nLab electronic band structure




By Bloch-Floquet theory the available energies of electron-excitations in crystals depends smoothly on the momentum/wave number kk of the excitation, taking values in the Brillouin torus, and otherwise only on a discrete index set labelling the remaining “quantum numbers?” (the electron‘s orbitals?, spin. etc.). Fixing the latter values (and jointly denoting them n\mathbf{n}, say), the given function kE n(k)k \mapsto E_{\mathbf{n}}(k) (or rather its graph) is called the n\mathbf{n}th electronic band (or just n\mathbf{n}th band, for short) of the material.

Often, the values of E n()E_{\mathbf{n}}(-) are closely spaced as n\mathbf{n} varies in certain subsets of its allowed values. For instance if n\mathbf{n} includes the spin of the electrons and if there is a (typically small) spin-orbit coupling and no sizeable external magnetic field, then the energies E n,()E_{n,\uparrow}(-) and E n,E_{n, \downarrow} differ (only) slightly. In these cases the graphs of these values jointly look approximately like a single but thickened curve (as such shown in the following schematic graphics), which is where the name “band” originates from.

The band geometry around the electron chemical potential of a material controls its electrical conductivity:

(graphics from SS 22)
metal/conductorthe electron chemical potential is inside the valence band
insulatorthe electron chemical potential is inside a large gap between (what is then) the valence- and conduction-band
semi-conductorthe electron chemical potential is inside a small gap between valence and conduction band
semi-metalthere is a large gap between valence and conduction band, except over a codimension2\geq 2 locus, where the gap closes right at the chemical potential
(graphics from SS 22)



Textbook accounts:

Lecture notes:

Account with focus on topological phases of matter (topological insulators, semimetals etc.):

See also:

Discussion of bands of metamaterials over hyperbolic spaces by a hyperbolic variant of Bloch's theorem:


The electronic band structure of graphene was predicted in

For interacting electrons

  • Yuejin Guo, Jean-Marc Langlois and William A. Goddard , Electronic Structure and Valence-Bond Band Structure of Cuprate Superconducting Materials, New Series, 239 4842 (1988) 896-899 [jstor:1700316]

  • Jingsan Hu, Jianfei Gu, Weiyi Zhang, Bloch’s band structures of a pair of interacting electrons in simple one- and two-dimensional lattices, Physics Letters A 414 (2021) 127634 [doi:10.1016/j.physleta.2021.127634]

Last revised on June 15, 2024 at 19:16:55. See the history of this page for a list of all contributions to it.