Intrinsic & Extrinsic Semiconductors: Doping and Carrier Concentration

Controlling carriers with impurities. Learn intrinsic (pure) semiconductors, thermal generation of electron–hole pairs, intrinsic carrier concentration n_i and its strong temperature dependence, the mass-action law np = n_i², donors (P, As, Sb) and acceptors (B), N-type and P-type material, majority and minority carriers, donor/acceptor ionisation energies, freeze-out, extrinsic and intrinsic temperature regions, compensation doping, heavy doping and degeneracy — with Python solving charge neutrality with incomplete ionisation from 50 K to 700 K.

Pure silicon at room temperature has only about 10¹⁰ free electrons per cm³ — far too few to be useful. Adding a tiny amount of a group-V element (donor) or group-III element (acceptor) sets the carrier concentration precisely, turning silicon into n-type or p-type material.

Because np = n_i² in equilibrium, raising one carrier type suppresses the other: an n-type region has abundant electrons and very few holes — the minority carriers that govern junction and bipolar behaviour.

An analogy

Intrinsic silicon is a quiet hall where a few people wander around. Doping with donors is like adding people who each bring an extra ticket (electron) to hand out; acceptors are people who each take one (creating holes). The product of 'ticket holders' and 'empty seats' stays fixed at a given temperature.

Intrinsic semiconductor

At absolute zero every bond is intact: no free carriers, the crystal insulates. As temperature rises, lattice vibrations break some bonds, creating equal numbers of electrons and holes (n = p = n_i). n_i roughly doubles every 8–10 °C in silicon, which is why leakage currents grow so quickly with temperature. The intrinsic Fermi level lies near mid-gap.

N-type and P-type

Doping sets majority carriers:

Temperature regions

Freeze-out (low T): carriers remain bound to dopants. Extrinsic (normal range): all dopants ionised, n ≈ N_D, nearly temperature-independent. Intrinsic (high T): n_i exceeds the doping, carrier density rises exponentially and junctions stop working — the upper temperature limit of silicon devices (~150–200 °C practically, set by leakage), much higher for wide-gap SiC and GaN.

Compensation, heavy doping and degeneracy

Adding both donors and acceptors compensates: the net doping N_D − N_A determines carriers, but ionised-impurity scattering depends on the total, lowering mobility. Above ~10¹⁸–10¹⁹ cm⁻³ dopant levels merge into a band, the gap narrows (bandgap narrowing) and E_F enters the band (degenerate): Boltzmann statistics fail and contacts become ohmic by tunnelling.

Formulas

Worked example: Minority carriers

Given: P-type Si, N_A 10¹⁷, n_i 10¹⁰

Find: n₀.

Answer: 10³ cm⁻³

Worked example: Compensation

Given: N_D 5 × 10¹⁶, N_A 2 × 10¹⁶

Find: Type and majority density.

Answer: N-type, n ≈ 3 × 10¹⁶ cm⁻³

Worked example: Near-intrinsic

Given: N_D = n_i = 10¹⁰

Find: n₀.

Answer: 1.62 × 10¹⁰ cm⁻³

Donor and acceptor doping

  N-type (phosphorus, 5 valence e⁻)      P-type (boron, 3 valence e⁻)
   Si─Si─Si                                Si─Si─Si
   │  │  │    extra e⁻ weakly bound         │  │  │   missing bond = hole
   Si─P⁺─Si  ← (E_d ≈ 45 meV below E_c)    Si─B⁻─Si ← (E_a ≈ 45 meV above E_v)
   │  │  │    ionised at room temperature   │  │  │
  Majority: electrons, minority: holes     Majority: holes, minority: electrons

The fifth electron of phosphorus is bound by only ~45 meV, so thermal energy (kT = 26 meV at 300 K, with many phonons) frees nearly all of them. Boron's missing electron is filled by a neighbour, creating a mobile hole. The ionised dopant atoms stay fixed in the lattice as charged ions — they matter later in depletion regions.

Carrier concentration vs temperature

import math
# Carrier concentration vs temperature for N-type Si (Nd = 1e15): freeze-out, extrinsic and intrinsic regions
k = 8.617e-5
def eg(T): return 1.170 - 4.73e-4 * T * T / (T + 636)
def ni(T):
    Nc = 2.8e19 * (T / 300) ** 1.5
    Nv = 1.04e19 * (T / 300) ** 1.5
    return math.sqrt(Nc * Nv) * math.exp(-eg(T) / (2 * k * T))
def n_of(T, Nd=1e15, Ed=0.045):
    Nc = 2.8e19 * (T / 300) ** 1.5
    lo, hi = 1.0, 1e20                            # solve charge neutrality n = p + Nd+ by bisection (log space)
    for _ in range(200):
        n = math.sqrt(lo * hi)
        EF_minus_Ec = k * T * math.log(n / Nc)     # Boltzmann
        Nd_plus = Nd / (1 + 2 * math.exp((EF_minus_Ec + Ed) / (k * T)))
        p = ni(T) ** 2 / n
        if n > p + Nd_plus: hi = n
        else: lo = n
    return n, Nd_plus / Nd
for T in (50, 100, 300, 500, 700):
    n, frac = n_of(T)
    print(f"T = {T:3d} K: ni = {ni(T):8.2e}, n = {n:8.2e} cm^-3, donors ionised {frac*100:5.1f}%")
print(f"300 K minority holes: p = ni^2/n = {ni(300)**2/1e15:.2e} cm^-3")
print(f"compensated Si (Nd 1e16, Na 4e15): n ~ Nd - Na = {1e16-4e15:.1e} cm^-3")
# T =  50 K: ni = 1.53e-41, n = 1.53e+14 cm^-3, donors ionised  15.3%
# T = 100 K: ni = 1.57e-11, n = 9.39e+14 cm^-3, donors ionised  93.9%
# T = 300 K: ni = 6.11e+09, n = 1.00e+15 cm^-3, donors ionised 100.0%
# T = 500 K: ni = 1.56e+14, n = 1.02e+15 cm^-3, donors ionised 100.0%
# T = 700 K: ni = 1.57e+16, n = 1.62e+16 cm^-3, donors ionised  99.9%
# 300 K minority holes: p = ni^2/n = 3.74e+04 cm^-3
# compensated Si (Nd 1e16, Na 4e15): n ~ Nd - Na = 6.0e+15 cm^-3

At 50 K most donors are still bound (freeze-out, 15% ionised). From ~100 K to ~500 K the electron concentration equals N_D (extrinsic region) — the useful operating range. By 700 K, n_i exceeds N_D and the material behaves intrinsically: devices lose control. Note: simple parameter sets give n_i(300 K) of 6–9 × 10⁹ cm⁻³; the widely used modern value is 9.65 × 10⁹ (older texts use 1.5 × 10¹⁰).

Common mistakes

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Key takeaways

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