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A biexciton is a two-electron, two-electron hole excitation. If $E_X$ and $E_{XX}$ are total excitation energies relative to the empty emitter, the cascade $XX\to X\to 0$ emits photons of energies $E_{XX}-E_X$ and $E_X$. Define its binding energy by $B_{XX}=2E_X-E_{XX}$. The first cascade photon then has energy $E_X-B_{XX}$. Positive binding shifts that line below the single-exciton line; confinement can also produce negative binding, shifting it above. The cascade does not emit a single photon of twice the exciton energy.

An interpenetrating donor-acceptor morphology in a photovoltaic absorber. Small domains facilitate exciton access to interfaces, while continuous pathways in each phase allow separated charges to reach selective contacts.

The initially empty energy band above the band gap in the simple semiconductor description. Excited electrons in it can carry electric current; they can also bind to valence band electron holes as excitons.

Parabolic electron-hole continua with $d$ unconfined directions have cumulative state count proportional to $(E-E_{\mathrm{th}}{)}^{d/2}$. Differentiating gives the threshold joint density proportional to $(E-E_{\mathrm{th}}{)}^{d/2-1}$: a square root for bulk, a step for a well, and an inverse square root for a wire. A fully confined dot instead has discrete lines. Optical matrix elements, occupations, excitons and broadening determine the actual absorption beyond this counting argument.

An electron hole is an unoccupied electron quantum state in an otherwise nearly full energy band, often the valence band. “Filled” refers to occupation of the allowed quantum states, not to filling all space with a continuous fluid of electrons. Each state, including its spin label, can hold at most one electron by the Pauli exclusion principle. The almost full energy band and its few missing occupants are therefore simpler to describe as a background plus a few electron holes. The electron hole is a quasiparticle, not a Positron inserted into the crystal.

Removing an electron of electric charge $-e$ changes the background's electric charge by $+e$. Thus an electron hole carries positive electric charge relative to that background. Promoting an electron into the conduction band creates an electron hole without changing the whole crystal's electric charge: $(-e)+(+e)=0$. An unbound pair still consists of two separate charge carriers; only a bound state of their relative motion is an exciton.

The local cartoon explains why a vacancy can move: when a neighboring electron fills it, another vacancy appears where that electron was. The vacancy has moved opposite to that particular transfer of an electron. In an extended Bloch state, use crystal momentum instead of imagining a permanently empty atomic site. For a completely filled energy band the summed electric current vanishes. Omitting one electron with group velocity ${\mathbf{v}}_v$ removes its contribution $-e{\mathbf{v}}_v$, leaving a contribution $+e{\mathbf{v}}_v$. This fixes both the positive electric charge and the velocity convention of the electron hole.

More precisely, near a parabolic valence band maximum, set its energy $E_v(0)$ as reference and define the electron hole wave vector by ${\mathbf{k}}_h=-{\mathbf{k}}_v$. ThenThe occupied electron branch has negative curvature there, while the electron hole branch has positive effective mass $m_h^{\ast }$. The velocity equality compares the electron hole with the missing Bloch state's group velocity; it does not contradict the opposite vacancy motion in a local transfer cartoon. The sign conventions are listed in Phys 460 Lecture 17.

The attractive electrostatic interaction between an electron and an electron hole. In a screened effective-mass approximation its potential has the displayed form. It can bind an exciton and reduce its optical excitation energy below the corresponding unbound-pair threshold.

An electron-hole pair consists of an excited electron and the unoccupied valence state described as an electron hole. Promoting a valence electron across a semiconductor band gap creates such a pair without changing the total electric charge. The carriers can separate under an electric field, or undergo electron-hole recombination. A bound electron-hole pair is an exciton; an arbitrary unbound pair need not be an exciton.

An electron filling an electron hole removes both excitations relative to the filled-band background. Their electric charges cancel, consistent with charge conservation; the remaining electrons are still present. The released energy can leave through photon emission or other channels, including phonons. Formation of an exciton instead retains the excited electron and electron hole as a bound pair; recombination removes the pair. A valence band electron filling a nearby vacancy merely moves the electron hole and does not remove the total vacancy count.

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]

In Giant Rydberg Excitons in Cuprous Oxide by Tomasz Kazimierczuk and others, a polished $34\mu \mathrm{m}$ copper(I) oxide crystal was mounted without strain at $1.2\mathrm{K}$. A tunable single-frequency dye laser scanned photon energy; balanced detection compared transmitted light with a reference beam. Transmission dips resolved spectral lines assigned to bound excitons through principal quantum number $n=25$. Their convergence towards an unbound threshold supports a hydrogen-like internal series, with small systematic deviations.^[ref]

Controls mattered: restricting broadband illumination improved resolution of high-$n$ excitons; lowering laser power or expanding its spot reduced interaction-induced suppression. Balanced detection suppressed source fluctuations. The measurement probes optical resonances, not individual electron-electron hole trajectories or directly imaged radii. Electric-dipole selection rules favor P-envelope states in this material, unlike the hypothetical permitted lowest-state transition above. Finite resolution and crystal inhomogeneity affect spectral lines; the inferred binding series requires a continuum threshold and a model.^[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.

A characteristic transport length before exciton decay, commonly $L_D=\sqrt{D_X{\tau }_X}$ in a diffusion-decay convention. A root-mean-square displacement over a fixed time has dimension-dependent factors and is not automatically this same convention.

An exciton-polariton condensate has coherent macroscopic occupation of an exciton–photon hybrid mode. Finite photon lifetime makes it an open system: pumping a noncondensed reservoir can replenish condensate losses. A condensate equation coupled to a reservoir-rate equation describes gain saturation, interaction shifts and reservoir feedback; it need not possess a conserved Hamiltonian.

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]

An optical excitation between nanotube electronic subbands, often labelled $E_{ii}$ for prominent axial-polarization transitions. One-dimensional Van Hove singularities enhance the joint density of states, but selection rules, oscillator strengths and strong excitons interactions determine actual absorption peaks. Dielectric environment, bundling, doping and defects can change those peaks.

For a one-dimensional subband $E=E_0+\alpha k^2$ with $\alpha >0$, the density of states varies as $(E-E_0{)}^{-1/2}$ just above its edge. Optical peaks also depend on the joint density of states, selection rules and excitons.

Transfer of energy from an optical electromagnetic wave into matter. Electronic absorption depends on transition matrix elements, occupied and empty states, and the joint density of states. Bound excitons can create resonances below an independent-particle continuum edge.

The joint density of states counts vertical valence-to-conduction transitions at photon energy $\hslash \omega$. Optical absorption weights it by dipole or momentum matrix elements, occupation factors and optical-frequency prefactors. A Van Hove singularity occurs at a stationary transition energy ${\nabla }_{\mathbf{k}}(E_c-E_v)=0$. It can create a strong peak, edge or cusp in an absorption spectrum and thereby reveal electronic band structure. The relevant stationary quantity is the band difference, not necessarily either band separately. Forbidden optical transitions can suppress the feature; excitons, finite lifetimes and disorder can shift or broaden it. A two-dimensional saddle gives a logarithm, whereas a parabolic two-dimensional minimum gives a step. ^[ref]

Put $\delta =\hslash \omega -E_g$, $A={\hslash }^2/(2m^{\ast })>0$ and $\eta =\delta /A$. For $\delta >0$, the saddle-point optical transition contour is the two-branch hyperbolaFor $\delta <0$ it opens along y instead; at zero the limiting contour is $k_x=\pm k_y$. On either positive-energy branch,Using the opening spin factor2, one branch therefore contributesThe displayed integral in the question is this one-branch expression with ${\Lambda }_y=\pi /a$. With both hyperbola branches and the opening spin factor2, the full joint density of states is $J_{\mathrm{full}}=2J_{\mathrm{branch}}$. Equivalently, the printed coefficient describes the full two-branch contour if the spin degeneracy is1. The PDF itself has this factor-of-two mismatch; it cannot be removed by reparametrizing the complete contour.

There is also a cutoff qualification. For the strict square first Brillouin zone, $|k_x|,|k_y|\le \Lambda =\pi /a$, both coordinates must remain inside, giving ${\Lambda }_y=\sqrt{{\Lambda }^2-\eta }$ for $0<\eta <{\Lambda }^2$. Taking ${\Lambda }_y=\Lambda$ is a convenient local-saddle cutoff with the same leading singularity, not the exact intersection with that square. The quadratic dispersion is only a local band model, so the nonsingular constant depends on how the saddle region joins the actual bands.

For the source's cutoff, the logarithmic two-dimensional saddle joint density of states follows directly:Thus, in the one-branch normalization of the requested integral,The strict square cutoff has the same leading logarithm. This establishes the requested $C-\ln (\hslash \omega -E_g)$ shape, with a dimensionless logarithm after choosing the cutoff energy. Here $E_g$ is the saddle transition energy; it need not be the global minimum optical gap. The definition in the question has units of states per area per energy, despite being written as a function of frequency.

An exciton is an electron-hole bound state and behaves approximately as a boson when the gas is dilute compared with the inverse exciton volume. A condensation experiment needs a long enough lifetime for cooling, high phase-space density and evidence of spontaneous coherence; excessive density causes ionization or a Mott transition instead.

Bulk Cu₂O offers long-lived excitons, but heating, losses and Auger recombination complicate interpretation. Spectral narrowing or anomalous transport alone is insufficient: early claims required reinterpretation. Coupled quantum wells separate electron and hole layers, increasing lifetime and enabling cold indirect-exciton gases, though disorder, trapping and nonequilibrium flow must be distinguished from condensation.^[ref]

In a semiconductor microcavity, coherent strong coupling mixes an exciton with a light cavity photon to form polaritons with very small effective mass. Nonresonant pumping supplies particles without imposing the condensate's phase. The 2006 CdTe experiment reported a threshold, strong low-momentum occupation, increased temporal coherence and long-range spatial coherence. These combined signatures support an exciton-polariton condensate; a threshold or bright emission alone does not distinguish it from photon lasing. Check that strong coupling persists above threshold. Finite lifetime and pumping make this an open, driven system, so thermalization must be demonstrated rather than inferred merely from the word condensate.^[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.