Editing Electronic band structure (section)

=== 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.