Editing Electronic band structure (section)

== Basic concepts ==

=== 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.

=== 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.

=== Density of states ===

{{Main|Density of states}}

The density of states function {{math|''g''(''E'')}} is defined as the number of electronic states per unit volume, per unit energy, for electron energies near {{math|''E''}}.

The density of states function is important for calculations of effects based on band theory.
In [[Fermi's golden rule|Fermi's Golden Rule]], a calculation for the rate of [[optical absorption]], it provides both the number of excitable electrons and the number of final states for an electron. It appears in calculations of [[electrical conductivity]] where it provides the number of mobile states, and in computing electron scattering rates where it provides the number of final states after scattering.{{Citation needed|date=October 2015}}

For energies inside a band gap, {{math|1=''g''(''E'') = 0}}.

=== Filling of bands ===

{{Main|Fermi level|Fermi–Dirac statistics}}
{{Band structure filling diagram}}

At [[thermodynamic equilibrium]], the likelihood of a state of energy {{math|''E''}} being filled with an electron is given by the [[Fermi–Dirac distribution]], a thermodynamic distribution that takes into account the [[Pauli exclusion principle]]:
<math display="block">f(E) = \frac{1}{1 + e^{{(E-\mu)}/{k_\text{B} T}}}</math>
where:
* {{math|''k''<sub>B</sub>''T''}} is the product of the [[Boltzmann constant]] and [[temperature]], and
* {{math|''µ''}} is the [[total chemical potential]] of electrons, or ''Fermi level'' (in [[semiconductor physics]], this quantity is more often denoted {{math|''E''<sub>F</sub>}}). The Fermi level of a solid is directly related to the voltage on that solid, as measured with a voltmeter. Conventionally, in band structure plots the Fermi level is taken to be the zero of energy (an arbitrary choice).

The density of electrons in the material is simply the integral of the Fermi–Dirac distribution times the density of states:
<math display="block">N/V = \int_{-\infty}^{\infty} g(E) f(E)\, dE</math>

Although there are an infinite number of bands and thus an infinite number of states, there are only a finite number of electrons to place in these bands.
The preferred value for the number of electrons is a consequence of electrostatics: even though the surface of a material can be charged, the internal bulk of a material prefers to be charge neutral.
The condition of charge neutrality means that {{math|''N''/''V''}} must match the density of protons in the material. For this to occur, the material electrostatically adjusts itself, shifting its band structure up or down in energy (thereby shifting {{math|''g''(''E'')}}), until it is at the correct equilibrium with respect to the Fermi level.

==== Names of bands near the Fermi level (conduction band, valence band) ====

A solid has an infinite number of allowed bands, just as an atom has infinitely many energy levels. However, most of the bands simply have too high energy, and are usually disregarded under ordinary circumstances.<ref>High-energy bands are important for [[electron diffraction]] physics, where the electrons can be injected into a material at high energies, see {{Cite journal | last1 = Stern | first1 = R. | last2 = Perry | first2 = J. | last3 = Boudreaux | first3 = D. | doi = 10.1103/RevModPhys.41.275 | title = Low-Energy Electron-Diffraction Dispersion Surfaces and Band Structure in Three-Dimensional Mixed Laue and Bragg Reflections | journal = Reviews of Modern Physics | volume = 41 | issue = 2 | pages = 275 | year = 1969 |bibcode = 1969RvMP...41..275S }}.</ref>
Conversely, there are very low energy bands associated with the core orbitals (such as [[1s electron]]s). These low-energy ''core band''s are also usually disregarded since they remain filled with electrons at all times, and are therefore inert.<ref>Low-energy bands are however important in the [[Auger effect]].</ref>
Likewise, materials have several band gaps throughout their band structure.

The most important bands and band gaps—those relevant for electronics and optoelectronics—are those with energies near the Fermi level.
The bands and band gaps near the Fermi level are given special names, depending on the material:
* In a [[semiconductor]] or [[Insulator (electricity)|band insulator]], the Fermi level is surrounded by a band gap, referred to as ''the'' band gap (to distinguish it from the other band gaps in the band structure). The closest band above the band gap is called ''the [[conduction band]]'', and the closest band beneath the band gap is called ''the [[valence band]]''. The name "valence band" was coined by analogy to chemistry, since in semiconductors (and insulators) the valence band is built out of the [[valence orbital]]s.
* In a metal or [[semimetal]], the Fermi level is inside of one or more allowed bands. In semimetals the bands are usually referred to as "conduction band" or "valence band" depending on whether the charge transport is more electron-like or hole-like, by analogy to semiconductors. In many metals, however, the bands are neither electron-like nor hole-like, and often just called "valence band" as they are made of valence orbitals.<ref>In copper, for example, the [[Effective mass (solid-state physics)|effective mass]] is a [[tensor]] and also changes sign depending on the wave vector, as can be seen in the [[De Haas–Van Alphen effect]]; see https://www.phys.ufl.edu/fermisurface/</ref> The band gaps in a metal's band structure are not important for low energy physics, since they are too far from the Fermi level.