N-type doping with donor atoms (phosphorus, arsenic)

2.2.1 · Hardware › Doping & PN Junctions

Hum silicon ko dope kyun karte hain?

Dope kyun karein? Conductivity ko by design control karne ke liye. Hum deliberately impurity atoms add karte hain taaki exactly utne free carriers milein jitne chahiye. Doping hi transistors aur diodes ko possible banata hai.

Donor atom KIYA hota hai?

Yeh free electron kaise produce karta hai (first principles se):

Donor ek silicon site par baithta hai. Use sirf 4 electrons chahiye bond karne ke liye, isliye 5th unbonded rehta hai.
Yeh 5th electron weakly attract hota hai us +1 core ki taraf jo peeche reh jaata hai (donor ek electron worth of screening khota hai → net +1 charge). Yeh crystal ke andar ek mini hydrogen atom ki tarah orbit karta hai.
Hydrogen-model binding energy use karke hum dikhaa sakte hain ki ise free karne mein almost koi energy nahi lagti (derivation neeche), isliye room temperature par yeh already free hota hai.
Result: ek extra mobile electron per donor, aur ek fixed positive donor ion $N_{D}^{+}$ peeche rehta hai (yeh MOVE nahi karta).

Donor binding energy derive karna (Feynman-style, scratch se)

Derive kyun karein? Taaki aap believe karo ki electron almost free hai — sirf memorise na karo.

5th electron + donor core ko ek hydrogen atom ki tarah model karo, lekin silicon ke andar. Vacuum ke comparison mein do cheezein badal jaati hain:

Coulomb attraction silicon ki dielectric constant $ε_{r} \approx 11.7$ se screen hota hai, isliye $ε_{0} \to ε_{r} ε_{0}$ replace karo.
Electron lattice mein effective mass $m^{*} \approx 0.26 m_{0}$ ke saath move karta hai, free mass se nahi.

Vacuum mein Bohr ground-state energy hai $E_{H} = \frac{m _{0} e ^{4}}{8 ε _{0}^{2} h ^{2}} = 13.6 eV .$

Dono substitutions apply karne par, har $ε_{0} \to ε_{r} ε_{0}$ (squared appear hota hai) aur $m_{0} \to m^{*}$ :

$E_{D} = 13.6 eV \times \frac{m ^{*} / m _{0}}{ε _{r}^{2}}$

Interpret karo: room temperature par thermal energy $k_{B} T \approx 0.026 eV$ hoti hai. Kyunki $E_{D} \approx k_{B} T$ , practically har donor room temperature par ionised hota hai → har donor ek free electron deta hai. Yahi regime hum hamesha assume karte hain.

(Real measured values: P in Si ≈ 45 meV, As ≈ 54 meV — same order; hamaara hydrogen model ek-line estimate ke liye beautifully close hai.)

Band-diagram / energy picture

Figure — N-type doping with donor atoms (phosphorus, arsenic)

Carrier concentration — woh numbers jo aap actually use karte hain

Concentrations kaise milte hain (mass-action law se):

Generation & recombination ka detailed balance temperature-only mass-action law deta hai: $n p = n_{i}^{2} .$ Fully ionised donors ke saath charge neutrality ( $N_{D}^{+} \approx N_{D}$ , aur $N_{A} = 0$ ): $n = p + N_{D} .$

Kyunki typically $N_{D} ≫ n_{i}$ , hum paate hain $n \approx N_{D}$ , phir $p = n_{i}^{2} / n$ :

Common mistakes (steel-manned)

Active recall

Recall Khud test karo (answers hide hain)

Group-V atom electron kyun donate karta hai? → Use bonds ke liye sirf 4 chahiye; 5th spare aur weakly bound hai.
Electron jaane ke baad kya peeche rehta hai? → Ek fixed, immobile positive donor ion $N_{D}^{+}$ .
$E_{F}$ kis direction mein move karta hai? → Upar, $E_{C}$ ki taraf.
N-type mein majority carrier? → Electrons.
$p$ estimate karo agar $N_{D} = 1 0^{17}$ , $n_{i} = 1.5 \times 1 0^{10}$ ho. → $n_{i}^{2} / N_{D} = 2.25 \times 1 0^{3} cm^{- 3}$ .

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek badi dance party hai jahan har silicon person apne 4 neighbours ke saath haath pakde hua hai — kisi ke paas free haath nahi, toh koi move nahi kar sakta: koi "flow" nahi. Ab kuch phosphorus log sneakily andar aa jaate hain jinke paas 5 haath hain. Woh 4 haath sabki tarah pakad lete hain, lekin ek haath free waiving rehta hai. Woh free haath room mein wander off kar jaata hai aur messages (electricity) carry kar sakta hai. Humne room ko charged nahi kiya — har wandering haath ke liye, phosphorus person ke paas "+" sticker rehti hai jahan haath hua karta tha. Bahut saare wandering haath = bijli aasaani se flow karti hai. Yahi N-type silicon hai.

Donor atoms kis group se hain aur kitne valence electrons hote hain?

Group V, 5 valence electrons (e.g. P, As, Sb).

Donor free electron kyun produce karta hai?

Uske 5 valence electrons mein se sirf 4 bonds banate hain; 5th weakly bound hota hai aur room temperature par thermally free ho jaata hai.

Donor ionise hone ke baad kya peeche rehta hai?

Ek fixed, immobile positive donor ion

N_{D}^{+}

(crystal overall neutral rehta hai).

Hydrogen model se donor ionisation energy ka formula?

E_{D} = 13.6 eV \cdot (m^{*} / m_{0}) / ε_{r}^{2} \approx 26

meV for Si.

Room temperature par almost har donor ionised kyun hota hai?

E_{D} \approx 26

meV

\approx k_{B} T

at 300 K, isliye thermal energy electron ko free karne ke liye kaafi hai.

Mass-action law batao.

n p = n_{i}^{2}

(fixed temperature par hold karta hai).

N-type ke liye charge-neutrality equation?

n = p + N_{D}

(with

N_{A} = 0

, donors fully ionised).

N-type mein

N_{D} ≫ n_{i}

hone par approximate n aur p?

n \approx N_{D}

p \approx n_{i}^{2} / N_{D}

N-type mein majority aur minority carriers?

Majority = electrons; minority = holes.

Kya N-type silicon electrically charged hota hai?

Nahi — overall neutral; mobile electrons fixed positive donor ions se balance hote hain.

Donor doping ke saath Fermi level kis taraf shift karta hai?

Upar, conduction band

E_{C}

ki taraf.

N_{D} = 1 0^{16}

n_{i} = 1.5 \times 1 0^{10}

cm⁻³ ke liye p calculate karo.

p = n_{i}^{2} / N_{D} = 2.25 \times 1 0^{4}

cm⁻³.

Connections

Intrinsic vs Extrinsic Semiconductors
P-type doping with acceptor atoms (boron)
Mass-action law and carrier concentrations
Fermi level and its shift with doping
PN Junction formation
Conductivity and carrier mobility