Bandindizes

Bandindizes are integer labels used to distinguish the energy bands in the electronic band structure of a crystalline solid. In a periodic potential, the Bloch theorem yields energy levels E_n(k) that depend on the crystal momentum k; the index n enumerates the bands, typically starting with n = 1 for the lowest-energy valence band and increasing for higher valence and conduction bands. The collection of functions {E_n(k)} for all n and k constitutes the band structure of the material.

In theoretical and computational work, the band index accompanies the Bloch state |n, k>. At a fixed k, energies are ordered to define the band sequence, and as k varies across the Brillouin zone, individual band dispersions can approach each other and undergo crossings or avoided crossings. To maintain a physically meaningful description, researchers often track bands continuously along paths in k-space to assign a consistent n(k).

Bandindizes play a central role in predicting and understanding material properties. They determine which states participate in electronic transport and optical transitions, with selection rules commonly involving changes between bands (for example, transitions between a valence band and a conduction band). The separation between bands defines band gaps and the density of states. In topological and correlated materials, the occupied bands (up to a valence index) and unoccupied bands above them are used to define invariants and to describe low-energy physics.

Notes: spin-orbit coupling, crystal symmetries, and degeneracies can modify band ordering and degeneracy, affecting band indexing. In practice, band indices are obtained from calculations such as density functional theory or tight-binding models and are often complemented by experimental probes like angle-resolved photoemission spectroscopy (ARPES) to map the band structure.