Minority vs majority carriers

2.1.12 · Hardware › Band Theory & Carrier Physics

HUM baat kar kya rahe hain?

Doping	Majority	Minority
n-type (donors, e.g. P in Si)	electrons $n$	holes $p$
p-type (acceptors, e.g. B in Si)	holes $p$	electrons $n$

HUM unhe count kaise karte hain? (Derivation from scratch)

Humein do facts chahiye, dono first principles se derivable hain.

Intrinsic (undoped) crystal se shuru karo jahan $n = p = n_{i}$ . Balance deta hai: $G = r n_{i}^{2}$ Usi temperature par ek doped crystal ke liye, $G$ unchanged hai, isliye: $r n p = r n_{i}^{2} \Rightarrow n p = n_{i}^{2}$

Yeh step kyun? Humne $r$ cancel kiya (same material, same $T$ ) aur use kiya ki $G$ sirf temperature se fix hoti hai. Yahi poora trick hai.

Fact 2: Charge neutrality

Positive: holes $p$ + ionized donors $N_{D}^{+}$ . Negative: electrons $n$ + ionized acceptors $N_{A}^{-}$ . $p + N_{D}^{+} = n + N_{A}^{-}$

Room temperature par full ionization assume karo ( $N_{D}^{+} \approx N_{D}$ , $N_{A}^{-} \approx N_{A}$ ).

Combine karein → carrier concentrations

Case n-type ( $N_{D} ≫ N_{A}$ , lo $N_{A} = 0$ ): neutrality deta hai $p + N_{D} = n$ . Kyunki aam taur par $N_{D} ≫ n_{i}$ , electrons $\approx$ donors: $n_{n} \approx N_{D} p_{n} = \frac{n _{i}^{2}}{N _{D}}$

Yeh step kyun? Agar $N_{D} ≫ n_{i}$ , toh chota $p$ neutrality mein negligible hai, isliye $n \approx N_{D}$ . Phir mass action force karta hai $p = n_{i}^{2} / n = n_{i}^{2} / N_{D}$ .

Case p-type ( $N_{A} ≫ N_{D}$ ): $p_{p} \approx N_{A} n_{p} = \frac{n _{i}^{2}}{N _{A}}$

Dhyan do: majority ko upar doping karna minority ko neeche push karta hai (inverse relationship). Same product $n_{i}^{2}$ .

Worked examples

Common mistakes

The 80/20 core

Flashcards

N-type Si mein majority aur minority carrier kaun sa hai?

Majority = electrons; minority = holes.

Law of mass action state karo aur batao kab hold karta hai.

n p = n_{i}^{2}

; sirf thermal equilibrium mein fixed temperature par.

Minority concentration kabhi zero kyun nahi hoti?

Electron–hole pairs ki thermal generation kabhi nahi rukti, isliye ek choti equilibrium population hamesha exist karti hai.

N-type mein donor density

N_{D}

ke saath minority holes ka formula.

p_{n} = n_{i}^{2} / N_{D}

(assuming

N_{D} ≫ n_{i}

Agar doping badhao, toh majority aur minority concentrations ka kya hoga?

Majority badhti hai (

\approx N_{D}

); minority girti hai (

\propto 1/ N_{D}

); product

n_{i}^{2}

fixed rehta hai.

n \approx N_{D}

kab valid NAHI hai?

Jab

N_{D}

n_{i}

≫

nahi hai (light doping / small band gap); tab quadratic solve karo.

Compensated material (

N_{A}, N_{D}

dono present) mein majority carrier kya set karta hai?

NET dopant,

∣ N_{A} - N_{D} ∣

Neutrality + mass action se exact electron conc. (n-type).

n = N_{D} /2 + (N_{D} /2)^{2} + n_{i}^{2}

Diodes ke liye minority carriers kyun matter karte hain?

Junction current, depletion region ke across minority-carrier injection aur diffusion se set hoti hai.

Recall Feynman: 12-saal ke bacche ko explain karo

Ek bade stadium (n-type) ki blue fans ki huge crowd imagine karo. Wahan red fans ki bhi ek tiny handful hai jo wander karke aa gayi. Blue fans "majority" hain, red "minority" hain. Chahe tum aur aur blue fans pack karo, red sirf thode hi rehte hain — aur red-count times blue-count ka product ek fixed number hai jo is baat se set hota hai ki stadium kitna warm hai (temperature). Funny baat yeh hai: exit gates par (ek diode ka junction) game un thode se red fans se decide hota hai, na ki us enormous blue crowd se. Chote par mighty.

Connections

Intrinsic carrier concentration $n_i$ — fixed product set karta hai.
Doping — donors and acceptors — decide karta hai kaun sa carrier majority hai.
Law of mass action — yahan use ki gayi equilibrium constraint.
Charge neutrality condition — $n, p$ solve karne ke liye doosri equation.
pn junction diode — jahan minority injection current produce karta hai.
Diffusion and drift currents — minority carriers zyaatar diffusion se move karte hain.
Fermi level position vs doping — majority type $E_{F}$ ko ek band edge ke paas set karta hai.