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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.

For standard deviations of position and momentum,Confinement to a region of size $R$ requires a momentum spread of order $\hslash /R$, hence kinetic energy of order ${\hslash }^2/(2m^{\ast }{R}^2)$. A sufficiently small semiconducting nanocrystal therefore has discrete energy levels and an optical edge shifted above the bulk band gap.

An illustrative strong-confinement model iswhere $\alpha$ is a geometry-dependent number of order unity. The positive confinement contribution grows as $R^{-2}$; electron-hole Coulomb interaction contributes an attractive term of order $-R^{-1}$. The uncertainty principle motivates the scale, not the exact coefficient or oscillator strength. Strong size dependence occurs near the exciton Bohr radius. Finite barriers, dielectric mismatch and surface states modify this model, while size distributions broaden ensemble spectra.

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.