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Modification of the interaction between electric charges by a medium's electrical response. In the simplest Wannier-Mott exciton model, the Coulomb potential is reduced by a constant relative permittivity; actual screening can depend on spatial and temporal scales.

An exciton is a bound state of an electron and an electron hole. Their opposite electric charges attract through the Coulomb potential, producing a neutral quasiparticle with internal relative motion and motion of its center of mass. In a semiconductor, promoting an electron from the valence band into the conduction band leaves an electron hole; the pair can occupy either an exciton bound state or an unbound scattering state. Merely creating the two charge carriers does not establish binding. A neutral exciton can transport excitation energy without transporting net electric charge. Coupling this material excitation strongly to a photon can form a polariton.

The striking optical consequence is that an exciton optical transition can absorb a photon with less energy than the threshold for creating unbound electrons and electron holes. The difference is the exciton binding energy, not an extra one-electron energy band inside the band gap. The Wannier-Mott exciton supplies a useful large-radius model; the Frenkel exciton describes the opposite, tightly localized limit.^[ref]

The characteristic relative-motion length of a hydrogenic Wannier-Mott exciton, with dielectric screening and exciton reduced mass ${\mu }^{\ast }$. Comparing a quantum dot's dimensions with this scale helps distinguish strong carrier confinement from confinement of a comparatively intact bound pair.

For the ideal Wannier-Mott exciton in energy bands whose edges occur at the same crystal momentum, the pair excitation energy, measured from the unexcited crystal, isHere $\hslash \mathbf{K}$ is the pair's total crystal momentum; the last term is center of mass kinetic energy. For a permitted photon absorption transition with negligible recoil energy, $\hslash \omega \simeq E_g-E_{B,n}$. An electric-dipole selection rule and a nonzero transition matrix element are still needed: a bound state need not produce an observable spectral line. If the relevant energy band edges have different values of crystal momentum, a phonon can also be needed to conserve crystal momentum; subtracting a binding scale from a band gap alone does not specify the optical threshold.

For example, the lowest internal bound state in this hypothetical model costs $1.489115\mathrm{eV}$ to create optically, yet needs only $0.010885\mathrm{eV}$ more to dissociate. The large creation energy and the small exciton binding energy answer different physical questions. An electron-hole recombination event can instead release excitation energy by photon emission or to other degrees of freedom; an emitted photon need not reproduce the original absorption energy if relaxation intervened.

A Frenkel exciton is an exciton whose internal excitation is concentrated on the scale of one molecule or unit cell. The local excitation can transfer between sites; localized internal structure does not require an immobile center of mass. The large-radius Wannier-Mott exciton approximation cannot simply be extrapolated to this limit.^[ref]

A Wannier-Mott exciton is a bound conduction-band electron and valence-band hole whose envelope extends over many lattice cells. That large radius permits an effective mass description and dielectric screening of the electron–hole attraction. With electron and hole masses $m_e^{\ast },m_h^{\ast }$, define reduced mass $\mu =m_e^{\ast }{m}_h^{\ast }/(m_e^{\ast }+m_h^{\ast })$ and center-of-mass mass $M=m_e^{\ast }+m_h^{\ast }$. The bulk relative-motion Hamiltonian is hydrogenic with an appropriate relative permittivity ${ϵ}_r$:A semiconductor quantum well confines the electron and hole perpendicular to the plane while leaving approximately free in-plane center-of-mass motion. If the well width is small compared with the bulk exciton Bohr radius, their relative motion becomes quasi-two-dimensional. The ideal zero-width, same-dielectric limit isIts ground-state binding is $4R_X$, compared with $R_X$ in three dimensions, and the ground envelope decays on the scale $a_X/2$. Stronger binding follows because confinement prevents the pair from separating in the third direction.

For a finite-width quantum-well exciton, average the Coulomb attraction over the normalized electron and hole confinement envelopes:This smooths the strict planar singularity and changes the binding. The ideal factor four is a useful reference, not a universal value for every real well. Finite barriers, dielectric mismatch, several confinement subbands and nonparabolic valence bands can require a coupled treatment.

Optical absorption creates a nearly zero in-plane center-of-mass wavevector exciton. For the lowest confinement subbands, its line lies approximately atThe discrete bound-state peaks occur below the free electron–hole continuum. Dipole-allowed envelope states have nonzero relative wavefunction at zero separation; the absorption strength also contains the confinement overlap $|\int {\chi }_e^{\ast }(z){\chi }_h(z)dz{|}^2$. Confinement can increase the overlap and the binding, making excitonic optical structure more prominent. A perpendicular electric field shifts and separates the confinement envelopes, changing the transition energy and usually reducing their optical overlap: this is the quantum-confined Stark effect.

Quantum wells tune both the continuum threshold and the bound-state energy; an optical line shift is not automatically a change in binding alone. The hydrogenic envelope model requires a Wannier-sized state, rather than an exciton localized to one lattice cell.

Relative electron–hole motion uses the exciton reduced mass, rather than either band mass alone:In a screened hydrogenic Wannier-Mott exciton, replace $m_e$ by ${\mu }^{\ast }$ and ${\epsilon }_0$ by ${\epsilon }_r{\epsilon }_0$ in the hydrogen binding-energy scale. ThereforeThe effective Rydberg is an energy here, not a spectroscopic inverse length. Its scale is reduced both by dielectric screening and by the small band reduced mass. The associated radius is about ${\epsilon }_r(m_e/{\mu }^{\ast })a_0=23.5$ Å, several lattice spacings, consistent with using an extended-pair model as a first approximation rather than a tightly localized molecular exciton.

A Frenkel exciton is a tightly bound electron–hole excitation localized on approximately one molecular or ionic unit. Its radius is comparable to the lattice spacing, its exciton binding energy can be large, and transport is described through transfer of the local excitation between units. Molecular solids such as crystalline anthracene provide a natural example: the optical excitation is largely associated with an individual molecule, so a broad parabolic-band hydrogen model is not the most useful starting point.

A Wannier-Mott exciton extends over many unit cells. It is described approximately by a conduction electron and a valence hole with effective masses and a dielectric-screened Coulomb attraction. Its binding is usually smaller, and its levels resemble a hydrogenic series. gallium arsenide is a suitable example because its direct-gap band-edge carriers have light effective masses and substantial dielectric dielectric screening, supporting spatially extended excitons. Localization length and dielectric screening, rather than whether a material happens to be called a semiconductor, distinguish the two limiting pictures. Real materials can lie between the limits, so neither family is a universal sharp classification.

An exciton is a Coulomb-correlated electron-hole pair, electrically neutral overall. Three useful properties are:

In many molecular organic semiconductors, limited screening and localization make exciton binding important, so absorption need not immediately produce collectable free charges.

A self-assembled InAs quantum-dot single-photon source uses small indium arsenide islands inside a larger-gap gallium arsenide host. Strain during epitaxial growth favors islands after a thin wetting layer has formed. Their characteristic diameter is of order $10$–$30\mathrm{nm}$ and their height a few nanometres, for example $2$–$5\mathrm{nm}$. Growth conditions determine the areal density: low-density samples suitable for isolating dots can be around $10^8$–$10^9\mathrm{c}{\mathrm{m}}^{-2}$, while ordinary dense layers can be much denser. The particular electrically driven device underlying the supplied image had about $5\times 10^8\mathrm{c}{\mathrm{m}}^{-2}$. These are characteristic scales, not dimensions deducible from the emission image.^[ref]

Quantum confinement in the dot traps both electrons and electron holes in three directions, giving discrete energy levels. In a junction device, injected carriers relax into these levels. Electron-hole recombination then emits a photon, and the dot must be repopulated before emitting again on that same transition. Spatial selection of one dot, spectral selection of one line, and well-separated injection pulses make this a single-photon source. Photon antibunching, rather than low average brightness alone, establishes suppression of simultaneous photons. In particular, collecting both photons of a cascade without selecting a line would not produce exactly one collected photon per excitation.

The lower-current feature $X$ is the transition from one confined exciton to the empty dot. The higher-current feature $X_2$ is the first step of the biexciton cascade $XX\to X\to 0$: it becomes strong when the dot can contain two electron-hole pairs. In the weak-pumping limit the probabilities of one and two pairs scale approximately as $I$ and $I^2$, respectively. Coulomb and confinement energies shift the two transitions relative to one another; $X_2$ is not a photon of twice the $X$ energy. In this particular image the $X_2$ photon is about $4$–$5\mathrm{meV}$ higher in energy. With $B_{XX}=2E_X-E_{XX}$, this is a negative biexciton binding energy, since $E_{XX}-E_X=E_X-B_{XX}$.

For the mass estimate, the original plot places the $X$ line at approximately $E_X=1.3945\mathrm{eV}$, with graph-reading precision of order $0.5\mathrm{meV}$. Under the prescribed identification of photon energy with the full lowest exciton excitation energy,The three-dimensional screened Wannier-Mott exciton has the hydrogenic exciton binding energyThereforeReading the line as $1.395\mathrm{eV}$ instead gives $0.331m_0$; about $0.33$–$0.34m_0$ is an appropriate precision. This inference uses the supplied pair-creation energy and a three-dimensional screened hydrogenic model. In a real dot, strain and confinement also alter transition energies, so it is an effective model estimate and does not independently determine either individual carrier mass.

The dimensionless ratio of a medium's permittivity to vacuum permittivity. The constant scalar value used in a Wannier-Mott exciton is an approximation to the relevant material response.

A Wannier-Mott exciton has an internal wave function extending over many unit cells. Its simplest model assumes a three-dimensional bulk crystal, approximately parabolic valence band and conduction band, positive isotropic scalar effective masses $m_e^{\ast },m_h^{\ast }$, a constant scalar relative permittivity ${ϵ}_r$, and a dilute pair population. Replace microscopic crystal forces by those effective masses and dielectric screening, and neglect exchange corrections and coupling to phonons. These assumptions must hold on the spatial and energy scales of the particular exciton.

With relative coordinate $\mathbf{r}={\mathbf{r}}_e-{\mathbf{r}}_h$, the internal Hamiltonian operator is the three-dimensional hydrogen atom problem with a reduced mass and a screened Coulomb potential:Here $e$ is the elementary charge, ${ϵ}_0$ the vacuum permittivity, and $\hslash$ the reduced Planck constant. The internal energy eigenvalues, radius scale, and exciton binding energies areThe principal quantum number is $n=1,2,\dots$. This reduction and these scales are derived in Excitons in Semiconductor Quantum Dots by Matjaž Humar (§3, printed pp. 5–6, equations (13)–(26)).

To check the scales by dimensional analysis, write $C=e^2/(4\pi {ϵ}_0{ϵ}_r)$, which has units of energy times length. Then $a_B^{\ast }={\hslash }^2/(\mu C)$ is a length and $R^{\ast }=\mu C^2/(2{\hslash }^2)$ is an energy. Rescaling $r=a_B^{\ast }\rho$ makes $H_{\mathrm{rel}}/R^{\ast }=-{\nabla }_{\rho }^2-2/\rho$, the dimensionless hydrogen atom operator. The Bohr radius $a_B^{\ast }$ is a wave function scale, not the radius of a classical orbit; for this model's lowest internal bound state, the mean separation is $3a_B^{\ast }/2$.

For the diagrams' hypothetical inputs $m_e^{\ast }=0.10m_0$, $m_h^{\ast }=0.40m_0$, ${ϵ}_r=10$ and $E_g=1.500\mathrm{eV}$, where $m_0$ is the free electron mass, independent substitution gives $\mu =0.080m_0$, $a_B^{\ast }\simeq 6.615\mathrm{nm}$ and $R^{\ast }\simeq 10.885\mathrm{meV}$. Here electronvolts express energy. The exciton binding energies for $n=1,2,3$ are approximately $10.885$, $2.721$ and $1.209\mathrm{meV}$. These are model calculations, not measured material parameters. The large-radius approximation additionally requires $a_B^{\ast }$ to be much larger than the crystal's unit cell length scale. In a thin layer, with anisotropic effective masses, or with spatially varying dielectric screening, this three-dimensional formula is not generally valid.