid becomes a free wanderer easily (n-type donor). Or sneak in a kid missing a hand, creating a permanent empty spot others rush to fill (p-type acceptor). By choosing which special kids to add, we design exactly how the class flows.
Dekho, ek pure (shuddh) semiconductor jaise Silicon room temperature pe thoda-bahut hi current chalata hai, kyunki current chalane ke liye electrons ko band gap Eg ke paar "jump" karna padta hai, aur ye jump sirf thermal energy se hota hai. Jab ek electron bond todke free hota hai, toh peeche ek khaali jagah bhi banti hai jise hole kehte hain. Isliye pure material mein electrons aur holes barabar hote hain: n=p=ni. Isko intrinsic semiconductor bolte hain.
Ab magic ye hai — hum doping karte hain, yaani jaan-boojhke thodi si impurity milate hain. Agar group-V atom (jaise Phosphorus) daalein toh ek extra electron milta hai bina hole banaye — ye n-type ho gaya, majority carrier electrons. Agar group-III atom (jaise Boron) daalein toh ek hole milta hai — ye p-type, majority carrier holes. Yaad rakhna: doping se crystal charged nahi hota, kyunki dopant ke nucleus mein extra proton bhi hota hai jo balance kar deta hai.
Sabse important tool hai Law of Mass Action: np=ni2, jo har condition mein sach hai. Isse hum minority carrier nikaal sakte hain — jaise n-type mein p=ni2/ND. Iska matlab jitne zyada electrons, utne kam holes; balance hamesha ni2 pe fixed rehta hai.
Ye kyun important hai? Kyunki poori electronics — diode, transistor, chip — isi controlled doping pe khadi hai. Aur ek warning: high temperature pe ni tezi se badhta hai (e−Eg/2kT), toh agar ni doping ke barabar ho jaaye toh device "intrinsic" ban jaata hai aur apna designed behaviour kho deta hai. Isliye chips ki ek maximum operating temperature hoti hai.
