Editing Exciton (section)

== Types ==

=== Frenkel exciton ===
In materials with a relatively small [[dielectric constant]], the Coulomb interaction between an electron and a hole may be strong and the excitons thus tend to be small, of the same order as the size of the unit cell. Molecular excitons may even be entirely located on the same molecule, as in [[fullerene]]s. This ''Frenkel exciton'', named after [[Yakov Frenkel]], has a typical binding energy on the order of 0.1 to 1 [[electron volt|eV]]. Frenkel excitons are typically found in alkali halide crystals and in organic molecular crystals composed of aromatic molecules, such as [[anthracene]] and [[tetracene]]. Another example of Frenkel exciton includes on-site ''d''-''d'' excitations in transition metal compounds with partially filled ''d''-shells. While ''d''-''d'' transitions are in principle forbidden by symmetry, they become weakly-allowed in a crystal when the symmetry is broken by structural relaxations or other effects. Absorption of a photon resonant with a ''d''-''d'' transition leads to the creation of an electron-hole pair on a single atomic site, which can be treated as a Frenkel exciton.

=== Wannier–Mott exciton ===
In semiconductors, the dielectric constant is generally large. Consequently, [[electric field screening]] tends to reduce the Coulomb interaction between electrons and holes. The result is a ''Wannier–Mott exciton'',<ref>{{cite journal|doi=10.1103/PhysRev.52.191|title=The Structure of Electronic Excitation Levels in Insulating Crystals|year=1937|last1=Wannier|first1=Gregory|journal=Physical Review|volume=52|page=191|bibcode = 1937PhRv...52..191W|issue=3 }}</ref> which has a radius larger than the lattice spacing. Small effective mass of electrons that is typical of semiconductors also favors large exciton radii. As a result, the effect of the lattice potential can be incorporated into the effective masses of the electron and hole. Likewise, because of the lower masses and the screened Coulomb interaction, the binding energy is usually much less than that of a hydrogen atom, typically on the order of {{gaps|0.01|eV}}. This type of exciton was named for [[Gregory Wannier]] and [[Nevill Francis Mott]]. Wannier–Mott excitons are typically found in semiconductor crystals with small energy gaps and high dielectric constants, but have also been identified in liquids, such as liquid [[xenon]]. They are also known as ''large excitons''.

In single-wall [[carbon nanotubes]], excitons have both Wannier–Mott and Frenkel character. This is due to the nature of the Coulomb interaction between electrons and holes in one-dimension. The dielectric function of the nanotube itself is large enough to allow for the spatial extent of the [[wave function]] to extend over a few to several nanometers along the tube axis, while poor screening in the vacuum or dielectric environment outside of the nanotube allows for large (0.4 to {{gaps|1.0|eV}}) binding energies.

Often more than one band can be chosen as source for the electron and the hole, leading to different types of excitons in the same material. Even high-lying bands can be effective as [[femtosecond]] two-photon experiments have shown. At cryogenic temperatures, many higher excitonic levels can be observed approaching the edge of the band,<ref>{{Cite journal | doi=10.1038/nature13832| pmid=25318523| title=Giant Rydberg excitons in the copper oxide Cu2O| journal=Nature| volume=514| issue=7522| pages=343–347| year=2014| last1=Kazimierczuk| first1=T.| last2=Fröhlich| first2=D.| last3=Scheel| first3=S.| last4=Stolz| first4=H.| last5=Bayer| first5=M.| arxiv=1407.0691| bibcode=2014Natur.514..343K| s2cid=4470179}}</ref> forming a series of spectral absorption lines that are in principle similar to [[hydrogen spectral series]].

==== 3D semiconductors ====
In a bulk semiconductor, a Wannier exciton has an energy and radius associated with it, called '''exciton Rydberg energy''' and '''exciton Bohr radius''' respectively.<ref>{{cite book |last=Fox |first=Mark |date= 2010-03-25|title=Optical Properties of Solids |edition=2 |url=https://global.oup.com/academic/product/optical-properties-of-solids-9780199573370?q=max%20fox%20optical%20properties%20of%20solids&lang=en&cc=no |publisher=[[Oxford University Press]] |page=97 |isbn=978-0199573363 |series=Oxford Master Series in Physics }}</ref> For the energy, we have

:<math>E(n)=- \frac{ \left( \frac{\mu}{m_0 \varepsilon_r^2}\text{Ry} \right)}{n^2}  \equiv -\frac{R_\text{X}}{n^2}</math>

where <math>\text{Ry}</math> is the Rydberg unit of energy (cf. [[Rydberg constant]]), <math>\varepsilon_r</math> is the (static) relative permittivity, <math>\mu= (m^*_e m^*_h)/(m^*_e+m^*_h)</math> is the reduced mass of the electron and hole, and <math>m_0</math> is the electron mass. Concerning the radius, we have

:<math>r_n = \left(\frac{m_0 \varepsilon_r a_\text{H}}{\mu}  \right)n^2 \equiv a_\text{X}n^2</math>

where <math>a_\text{H}</math> is the [[Bohr radius]].

For example, in [[Gallium arsenide|GaAs]], we have relative permittivity of 12.8 and effective electron and hole masses as 0.067''m<sub>0</sub>'' and 0.2''m<sub>0</sub>'' respectively; and that gives us <math>R_\text{X}=4.2</math> meV and <math>a_\text{X}=13</math> nm.

==== 2D semiconductors ====
In [[Two-dimensional materials|two-dimensional (2D) materials]], the system is [[quantum confinement|quantum confined]] in the direction perpendicular to the plane of the material. The reduced dimensionality of the system has an effect on the binding energies and radii of Wannier excitons. In fact, excitonic effects are enhanced in such systems.<ref name="ChernikovBerkelbach2014">{{cite journal|last1=Chernikov|first1=Alexey|last2=Berkelbach|first2=Timothy C.|last3=Hill|first3=Heather M.|last4=Rigosi|first4=Albert|last5=Li|first5=Yilei|last6=Aslan|first6=Ozgur Burak|last7=Reichman|first7=David R.|last8=Hybertsen|first8=Mark S.|last9=Heinz|first9=Tony F.|title=Exciton Binding Energy and Nonhydrogenic Rydberg Series in MonolayerWS2|journal=Physical Review Letters|volume=113|issue=7|year=2014|issn=0031-9007|doi=10.1103/PhysRevLett.113.076802|doi-access=free|bibcode=2014PhRvL.113g6802C|pmid=25170725|page=076802|arxiv=1403.4270}}</ref>

For a simple screened Coulomb potential, the binding energies take the form of the 2D hydrogen atom<ref>
{{cite journal |last1=Yang |first1=X. L. |title=Analytic solution of a two-dimensional hydrogen atom. I. Nonrelativistic theory |journal=Physical Review A |date=1 February 1991 |volume=43 |issue=3 |pages=1186–1196 |doi=10.1103/PhysRevA.43.1186 |pmid=9905143 |bibcode=1991PhRvA..43.1186Y |url=https://link.aps.org/doi/10.1103/PhysRevA.43.1186}}</ref>

:<math>E(n)= -\frac{R_\text{X}}{\left(n-\tfrac{1}{2}\right)^2}</math>.

In most 2D semiconductors, the Rytova–Keldysh form is a more accurate approximation to the exciton interaction<ref>{{cite journal |last1=Rytova |first1=N S. |title=The screened potential of a point charge in a thin film |journal=Proc. MSU Phys. Astron. |date=1967 |volume=3 |page=30}}</ref><ref>{{cite journal |last1=Keldysh |first1=LV |title=Coulomb interaction in thin semiconductor and semimetal films |journal=JETP Lett. |date=1979 |volume=29 |page=658|bibcode=1979JETPL..29..658K }}</ref><ref>{{cite journal |last1=Trolle |first1=Mads L. |last2=Pedersen |first2=Thomas G. |last3=Véniard |first3=Valerie |title=Model dielectric function for 2D semiconductors including substrate screening |journal=Sci. Rep. |date=2017 |volume=7 |page=39844 |doi=10.1038/srep39844|pmid=28117326 |pmc=5259763 |bibcode=2017NatSR...739844T |doi-access=free }}</ref>

:<math>V(r)= -\frac{e^2}{8 \epsilon_0 r_0}\left[\text{H}_0\left(\frac{\kappa r}{r_0}\right)-Y_0\left(\frac{\kappa r}{r_0}\right)\right].</math>

where <math>r_0</math> is the so-called screening length, <math>\epsilon_0</math> is the [[vacuum permittivity]], <math>e</math> is the [[elementary charge]], <math>\kappa</math> the average dielectric constant of the surrounding media, and <math>r</math> the exciton radius. For this potential, no general expression for the exciton energies may be found. One must instead turn to numerical procedures, and it is precisely this potential that gives rise to the nonhydrogenic Rydberg series of the energies in 2D semiconductors.<ref name="ChernikovBerkelbach2014"/>

===== Example: excitons in transition metal dichalcogenides (TMDs) =====
Monolayers of a transition metal dichalcogenide (TMD) are a good and cutting-edge example where excitons play a major role. In particular, in these systems, they exhibit a bounding energy of the order of 0.5 eV<ref name=":0"/> with a Coulomb attraction between the hole and the electrons stronger than in other traditional quantum wells. As a result, optical excitonic peaks are present in these materials even at room temperatures.<ref name=":0" />

==== 0D semiconductors ====
{{See also|Quantum dot#Quantum confinement in semiconductors}}
In [[nanoparticles]] which exhibit quantum confinement effects and hence behave as quantum dots (also called 0-dimensional semiconductors), excitonic radii are given by<ref>
{{cite journal |last1=Brus |first1=Louis |date=1986 |title=Electronic wave functions in semiconductor clusters: experiment and theory |journal=The Journal of Physical Chemistry |volume=90 |issue=12 |pages=2555–2560 |doi= 10.1021/j100403a003|publisher=ACS Publications}}
</ref><ref name="Edvinsson2018">
{{cite journal|last1=Edvinsson|first1=T.|title=Optical quantum confinement and photocatalytic properties in two-, one- and zero-dimensional nanostructures|journal=Royal Society Open Science|volume=5|issue=9|year=2018|pages=180387|issn=2054-5703|doi=10.1098/rsos.180387|pmid=30839677|doi-access=free|pmc=6170533|bibcode=2018RSOS....580387E}}</ref>
:<math>a_\text{X} = \frac{\varepsilon_r}{\mu/m_0}a_0</math>

where <math>\varepsilon_r</math> is the [[relative permittivity]], <math>\mu \equiv (m_e^*m_h^*)/(m_e^*+m_h^*)</math> is the reduced mass of the electron-hole system, <math>m_0</math> is the electron mass, and <math>a_0</math> is the [[Bohr radius]].
<!--
THE FOLLOWING IS INCORRECT
:<math>a_\text{X} = \frac{\varepsilon_\infty}{\mu/m_0}a_0</math>

where <math>\varepsilon_\infty</math> is the [[Permittivity#Complex permittivity|high-frequency dielectric constant]], <math>m^*_e</math> is the effective electron mass, <math>m_e</math> is the electron mass, and <math>a_0</math> is the [[Bohr radius]].
-->
=== Hubbard exciton ===
{{Main|Hubbard model}}
Hubbard excitons are linked to electrons not by a [[Coulomb's law|Coulomb's interaction]], but by a [[magnetic force]]. Their name derives by the English physicist [[John Hubbard (physicist)|John Hubbard]].

Hubbard excitons were observed for the first time in 2023 through the [[Terahertz time-domain spectroscopy]]. Those particles have been obtained by applying a light to a [[Mott insulator|Mott antiferromagnetic insulator]].<ref>{{cite web|url=https://phys.org/news/2023-09-scientists-hubbard-exciton-strongly-insulators.html|publisher=[[Phys.org]]|title=Scientists observe Hubbard exciton in strongly correlated insulators|date=September 25, 2023|access-date=October 11, 2023|doi=10.1038/s41567-023-02204-2|archive-date=October 11, 2023|archive-url=https://archive.today/20231011073541/https://phys.org/news/2023-09-scientists-hubbard-exciton-strongly-insulators.html|url-status=bot: unknown}}</ref>

=== Charge-transfer exciton ===
An intermediate case between Frenkel and Wannier excitons is the ''charge-transfer (CT) exciton''. In molecular physics, CT excitons form when the electron and the hole occupy adjacent molecules.<ref>{{cite book |author=Wright |first=J. D. |url={{Google books|7sroAgMASIEC|Molecular Crystals|page=108|plainurl=yes}} |title=Molecular Crystals |publisher=Cambridge University Press |year=1995 |isbn=978-0-521-47730-7 |edition=2nd |page=108 |language=en-uk |orig-year=First published 1987}}</ref> They occur primarily in organic and molecular crystals;<ref>{{cite book |author=Lanzani |first=Guglielmo |url={{Google books|RVyvgKo0lGQC|The Photophysics Behind Photovoltaics and Photonics|page=82|plainurl=yes}} |title=The Photophysics Behind Photovoltaics and Photonics |publisher=Wiley-VCH Verlag |year=2012 |page=82 |language=en}}</ref> in this case, unlike Frenkel and Wannier excitons, CT excitons display a static [[electric dipole moment]]. CT excitons can also occur in transition metal oxides, where they involve an electron in the transition metal 3''d'' orbitals and a hole in the oxygen 2''p'' orbitals. Notable examples include the lowest-energy excitons in correlated cuprates<ref>{{cite journal|doi=10.1103/PhysRevB.77.060501|title=Charge-transfer exciton in La<sub>2</sub>CuO<sub>4</sub> probed with resonant inelastic x-ray scattering|year=2008|journal=Physical Review B|volume=77|issue=6|pages=060501(R)|last1=Ellis|first1=D. S.|last2=Hill|first2=J. P.|last3=Wakimoto|first3=S.|last4=Birgeneau|first4=R. J.|last5=Casa|first5=D.|last6=Gog|first6=T.|last7=Kim|first7=Young-June|arxiv=0709.1705|bibcode=2008PhRvB..77f0501E|s2cid=119238654}}</ref> or the two-dimensional exciton of TiO<sub>2</sub>.<ref>{{cite journal|doi=10.1038/s41467-017-00016-6|title=Strongly bound excitons in anatase TiO<sub>2</sub> single crystals and nanoparticles|year=2017|journal=Nature Communications|volume=8|issue=13|last1=Baldini|first1=Edoardo|last2=Chiodo|first2=Letizia|last3=Dominguez|first3=Adriel|last4=Palummo|first4=Maurizia|last5=Moser|first5=Simon|last6=Yazdi-Rizi|first6=Meghdad|last7=Aubock|first7=Gerald|last8=Mallett|first8=Benjamin P P|last9=Berger|first9=Helmuth|last10=Magrez|first10=Arnaud|last11=Bernhard|first11=Christian|last12=Grioni|first12=Marco|last13=Rubio|first13=Angel|last14=Chergui|first14=Majed|page=13|pmid=28408739|pmc=5432032|arxiv=1601.01244|bibcode=2017NatCo...8...13B|doi-access=free}}</ref> Irrespective of the origin, the concept of CT exciton is always related to a transfer of charge from one atomic site to another, thus spreading the wave-function over a few lattice sites.

=== Surface exciton ===
At surfaces it is possible for so called ''image states'' to occur, where the hole is inside the solid and the electron is in the vacuum. These electron-hole pairs can only move along the surface.

=== Dark exciton ===
Dark excitons are those where the electrons have a different momentum from the holes to which they are bound that is they are in an optically [[forbidden transition]] which prevents them from photon absorption and therefore to reach their state they need [[phonon scattering]]. They can even outnumber normal bright excitons formed by absorption alone.<ref>{{Cite journal |last1=Madéo |first1=Julien |last2=Man |first2=Michael K. L. |last3=Sahoo |first3=Chakradhar |last4=Campbell |first4=Marshall |last5=Pareek |first5=Vivek |last6=Wong |first6=E. Laine |last7=Al-Mahboob |first7=Abdullah |last8=Chan |first8=Nicholas S. |last9=Karmakar |first9=Arka |last10=Mariserla |first10=Bala Murali Krishna |last11=Li |first11=Xiaoqin|author11-link=Xiaoqin Li |last12=Heinz |first12=Tony F. |last13=Cao |first13=Ting |last14=Dani |first14=Keshav M. |date=2020-12-04 |title=Directly visualizing the momentum-forbidden dark excitons and their dynamics in atomically thin semiconductors |url=https://www.science.org/doi/10.1126/science.aba1029 |journal=Science |language=en |volume=370 |issue=6521 |pages=1199–1204 |doi=10.1126/science.aba1029 |pmid=33273099 |issn=0036-8075|arxiv=2005.00241 |bibcode=2020Sci...370.1199M }}</ref><ref>{{Cite web |date=2020-12-04 |title=Dark excitons hit the spotlight |url=https://www.oist.jp/news-center/news/2020/12/4/dark-excitons-hit-spotlight |access-date=2023-12-02 |website=Okinawa Institute of Science and Technology OIST |language=en}}</ref><ref>{{Cite journal |date=2021-01-07 |title=Dark excitons outnumber bright ones |url=https://pubs.aip.org/physicstoday/online/30231 |journal=Physics Today |language=en |volume=2021 |issue=1 |pages=0107a |doi=10.1063/PT.6.1.20210107a|bibcode=2021PhT..2021a.107. |last1=Middleton |first1=Christine }}</ref> The first direct measurement of the dynamics of momentum-forbidden dark excitons have been performed using time-resolved photoemission from monolayer WS<sub>2</sub>.<ref>{{Cite journal |date=2024-11-08 |title=Sub-100 fs Formation of Dark Excitons in Monolayer WS<sub>2</sub> |journal=Nano Letters |language=en |volume=24 |issue=46 |pages=14663–14670 |doi=10.1021/acs.nanolett.4c03807|last1=Kolesnichenko |first1=Pavel |last2=Wittenbecher |first2=Lukas |last3=Zhang |first3=Qianhui |last4=Yan Teh |first4=Run |last5=Babu |first5=Chandni |last6=S Fuhrer |first6=Michael |last7=Mikkelsen |first7=Anders |last8=Zigmantas |first8=Donatas|pmid=39516189 |pmc=11583335 |arxiv=2403.08390 |bibcode=2024NanoL..2414663K }}</ref>

=== Atomic and molecular excitons ===
Alternatively, an exciton may be described as an excited state of an atom, [[ion]], or molecule, if the excitation is wandering from one cell of the lattice to another.

When a molecule absorbs a quantum of energy that corresponds to a transition from one [[molecular orbital]] to another molecular orbital, the resulting electronic excited state is also properly described as an exciton. An [[electron]] is said to be found in the [[HOMO/LUMO|lowest unoccupied orbital]] and an [[electron hole]] in the [[HOMO/LUMO|highest occupied molecular orbital]], and since they are found within the same molecular orbital manifold, the electron-hole state is said to be bound. Molecular excitons typically have characteristic lifetimes on the order of [[nanoseconds]], after which the ground electronic state is restored and the molecule undergoes photon or [[phonon]] emission. Molecular excitons have several interesting properties, one of which is energy transfer (see [[Förster resonance energy transfer]]) whereby if a molecular exciton has proper energetic matching to a second molecule's spectral absorbance, then an exciton may transfer (''hop'') from one molecule to another. The process is strongly dependent on intermolecular distance between the species in solution, and so the process has found application in sensing and ''molecular rulers''.

The hallmark of molecular excitons in organic molecular crystals are doublets and/or triplets of exciton absorption bands strongly polarized along crystallographic axes. In these crystals an elementary cell includes several molecules sitting in symmetrically identical positions, which results in the level degeneracy that is lifted by intermolecular interaction. As a result, absorption bands are polarized along the symmetry axes of the crystal. Such multiplets were discovered by [[Antonina Prikhot'ko]]<ref>A. Prikhotjko, Absorption Spectra of Crystals at Low Temperatures, J. Physics USSR '''8''', p. 257 (1944).</ref><ref>A. F. Prikhot'ko, Izv, AN SSSR Ser. Fiz. '''7''', p. 499 (1948) http://ujp.bitp.kiev.ua/files/journals/53/si/53SI18p.pdf {{Webarchive|url=https://web.archive.org/web/20160305193404/http://ujp.bitp.kiev.ua/files/journals/53/si/53SI18p.pdf|date=2016-03-05}}.</ref> and their genesis was proposed by Alexander Davydov. It is known as 'Davydov splitting'.<ref>A. S. Davydov, ''Theory of Molecular Excitons'' (Plenum, New York, New York) 1971.</ref><ref>V. L. Broude, E. I. Rashba, and E. F. Sheka, ''Spectroscopy of molecular excitons'' (Springer, New York, New York) 1985.</ref>

=== Giant oscillator strength of bound excitons ===
Excitons are lowest excited states of the electronic subsystem of pure crystals. Impurities can bind excitons, and when the bound state is shallow, the oscillator strength for producing bound excitons is so high that impurity absorption can compete with intrinsic exciton absorption even at rather low impurity concentrations. This phenomenon is generic and applicable both to the large radius (Wannier–Mott) excitons and molecular (Frenkel) excitons. Hence, excitons bound to impurities and defects possess [[giant oscillator strength]].<ref>E. I. Rashba, ''Giant Oscillator Strengths Associated with Exciton Complexes'', Soviet Physics Semiconductors '''8''', 807–816 (1975).</ref>