Exciton
Encyclopedia
An exciton is a bound state
of an electron
and hole
which are attracted to each other by the electrostatic Coulomb force
. It is an electrically neutral quasiparticle
that exists in insulators
, semiconductor
s and some liquids. The exciton is regarded as an elementary excitation of condensed matter
that can transport energy without transporting net electric charge.
An exciton forms 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 localized positively-charged hole (holes actually don't exist, the term is an abstraction for the location an electron moved from; they have no charge in and of themselves). The electron in the conduction band is then attracted to this localized hole by the Coulomb force. 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 spin
s. The spins are coupled by the exchange interaction
, giving rise to exciton fine structure
. In periodic lattices, the properties of 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.
, 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 eV. Frenkel excitons are typically found in alkalihalide crystals and in organic molecular crystals composed of aromatic molecules, such as anthracene
and tetracene
.
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 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 constant, but have also been identified in liquids, such as liquid xenon
.
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 to 1.0 eV.
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.
, 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 fluorescence
. 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 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. Combinations of the two have also been studied.
Much like molecular systems that have well defined resonances, excitons can undergo internal conversion
s 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 crossing
s 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.
(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, and indeed such condensate has been already observed in recent experiments. The inference was obtained by cooling an exciton state below 5 K
and further observing coherent light emission (with interference patterns) from it.
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
of an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
and hole
Electron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
which are attracted to each other by the electrostatic Coulomb force
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
. It is an electrically neutral quasiparticle
Quasiparticle
In physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
that exists in insulators
Electrical insulation
thumb|250px|[[Coaxial Cable]] with dielectric insulator supporting a central coreThis article refers to electrical insulation. For insulation of heat, see Thermal insulation...
, semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s and some liquids. The exciton is regarded as an elementary excitation of condensed matter
Condensed Matter
Condensed matter may refer to several things*Condensed matter physics, the study of the physical properties of condensed phases of matter*European Physical Journal B: Condensed Matter and Complex Systems, a scientific journal published by EDP sciences...
that can transport energy without transporting net electric charge.
An exciton forms when a photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
is absorbed by a semiconductor. This excites an electron from the valence band
Valence band
In solids, the valence band is the highest range of electron energies in which electrons are normally present at absolute zero temperature....
into the conduction band
Conduction band
In the solid-state physics field of semiconductors and insulators, the conduction band is the range of electron energies, higher than that of the valence band, sufficient to free an electron from binding with its individual atom and allow it to move freely within the atomic lattice of the material...
. In turn, this leaves behind a localized positively-charged hole (holes actually don't exist, the term is an abstraction for the location an electron moved from; they have no charge in and of themselves). The electron in the conduction band is then attracted to this localized hole by the Coulomb force. This attraction provides a stabilizing energy balance. Consequently, the exciton has slightly less energy than the unbound electron and hole. The wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
of the bound state is said to be hydrogenic
Hydrogen-like atom
A hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself , these ions carry the positive charge e, where Z is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+...
, an exotic atom
Exotic atom
An exotic atom is an otherwise normal atom in which one or more sub-atomic particles have been replaced by other particles of the same charge. For example, electrons may be replaced by other negatively charged particles such as muons or pions...
state akin to that of a hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
. However, the binding energy
Binding energy
Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together—often this means that energy is released upon the creation of a bound state...
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
Dielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
), 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 spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
s. The spins are coupled by the exchange interaction
Exchange interaction
In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap...
, giving rise to exciton fine structure
Fine structure
In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy...
. In periodic lattices, the properties of exciton show momentum (k-vector) dependence.
The concept of excitons was first proposed by Yakov Frenkel
Yakov Frenkel
Yakov Il'ich Frenkel, was a Soviet physicist renowned for his works in the field of solid-state physics. He is also known as Jacov 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.
Classification
Excitons may be treated in two limiting cases, depending on the properties of the material in question.Frenkel excitons
In materials with a small dielectric constantDielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
, 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
Fullerene
A fullerene is any molecule composed entirely of carbon, in the form of a hollow sphere, ellipsoid, or tube. Spherical fullerenes are also called buckyballs, and they resemble the balls used in association football. Cylindrical ones are called carbon nanotubes or buckytubes...
s. This Frenkel exciton, named after Yakov Frenkel
Yakov Frenkel
Yakov Il'ich Frenkel, was a Soviet physicist renowned for his works in the field of solid-state physics. He is also known as Jacov Frenkel....
, has a typical binding energy on the order of 0.1 to 1 eV. Frenkel excitons are typically found in alkalihalide crystals and in organic molecular crystals composed of aromatic molecules, such as anthracene
Anthracene
Anthracene is a solid polycyclic aromatic hydrocarbon consisting of three fused benzene rings. It is a component of coal-tar. Anthracene is used in the production of the red dye alizarin and other dyes...
and tetracene
Tetracene
Tetracene, also called naphthacene, is a polycyclic aromatic hydrocarbon. It has the appearance of a pale orange powder. Tetracene is the four-ringed member of the series of acenes, the previous one being anthracene and the next one being pentacene.Tetracene is a molecular organic semiconductor,...
.
Wannier-Mott excitons
In semiconductors, the dielectric constant is generally large. Consequently, electric field screeningElectric field screening
Screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases and conduction electrons in semiconductors and metals....
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 a hydrogen atom, typically on the order of . This type of exciton was named for Gregory Wannier
Gregory Wannier
Gregory Hugh Wannier was a Swiss physicist.He attended Princeton as a graduate student and later taught at several American universities before a stint in industry....
and Nevill Francis Mott
Nevill Francis Mott
Sir Nevill Francis Mott, CH, FRS was an English physicist. He won the Nobel Prize for Physics in 1977 for his work on the electronic structure of magnetic and disordered systems, especially amorphous semiconductors. The award was shared with Philip W. Anderson and J. H...
. Wannier-Mott excitons are typically found in semiconductor crystals with small energy gaps and high dielectric constant, but have also been identified in liquids, such as liquid xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...
.
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 to 1.0 eV.
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.
Surface excitons
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 thought of as an excited state of an atom, ionIon
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...
, 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
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
to another molecular orbital, the resulting electronic excited state is also properly described as an exciton. An electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
is said to be found in the lowest unoccupied orbital
HOMO/LUMO
HOMO and LUMO are acronyms for highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively. The energy difference between the HOMO and LUMO is termed the HOMO-LUMO gap...
and an electron hole
Electron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
in the highest occupied molecular orbital
HOMO/LUMO
HOMO and LUMO are acronyms for highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively. The energy difference between the HOMO and LUMO is termed the HOMO-LUMO gap...
, 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 fluorescence
Fluorescence
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation of a different wavelength. It is a form of luminescence. In most cases, emitted light has a longer wavelength, and therefore lower energy, than the absorbed radiation...
. 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.
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
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
. The other one is energy carried away by radiation. Combinations of the two have also been studied.
Much like molecular systems that have well defined resonances, excitons can undergo internal conversion
Internal conversion
Internal conversion is a radioactive decay process where an excited nucleus interacts with an electron in one of the lower atomic orbitals, causing the electron to be emitted from the atom. Thus, in an internal conversion process, a high-energy electron is emitted from the radioactive atom, but...
s 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 crossing
Intersystem crossing
Intersystem crossing is a radiationless process involving a transition between two electronic states with different spin multiplicity.-Singlet and triplet states:...
s are possible when adequate spin orbit interactions
Spin-orbit interaction
In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
are present in the material, and usually take place on a time scale of a few to hundreds of picoseconds.
Interaction
Excitons are the main mechanism for light emission in semiconductors at low temperatureTemperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
(when the characteristic thermal energy kT is less than the exciton binding energy
Binding energy
Binding energy is the mechanical energy required to disassemble a whole into separate parts. A bound system typically has a lower potential energy than its constituent parts; this is what keeps the system together—often this means that energy is released upon the creation of a bound state...
), 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
Band gap
In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...
.
When excitons interact with photons a so-called polariton
Polariton
In physics, polaritons are quasiparticles resulting from strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation. They are an expression of the common quantum phenomenon known as level repulsion, also known as the anti-crossing principle...
(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
Biexciton
Biexcitons are created from two free excitons.- Formation of biexcitons :In quantum information and computation, it is essential to construct coherent combinations of quantum states....
, analogous to a dihydrogen molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
. If a large density of excitons is created in a material, they can interact with one another to form an electron-hole
Carrier generation and recombination
In the solid state physics of semiconductors, carrier generation and recombination are processes by which mobile charge carriers are created and eliminated. Carrier generation and recombination processes are fundamental to the operation of many optoelectronic semiconductor devices, such as...
liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
, 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. The inference was obtained by cooling an exciton state below 5 K
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
and further observing coherent light emission (with interference patterns) from it.