Exciton


An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force resulting from their opposite charges. It is an electrically neutral quasiparticle regarded as an elementary excitation primarily in condensed matter, such as insulators, semiconductors, some metals, and in some liquids. It transports energy without transporting net electric charge.
An exciton can form when an electron from the valence band of a crystal is promoted in energy to the conduction band, for instance when a material absorbs a photon. Promoting the electron to the conduction band leaves a positively charged hole in the valence band. Here 'hole' represents the unoccupied quantum mechanical electron state with a positive charge, an analogue in crystal of a positron. Because of the attractive Coulomb force between the electron and the hole, a bound state is formed, akin to that of the electron and proton in a hydrogen atom or the electron and positron in positronium. Excitons are composite bosons since they are formed from two fermions which are the electron and the hole.

Occurrences

Excitons are often treated in two limiting cases, namely small-radius excitons, named Frenkel exciton, and large-radius excitons, often called Wannier-Mott excitons.
A Frenkel exciton occurs when the distance between electron and hole is restricted to one or only a few nearest neighbour unit cells. Frenkel excitons typically occur in insulators and organic semiconductors with relatively narrow allowed energy bands and accordingly, rather heavy effective mass.
In the case of Wannier-Mott excitons, the relative motion of electron and hole in the crystal covers many unit cells. Wannier-Mott excitons are considered as hydrogen-like quasiparticles. The wavefunction of the bound state then is said to be hydrogenic, resulting in a series of energy states in analogy to a hydrogen atom. Compared to a hydrogen atom, the exciton binding energy in a crystal is much smaller and the exciton's size is much larger. This is mainly because of two effects: Coulomb forces are screened in a crystal, which is expressed as a relative permittivity εr significantly larger than 1 and the effective mass of the electron and hole in a crystal are typically smaller compared to that of free electrons. Wannier-Mott excitons with binding energies ranging from a few to hundreds of meV, depending on the crystal, occur in many semiconductors including Cu2O, GaAs, other III-V and II-VI semiconductors, transition metal dichalcogenides such as MoS2.
Excitons give rise to spectrally narrow lines in optical absorption, reflection, transmission and luminescence spectra with the energies below the free-particle band gap of an insulator or a semiconductor. Exciton binding energy and radius can be extracted from optical absorption measurements in applied magnetic fields.
The exciton as a quasiparticle is characterized by the momentum describing free propagation of the electron-hole pair as a composite particle in the crystalline lattice in agreement with the Bloch theorem. The exciton energy depends on K and is typically parabolic for the wavevectors much smaller than the reciprocal lattice vector of the host lattice. The exciton energy also depends on the respective orientation of the electron and hole spins, whether they are parallel or anti-parallel. The spins are coupled by the exchange interaction, giving rise to exciton energy fine structure.
In metals and highly doped semiconductors a concept of the Gerald Mahan exciton is invoked where the hole in a valence band is correlated with the Fermi sea of conduction electrons. In that case no bound state in a strict sense is formed, but the Coulomb interaction leads to a significant enhancement of absorption in the vicinity of the fundamental absorption edge also known as the Mahan or Fermi-edge singularity.

History

The concept of excitons was first proposed by Yakov Frenkel in 1931, when he described the excitation of an atomic lattice considering what is now called the tight-binding description of the band structure. In his model the electron and the hole bound by the coulomb interaction are located either on the same or on the nearest neighbouring sites of the lattice, but the exciton as a composite quasi-particle is able to travel through the lattice without any net transfer of charge, which led to many propositions for optoelectronic devices.

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 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. 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, 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. 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 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, 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. For the energy, we have
where is the Rydberg unit of energy, is the relative permittivity, is the reduced mass of the electron and hole, and is the electron mass. Concerning the radius, we have
where is the Bohr radius.
For example, in GaAs, we have relative permittivity of 12.8 and effective electron and hole masses as 0.067m0 and 0.2m0 respectively; and that gives us meV and nm.

2D semiconductors

In two-dimensional materials, the system is 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.
For a simple screened Coulomb potential, the binding energies take the form of the 2D hydrogen atom
In most 2D semiconductors, the Rytova–Keldysh form is a more accurate approximation to the exciton interaction
where is the so-called screening length, is the vacuum permittivity, is the elementary charge, the average dielectric constant of the surrounding media, and 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.
Example: excitons in transition metal dichalcogenides (TMDs)
Monolayers of a transition metal dichalcogenide 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 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.

0D semiconductors

In nanoparticles which exhibit quantum confinement effects and hence behave as quantum dots, excitonic radii are given by
where is the relative permittivity, is the reduced mass of the electron-hole system, is the electron mass, and is the Bohr radius.

Hubbard exciton

Hubbard excitons are linked to electrons not by a Coulomb's interaction, but by a magnetic force. Their name derives by the English 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 antiferromagnetic insulator.