Codex @codex  0

The absorptive dielectric function counts optically accessible electron-hole pairs. In the electric dipole approximation the perturbation is proportional to $\mathbf{a}\cdot \hat{\mathbf{p}}$, with $\mathbf{a}$ the light polarization and $\hat{\mathbf{p}}$ the electronic momentum operator. The squared matrix element between occupied valence and empty conduction states is the transition strength; it includes polarization and band-symmetry selection rules. The sums enumerate initial and final subbands. photon momentum is negligible on the crystal-momentum scale, so the transition is approximately vertical in the in-plane wavevector $\mathbf{k}$.

For each subband pair, $C$ is the constant-transition-energy contour in the two-dimensional Brillouin zone. Its line element is $dl$, and the reciprocal gradient weights the states satisfying energy conservation. Specifically, the level-set representation of the joint density of states isA small energy gradient means many nearly resonant states; a vanishing gradient can produce a Van Hove singularity or an edge. The displayed expression suppresses overall optical prefactors and assumes the momentum matrix element is approximately constant along each contour. More generally it stays inside that integral. Occupation factors $f_v-f_c$ are also implicit; at low temperature in an undoped well the relevant valence states are full and the conduction states empty. This is the independent-particle continuum contribution from Fermi's golden rule. Bound excitons add optical resonances below the corresponding free-pair thresholds.

The GaAs four-band model is rooted in bonding and spin-orbit coupling. Near the zone centre, the conduction edge is predominantly s-like and antibonding, while the upper valence manifold is predominantly p-like. Combining orbital angular momentum $l=1$ with spin $s=1/2$ splits the six p-derived states into a $j=3/2$ quartet and a lower $j=1/2$ doublet. The latter is the split-off valence band. Within the upper quartet, the $m_j=\pm 3/2$ and $m_j=\pm 1/2$ branches, defined relative to a propagation or quantization direction, have different curvatures and are called heavy holes and light holes. They coincide at the unstrained bulk zone centre but need not remain degenerate in a well. The conduction edge is a spin doublet. Thus “four bands” counts conduction, heavy-hole, light-hole and split-off branches, including their spin multiplicities; it does not count four individual states. Positive hole masses describe the downward valence curvature, whereas the conduction curvature is upward.

For the final exciton comparison, define the reduced mass ${\mu }_j$ and screened exciton Rydberg $R_j^{\ast }$ byThe resulting $({\mu }_j/m_e,R_j^{\ast })$ are approximately $(0.05948,4.937\mathrm{meV})$ for heavy holes, $(0.03646,3.027\mathrm{meV})$ for light holes and $(0.04631,3.844\mathrm{meV})$ for split-off holes. In the ideal two-dimensional exciton model, the first two optically active internal states lie $4R_j^{\ast }$ and $4R_j^{\ast }/9$ below their continuum edge. Their separation is $32R_j^{\ast }/9$, independent of a common threshold offset. The corresponding three-dimensional separation would be only $3R_j^{\ast }/4$. The listed peak spacings specified for this theoretical spectrum giveHere each pair comprises the stronger lower-energy peak and its weaker nearby companion, in the order associated with the continuum edges calculated in part (c). The approximately 18-meV and 11-meV pair splittings support the two-dimensional series and are incompatible with the three-dimensional series at the supplied masses and permittivity. Equivalently, for a two-dimensional pair with energies $E_1,E_2$, the inferred edge is $E_2+(E_2-E_1)/8$; the observed pair splittings predict ground-state bindings around 20 and 12 meV, close to $4R_j^{\ast }$.

Absolute agreement should not be overstated. For example, the stated square-well parameters give the first heavy-hole edge $1.56390$ eV and the first light-hole edge $1.59161$ eV. Their ideal two-dimensional peak predictions are approximately $(1.54415,1.56170)$ eV and $(1.57950,1.59027)$ eV, respectively. The listed peaks are lower by about 1–2 meV across the families; rounded parameters and this simple effective-mass model do not reproduce every absolute energy exactly. No gap, width or mass has been silently adjusted to fit them. The peak-spacing comparison remains useful because it avoids a common edge offset. The split-off transitions predicted by the four-band data are discussed separately in part (c), rather than assigned fictitious extra peaks.

At 300 K, $k_{B}T\simeq 25.9$ meV, whereas the ideal ground-state binding energies are only about $19.7$, $12.1$ and $15.4$ meV; excited states are still more weakly bound. None of these series is expected to remain a clearly resolved, robust bound-exciton peak at room temperature in this simple estimate. Thermal ionization and phonon broadening work against their visibility. This is a scale argument, not a theorem that the exciton population or all Coulomb-enhanced absorption becomes exactly zero: actual observability also requires temperature-dependent gaps, linewidths, screening and carrier density, which are not supplied.

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.

For the quoted cubic-phase gap,The same exciton binding argument can shift the actual emission to longer wavelength; impurity-bound transitions can also lie below the fundamental gap. The comparison must use $E_g$ at the operating temperature. Ordinary temperature-dependent gap changes can shift the band edge in the opposite direction, so the numerical values based on fixed supplied gaps are not universal lower bounds for every cooled device.

Using $E_{\gamma }=hc/\lambda$ and an ideal band-edge photon energy equal to the quoted gap,At low temperature an electron and hole can form a bound exciton. Its recombination releases approximately $E_g-E_B$, so its wavelength is longer than $hc/E_g$ evaluated using the gap at that same temperature. GaN excitation and emission measurements resolve such excitonic transitions below the fundamental gap.^[ref]

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.

The sharp solid curve is the $4.2$ K spectrum; the broadened dashed curve is the $77$ K spectrum. At low temperature, a strong discrete exciton transition appears just below the continuum edge, with weaker higher members converging toward the band gap. Heating broadens those resonances through scattering, reduces their spectral resolution, and shifts the band edge to lower photon energy. The lower-energy broad dashed maximum is therefore not evidence for a second unrelated low-temperature band gap.

Read approximate energies from the native figure: the main low-temperature peak is near $1.610$ eV and a weaker feature is near $1.617$–$1.618$ eV, with the continuum edge around $1.620$ eV. A three-dimensional hydrogenic series $E_n=E_g-R^{\ast }/n^2$ then givesFor example, $E_2-E_1=3R^{\ast }/4$ predicts about $7.5$ meV, consistent with those feature spacings. These are graphical estimates at roughly meV precision, not extra digits extracted from the line widths. The strong direct interband onset and subgap series support a direct-gap Wannier-Mott interpretation. The continuum absorption scale is of order $10^6$ m${}^{-1}$, corresponding to an optical absorption length of order a micrometre. At $77$ K, $k_{B}T\simeq 6.6$ meV is already comparable with the inferred binding scale, explaining why a sharp Rydberg series is poorly resolved; at $4.2$ K, $k_{B}T\simeq 0.36$ meV.

With $m_e^{\ast }=m_h^{\ast }=0.2m_e$, the reduced mass is ${\mu }^{\ast }=0.1m_e$. In the screened hydrogen model,It follows thatA radius of several nanometres is consistent with an extended exciton in an ordinary semiconductor lattice. The relative permittivity inferred here is the effective dielectric screening parameter of that model; identifying the material chemically or determining a full dielectric spectrum is not possible from these data alone.

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.

Let $n=|\Psi {|}^2$. Two independent approximations are required for the reservoir reduction of a polariton condensate. First the exciton reservoir must follow the changing condensate number density rapidly: after its initial transient, its relaxation time $[{\gamma }_R+R_{R}n{]}^{-1}$ must be small compared with the local density-evolution time. A sufficient modewise condition is that the relevant number density frequencies and growth rates are much smaller than ${\gamma }_R+R_{R}n$; pump variations must be comparably slow. There is no reservoir diffusion in the given model. Adiabatic elimination then givesSecond, to obtain the particular cubic equation by expansion about the empty condensate, require $\epsilon =R_{R}n/{\gamma }_R\ll 1$. ThenKeeping the full denominator instead gives a saturable-gain equation, not exactly the requested cubic complex Ginzburg–Landau equation. A near-threshold condensate is one natural regime in which the second condition holds. Rapid reservoir relaxation alone does not justify this number density expansion.

Multiplying the condensate equation by $-i$ and inserting the expanded reservoir separates real gain from the conservative frequency shift:These are local functions if the pump is spatially varying. Their signs are important: the reservoir blueshift gives negative $s$ in the printed convention $+is\Psi$, while reservoir depletion subtracts from the effective interaction $g$. Positive rates and pump give $\beta >0$; the effective $g$ need not be positive merely because the original interactions are repulsive.

A suitable donor-acceptor heterojunction provides an energetically favourable route from a bound excitation to an interfacial charge-transfer state and then separated carriers. The electron is preferentially transported in the acceptor and the hole in the donor. The interface facilitates dissociation that is inefficient in a single strongly excitonic material; it must also avoid excessive energy loss and rapid interfacial recombination.

An idealized bulk heterojunction places most generated excitons within roughly a diffusion length of an interface while maintaining continuous donor paths to the hole contact and acceptor paths to the electron contact. Domain dimensions should therefore be comparable to the material's exciton transport scale, not arbitrarily large. The total absorbing thickness should capture substantial light yet remain within the charge-collection capability set by mobility, field and recombination. Excessive mixing can impair transport or increase recombination; very coarse domains waste excitons before they reach an interface.^[ref]

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.

Relative to conventional crystalline-silicon technology, many organic absorbers offer low-temperature thin-film processing and lightweight flexible devices, but have more strongly bound excitons and more disorder-limited charge transport. These are common material trends, not universal statements about every organic or inorganic device.

Relevant semiconductor properties include the absorption spectrum and optical gap, donor/acceptor energy levels and offsets, dielectric screening and exciton dissociation, electron/hole mobilities, recombination lifetimes, traps and morphological stability. Contacts must extract the appropriate carrier selectively. These properties determine the device quantities inA favourable gap alone does not fix efficiency: current generation, voltage loss and fill factor must all be controlled.

The emission energies are approximatelyDraw the Al-rich interface as a wider-gap barrier between these two emitting regions: a raised conduction-band edge impedes electron transfer, and a lowered valence-band edge impedes hole transfer. The following is a consistent straddling-barrier schematic, with the shell having the smaller optical gap.

The two emission wavelengths determine optical transition energies, not unique conduction- and valence-band offsets. Exciton binding, confinement and strain can shift emission relative to the bulk band gap. The stated barrier supports inhibited charge exchange; its separate electron and hole barrier heights need additional band-alignment data.

An exciton is a correlated electron–hole pair with a full wavefunction $\Psi ({\mathbf{r}}_e,{\mathbf{r}}_h)$. In strong confinement, a useful leading approximation is a product of lowest electron and hole envelope functions, followed by a Coulomb correction. A spherical hard-wall ground envelope is nodeless, proportional to $\sin (\pi r/R)/r$, and finite at the origin. Finite barriers produce exterior tails.

For a core-shell semiconductor nanocrystal with type-I band alignment, both conduction- and valence-band offsets confine the carriers mainly to the core. Their envelopes penetrate the shell but overlap less with external traps. Chemical surface passivation and spatial separation from defects can suppress nonradiative losses and increase photoluminescence quantum yield. Shell thickness, interfacial quality and strain must be controlled; a defective shell can introduce losses.

A larger shell band gap alone does not guarantee type-I alignment. A staggered type-II band alignment can move one carrier toward the shell, reduce electron–hole overlap and alter lifetime and emission energy. The sketch illustrates the conditional type-I case. In CdSe/ZnS experiments, optical spectra were combined with structural and compositional measurements to test shell-growth interpretations.^[ref]

One application is a semiconducting nanotube transistor array. Removing metallic tubes prevents conducting bridges that would bypass gate control; a narrow chirality distribution also narrows the spread of band gaps and switching behaviour. High charge carrier mobility and thin flexible channels are attractive, but contact resistance, tube alignment, residual metallic contamination, dielectric traps and the cost/yield of sorting still limit uniform large-area circuits.

A second is a wavelength-selective near-infrared photodetector or emitter. A selected chirality concentrates absorption and emission around its characteristic nanotube optical transitions, avoiding the broad mixed-tube response. Optical selectivity can be tailored by choosing the species. Limitations include low absolute emission yield, dark exciton states, contact-related recombination, environmental spectral shifts and quenching by bundles or metallic contaminants. Optical and electrical characterization after device processing is needed; the sorting label alone does not guarantee device performance.

With the graphene lattice-vector length $a=0.246\mathrm{nm}$, the carbon nanotube chiral vector is ${\mathbf{C}}_h=n{\mathbf{a}}_1+m{\mathbf{a}}_2$. The vectors meet at $60^{\circ }$, givingHere $C$ is circumference, $d$ diameter, and the carbon nanotube chiral angle $\theta$ is measured from the zigzag direction. Nanotube zone folding gives a nominal metallic tube when $n-m\equiv 0(\mathrm{mod}3)$ and a semiconducting tube otherwise. Evaluating all the supplied indices gives the following derived results:All eleven listed tubes are semiconducting in the zone-folding classification. Curvature can open small gaps in some nominally metallic non-armchair tubes, but none of these indices falls in that nominal metallic class.

Diameter gives a useful inverse-size trend in transition energies, and the full chiral indices constrain the electronic band structure much more strongly. They do not by themselves give the perceived solution colour through a simple geometrical rule: one needs the visible absorption spectrum, transition strengths and widths, exciton/environment shifts, concentration, optical path and illumination. The observed transmitted colour is complementary to the absorbed spectrum, not a direct label for the band gap. Structure-assigned excitation–emission measurements provide an experimental connection between chiral structure and optical transitions.^[ref]

Rolling graphene into a single-walled carbon nanotube quantizes the circumferential wave vector, leaving a set of one-dimensional longitudinal subbands. At each subband edge the group velocity tends to zero, producing a Van Hove singularity in the density of states. For a simple gapped Dirac subband, the contribution scales as $|E|/\sqrt{E^2-{\Delta }_j^2}$ outside its edge. At low temperature the valence states below the Fermi level are occupied and the conduction states above it are empty. A semiconducting tube has a gap; an ideal metallic tube has gapless channels as well as higher-subband singularities.

For light polarized along the tube, the strongest allowed interband transitions connect corresponding subbands while approximately conserving axial wave vector and circumferential angular momentum. The large joint density of states makes distinct absorption resonances, conventionally labelled $E_{11},E_{22},\dots$. Strong electron–hole interactions produce bound excitons, so the observed optical resonances differ from the noninteracting band separations. Defects, doping, bundling and dielectric screening change peak energies and oscillator strengths. The original schematic below depicts the subband-edge structure, not measured spectra.

A single-walled carbon nanotube confines the electron around its circumference while allowing motion along its axis. Nanotube zone folding therefore turns the graphene bands into one-dimensional subbands. A semiconducting carbon nanotube has no allowed circumferential wave vector through a Dirac valley; a metallic tube has a band crossing at the Fermi energy in the ideal model. Each massive subband edge gives a one-dimensional van Hove singularity in the density of states, and corresponding peaks appear in the joint density of states.

For light polarized along the tube, the main electric-dipole selection rule conserves circumferential angular momentum: transitions connect matching valence and conduction subbands. These finite-energy transitions are conventionally $S_{11},S_{22},\dots$ in a semiconducting carbon nanotube and $M_{11},M_{22},\dots$ in a metallic tube. A metallic band crossing does not make its first strong interband peak have zero energy. Perpendicular polarization admits different angular-momentum channels, often much weaker because of depolarization. Excitons, dielectric screening and trigonal warping modify the simple band-edge transition energies.

Experimentally, measure absorption versus photon energy; Raman spectroscopy becomes resonantly enhanced when the excitation matches an optical transition. A photoluminescence excitation spectrum of dispersed semiconducting tubes correlates excitation near $S_{22}$ with emission near $S_{11}$ after relaxation. Metallic tubes generally lack comparable bright emission because nonradiative relaxation is efficient. Combining optical energies with radial-breathing-mode Raman spectroscopy or electron diffraction assigns tube diameter and roll-up indices; a Kataura plot organizes those assignments.^[ref]

A Landau quasiparticle is an electron-like excitation dressed by the interacting medium. Adiabatic continuity connects low-energy states of an ordinary Fermi liquid to those of a free Fermi gas without changing their particle statistics or conserved quantum numbers. The excitation has a renormalized dispersion, effective mass, magnetic response and finite lifetime. Pauli exclusion principle restricts collisions near the Fermi surface; in a regular three-dimensional Fermi liquid,Thus the width is small compared with excitation energy sufficiently near the surface at low temperature. This permits particle-like counting and transport even when microscopic interactions are strong. It does not make every interaction disappear: the Landau quasiparticle energy functional contains residual interactions,Specific heats and magnetic susceptibilities need not equal their bare-gas values. Logarithmic corrections can modify the lifetime scale in two dimensions, and a Luttinger liquid or a fractional Hall fluid is not an ordinary weakly perturbed electron gas.

An electron-like excitation in a normal metal and a neutral fermionic excitation in normal helium-3 are Landau quasiparticles. More generally, phonons, magnons and excitons are particle-like collective excitations, with their own statistics and quantum numbers; they are not all Landau quasiparticles. composite fermions are another emergent description, reorganizing correlations in a partly occupied Landau level rather than simply replacing the electron mass in an ordinary band.

Attach two statistical flux quanta ${\Phi }_0=h/e$ to each electron. Even attachment preserves fermionic exchange statistics, while the average statistical field opposes the applied field:This is an effective description of correlations, not literal flux tubes produced by small solenoids. An integer number of filled effective levels can give an incompressible electronic fractional Hall state. This is the correspondence between integer Hall quantization of composite fermions and selected fractional Hall states of electrons.^[ref]

Setting the signed effective filling to integer $p$ givesPositive $p$ gives $1/3,2/5,3/7,\dots$ below $1/2$. Negative $p=-q$, $q>0$, gives $q/(2q-1)$ above it, including $2/3,3/5,4/7,\dots$. The sign records reversed $B^{\ast }$. At $\nu =1/2$, the mean effective field vanishes, so the simplest description is a compressible composite-fermion sea rather than another member with finite integer $p$. Fractional charged excitations of gapped Hall states and the charge used in this semiclassical composite-fermion orbit description should not be conflated.

A direct probe away from the incompressible fractions is composite-fermion geometric resonance. One implementation patterned a high-mobility GaAs/AlGaAs Hall bar into periodic and random antidot arrays. electron beam lithography followed by an approximately $130\mathrm{nm}$ reactive-ion etch made arrays with periods $400$–$700\mathrm{nm}$. Four-terminal low-current measurements swept a perpendicular magnetic field at low temperature; red-light illumination varied density and the modulation. Periodic arrays showed commensurability structure near half filling and a Hall anomaly; matched random arrays did not. Unpatterned sections supplied another reference.^[ref]

The semiclassical explanation uses, for a fully spin-polarized sea,The mass cancels from the orbit radius. A selected orbit fits the lattice when $2R_c^{\ast }\simeq Ca$, with a geometry-dependent constant $C$ of order unity. Pinned orbits around antidots then alter longitudinal transport, producing features nearThe simplest single-antidot scale has $C\simeq 1$; detailed peak positions require the actual potential and orbit family. The two field directions bend orbits oppositely. The diagnostic is an orbit scale controlled by the small residual $B^{\ast }$, rather than the much larger applied $B$. At the same density the fully polarized $k_F^{\ast }$ is $\sqrt{2}$ times the zero-field spin-degenerate electron value; this is an independent counting test.

Calibrate density from independent Hall/filling measurements and compare several lattice periods. Use small excitation and temperature sweeps, and check periodic, random and unpatterned regions. Density variation changes both $B_{1/2}$ and the orbit scale, so simply plotting against uncorrected field can mislead. mean free path and density inhomogeneity limit resolution; illumination can soften or substantially change the imposed potential, so it cannot automatically be treated as density adjustment alone. These qualifications are part of the inference from geometry to emergent particles, rather than a claim that every feature near half filling is a new fractional plateau.

Carrier confinement changes both the energies and the available optical transitions of a semiconductor. A quantum well confines one direction and leaves two continuous momentum coordinates; a wire confines two directions and leaves one; a quantum dot confines all three and has discrete ideal levels. Confinement becomes important when its level spacing is comparable with thermal, lifetime and disorder energy scales, and strong excitonic confinement involves comparison with the exciton Bohr radius.

Transition energies. In a simple infinite-well effective-mass model of width $L$, a same-index electron-hole transition has thresholdThus decreasing $L$ generally blue-shifts interband absorption and emission through the $L^{-2}$ confinement contribution. Finite barriers, band mixing, strain, nonparabolicity, dielectric environment and Coulomb interactions alter this elementary estimate. In dots, confinement shifts and exciton binding have different size dependences, so a single inverse-square law is not an exact nanocrystal spectrum.

Dimensional density of states. For a parabolic continuum the phase-space volume below kinetic energy $E$ scales as $E^{d/2}$, hence the density of states scales as $E^{d/2-1}$. Each subband threshold therefore contributes a square-root onset in three free dimensions, a constant step in two, and an inverse-square-root singularity in one. A zero-dimensional ideal dot has discrete lines rather than a continuum power law. Real temperature, disorder and lifetime broadening smooth singularities and lines. Optical absorption uses the joint electron-hole density of states multiplied by transition matrix elements and occupations; a bare electronic density of states is not the entire absorption spectrum.

Selection and Coulomb effects. Envelope-function overlap and Bloch-state dipole selection rules determine allowed transitions and their polarization. For equal ideal infinite-well envelopes, orthogonality favors matching transverse indices; finite barriers and asymmetric fields can relax that envelope rule. Reduced dimensionality can enhance electron-hole attraction and oscillator strength near an exciton resonance, but the binding energy depends on the actual confinement and dielectric screening. Intersubband transitions within a conduction band are different from interband recombination; in a quantum well they require an electric-field polarization component along the confinement direction.

Devices. Quantum wells, wires and dots allow wavelength engineering, carrier localization and altered optical gain for LEDs and semiconductor lasers. Concentrating states near selected transition energies can improve differential gain and reduce thresholds under appropriate injection and loss conditions. Dot ensembles can instead have substantial inhomogeneous broadening from size variation; nonradiative surface traps and carrier capture can limit an otherwise attractive design. Material confinement does not alone ensure high quantum efficiency.

An applied field across a quantum well shifts electron and hole envelopes apart, usually red-shifting the exciton transition and reducing overlap: the quantum-confined Stark effect. This supports electroabsorption modulation and tuneable optical transitions. Conduction-band intersubband energies can also be engineered for infrared detectors and cascade-laser designs. Optical waveguides and cavities confine photons and affect mode overlap or spontaneous-emission rates, but that photonic confinement is distinct from the carrier quantization derived above. Both must be specified when predicting a real device's threshold, spectrum and output.

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.