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|Condensed matter physics|
|Phases · Phase transition|
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and in some liquids. The exciton is regarded as an elementary excitation of condensed matter that can transport energy without transporting net electric charge.
An exciton can form when a photon is absorbed by a semiconductor. This excites an electron from the valence band into the conduction band. In turn, this leaves behind a positively charged electron hole (an abstraction for the location from which an electron was moved). The electron in the conduction band is then effectively attracted to this localized hole by the repulsive Coulomb forces from large numbers of electrons surrounding the hole and excited electron. This attraction provides a stabilizing energy balance. Consequently, the exciton has slightly less energy than the unbound electron and hole. The wavefunction of the bound state is said to be hydrogenic, an exotic atom state akin to that of a hydrogen atom. However, the binding energy is much smaller and the particle's size much larger than a hydrogen atom. This is because of both the screening of the Coulomb force by other electrons in the semiconductor (i.e., its dielectric constant), and the small effective masses of the excited electron and hole. The recombination of the electron and hole, i.e. the decay of the exciton, is limited by resonance stabilization due to the overlap of the electron and hole wave functions, resulting in an extended lifetime for the exciton.
The electron and hole may have either parallel or anti-parallel spins. The spins are coupled by the exchange interaction, giving rise to exciton fine structure. In periodic lattices, the properties of an exciton show momentum (k-vector) dependence.
The concept of excitons was first proposed by Yakov Frenkel in 1931, when he described the excitation of atoms in a lattice of insulators. He proposed that this excited state would be able to travel in a particle-like fashion through the lattice without the net transfer of charge.
Excitons may be treated in two limiting cases, depending on the properties of the material in question.
In materials with a 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 fullerenes. This Frenkel exciton, named after Yakov Frenkel, has a typical binding energy on the order of 0.1 to 1 eV. Frenkel excitons are typically found in alkali halide crystals and in organic molecular crystals composed of aromatic molecules, such as anthracene and tetracene.
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 exciton, which has a radius larger than the lattice spacing. 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 0.01eV. 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 1.0eV) binding energies.
Often there is more than one band to choose from 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, forming a series of spectral absorption lines that are in principle similar to hydrogen spectral series.
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An intermediate case between Frenkel and Wannier excitons, charge-transfer excitons (sometimes called simply CT excitons) occur when the electron and the hole occupy adjacent molecules. They occur primarily in ionic crystals. Unlike Frenkel and Wannier excitons they display a static electric dipole moment.
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.
Atomic and molecular excitons
Alternatively, an exciton may be an excited state of an atom, ion, or molecule, the excitation 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 lowest unoccupied orbital and an electron hole in the 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 and their genesis was proposed by Alexander Davydov. It is known as 'Davydov splitting'.
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. For large excitons in direct-gap crystals, the oscillator strength of exciton absorption is proportional to , where is the exciton wave function, and and are the electron and hole coordinate, respectively. The expression for reflects the fact that the electron-hole pair is created inside a unit cell, at electron-hole separation much less than the exciton radius. The oscillator strength for producing a bound exciton can be expressed in terms of when the exciton is weakly bound to impurity, hence, its wave function of internal motion is only slightly distorted. Then 
Here is the exciton effective mass, is its reduced mass, is the exciton ionization energy, is the binding energy of the exciton to impurity, is the exciton radius, and is is the volume of a unit cell. In this equation is calculated per a single impurity and per a unit cell. For large excitons , and therefore bound excitons possess giant oscillator strength. As a result, shallow trapped excitons also possess short radiative lifetimes. The physical mechanism behind the giant oscillator strength is coherently oscillating electron polarization inside a bound exciton.
Similarly, spectra of weakly trapped molecular excitons are also strongly influenced by adjacent exciton bands. Near strongly polarized exciton multiplets, intensity and polarization of bound exciton states strongly depends on the binding energy. Changing it, e.g., by isotopic substitution of molecules, allows distinguishing intrinsic and extrinsic exciton spectra and investigating the structure of exciton energy bands. This technique is also used in spectroscopy of biological systems.
Excitons are the main mechanism for light emission in semiconductors at low temperature (when the characteristic thermal energy kT is less than the exciton binding energy), replacing the free electron-hole recombination at higher temperatures.
The existence of exciton states may be inferred from the absorption of light associated with their excitation. Typically, excitons are observed just below the band gap.
When excitons interact with photons a so-called polariton (also exciton-polariton) is formed. These excitons are sometimes referred to as dressed excitons.
Provided the interaction is attractive, an exciton can bind with other excitons to form a biexciton, analogous to a dihydrogen molecule. If a large density of excitons is created in a material, they can interact with one another to form an electron-hole liquid, a state observed in k-space indirect semiconductors.
Additionally, excitons are integer-spin particles obeying Bose statistics in the low-density limit. In some systems, where the interactions are repulsive, a Bose–Einstein condensed state is predicted to be the ground state. Exciton condensates have been seen in a double quantum well systems.
Spatially Direct and Indirect Excitons
Normally, excitons in a semiconductor have a very short lifetime due to the close proximity of the electron and hole. However, by placing the electron and hole in spatially separated quantum wells with an insulating barrier layer in between so called 'spatially indirect' excitons can be created. In contrast to ordinary (spatially direct), these spatially indirect excitons can have large spatial separation between the electron and hole, and thus possess a much longer lifetime. This is often used to cool excitons to very low temperatures in order to study Bose Einstein condensation (or rather it's 2 dimensional analog).
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