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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 exciton binding energy is the positive energy needed to separate a bound electron and electron hole into the relevant unbound scattering states. Compare the exciton energy $E_X$ with the continuum threshold $E_{\mathrm{cont}}$ at the same total crystal momentum and for the same energy bands. The negative internal energy eigenvalue $-E_B$ and the positive separation cost $E_B$ are different sign conventions for related quantities. An exciton may move freely through a perfect crystal while remaining internally bound: “free exciton” does not mean an unbound 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.

An organic semiconductor has conjugated molecular units with strong intramolecular bonding and comparatively weak intermolecular electronic overlap. Optical excitation commonly promotes a $\pi$ electron from an occupied molecular orbital toward an unoccupied one. Weak dielectric screening and narrow bands favour bound Frenkel excitons, sometimes mixed with intermolecular charge-transfer states, rather than immediate production of free carriers. The optical excitation energy is below the unbound electron–hole threshold by the exciton binding energy, with relaxation conventions specified when comparing measured gaps.

The electric-dipole interaction selects bright states through their transition dipoles, producing polarization-dependent absorption in oriented samples. From a singlet ground state, strong electric-dipole absorption normally preserves spin and reaches singlet excitons; direct triplet excitation is weak without spin mixing. The Franck-Condon principle treats excitation as vertical on the nuclear-motion timescale. For one displaced harmonic vibration at low temperature, the displaced-oscillator Franck-Condon progression has relative strengthsHere the Huang-Rhys factor $S$ measures the displacement and $E_{00}$ is the zero-vibration transition energy. Vibrational relaxation before emission produces a Stokes shift. Molecular packing and excitonic coupling shift or split bright features and change their intensities; disorder and vibrations broaden them. Consequently a universal parabolic-band absorption edge is inappropriate for a localized molecular excitation.^[ref]

The Exciton Rydberg series lies below the unbound electron–hole threshold:Using $E_g=1.240$ eV and $R^{\ast }=0.038241$ eV givesTheir positive exciton binding energies relative to the continuum are $38.24$ meV and $9.56$ meV, respectively.

These are optical electron–hole excitation energies for negligible centre-of-mass momentum, not new occupied one-electron states inside the valence band. The simple model predicts their locations; observing a particular line also requires a nonzero optical transition matrix element and adequate linewidth.

At cryogenic temperatures, the intrinsic carrier concentration is exponentially small, while a low phonon population permits relatively sharp band-edge and exciton spectral features. Near room temperature, thermal electron–hole generation increases strongly:This generally raises intrinsic conductivity, although increased phonon scattering tends to lower the individual carrier mobilities. With a gap of $1.24$ eV, the material remains far from a degenerate metal at room temperature.

For optical structure, compare $k_{\mathrm{B}}T\simeq 25.9$ meV at 300 K with the exciton binding energies. The $n=2$ binding of $9.56$ meV is substantially smaller, so excited bound states are readily thermally disrupted and broadened. The ground binding of $38.24$ meV is larger but only about $1.48k_{\mathrm{B}}T$: the ground exciton need not disappear completely, yet strong thermal ionization and broadening can weaken its sharp line. A comparison with $k_{\mathrm{B}}T$ is a useful scale argument, not an exact bound-pair fraction independent of density or lifetime.

Thermal lattice expansion and electron–phonon interactions can also shift a real material's gap and optical edge. The given fixed tight-binding parameters contain no temperature dependence, so they do not determine the magnitude or even a universal sign of that extra gap shift. Within the stated model, the dependable trends are more thermally generated carriers and less distinct excitonic structure on warming.

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.

Assume type-I confinement of both carriers, and first use an infinite square well as an explicit estimate. For thickness $L=5$ nm and each carrier mass $0.2m_e$, confinement measured from the corresponding well band edge isThus the first three infinite-well levels are approximately $75$, $301$ and $677$ meV for each carrier. Interband envelope overlap favors matching quantum numbers when the electron and hole well profiles are the same. Ignoring exciton binding energy initially, their absorption thresholds areThe first two lie in the requested optical window. Two-dimensional subband joint densities of states produce step-like onsets, with excitonic peaks just below them, rather than only a bulk three-dimensional onset. Confinement strengthens electron–hole binding: the ideal strict-two-dimensional hydrogen ground binding is $4R^{\ast }\simeq 40$ meV, while a finite-width well interpolates toward the bulk value near $10$ meV. The $5$ nm width is comparable to the bulk exciton radius, so neither limiting binding value is exact. A schematic based on the infinite-well estimate therefore has first exciton peaks around $1.73$–$1.76$ eV and $2.18$–$2.21$ eV, followed by their subband continua; the $2.3$ eV barrier absorption is another feature near the top of the requested window.

The barrier gap alone does not specify the finite confinement energies. Under type-I alignment it supplies only $\Delta E_c+\Delta E_v\simeq 2.3-1.62=0.68$ eV. It does not give their division or the barrier effective masses, and a larger gap by itself does not prove type-I alignment. Infinite-well levels are consequently estimates, not uniquely determined measured levels.

For a concrete conditional comparison, assigning equal $0.34$ eV electron and hole barriers and taking the same mass on both sides gives finite square well levels $44.1$, $169.4$ and $331.7$ meV per carrier. These follow from $z\tan z=\sqrt{z_0^2-z^2}$ for even levels and $-z\cot z=\sqrt{z_0^2-z^2}$ for odd levels, with $z=kL/2$, $z_0=(L/2)\sqrt{2m^{\ast }{V}_0}/\hslash$. The corresponding unbound-pair optical thresholds would be $1.708$, $1.959$ and $2.283$ eV before exciton shifts. This example is deliberately conditional and demonstrates why band offsets must not be invented to make a unique finite square well spectrum. The requested schematic below states which approximation it uses.

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.

Thermal energy comparable with an exciton binding energy promotes dissociation and often washes out narrow bound-exciton resonances. Comparing $k_{B}T$ with binding energy is a scale estimate, not an exact disappearance criterion: densities, screening, oscillator strength and linewidths also matter.

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.