Exciton
Condensed matter physics |
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An
An exciton can form when an electron from the valence band of a crystal is promoted in energy to the
The concept of excitons was first proposed by
Excitons are often treated in the two limiting cases:
(i) The small radius excitons, or Frenkel excitons, where the electron-hole relative distance is restricted to one or only a few nearest neighbour unit cells. Frenkel excitons typically occur in
(ii) the large radius excitons are called Wannier-Mott excitons, for which the relative motion of electron and hole in the crystal covers many unit cells. Wannier-Mott excitons are considered as hydrogen-like quasiparticles. The
The exciton as a quasiparticle is characterized by the momentum (or
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.
Frenkel exciton
In materials with a relatively small
Wannier–Mott exciton
In semiconductors, the dielectric constant is generally large. Consequently,
In single-wall
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,[8] 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.[9] For the energy, we have
where is the Rydberg unit of energy (cf. Rydberg constant), is the (static) 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
For a simple screened Coulomb potential, the binding energies take the form of the 2D hydrogen atom[11]
- .
In most 2D semiconductors, the Rytova–Keldysh form is a more accurate approximation to the exciton interaction[12][13][14]
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.[10]
Example: excitons in transition metal dichalcogenides (TMDs)
Monolayers of a transition metal dichalcogenide (TMD) 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[2] 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.[2]
0D semiconductors
In
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
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.[17]
Charge-transfer exciton
An intermediate case between Frenkel and Wannier excitons is the charge-transfer (CT) exciton. In molecular physics, CT excitons form when the electron and the hole occupy adjacent molecules.[18] They occur primarily in organic and molecular crystals;[19] in this case, unlike Frenkel and Wannier excitons, CT excitons display a static electric dipole moment. CT excitons can also occur in transition metal oxides, where they involve an electron in the transition metal 3d orbitals and a hole in the oxygen 2p orbitals. Notable examples include the lowest-energy excitons in correlated cuprates[20] or the two-dimensional exciton of TiO2.[21] Irrespective of the origin, the concept of CT exciton is always related to a transfer of charge from one atomic site to another, thus spreading the wave-function over a few lattice sites.
Surface exciton
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.
Dark exciton
Dark excitons are those where the electrons have a different momentum from the holes to which they are bound that is they are in an optically
Atomic and molecular excitons
Alternatively, an exciton may be described as an excited state of an atom, ion, or molecule, if the excitation is wandering from one cell of the lattice to another.
When a molecule absorbs a quantum of energy that corresponds to a transition from one
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
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. This phenomenon is generic and applicable both to the large radius (Wannier–Mott) excitons and molecular (Frenkel) excitons. Hence, excitons bound to impurities and defects possess giant oscillator strength.[29]
Self-trapping of excitons
In crystals, excitons interact with phonons, the lattice vibrations. If this coupling is weak as in typical semiconductors such as GaAs or Si, excitons are scattered by phonons. However, when the coupling is strong, excitons can be self-trapped.[30][31] Self-trapping results in dressing excitons with a dense cloud of virtual phonons which strongly suppresses the ability of excitons to move across the crystal. In simpler terms, this means a local deformation of the crystal lattice around the exciton. Self-trapping can be achieved only if the energy of this deformation can compete with the width of the exciton band. Hence, it should be of atomic scale, of about an electron volt.
Self-trapping of excitons is similar to forming strong-coupling polarons but with three essential differences. First, self-trapped exciton states are always of a small radius, of the order of lattice constant, due to their electric neutrality. Second, there exists a self-trapping barrier separating free and self-trapped states, hence, free excitons are metastable. Third, this barrier enables coexistence of free and self-trapped states of excitons.[32][33][34] This means that spectral lines of free excitons and wide bands of self-trapped excitons can be seen simultaneously in absorption and luminescence spectra. While the self-trapped states are of lattice-spacing scale, the barrier has typically much larger scale. Indeed, its spatial scale is about where is effective mass of the exciton, is the exciton-phonon coupling constant, and is the characteristic frequency of optical phonons. Excitons are self-trapped when and are large, and then the spatial size of the barrier is large compared with the lattice spacing. Transforming a free exciton state into a self-trapped one proceeds as a collective tunneling of coupled exciton-lattice system (an instanton). Because is large, tunneling can be described by a continuum theory.[35] The height of the barrier . Because both and appear in the denominator of , the barriers are basically low. Therefore, free excitons can be seen in crystals with strong exciton-phonon coupling only in pure samples and at low temperatures. Coexistence of free and self-trapped excitons was observed in rare-gas solids,[36][37] alkali-halides,[38] and in molecular crystal of pyrene.[39]
Interaction
Excitons are the main mechanism for
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 (or more specifically 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
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.[43] This is often used to cool excitons to very low temperatures in order to study Bose–Einstein condensation (or rather its two-dimensional analog).[44]
See also
References
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