Band theory

Band theory describes electrons in solids as occupying allowed energy ranges separated by forbidden gaps. The bands’ occupancy and structure help determine a solid’s electrical and optical properties.

Bloch's theorem: A theorem stating that wavefunctions in a periodic potential are plane waves multiplied by functions with the lattice periodicity. It describes the form of electron states in the periodic crystal potential that generates bands.
Metal: A material whose electrons can move readily through its solid structure and conduct electricity and heat. Partially filled bands or overlapping bands account for its mobile charge carriers.
Tight-binding model: An approximation that builds electronic states from localized atomic orbitals coupled between neighboring sites. It derives bands from atomic-like states, complementing nearly-free-electron descriptions.
Crystal lattice: A periodic arrangement of points representing the repeating structure of a crystalline solid. Its repeating potential shapes the electron states from which bands emerge.
Brillouin zone: A primitive cell in reciprocal space that represents the distinct wave vectors of a crystal lattice. Band energies are plotted across this reciprocal-space region to show their dispersion.
Semiconductor: A material whose electrical conductivity can be controlled by temperature, light, or added impurities. Its modest band gap enables conductivity to be tuned for electronic and optical devices.
Nearly-free electron model: An approximation that treats electrons as nearly free particles weakly perturbed by a crystal’s periodic potential. It explains how a weak lattice potential opens gaps at Brillouin-zone boundaries.
Quantum state: A mathematical description of the possible properties of a quantum system. Each allowed electron state contributes to the spectrum that forms solid-state bands.
Energy band: A continuous range of electron energies available in a solid, arising from the states of many interacting atoms. These allowed ranges are the central structures whose widths and positions determine electronic behavior.
Insulator: A material that strongly resists electrical current under ordinary conditions. A large gap between occupied and empty bands suppresses mobile charge carriers.

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