Editing Electronic band structure (section)

== Why bands and band gaps occur ==
[[File:Solid state electronic band structure.svg|thumb|upright=2.0|A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. The right graph shows the energy levels as a function of the spacing between atoms. When the atoms are far apart (right side of graph) the orbitals of each atom have the same energy, given by the atomic orbitals of carbon. When the atoms come close enough (left side) that the orbitals begin to overlap, they hybridize into molecular orbitals with different energies. Each atomic orbital splits into ''N'' molecular orbitals, where ''N'' is the number of atoms in the crystal.  Since there are so many atoms, the orbitals are very close in energy, and form continuous bands. The [[Pauli exclusion principle]] limits the number of electrons in a single orbital to two, and the bands are filled beginning with the lowest energy. At the actual diamond crystal cell size denoted by ''a'', two bands are formed, separated by a 5.5&nbsp;eV band gap.]]
[[File:Metals and insulators, quantum difference from band structure.ogv|thumb|upright=1.65|Animation of band formation and how electrons fill them in a metal and an insulator]]

The formation of electronic bands and band gaps can be illustrated with two complementary models for electrons in solids.<ref>{{Cite book |last=Simon |first=Steven H. |url=http://archive.org/details/oxfordsolidstate0000simo |title=The Oxford Solid State Basics |date=2013 |publisher= Oxford University Press |location=Oxford|isbn=978-0-19-150210-1}}</ref>{{rp|161}} The first one is the [[nearly free electron model]], in which the electrons are assumed to move almost freely within the material. In this model, the electronic states resemble [[Free_electron_model|free electron plane waves]], and are only slightly perturbed by the crystal lattice. This model explains the origin of the electronic dispersion relation, but the explanation for band gaps is subtle in this model.<ref name="girvin">{{Cite book |last1=Girvin |first1=Steven M. |title=Modern Condensed Matter Physics |last2=Yang |first2=Kun |date=2019 |publisher=Cambridge University Press |isbn=978-1-107-13739-4 |location=Cambridge}}</ref>{{rp|121}}

The second model starts from the opposite limit, in which the electrons are tightly bound to individual atoms. The electrons of a single, isolated atom occupy [[atomic orbital]]s with discrete [[energy level]]s. If two atoms come close enough so that their atomic orbitals overlap, the electrons can [[Quantum tunnelling|tunnel]] between the atoms. This tunneling splits ([[Orbital hybridisation|hybridizes]]) the atomic orbitals into [[molecular orbital]]s with different energies.<ref name="girvin" />{{rp|117-122}}

Similarly, if a large number {{math|''N''}} of identical atoms come together to form a solid, such as a [[crystal lattice]], the atoms' atomic orbitals overlap with the nearby orbitals.<ref name="Holgate">
{{cite book
 | last1  = Holgate
 | first1 = Sharon Ann
 | title  = Understanding Solid State Physics
 | publisher = CRC Press
 | date   = 2009
 | pages  = 177–178
 | url    = https://books.google.com/books?id=eefKBQAAQBAJ&pg=PA178
 | isbn   = 978-1-4200-1232-3
}}</ref> Each discrete energy level splits into {{math|''N''}} levels, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number ({{nowrap|{{math|''N''}} ≈ 10<sup>22</sup>}}), the number of orbitals that hybridize with each other is very large. For this reason, the adjacent levels are very closely spaced in energy (of the order of {{val|e=-22|u=eV}}),<ref name="Halliday">{{cite book
  | last1  = Halliday
  | first1 = David
  | last2  = Resnick
  | first2 = Robert 
  | last3  = Walker
  | first3 = Jearl 
  | title = Fundamentals of Physics, Extended, 10th Ed.
  | publisher = John Wiley and Sons
  | date = 2013
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  | pages = 1254
  | language = 
  | url = https://books.google.com/books?id=RReJCgAAQBAJ&dq=band+%22band+gap%22++%22pauli+exclusion+principle%22&pg=PA1254
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  | isbn = 9781118230619
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  | jfm =}}</ref><ref name="Cai">{{cite book
  | last1  = Cai
  | first1 = Wenshan
  | last2  = Shalaev
  | first2 = Vladimir
  | title = Optical Metamaterials: Fundamentals and Applications
  | publisher = Springer Science and Business Media
  | date = 2009
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  | pages = 12
  | language = 
  | url = https://books.google.com/books?id=q8gDF2pbKXsC&dq=22pauli%20exclusion%20principle%22&pg=PA12
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  | isbn = 9781441911513
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  | last1 = Ibach
  | first1 = Harald
  | last2  = Lüth
  | first2 = Hans 
  | title = Solid-State Physics: An Introduction to Principles of Materials Science, 4th Ed.
  | publisher = Springer Science and Business Media
  | date = 2009
  | location = 
  | pages = 2
  | language = 
  | url = https://books.google.com/books?id=qjxv68JFe3gC&q=%22pauli%20exclusion%20principle%22&pg=PA2
  | archive-url= 
  | archive-date=
  | doi = 
  | id = 
  | isbn = 9783540938040
  | mr = 
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  | jfm =}}</ref> and can be considered to form a continuum, an energy band.

This formation of bands is mostly a feature of the outermost electrons ([[valence electron]]s) in the atom, which are the ones involved in chemical bonding and [[electrical conductivity]]. The inner electron orbitals do not overlap to a significant degree, so their bands are very narrow.

[[Band gap]]s are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in the [[atomic orbital]]s from which they arise. Two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the bands associated with core orbitals (such as [[1s electron]]s) are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands. Higher bands involve comparatively larger orbitals with more overlap, becoming progressively wider at higher energies so that there are no band gaps at higher energies.