Donor - acceptor energy levels in the band gap

Dekho, ek pure semiconductor jaise silicon mein ek band gap hota hai — yeh ek "forbidden zone" hai jahan koi electron state nahi hoti. Iska matlab electrons wahan reh hi nahi sakte. Ab jab hum doping karte hain (jaise phosphorus daalte hain n-type ke liye, ya boron p-type ke liye), tab hum gap ko bharte nahi hain — balki har dopant atom ek chhota localized energy level banata hai jo band edge ke bilkul paas hota hai. Donor level $E_C$ ke thoda neeche baithta hai, aur acceptor level $E_V$ ke thoda upar.

Yeh "thoda" kitna hai? Sirf ~45 meV (Si mein). Aur room temperature par thermal energy $kT \approx 26$ meV hoti hai — matlab garmi ka nudge kaafi hai in carriers ko free karne ke liye. Isiliye room temperature par almost saare shallow dopants ionize ho jaate hain (complete ionization). Yahi wajah hai ki n-type material mein bahut saare free electrons aur p-type mein bahut saare holes mil jaate hain.

Ionization energy itni chhoti kyun hai? Kyunki extra electron ek "hydrogen atom" ki tarah behave karta hai, lekin crystal ke andar. Yahan Coulomb pull dielectric constant $\varepsilon_r$ se screen ho jaata hai aur electron ka effective mass $m^*$ kam hota hai. Bohr formula mein yeh dono substitute karo to binding energy $13.6\,\text{eV}\times (m^*/m_0)/\varepsilon_r^2$ ban jaati hai — jo hydrogen se ~1000 guna chhoti hoti hai. Bas isiliye level itna shallow hai.

Ek important galti se bacho: donor level middle of gap mein nahi, edge ke paas hota hai. Aur $E_F$ (Fermi level) alag cheez hai — woh statistics se decide hota hai aur temperature/doping ke saath move karta hai, jabki $E_D$ fixed structural property hai. Yeh distinction exam aur real device design dono mein bahut kaam aata hai.