Band Theory: Conductors, Insulators, Semiconductors

Band theory describes solids by merging atomic orbitals into continuous energy bands. Whether a solid conducts electricity depends on the gap between its filled valence band and its empty conduction band.

It matters because it explains in one picture why copper wires conduct, rubber insulates, and silicon, the heart of every chip, sits in between and can be tuned by doping.

Key Concepts

• Valence band: the band of filled energy levels formed from valence atomic orbitals; electrons here are bound to atoms.
• Conduction band: the next higher band, empty in insulators; electrons here move freely and carry current.
• Band gap (Eg): the energy gap between the bands. Conductors: zero gap (bands overlap). Semiconductors: small gap, about 0.1 to 3 eV. Insulators: large gap, above about 3 eV.
• Intrinsic semiconductor: pure silicon (Eg = 1.1 eV) or germanium (Eg = 0.66 eV); conductivity rises with temperature as heat lifts electrons across the gap.
• Extrinsic semiconductor: doped silicon. n-type (phosphorus doping) adds free electrons; p-type (boron doping) adds holes. Doping raises conductivity enormously.

Formulas

• Photon energy: E = hc / lambda. A photon can lift an electron across the gap only if its energy exceeds Eg.
• Convenient form: lambda (nm) = 1240 / E (eV). This converts a band gap directly into the longest wavelength of light the material absorbs.
• Silicon check: Eg = 1.1 eV gives lambda = 1240 / 1.1 = 1127 nm, in the infrared. Visible light (400-700 nm) has higher energy, so silicon absorbs all visible light and looks grey and shiny.
• Temperature rule: metal conductivity falls as temperature rises (vibrating atoms scatter electrons); semiconductor conductivity rises (more electrons jump the gap).

Worked Example

Question: What is the minimum photon energy needed to make silicon conduct, and what colour of light is it?

Step 1: Silicon's band gap is Eg = 1.1 eV. A photon needs at least this energy to promote an electron from the valence band to the conduction band.

Step 2: Convert to wavelength: lambda = 1240 / 1.1 = 1127 nm.

Step 3: 1127 nm is infrared, invisible to the eye. Any photon with a shorter wavelength (higher energy), including all visible light, can also do the job.

Step 4: This is why solar cells work in sunlight: visible photons easily clear silicon's 1.1 eV gap, creating the electron-hole pairs that become electric current.

Common Misconceptions

• "Semiconductors conduct because they are half metal." Pure silicon barely conducts at room temperature; useful conductivity comes from tiny, deliberate doping (about 1 dopant atom per million silicon atoms).
• "Heating always reduces conductivity." True for metals, false for semiconductors, whose conductivity rises with temperature as more electrons cross the gap.
• "Insulators never conduct." A large enough voltage forces breakdown: lightning crosses air's huge band gap, and diamond conducts under extreme fields.

Real-World Applications

• Every transistor, diode, and solar cell is band-gap engineering: silicon chips, GaAs (Eg = 1.43 eV) LEDs and lasers, and wide-gap materials for power electronics.
• Photodetectors and night-vision sensors use narrow-gap semiconductors (like Ge, 0.66 eV) to catch infrared photons.
• Copper and aluminium wiring exploit zero-gap bands; ceramic and polymer insulators exploit gaps above 3 eV.
• Thermistors use the steep temperature dependence of semiconductor conductivity to measure temperature precisely.

Band theory sorts solids by their band gap: zero for conductors, small for semiconductors, large for insulators. Doping and temperature tune semiconductors, and the photon relation lambda = 1240/E connects the gap to light. One diagram explains wires, chips, and solar cells.