exciton [′ek·sə‚tän]
Exciton
A fundamental quantum of electronic excitation in condensed matter, consisting of a negatively charged electron and a positively charged hole bound to each other by electrostatic attraction. Excitons exist in all kinds of condensed matter, whenever it is possible for an electron to be excited from a filled energy level to an empty one, leaving behind a hole. Unlike an excitation in a single atom or molecule, an exciton can in general move through the solid like a particle. Excitons transport energy, not charge or mass. Typically, an exciton is created when a photon is absorbed in a solid; the exciton then moves through the crystal; and finally the electron and hole recombine, resulting in the emission of another photon, often at a wavelength different from that of the original photon. Excitons can also be created by injection of free electrons into excited states via an electric current. See Electron-hole recombination, Hole states in solids, Luminescence
Excitons fall into two broad classes, Wannier (or Wannier-Mott) excitons and Frenkel excitons, based on their size relative to the interatomic or intermolecular distances in the material. In Wannier excitons, typically observed in covalent semiconductors and insulators, the electron and hole are separated by a distance much larger than the atomic spacing, so that the effect of the crystal lattice on the exciton can be taken into account primarily via an average permittivity. In Frenkel excitons, typically seen in molecular or rare-gas crystals, the electron and hole are separated by a distance comparable to the atomic spacing, so that the exciton is localized to a single site at any given time. Wannier excitons move essentially like free particles, while motion of Frenkel excitons is envisioned as hopping from one site to another. See Electric insulator, Semiconductor
Exciton
a quasiparticle that represents an electronic excitation in a dielectric or a semiconductor. Excitons propagate through a crystal and are not associated with the transfer of electric charge or of mass. The concept of the exciton was introduced in 1931 bv Ia. I. Frenkel’ (Frenkel), who hypothesized that photoconductivity does not occur in dielectrics upon the absorption of light because the energy absorbed is expended on the formation of excitons rather than the production of charge carriers.
In molecular crystals, an exciton is an elementary excitation of the electron system of an individual molecule. Such elementary excitations are known as Frenkel excitons. Owing to molecular interaction, a Frenkel exciton propagates through a crystal as a wave. Frenkel excitons are manifested in the absorption and emission spectra of molecular crystals (seeSPECTROSCOPY, CRYSTAL). If a unit cell of such a crystal contains several molecules, molecular interaction results in the splitting of exciton lines. This effect, called Davydov splitting, is associated with the possible transfer of a Frenkel exciton from one group of molecules to another group within the unit cell. Davydov splitting has been observed experimentally in a number of molecular crystals, such as naphthalene, anthracene, and benzene.
In semiconductors, an exciton is a hydrogen-like bound state consisting of a conduction electron and a hole. Such bound states are referred to as Mott-Wannier excitons. The binding energies ℰ* and effective radii a* of Mott-Wannier excitons may be estimated on the basis of N. Bohr’s equations for the hydrogen atom if two circumstances are taken into account. First, the effective mass me of a conduction electron and the effective mass mh of a hole differ from the mass m0 of a free electron. Second, the Coulomb interaction between an electron and a hole in a crystal is reduced by the permittivity ∊ of the medium. Therefore,
where m* = (memh)/(me + mh), ħ is Planck’s constant, and e is the charge of the electron.
Equations (1) do not take into account the effect of the complex band structure of a crystal or the interaction of electrons and holes with phonons. However, allowance for these factors does not change the order of magnitude of ℰ* and a*. For Ge, Si, and III-V and II-VI semiconductors, m* ~ 0.1 m0 and ∊ ~ 10, yielding ℰ* ~ 10–2 eV and a* ~ 10–6 cm. Thus, the binding energies of Mott-Wannier excitons are many times lower than the binding energy of the electron and the proton in a hydrogen atom, and the radii of such excitons are many times larger than the interatomic distances in a crystal. The high values of a* indicate that an exciton in a semiconductor crystal is a macroscopic formation; hence, the crystal structure determines only the parameters m* and ℰ*. Therefore, a Mott-Wannier exciton may be regarded as a quasiatom that moves in a vacuum. The distortions of the crystal structure that are caused by a single exciton or by even a large number of excitons are negligibly small. In alkali halide and inert-gas crystals, ℰ* ~ 0.1–1 eV, a* ~ 10–7 –10–8 cm, and the formation of an exciton is accompanied by the deformation of a unit cell.
Mott-Wannier excitons are clearly apparent in the absorption spectra of semiconductors as narrow lines that are shifted below the optical absorption edge by a quantity equal to ℰ*. The hydrogen-like spectrum of such excitons was first observed in the absorption spectrum of Cu2O and later in other semiconductors. Excitons are also manifested in luminescence spectra, photoconductivity, the Stark effect, and the Zeeman effect. The lifetime of excitons is short; the electron and the hole that constitute an exciton can recombine, emitting a photon. For example, the lifetime of an exciton in Ge is of the order of 10–5 sec. An exciton may decay upon colliding with lattice defects.
When excitons interact with photons having frequencies ѡ = ℰ*/ħ, new quasiparticles called polaritons are formed. Polari-tons are mixed exciton-photon states. Their properties, such as their dispersion relation, differ considerably from the properties of both excitons and photons. Polaritons play an important role in the processes by which electronic excitation energy is transferred in a crystal and are responsible for various effects, such as peculiar features in the optical spectra of semiconductors in the exciton-band region.
At low densities, excitons behave in a crystal like a quasiparticle gas. At high densities, the interaction of excitons becomes substantial. For example, a bound state consisting of two excitons may be formed; such a bound state is called an exciton molecule, or a biexciton. However, in contrast to the hydrogen molecule, the dissociation energy of a biexciton is much lower than its binding energy, since the effective masses of electrons and holes in semiconductors are of the same order of magnitude.
If the density of excitons is increased, the distance between the quasiparticles may become of the order of their radius, leading to their collapse. The collapse may be accompanied by the formation of droplets of an electron-hole plasma. The formation of electron-hole droplets in such semiconductors as Ge and Si is manifested in the appearance of a new broad luminescence line that is shifted toward lower photon energies. Electron-hole droplets have a number of interesting properties, including a high concentration of electrons and holes at a low density (averaged over volume) and a high mobility in nonuniform fields.
At low exciton densities, an exciton, which consists of two fer-mions (a conduction electron and a hole), may be regarded as a boson. Consequently, Bose-Einstein condensation of excitons is possible; that is, a large number of excitons may accumulate in the lowest-lying energy level. Bose-Einstein condensation of excitons may lead to unattenuated energy fluxes in a crystal. However, in contrast to superfluid helium or a superconductor, a superfluid flux of excitons cannot exist for an arbitrarily long time. Such an exciton flux can exist only for the lifetime of the excitons.
REFERENCES
Gross, E. F. “Eksiton i ego dvizhenie v kristallicheskoi reshetke.” Uspekhiftzicheskikh nauk, 1962, vol. 76, issue 3.Knox, R. Teoriia eksitonov. Moscow, 1966. (Translated from English.)
Agranovich, V. M. Teoriia eksitonov. Moscow, 1968.
Davydov, A. S. Teoriia molekuliarnykh eksitonov. Moscow, 1968.
Eksitony v poluprovodnikakh (collection of articles). Moscow, 1971.
A. P. SILIN
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