Carrier concentration equations (n, p, ni) — Hinglish notes

2.1.6 · Hardware › Band Theory & Carrier Physics

1. First principles se: hum integrate kya kar rahe hain?

Limits KYUN yahan hain? Electrons tabhi conduct karte hain jab wo $E_{c}$ ke upar baithe hon (conduction band). Holes $E_{v}$ ke neeche missing electrons hain (valence band). Isliye har integral sirf apni band ko cover karta hai.

2. Density of states $g_{c} (E)$ derive karna

Band edge ke paas electron ek free particle ki tarah behave karta hai jisme effective mass $m_{n}^{*}$ hoti hai: $E - E_{c} = \frac{ℏ ^{2} k ^{2}}{2 m _{n}^{*}}$

KYUN? Parabolic approximation: kisi band ka bottom hamesha ek parabola jaisa dikhta hai, aur curvature $\to$ effective mass.

Radius $k$ ke sphere mein $k$ -states count karo (spin factor 2, volume $V$ ): $N (k) = 2 \cdot \frac{V}{( 2 π ) ^{3}} \cdot \frac{4}{3} π k^{3}$

$k = 2 m_{n}^{*} (E - E_{c}) /ℏ$ use karke energy mein convert karo aur $g_{c} (E) = \frac{1}{V} \frac{d N}{d E}$ differentiate karo:

$g_{c} (E) = \frac{1}{2 π ^{2}} (\frac{2 m _{n}^{*}}{ℏ ^{2}})^{3/2} E - E_{c}$

3. Boltzmann approximation (KYUN yeh sab kuch simplify kar deta hai)

Ek non-degenerate semiconductor ke liye, $E_{F}$ gap mein deep baith a hai, isliye $E \geq E_{c}$ ke liye $E - E_{F} ≫ k T$ hota hai: $f (E) = \frac{1}{1 + e ^{(E - E_{F}) / k T}} \approx e^{- (E - E_{F}) / k T}$

4. Master result: $n$ aur $N_{c}$

$g_{c}$ aur Boltzmann $f$ ko integral mein daalo: $n = \int_{E_{c}}^{\infty} \frac{1}{2 π ^{2}} (\frac{2 m _{n}^{*}}{ℏ ^{2}})^{3/2} E - E_{c} e^{- (E - E_{F}) / k T} d E$

$x = (E - E_{c}) / k T$ substitute karo, $\int_{0}^{\infty} x e^{- x} d x = \frac{π}{2}$ use karo:

$n = N_{c} e^{- (E_{c} - E_{F}) / k T}, N_{c} = 2 (\frac{2 π m _{n}^{*} k T}{h ^{2}})^{3/2}$

Holes ke liye bilkul same argument se (using $1 - f \approx e^{- (E_{F} - E_{v}) / k T}$ ):

$p = N_{v} e^{- (E_{F} - E_{v}) / k T}, N_{v} = 2 (\frac{2 π m _{p}^{*} k T}{h ^{2}})^{3/2}$

Figure — Carrier concentration equations (n, p, ni)

5. Intrinsic concentration $n_{i}$ aur mass-action law

$n$ aur $p$ multiply karo — $E_{F}$ terms cancel ho jaate hain: $n p = N_{c} N_{v} e^{- (E_{c} - E_{v}) / k T} = N_{c} N_{v} e^{- E_{g} / k T}$

Yeh $E_{F}$ se independent hai → kisi bhi doping ke liye sach. Ek intrinsic crystal mein $n = p \equiv n_{i}$ , isliye:

$n_{i} = N_{c} N_{v} e^{- E_{g} /2 k T} \Rightarrow n p = n_{i}^{2}$

Intrinsic level $E_{i}$ ki location ( $n = p$ set karo): $E_{i} = \frac{E _{c} + E _{v}}{2} + \frac{k T}{2} ln \frac{N _{v}}{N _{c}}$ Isliye $E_{i}$ midgap ke paas baith ta hai, chhote $ln (N_{v} / N_{c})$ term se thoda shift hokar.

$E_{i}$ ke around $n, p$ rewrite karna (bahut handy form): $n = n_{i} e^{(E_{F} - E_{i}) / k T}, p = n_{i} e^{(E_{i} - E_{F}) / k T}$

6. Worked examples

7. Feynman + Mnemonic

Recall Ek 12-saal ke bachche ko explain karo (click to reveal)

Ek theater imagine karo. Balcony (conduction band) mein khaali seats hain; main floor (valence band) packed hai. Garmi kuch floor-logon ko itni energy deti hai ki wo balcony tak jump kar sakein — har koi ek khaali seat (ek "hole") neeche chhodta hai. $N_{c}$ hai kitni balcony seats actually hain, aur exponential $e^{- E_{g} /2 k T}$ hai kitni probability hai ki koi jump afford kar sake. Jitna bada jump ( $E_{g}$ ), utne kam jump kar paate hain; jitna garm kamra ( $T$ ), utne zyada. Aur magically, (upar waale log) × (neeche ki khaali seats) hamesha ek fixed number $n_{i}^{2}$ ke barabar hote hain.

8. Forecast-then-Verify checkpoints

Flashcards

n

define karne wala integral kya hai?

n = \int_{E_{c}}^{\infty} g_{c} (E) f (E) d E

— density of states × occupancy conduction band ke upar.

f (E) \to e^{- (E - E_{F}) / k T}

KYUN hota hai?

E - E_{F} ≫ k T

ke liye (non-degenerate), Fermi–Dirac mein "+1" negligible hai → Boltzmann tail.

N_{c}

ke terms mein

n

batao.

n = N_{c} e^{- (E_{c} - E_{F}) / k T}

N_{c}

physically kya hai?

Conduction edge par effective density of states,

N_{c} = 2 (2 π m_{n}^{*} k T / h^{2})^{3/2}

Mass-action law derive karo.

n p = N_{c} N_{v} e^{- (E_{c} - E_{v}) / k T} = n_{i}^{2}

;

E_{F}

cancel ho jaata hai, isliye kisi bhi doping ke liye hold karta hai.

n_{i}

ka formula.

n_{i} = N_{c} N_{v} e^{- E_{g} /2 k T}

n_{i}

mein

E_{g}

nahi

E_{g} /2

KYUN?

Kyunki

n_{i} = n p

, aur

n p \propto e^{- E_{g} / k T}

; square root exponent ko half kar deta hai.

E_{i}

ke around compact forms.

n = n_{i} e^{(E_{F} - E_{i}) / k T}

p = n_{i} e^{(E_{i} - E_{F}) / k T}

N-type mein

N_{D} ≫ n_{i}

ke saath

p

dhundho.

n \approx N_{D}

, phir

p = n_{i}^{2} / N_{D}

Kya doping

n_{i}

change karta hai?

Nahi —

n_{i}

sirf

N_{c}, N_{v}, E_{g}, T

par depend karta hai.

E_{i}

kahan baith ta hai?

Near midgap:

E_{i} = \frac{E _{c} + E _{v}}{2} + \frac{k T}{2} ln (N_{v} / N_{c})

g_{c} \propto E - E_{c}

KYUN?

Spheres par

k

-states count karna; energy

\propto k^{2}

density

\propto E - E_{c}

deta hai.

Connections

Fermi-Dirac distribution — occupancy factor $f (E)$ jo humne approximate kiya.
Density of states — $g_{c} (E) \propto E - E_{c}$ ki origin.
Effective mass — $N_{c}, N_{v}$ ke andar $m_{n}^{*}, m_{p}^{*}$ set karta hai.
Doping and charge neutrality — $n p = n_{i}^{2}$ ke saath combine hokar $n, p$ solve karta hai.
Fermi level position — $n$ vs $N_{c}$ se compute hota hai.
Intrinsic vs extrinsic semiconductors — regimes jahan ye equations apply hoti hain.
Band gap Eg — $n_{i}$ mein exponential control karta hai.