Editing Electronic band structure (section)

=== Assumptions and limits of band structure theory ===
Band theory is only an approximation to the quantum state of a solid, which applies to solids consisting of many identical atoms or molecules bonded together. These are the assumptions necessary for band theory to be valid:
* ''Infinite-size system'': For the bands to be continuous, the piece of material must consist of a large number of atoms. Since a macroscopic piece of material contains on the order of 10<sup>22</sup> atoms, this is not a serious restriction; band theory even applies to microscopic-sized [[transistor]]s in [[integrated circuit]]s. With modifications, the concept of band structure can also be extended to systems which are only "large" along some dimensions, such as [[2DEG|two-dimensional electron systems]].
* ''Homogeneous system'': Band structure is an intrinsic property of a material, which assumes that the material is homogeneous. Practically, this means that the chemical makeup of the material must be uniform throughout the piece.
* ''Non-interactivity'': The band structure describes "single electron states". The existence of these states assumes that the electrons travel in a static potential without dynamically interacting with [[lattice vibration]]s, other electrons, [[photon]]s, etc.

The above assumptions are broken in a number of important practical situations, and the use of band structure requires one to keep a close check on the limitations of band theory:
* Inhomogeneities and interfaces: Near surfaces, junctions, and other inhomogeneities, the bulk band structure is disrupted. Not only are there local small-scale disruptions (e.g., [[surface states]] or [[dopant]] states inside the band gap), but also local charge imbalances. These charge imbalances have electrostatic effects that extend deeply into semiconductors, insulators, and the vacuum (see [[doping (semiconductor)|doping]], [[band bending]]).
* Along the same lines, most electronic effects ([[capacitance]], [[electrical conductance]], [[electric-field screening]]) involve the physics of electrons passing through surfaces and/or near interfaces. The full description of these effects, in a band structure picture, requires at least a rudimentary model of electron-electron interactions (see [[space charge]], [[band bending]]).
* Small systems: For systems which are small along every dimension (e.g., a small [[molecule]] or a [[quantum dot]]), there is no continuous band structure. The crossover between small and large dimensions is the realm of [[mesoscopic physics]].
* [[Strongly correlated material]]s (for example, [[Mott insulator]]s) simply cannot be understood in terms of single-electron states. The electronic band structures of these materials are poorly defined (or at least, not uniquely defined) and may not provide useful information about their physical state.