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{{Quantum book backlink|Condensed matter and solid-state physics}}

The '''electronic band structure''' (or simply '''band structure''') of a [[solid]] describes the range of [[energy level]]s that electrons may have within it, as well as the ranges of energy that they may not have, called ''[[band gap]]s'' or forbidden bands. Band theory explains these bands and gaps by examining the allowed quantum-mechanical [[wave function]]s for an electron in a large, periodic lattice of atoms or molecules. It provides the foundation for understanding the electrical and optical properties of solids and underlies the operation of [[Solid-state (electronics)|solid-state devices]] such as transistors and solar cells.<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><ref name="girvin">{{Cite book |last1=Girvin |first1=Steven M. |last2=Yang |first2=Kun |title=Modern Condensed Matter Physics |date=2019 |publisher=Cambridge University Press |isbn=978-1-107-13739-4 |location=Cambridge}}</ref>

<div style="float:right; background:#fff8cc; border:1px solid #e0d890; padding:8px; margin:0 0 1em 1em; width:420px; text-align:center;">
[[File:Solid state electronic band structure.svg|400px|A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.]]
<div style="font-size:90%; line-height:1.4; margin-top:6px;">A hypothetical example of band formation when a large number of carbon atoms is brought together to form a diamond crystal. As the atoms approach one another, atomic orbitals split into many closely spaced levels that merge into bands. At the diamond lattice spacing, two bands are separated by a band gap of about 5.5 eV.</div>
</div>
== Introduction ==
In an isolated atom, electrons occupy discrete [[atomic orbital]]s with specific energies. When many identical atoms are brought together in a crystal, these orbitals overlap and split into a very large number of closely spaced levels. Since a macroscopic solid contains an enormous number of atoms, the levels become so densely packed that they can be treated as continuous energy bands.<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>

This band formation is especially important for the outermost, or [[valence electron|valence]], electrons, which participate in chemical bonding and electrical conduction. Inner orbitals overlap much less strongly and therefore form very narrow bands. Between bands there may be ranges of energy that contain no allowed electron states; these are the [[band gap]]s.<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
| pages = 1254
| url = https://books.google.com/books?id=RReJCgAAQBAJ&pg=PA1254
| isbn = 9781118230619
}}</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
| pages = 12
| url = https://books.google.com/books?id=q8gDF2pbKXsC&pg=PA12
| isbn = 9781441911513
}}</ref><ref name="Ibach">{{cite book
| 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
| pages = 2
| url = https://books.google.com/books?id=qjxv68JFe3gC&pg=PA2
| isbn = 9783540938040
}}</ref>

== Why bands and band gaps occur ==
Two complementary pictures are often used to explain the origin of bands and band gaps.<ref name="girvin" />

In the [[nearly free electron model]], electrons are treated as moving almost freely through the crystal, with the periodic lattice acting as a weak perturbation. In this case the electron states resemble plane waves, and the periodic lattice gives rise to the characteristic dispersion relations and gaps at special points in reciprocal space.<ref name="girvin" />

In the opposite limit, the [[tight binding]] picture begins with electrons strongly bound to individual atoms. If neighbouring atomic orbitals overlap, electrons can [[Quantum tunnelling|tunnel]] between atoms, and the original atomic levels split into many different energies. For a crystal containing {{math|''N''}} atoms, each atomic level gives rise to {{math|''N''}} closely spaced levels, which together form a band.<ref name="Holgate" />

The widths of these bands depend on how strongly the orbitals overlap. Narrow overlap produces narrow bands, while strong overlap produces wider bands. Band gaps arise when neighbouring bands do not extend enough in energy to overlap. Core-electron bands are especially narrow, so they are often separated by large gaps, whereas higher-energy bands broaden and may overlap more strongly.<ref name="Halliday" />

== Basic concepts ==

=== Assumptions and limits of band theory ===
Band structure is an approximation that works best for solids made of many identical atoms or molecules arranged in a regular way. It generally assumes:
* a large or effectively infinite system, so that the allowed levels form nearly continuous bands;
* a chemically homogeneous material;
* electrons moving in an average static potential, without fully treating interactions with other electrons, phonons, or photons.

These assumptions can break down in important cases. Near surfaces, interfaces, and junctions, the bulk band structure is altered and effects such as [[surface states]], [[dopant]] states, and [[band bending]] become important. In very small systems such as molecules or [[quantum dot]]s, the concept of a continuous band structure no longer applies. Strongly correlated materials such as [[Mott insulator]]s also fall outside the normal single-electron band picture.<ref name="girvin" />

=== Crystalline symmetry and wavevectors ===
For a crystal, band structure calculations take advantage of lattice periodicity. The single-electron [[Schrödinger equation]] in a periodic potential has solutions of the Bloch form
<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 the wavevector and {{math|''n''}} labels the band. For each value of {{math|'''k'''}}, there are multiple allowed energies, and these vary smoothly with {{math|'''k'''}} to produce the dispersion relation {{math|''E''<sub>''n''</sub>('''k''')}}.<ref name=Kittel>{{cite book
|author=Charles Kittel
|title=[[Introduction to Solid State Physics (Kittel book)|Introduction to Solid State Physics]]
|year=1996
|edition=Seventh
|publisher=Wiley
|location=New York
|isbn=978-0-471-11181-8
}}</ref>

The relevant wavevectors lie inside the [[Brillouin zone]], the fundamental region of reciprocal space. Band structure plots typically show energy versus wavevector along lines connecting high-symmetry points such as Γ, X, L, or K. These diagrams make it possible to distinguish, for example, between [[direct band gap]] and [[indirect band gap]] materials.<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://web.archive.org/web/20180117001810/https://www.springer.com/cda/content/document/cda_downloaddocument/9783642007095-c1.pdf?SGWID=0-0-45-898341-p173918216|website=www.springer.com|publisher=Springer|page=24}}</ref>

[[File:Brillouin Zone (1st, FCC).svg|thumb|300px|[[Brillouin zone]] of a [[face-centered cubic lattice]], showing several special symmetry points.]]

=== Density of states ===
The [[density of states]] function {{math|''g''(''E'')}} gives the number of electronic states per unit volume and per unit energy near energy {{math|''E''}}. It is central in calculations of conductivity, optical absorption, and scattering processes. Inside a band gap, the density of states is zero:
<math display="block">g(E)=0.</math>

=== Filling of bands ===
At thermal equilibrium, the probability that a state of energy {{math|''E''}} is occupied is given by the [[Fermi–Dirac statistics|Fermi–Dirac distribution]]
<math display="block">f(E) = \frac{1}{1 + e^{(E-\mu)/(k_\text{B} T)}}</math>
where {{math|''\mu''}} is the chemical potential, usually called the [[Fermi level]], and {{math|''k''<sub>B</sub>''T''}} is the thermal energy. The number of electrons per unit volume is obtained by integrating the occupied density of states:
<math display="block">N/V = \int_{-\infty}^{\infty} g(E) f(E)\, dE</math>

In [[semiconductor]]s and insulators, the Fermi level lies in a band gap. The highest occupied band is called the [[valence band]], and the lowest unoccupied band above it is the [[conduction band]]. In metals, by contrast, the Fermi level lies within one or more allowed bands, so electrons can move easily and conduct electricity.<ref name="girvin" />

== Theoretical descriptions ==

=== Nearly free electron approximation ===
The [[nearly free electron model]] treats electrons as almost free, with only a weak periodic perturbation from the lattice. It is particularly useful for simple metals and can explain why gaps open at certain boundaries in reciprocal space. In some metals, such as [[aluminium]], the resulting bands are close to those of the [[empty lattice approximation]].<ref name=Kittel />

=== Tight binding model ===
The [[tight binding]] approach assumes that electrons remain largely localized around atoms and that crystal wavefunctions can be approximated as linear combinations of atomic orbitals. This model works especially well in systems where orbital overlap is limited, such as [[silicon]], [[gallium arsenide]], [[diamond]], and many insulators. A related and more refined formulation uses [[Wannier function]]s.<ref name=Kittel /><ref name=Mattis>{{cite book | author=Daniel Charles Mattis | title=The Many-Body Problem: Encyclopaedia of Exactly Solved Models in One Dimension | year = 1994 | publisher=World Scientific | page=340 | isbn=978-981-02-1476-0 | url=https://books.google.com/books?id=BGdHpCAMiLgC&pg=PA332}}</ref><ref name=Harrison>{{cite book
|author=Walter Ashley Harrison
|title=Electronic Structure and the Properties of Solids
|year=1989
|publisher=Dover Publications
|url=https://books.google.com/books?id=R2VqQgAACAAJ
|isbn=978-0-486-66021-9
}}</ref>

=== KKR model ===
The [[Korringa–Kohn–Rostoker method]] (KKR), also known as multiple scattering theory, reformulates the problem using scattering matrices and Green's functions. It is particularly useful for alloys and disordered systems, and often uses the ''muffin-tin'' approximation for the crystal potential.<ref name=Galsin>{{cite book |title=Impurity Scattering in Metal Alloys |author=Joginder Singh Galsin |page=Appendix C |url=https://books.google.com/books?id=kmcLT63iX_EC&pg=PA498 |isbn=978-0-306-46574-1 |year=2001 |publisher=Springer}}</ref><ref name=Ohtaka>{{cite book |title=Photonic Crystals |author=Kuon Inoue, Kazuo Ohtaka |page=66 |url=https://books.google.com/books?id=GIa3HRgPYhAC&pg=PA66 |isbn=978-3-540-20559-3 |year=2004 |publisher=Springer}}</ref>

=== Density-functional theory ===
Many modern band calculations are carried out using [[density-functional theory]] (DFT). DFT often reproduces the overall shape of experimentally measured bands quite well, but it commonly underestimates the size of band gaps in semiconductors and insulators by about 30–40%.<ref>{{Cite journal|last1=Assadi|first1=M. Hussein. N.|last2=Hanaor|first2=Dorian A. H.|date=2013-06-21|title=Theoretical study on copper's energetics and magnetism in TiO<sub>2</sub> polymorphs|journal=Journal of Applied Physics|volume=113|issue=23|pages=233913–233913–5|arxiv=1304.1854|doi=10.1063/1.4811539|bibcode=2013JAP...113w3913A|s2cid=94599250|issn=0021-8979}}</ref> Although DFT is formally a ground-state theory, its Kohn–Sham solutions are widely used as practical approximations to real band structures.<ref>{{cite journal | last=Hohenberg|first=P | author2=Kohn, W.|title=Inhomogeneous Electron Gas|journal=Phys. Rev.|date=1964|volume=136|issue=3B|pages=B864–B871|doi=10.1103/PhysRev.136.B864|bibcode=1964PhRv..136..864H|doi-access=free}}</ref><ref name=Paier>{{Cite journal | last1 = Paier | first1 = J. | last2 = Marsman | first2 = M. | last3 = Hummer | first3 = K. | last4 = Kresse | first4 = G. | last5 = Gerber | first5 = I. C. | last6 = Angyán | first6 = J. G. | title = Screened hybrid density functionals applied to solids | journal = J Chem Phys | volume = 124 | issue = 15 | pages = 154709 | date=2006 | doi = 10.1063/1.2187006 | pmid = 16674253 | bibcode = 2006JChPh.124o4709P }}</ref>

=== Green's function and GW methods ===
To include many-body effects more accurately, one can use [[Green's function (many-body theory)|Green's function]] methods. In these approaches, the poles of the Green's function correspond to the [[quasiparticle]] energies of the solid. The [[GW approximation]] is especially important because it often corrects the systematic band-gap underestimation of standard DFT and gives results in closer agreement with experiment.

=== Dynamical mean-field theory ===
Some materials, such as [[Mott insulator]]s, cannot be understood in terms of ordinary single-electron bands. In these cases, strong electron-electron interactions dominate. Methods such as the [[Hubbard model]] and [[dynamical mean-field theory]] are used to describe the resulting behaviour.

== Band diagrams ==
In practical device physics, the full band structure is often simplified into a [[band diagram]], where energy is plotted vertically and real-space position horizontally. In such diagrams, energy levels or bands may tilt with position, indicating the presence of an [[electric field]]. Band diagrams are especially useful for understanding junctions, interfaces, and semiconductor devices.

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Band structure|1}}

Physics:Quantum Band structure - Revision history

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