Exciton

The picture: a hydrogen atom inside a crystal

Start with a perfect, cold semiconductor: every valence-band state is full, every conduction-band state empty, and the material is transparent to photons whose energy is below the band gap E_g. Now shine in a photon with energy just at the gap. It lifts one electron from the valence band into the conduction band. That electron leaves behind a vacancy — a hole — which behaves like a real particle of charge +e and positive effective mass.

The freed electron and the hole are oppositely charged, so they attract each other through the Coulomb force, screened by the crystal's dielectric constant. If the attraction wins over the thermal jostling that would tear them apart, they form a bound, orbiting pair — an exciton. It is the solid-state analog of a hydrogen atom: a light negative particle (the electron) orbiting a heavier positive one (the hole), except both move through a polarizable medium with effective masses set by the band structure.

The crucial consequence: the exciton's two charges cancel. It is electrically neutral. An applied voltage exerts no net force on it, so it cannot carry a current. What it does carry is the photon's energy and momentum, which it can ferry across the crystal — tens to hundreds of nanometers — before the electron and hole annihilate.

Binding energy and the hydrogen analogy

Because an exciton is hydrogen-like, the same Bohr-model algebra applies, with two substitutions. Replace the electron mass m_e by the reduced effective mass μ of the electron–hole pair, and replace the vacuum permittivity by the crystal's relative dielectric constant ε_r (which screens the attraction):

1/μ = 1/m_e* + 1/m_h*          (reduced effective mass)

E_b(n) = 13.6 eV · (μ/m_e) / (ε_r² · n²)     (Rydberg series, n = 1, 2, 3 …)

a_X = 0.0529 nm · ε_r · (m_e/μ)              (exciton Bohr radius)

The binding energy is the depth of the n = 1 ground state below the band gap. The exciton absorption lines therefore sit just below E_g, forming a hydrogen-like Rydberg ladder that converges onto the gap from below. The total energy of a free exciton with center-of-mass momentum ℏK is:

E(n, K) = E_g − E_b/n² + ℏ²K² / (2(m_e* + m_h*))

Two numbers dominate the physics: a large dielectric constant ε_r and a small reduced mass μ both make the binding weak and the orbit large. Conventional inorganic semiconductors (high ε_r) give tiny binding energies; molecular and low-dimensional materials (low ε_r, poor screening) give huge ones.

Two limits: Wannier–Mott and Frenkel

Excitons span a continuum, but two textbook limits anchor it.

	Wannier–Mott exciton	Frenkel exciton
Found in	Inorganic semiconductors (GaAs, Si, Cu₂O)	Molecular crystals, alkali halides, organics
Dielectric constant	High (ε_r ≈ 10–13)	Low (ε_r ≈ 2–4)
Radius	Large — many lattice sites (~5–15 nm)	Small — one molecule (~0.5–1 nm)
Binding energy	A few meV to tens of meV	0.1–1 eV
Stable at room T?	Often ionizes (kT ≈ 26 meV)	Yes — very robust
Model	Effective-mass / screened hydrogen	Tight-binding, on-site excitation

The intermediate charge-transfer exciton, where the electron and hole sit on adjacent molecules, is the workhorse of organic photovoltaics. And a new regime has dominated the last decade: two-dimensional excitons in monolayer transition-metal dichalcogenides such as MoS₂ and WSe₂. Confining carriers to a single atomic plane, with the surrounding vacuum providing almost no screening, pushes binding energies to ~0.3–0.5 eV — an order of magnitude beyond bulk values, and easily stable at room temperature.

Real numbers

Material	Binding energy E_b	Exciton radius a_X	Notes
GaAs	~4.2 meV	~12 nm	Classic Wannier–Mott; visible only below ~100 K
Silicon	~15 meV	~4 nm	Indirect gap → slow, phonon-assisted recombination
GaN	~25 meV	~3 nm	Blue/UV LEDs; near room-temperature stability
ZnO	~60 meV	~2 nm	Robust at 300 K (kT ≈ 26 meV)
Cu₂O	~150 meV (n=1)	~1 nm (n=1)	Rydberg "yellow series" resolved to n ≈ 25
Monolayer MoS₂	~440 meV	~1 nm	Giant 2D exciton; dominates optical spectrum
Organic (anthracene)	~0.5–1 eV	<1 nm	Frenkel; localized on a molecule

Compare the binding energy to room-temperature thermal energy, kT ≈ 26 meV. GaAs excitons (4 meV) are blown apart by heat and survive only in cryostats, which is why GaAs photoluminescence experiments are done cold. ZnO, GaN, Cu₂O, and especially the 2D materials sit comfortably above 26 meV and show clean excitonic features at room temperature.

Absorption: the exciton peak below the gap

Without excitonic effects, the optical absorption of a direct-gap semiconductor would simply switch on at E_g and rise as √(ℏω − E_g) from the joint density of states. Excitons rewrite this in two ways. First, they add discrete absorption lines below the gap at E_g − E_b/n², a hydrogen-like series. Second, the electron–hole attraction enhances the absorption above the gap as well (the Sommerfeld factor), so the continuum edge is steeper and stronger than the free-carrier prediction. In a material like Cu₂O these lines are so sharp that researchers have resolved a Rydberg staircase up to principal quantum number n ≈ 25 — giant excitons spanning micrometers, the solid-state cousins of Rydberg atoms.

Recombination: how the exciton dies

An exciton is a transient. The electron and hole eventually find each other and annihilate — recombination — releasing the stored energy by one of two channels:

• Radiative. The energy comes out as a photon at the exciton energy, E_g − E_b. This is photoluminescence and the light-emitting step of an LED or laser diode. In direct-gap materials (GaAs, GaN, MoS₂) it is fast and efficient, with lifetimes from picoseconds to nanoseconds.
• Non-radiative. The energy is dumped into lattice vibrations (phonons), trap states, or defects as heat, with no photon emitted. In indirect-gap silicon, radiative recombination requires a phonon to conserve momentum, so it is slow (microseconds to milliseconds) and non-radiative paths usually win — which is exactly why silicon is a poor light emitter but a fine solar absorber.

The competition between these channels sets the internal quantum efficiency of every light emitter. A high-quality GaN LED routes most excitons through the radiative channel; a defective crystal traps them and turns the energy into wasted heat.

Trions, biexcitons, and excitonic condensates

Add more carriers and excitons combine into richer bound states. A trion is a charged three-body complex — two electrons and a hole, or two holes and an electron — that does carry charge and shows up as a satellite peak a few tens of meV below the neutral exciton in gated 2D devices. A biexciton is two excitons bound together, an excitonic "molecule." At high densities and low temperatures, excitons (being composite bosons) can in principle undergo Bose–Einstein condensation; long-lived spatially indirect excitons in coupled quantum wells, with the electron and hole in separate layers, are the leading platform for hunting this excitonic condensate.

Where excitons matter

• LEDs and laser diodes. Radiative exciton recombination is the photon source; the exciton energy E_g − E_b sets the emission color.
• Solar cells. Absorbed photons make excitons; in organic and perovskite cells the exciton must diffuse to an interface and be split into free carriers before it recombines.
• Photoluminescence spectroscopy. Exciton peaks are a fingerprint of band gap, quality, strain, and doping in any new material.
• 2D materials and valleytronics. Tightly bound excitons in MoS₂/WSe₂ dominate the optics and enable spin/valley-selective light–matter coupling.
• Quantum dots. Confinement quantizes exciton energy; a single dot emits single photons on demand for quantum technologies.
• Photosynthesis and light harvesting. Molecular Frenkel excitons funnel absorbed sunlight from antenna pigments to reaction centers.

Common misconceptions

• "An exciton is just a free electron and hole." No — it is a bound pair. Above the binding energy you have free carriers; below it you have a discrete, hydrogen-like bound state with its own quantized energy ladder.
• "Excitons carry current." A neutral exciton carries energy, not charge. A field cannot drive it. (A charged trion is the exception.)
• "The absorption edge is at E_g." Excitonic lines absorb below the gap, at E_g − E_b/n², and the attraction also enhances absorption above the gap.
• "Excitons are always tiny." Wannier–Mott excitons can span ~10 nm in GaAs, and Cu₂O Rydberg excitons reach the micrometer scale — far larger than a unit cell.
• "Recombination always makes light." Only radiative recombination emits a photon; non-radiative paths turn the energy into heat, and in indirect-gap silicon that is the usual outcome.