Semiconductors

Semiconductors

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A semiconductor is a crystalline solid whose electrical conductivity lies between a conductor and an insulator (≈10⁻⁵ to 10² S/m), tuned by doping and temperature. Its behaviour is explained by the energy band model: a filled valence band (VB), an empty conduction band (CB), and a forbidden energy gap E_g between them (≈1 eV for Si, ≈0.67 eV for Ge). At T = 0 K the VB is full and the CB is empty, so the crystal behaves as an insulator; thermal or optical excitation promotes electrons across E_g, leaving behind mobile holes.

Two classes exist: intrinsic (pure Si/Ge, equal electron and hole concentrations) and extrinsic — n-type doped with pentavalent P/As/Sb (donor level just below CB, electrons are majority carriers) and p-type doped with trivalent B/Al/Ga/In (acceptor level just above VB, holes are majority). The minimum photon energy for band-to-band excitation is E_g = hν, and the intrinsic conductivity is σ = n_i q (μ_e + μ_h). CUET tests these definitions, the Si vs Ge gap difference, and the identification of majority carriers — keep those three ideas crisp.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Energy bands and the band gap

When N atoms of a crystal form a solid, their atomic orbitals split into N closely spaced levels, producing continuous energy bands. The topmost filled band is the valence band; the next higher band, partially or fully empty at room temperature, is the conduction band; the energy separation is the forbidden gap E_g. Conductors have overlapping or partially filled bands (E_g ≈ 0 eV), insulators have E_g > 3 eV with no thermal excitation possible, and semiconductors sit in between with E_g ≈ 1 eV so that kT ≈ 0.026 eV is enough to excite a measurable number of carriers.

Intrinsic behaviour and mass-action law

In a pure (intrinsic) semiconductor, every electron excited to the CB leaves a hole in the VB, so n_e = n_h = n_i. The intrinsic carrier density follows n_i² = n_e · n_h, with n_i = A T^(3/2) exp(−E_g / 2kT). Because the exponential dominates, n_i — and therefore σ — rises sharply with temperature, the opposite of metallic behaviour. Optical excitation obeys E_g = hν, defining the long-wavelength cutoff λ_c = hc/E_g.

Doping: n-type and p-type

Replacing a Si atom with a pentavalent impurity (P, As, Sb) donates a weakly bound extra electron, creating a donor level ≈ 0.05 eV below the CB. At 300 K this electron is ionised into the CB, giving n_e ≈ N_D and holes as minority carriers (n_h = n_i² / n_e). The Fermi level shifts upward, toward the CB. Conversely, a trivalent dopant (B, Al, Ga, In) creates an acceptor level ≈ 0.05 eB above the VB, accepts an electron, and releases a mobile hole — p-type with n_h ≈ N_A, electrons as minority carriers, Fermi level near the VB.

p-n junction essentials

Joining p- and n-regions causes holes to diffuse into n-side and electrons into p-side, leaving behind ionised dopants that form a depletion region with a built-in barrier potential V_b = (kT/q) ln(N_A N_D / n_i²) (≈0.7 V for Si). At equilibrium there is no net current. Forward bias (p-side positive) lowers the barrier and produces a large current; reverse bias widens the depletion region and allows only a tiny reverse saturation current.

Material	E_g (eV)	Dopant example	Majority carrier
Intrinsic Si	1.12	—	e⁻ = h⁺
n-type Si	1.12	P, As, Sb	electron
p-type Si	1.12	B, Al, Ga, In	hole
Intrinsic Ge	0.67	—	e⁻ = h⁺

CUET question patterns

Expect 1–2 items per paper on identifying majority carriers from a doping statement, choosing the correct band-gap order (insulator > semiconductor > conductor), applying σ ∝ exp(−E_g/2kT) to predict conductivity changes, and the basic forward/reverse bias behaviour of a junction.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

The hole as a quasiparticle

A hole is not a real particle — it is the absence of an electron in an otherwise full valence band. When neighbouring electrons shift to fill the vacancy, the vacancy moves in the opposite direction, behaving as a positive charge +q with its own mobility μ_h (typically smaller than μ_e in Si). This quasiparticle picture is what allows holes to be treated as independent carriers in drift, diffusion, and continuity equations.

Why n_i depends on T

From n_i = A T^(3/2) exp(−E_g/2kT), the exponential factor dominates. A common CUET trap: “Does conductivity rise or fall with temperature?” In metals, σ falls because lattice scattering increases; in semiconductors, the exponential growth in n_i overwhelms the mobility decrease, so σ increases with T. Memorising the qualitative shape of ln σ vs 1/T (a straight line of slope −E_g/2k) is a reliable problem-solving tool.

Fermi level motion

In an intrinsic crystal the Fermi level sits near the middle of the gap. n-doping shifts it upward toward the conduction band; p-doping shifts it downward toward the valence band. The amount of shift is roughly ΔE_F = kT ln(N_D/n_i) for n-type. Under forward bias the two Fermi levels split into quasi-Fermi levels separated by qV — the deeper reason current flows.

Worked example

A Si sample (E_g = 1.12 eV) is doped with N_D = 10¹⁶ cm⁻³ phosphorus at 300 K, where n_i ≈ 1.5 × 10¹⁰ cm⁻³. Majority electron density n_e ≈ N_D = 10¹⁶ cm⁻³; minority hole density n_h = n_i² / n_e = (1.5 × 10¹⁰)² / 10¹⁶ = 2.25 × 10⁴ cm⁻³ — six orders smaller, illustrating why doped material is overwhelmingly extrinsic. With μ_e = 1350 cm²/V·s, σ_e = n_e q μ_e ≈ 10¹⁶ × 1.6 × 10⁻¹⁹ × 1350 ≈ 2.16 S/cm. The hole contribution (μ_h ≈ 480 cm²/V·s) is negligible at 10⁻⁸ S/cm.

Common mistakes

• Calling a hole an actual particle rather than a vacant state in the valence band.
• Reversing majority and minority carriers: n-type has electrons as majority, p-type has holes as majority.
• Assuming semiconductor conductivity decreases with temperature like a metal — it increases.
• Forgetting that doping adds a shallow level (≈0.05 eV from the band edge), not a mid-gap state.

Adjacent topics to link

• Photodiodes and solar cells use the E_g = hν cutoff: photons with hν < E_g pass through; hν > E_g generate electron–hole pairs.
• Zener and avalanche breakdown occur under strong reverse bias once the field is high enough to tear carriers across the widened depletion region.
• Bipolar junction transistors are two back-to-back p-n junctions sharing a thin base; the emitter–base junction is forward-biased and the collector–base junction reverse-biased.

Two practice prompts

1. A Ge sample (E_g = 0.67 eV) and a Si sample (E_g = 1.12 eV) are at the same temperature. Which has higher intrinsic conductivity, and by approximately what factor (use the exp(−E_g/2kT) ratio at 300 K)?
2. An LED emits at 620 nm. Estimate the band gap of the semiconductor in eV and state whether this corresponds better to GaAsP (red) or GaAs (infra-red, ~870 nm).

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