band structure

Band structure is the master map of a solid: it charts the allowed energies of an electron as a function of its crystal momentum. Isolated atoms have sharp, discrete energy levels; bring 10^23 of them together into a crystal and each level broadens into a continuous band of allowed energies, separated by forbidden gaps where no electron state can exist. Whether a material is a shiny metal, a transparent insulator, or a switchable semiconductor is written entirely in this pattern of bands and gaps.

Formally, solving the Schrodinger equation for an electron in the periodic potential of the lattice, using Bloch's theorem, yields the energies E_n(k), a set of functions labelled by a band index n over wavevectors k in the Brillouin zone. Two complementary pictures explain the bands: from the tight-binding side, atomic orbitals overlap and their discrete levels spread into bands as atoms are brought together; from the nearly-free-electron side, the free-electron parabola is folded back into the zone and small gaps open at zone boundaries where Bragg reflection mixes plane waves. Completely filled bands are valence bands and carry no net current; a partly filled band is a conduction band. A metal has a partly filled band (a Fermi surface); an insulator or semiconductor has filled valence bands separated by a gap from empty conduction bands.

Band structure is the single most organizing idea in solid-state physics, dictating conductivity, optical absorption, thermoelectric response, and the behaviour of every semiconductor device. Its honest limit is that it rests on the independent-electron (single-particle) approximation, in which each electron moves in an average potential. Strongly correlated materials, such as Mott insulators, are predicted by simple band theory to be metals but are in fact insulating because of electron-electron repulsion that band structure alone cannot capture.