Band Theory of Solids: Energy Bands, Gap and Conduction

When atoms join to form a solid, their sharp atomic energy levels smear into broad ranges called energy bands, each holding a huge number of closely spaced electron states. Electrons fill these bands from the bottom up, following the Pauli exclusion principle. The highest band containing electrons is the valence band; the next band up, usually empty, is the conduction band; and the forbidden energy range between them is the band gap, E_g.

This one picture explains why copper conducts, glass insulates and silicon can do both on demand. Every transistor, solar cell and LED works by moving electrons between bands or across the gap.

Key Concepts

• Energy band: a continuous range of allowed electron energies formed from overlapping atomic orbitals; each band holds 2N states for N atoms (spin up and down).
• Band gap E_g: the forbidden energy between valence and conduction bands. Metals: E_g = 0 (bands overlap); semiconductors: E_g ≈ 0.1–3 eV; insulators: E_g > 3 eV.
• Fermi level E_F: the energy of the highest filled state at absolute zero; in metals it sits inside a band, in semiconductors inside the gap.
• Direct vs indirect gap: in a direct gap (GaAs) an electron can fall across the gap emitting light; in an indirect gap (Si) it also needs a phonon, making light emission inefficient.
• Doping: donor atoms (phosphorus) make n-type silicon with extra electrons; acceptors (boron) make p-type with extra holes.

Formulas

• Intrinsic carrier concentration: n_i ∝ T^{3/2}·exp(−E_g / 2k_B·T). The exponential is why warming a semiconductor boosts its conductivity enormously. For silicon at 300 K, n_i ≈ 1.0×10¹⁰ cm⁻³.
• Conductivity: σ = n·e·μ_e + p·e·μ_h, with n, p the electron and hole concentrations (m⁻³), e = 1.602×10⁻¹⁹ C, and μ the mobilities (m²/V·s). In intrinsic material n = p = n_i.
• Key gap values at 300 K: Ge 0.66 eV, Si 1.12 eV, GaAs 1.42 eV, diamond 5.5 eV. These numbers fix each material's electronic personality.

Worked Example

Estimate the resistivity of pure silicon at 300 K. Take n_i = 1.0×10¹⁰ cm⁻³, μ_e = 1400 cm²/V·s, μ_h = 450 cm²/V·s.

1. Convert to SI: n_i = 1.0×10¹⁶ m⁻³, μ_e = 0.14 m²/V·s, μ_h = 0.045 m²/V·s.
2. σ = n_i·e·(μ_e + μ_h) = 1.0×10¹⁶ × 1.602×10⁻¹⁹ × (0.14 + 0.045).
3. σ = 1.602×10⁻³ × 0.185 = 2.96×10⁻⁴ S/m.
4. Resistivity ρ = 1/σ ≈ 3.4×10³ Ω·m, as measured for ultra-pure silicon.

Common Misconceptions

• "Electrons jump the gap easily." A 1 eV gap is ~40 times room-temperature thermal energy (0.026 eV), so few electrons cross it — silicon conducts poorly until doped.
• "Insulators have no bands." Insulators have bands too; their bands are just completely full or empty with an enormous gap, so no current can flow.
• "Doping adds lots of atoms." One dopant atom per million silicon atoms is heavy doping; tiny chemical changes cause gigantic electrical changes.

Real-World Applications

• Transistors and chips: p–n junctions and MOSFETs are band-gap engineering in action, switching current by bending bands with voltage.
• Solar cells: photons with energy above E_g kick electrons into the conduction band; silicon's 1.12 eV gap is near the ideal for sunlight.
• LEDs: direct-gap semiconductors like GaAs emit light when electrons fall across the gap; the gap size sets the colour.

Band theory turns overlapping atomic orbitals into allowed energy bands separated by a gap E_g. Zero gap means a metal, a ~1 eV gap means a semiconductor and a multi-eV gap means an insulator. Carrier statistics and doping then let engineers tune conductivity over ten orders of magnitude.