Mass action law (np = ni²) — Hinglish notes

2.1.7 · Hardware › Band Theory & Carrier Physics

KYA claim kiya ja raha hai?

Law yeh kehta hai: product $n p$ barabar hota hai $n_{i}^{2}$ ke, chahe doping kuch bhi ho, jab tak hum equilibrium mein hain aur doping non-degenerate hai.

Product kyun constant hona chahiye? (Feynman-level reason)

HOW to derive it from band statistics (from scratch)

Hum equilibrium band populations (Boltzmann / non-degenerate limit) use karte hain.

Step 1 — Electrons in conduction band. $n = N_{C} e^{- (E_{C} - E_{F}) / k T}$ Yeh step kyun? $N_{C}$ conduction band edge par effective density of states hai; exponential woh Boltzmann probability hai ki ek electron energy $E_{C}$ tak pahunche jab Fermi level $E_{F}$ ho.

Step 2 — Holes in valence band. $p = N_{V} e^{- (E_{F} - E_{V}) / k T}$ Yeh step kyun? Ek hole ek empty state hai; iska population exponentially girta hai jab $E_{F}$ $E_{V}$ ke upar hota hai. Same statistics, mirrored.

Step 3 — Multiply. Notice karo ki $E_{F}$ dono exponents mein opposite sign ke saath aata hai, isliye yeh cancel ho jaata hai: $n p = N_{C} N_{V} e^{- (E_{C} - E_{V}) / k T} = N_{C} N_{V} e^{- E_{g} / k T}$ Yeh step kyun? Yahi toh poora point hai: Fermi level (jise doping move karti hai) nikal jaata hai. Jo bachta hai woh sirf $N_{C}, N_{V}, E_{g}, T$ par depend karta hai — pure material constants.

Step 4 — $n_{i}$ identify karo. Intrinsic material mein $n = p \equiv n_{i}$ , isliye $n p = n_{i}^{2}$ . Therefore $n_{i}^{2} = N_{C} N_{V} e^{- E_{g} / k T} \Rightarrow n p = n_{i}^{2}$

Charge neutrality ke saath combine karna (practical payoff)

Ek doped sample mein actual $n$ aur $p$ nikalne ke liye tumhe do equations chahiye:

Mass action: $n p = n_{i}^{2}$
Charge neutrality: $n + N_{A}^{-} = p + N_{D}^{+}$ (fully ionized ke liye: $n + N_{A} = p + N_{D}$ )

n-type example ( $N_{D} ≫ N_{A}$ , aur $N_{D} ≫ n_{i}$ ): neutrality se milta hai $n \approx N_{D}$ , phir $p = \frac{n _{i}^{2}}{n} \approx \frac{n _{i}^{2}}{N _{D}}$

Common mistakes (Steel-manned)

Flashcards

Mass action law aur uski condition state karo

n p = n_{i}^{2}

, thermal equilibrium par valid (non-degenerate).

n p

product kyun hai, sum kyun nahi?

Recombination ek two-body process hai,

R \propto n p

; isko generation

G (T)

ke saath balance karne par product constant rehne ki constraint aati hai.

Band populations se

n p

derive karo —

E_{F}

kyun vanish ho jaata hai?

n \propto e^{(E_{F} - E_{C}) / k T}

aur

p \propto e^{(E_{V} - E_{F}) / k T}

; multiply karne par

E_{F}

cancel ho jaata hai, sirf

N_{C} N_{V} e^{- E_{g} / k T}

bachta hai.

n_{i}^{2}

ka full expression

n_{i}^{2} = N_{C} N_{V} e^{- E_{g} / k T}

n_{i}

ka expression (factor of 2 dhyaan se)

n_{i} = N_{C} N_{V} e^{- E_{g} /2 k T}

n-type silicon mein

N_{D} = 1 0^{16}

n_{i} = 1 0^{10}

p

nikalo

p = n_{i}^{2} / N_{D} = 1 0^{20} /1 0^{16} = 1 0^{4}

/cm³.

Ek doped sample mein

n, p

solve karne ke liye do equations

Mass action (

n p = n_{i}^{2}

) aur charge neutrality (

n + N_{A} = p + N_{D}

n

ka general solution (net doping

N = N_{D} - N_{A}

)

n = \frac{N}{2} + (N /2)^{2} + n_{i}^{2}

Illumination/injection ke under

n p

ka kya hota hai?

n p > n_{i}^{2}

;

n p = n_{i}^{2} e^{(F_{n} - F_{p}) / k T}

use karo quasi-Fermi levels ke saath.

n_{i}

itna temperature sensitive kyun hai?

e^{- E_{g} /2 k T}

factor isko

T

ke saath steeply badhata hai (roughly har kuch K mein double ho jaata hai).

Recall Feynman: ek 12-saal ke bacche ko explain karo

Ek dance floor (crystal) imagine karo. Heat dancers ko stage par kick karti rehti hai (electrons), aur floor par khaali jagah chhodti hai (holes). Stage par dancers wapas kood ke khaali jagah fill karte rehte hain. Jab kick karna aur wapas aana same rate par ho, toh bheed "balanced" hai. Magic rule yeh hai: (stage par jinki ginti) × (khaali jagahon ki ginti) hamesha uss dance floor ke liye us temperature par same number hota hai. Toh agar tum extra dancers stage par ghusa do, toh floor par khaali jagahein lagbhag saari bhar jaati hain — unki ginti itni girti hai ki product same raha.

Connections

Intrinsic carrier concentration $n_i$
Fermi level & Fermi–Dirac statistics
Effective density of states $N_C$, $N_V$
Charge neutrality condition
Doping: donors and acceptors
Quasi-Fermi levels & non-equilibrium carriers
Band gap $E_g$ and its temperature dependence