Editing Electronic band structure (section)

=== Crystalline symmetry and wavevectors ===
[[File:Brillouin Zone (1st, FCC).svg|thumb|Fig 1. [[Brillouin zone]] of a [[face-centered cubic lattice]] showing labels for special symmetry points.]]
[[File:Bulkbandstructure.gif|thumb|300 px|Fig 2. Band structure plot for [[silicon|Si]], [[germanium|Ge]], [[gallium arsenide|GaAs]] and [[indium arsenide|InAs]] generated with tight binding model. Note that Si and Ge are indirect band gap materials, while GaAs and InAs are direct.]]

{{Main|Bloch's theorem|Brillouin zone}}
{{See also|Symmetry in physics|Crystallographic point group|Space group}}
Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single-electron [[Schrödinger equation]] is solved for an electron in a lattice-periodic potential, giving [[Bloch electron]]s as solutions
<math display="block">\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n\mathbf{k}}(\mathbf{r}),</math>
where {{math|'''k'''}} is called the wavevector. For each value of {{math|'''k'''}}, there are multiple solutions to the Schrödinger equation labelled by {{math|''n''}}, the band index, which simply numbers the energy bands.
Each of these energy levels evolves smoothly with changes in {{math|'''k'''}}, forming a smooth band of states. For each band we can define a function {{math|''E''<sub>''n''</sub>('''k''')}}, which is the [[dispersion relation]] for electrons in that band.

The wavevector takes on any value inside the [[Brillouin zone]], which is a polyhedron in wavevector ([[reciprocal lattice]]) space that is related to the crystal's lattice.
Wavevectors outside the Brillouin zone simply correspond to states that are physically identical to those states within the Brillouin zone.
Special high symmetry points/lines in the Brillouin zone are assigned labels like Γ, Δ, Λ, Σ (see Fig 1).

It is difficult to visualize the shape of a band as a function of wavevector, as it would require a plot in four-dimensional space, {{math|''E''}} vs. {{math|''k<sub>x</sub>''}}, {{math|''k<sub>y</sub>''}}, {{math|''k<sub>z</sub>''}}. In scientific literature it is common to see '''band structure plots''' which show the values of {{math|''E''<sub>''n''</sub>('''k''')}} for values of {{math|'''k'''}} along straight lines connecting symmetry points, often labelled Δ, Λ, Σ, or [[Miller index|[100], [111], and [110]]], respectively.<ref>{{Cite web | url=http://www.ioffe.ru/SVA/NSM/Semicond/AlGaAs/bandstr.html | title=NSM Archive - Aluminium Gallium Arsenide (AlGaAs) - Band structure and carrier concentration | website=www.ioffe.ru }}</ref><ref name="SpringerBandStructure">{{cite web|title=Electronic Band Structure|url=https://www.springer.com/cda/content/document/cda_downloaddocument/9783642007095-c1.pdf?SGWID=0-0-45-898341-p173918216 | website=www.springer.com | publisher=Springer | access-date=10 November 2016|page=24}}</ref> Another method for visualizing band structure is to plot a constant-energy [[isosurface]] in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to the [[Fermi level]] is known as the [[Fermi surface]].

Energy band gaps can be classified using the wavevectors of the states surrounding the band gap:
* [[Direct band gap]]: the lowest-energy state above the band gap has the same {{math|'''k'''}} as the highest-energy state beneath the band gap.
* [[Indirect band gap]]: the closest states above and beneath the band gap do not have the same {{math|'''k'''}} value.

==== Asymmetry: Band structures in non-crystalline solids ====

Although electronic band structures are usually associated with [[crystal]]line materials, [[quasi-crystal]]line and [[amorphous solid]]s may also exhibit band gaps. These are somewhat more difficult to study theoretically since they lack the simple symmetry of a crystal, and it is not usually possible to determine a precise dispersion relation. As a result, virtually all of the existing theoretical work on the electronic band structure of solids has focused on crystalline materials.