Exciton

Exciton

[ik-sahy-ton, ek-si-ton]
This page is about the quasiparticle. Exciton is also the title of a single by IDM composer Squarepusher.

An exciton is a bound state of an electron and an imaginary particle called an electron hole in an insulator or semiconductor, and such is a Coulomb-correlated electron-hole pair. It is an elementary excitation, or a quasiparticle of a solid. In current research, the bound electron and hole pairs (excitons) provide a means to transport energy without transporting net charge.

A vivid picture of exciton formation is as follows: a photon enters a semiconductor, exciting an electron from the valence band into the conduction band. The missing electron in the valence band leaves a hole (of opposite electric charge) behind, to which the electron is attracted by the Coulomb force. The exciton results from the binding of the electron with its hole. As a result, the exciton has slightly less energy than the unbound electron and hole. The wavefunction of the bound state is hydrogenic (an "exotic atom" state akin to that of a hydrogen atom). However, the binding energy is much smaller and the size much bigger than a hydrogen atom because of the effects of screening and the effective mass of the constituents in the material.

In a hydrogen atom the core and the electron can have parallel or antiparallel spin, the same is true for the exciton, and also for positronium, but not for the two electrons in the He atom. Often excitons were given names which look like hydrogen orbital names, but have the wrong numbering for angular momentum, or other quantum numbers.

Subtypes

Excitons can be treated in two limiting cases, which depend on the properties of the material in question. In semiconductors, the dielectric constant is generally large, and as a result, screening tends to reduce the Coulomb interaction between electrons and holes. The result is a Mott-Wannier exciton, which has a radius much 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, and because of the lower masses and the screened Coulomb interaction, the binding energy is usually much less than a hydrogen atom, typically on the order of 0.1 eV. This type of exciton was named for Sir Nevill Francis Mott and Gregory Wannier.

When a material's dielectric constant is very small, the Coulomb interaction between electron and hole become very strong and the excitons tend to be much smaller, of the same order as the unit cell (or on the same molecule as with fullerenes), so the electron and hole sit on the same cell. This Frenkel exciton, named after Yakov Frenkel, has typical binding energy on the order of 1.0 eV.

Alternatively, an exciton may be thought of as an excited state of an atom or ion, the excitation wandering from one cell of the lattice to another.

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 used as is seen in femtosecond two-photon experiments.

At surfaces so called image states may 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.

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 allow for significant binding energies of 0.4-1.0 eV.

Dynamics

The probability of the hole disappearing (the electron occupying the hole) is limited by the difficulty of losing the excess energy, and as a result excitons can have a relatively long lifetime. (Lifetimes up to several milliseconds have been observed in copper (I) oxide) Another limiting factor in the recombination probability is the spatial overlap of the electron and hole wavefunctions (roughly the probability for the electron to run into the hole). This overlap is smaller for lighter electrons and holes and for highly excited hydrogenic states.

The whole exciton can move through the solid. With this additional kinetic energy the exciton may lie above the band-gap.

The exciton propagating through molecular crystal is one that is of greatest concern. Several mechanisms have been proposed in the literature. Two are important. The first one is exciton energy dissipated due to interaction with phonon bath. The other one is energy carried away by radiation. Combination of the two has also been studied.

Much like molecular systems that have well defined resonances, excitons can undergo internal conversions from energetically higher lying states to lower lying states by coupling to vibrational or electronic degrees of freedom. Internal conversions usually take place of a time scale of a few to tens of femtoseconds. Also, intersystem crossings are possible when adequate spin orbit interactions are present in the material, and usually take place on a time scale of a few to hundreds of picoseconds.

Features

  • Since an exciton is a bound state of an electron and a hole, the overall charge for this quasiparticle is zero. Hence it carries no electric current.

Interaction

With other particles

Excitons are thus the main mechanism for light emission in semiconductors at low temperatures (where 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.

Excitons may also interact with phonons and lattice distortions to form polarons. In that case, the excitons are called dressed excitons.

With each other

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, and indeed such condensate has been already observed in recent experiments AIP Update 800 The inference was obtained by cooling an exciton state below 5 kelvins and further observing coherent light emission (with interference patterns) from it.

Notes

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