8.4 Excitons

Fundamental Concepts of Excitons

An exciton is a bound state of an electron and a hole, held together by the Coulomb attraction between them. Excitons govern much of the optical behavior of semiconductors and insulators: they shape absorption spectra, drive photoluminescence, and mediate energy transfer without net charge flow. Understanding excitons is central to designing optoelectronic devices like LEDs, solar cells, and laser systems.

Definition and Formation

When a photon with energy at or above the band gap strikes a semiconductor, it can promote an electron from the valence band into the conduction band. The missing electron left behind acts as a positively charged quasiparticle called a hole. Rather than flying apart, the electron and hole can bind together through their mutual Coulomb attraction, forming an exciton.

Because the electron and hole charges cancel, the exciton is electrically neutral. This means it can propagate through the crystal lattice as a single entity, carrying energy but not net charge.

Types of Excitons

Wannier-Mott excitons have a large spatial extent, spreading over many unit cells. They occur in materials with high dielectric constants (like inorganic semiconductors such as GaAs), where the Coulomb interaction is strongly screened. Their binding energies are small, typically 1–100 meV.

Frenkel excitons are tightly localized, often confined to a single atom or molecule. They're common in organic materials and wide-gap insulators, where low dielectric constants leave the Coulomb interaction largely unscreened. Binding energies can exceed 1 eV.

Charge-transfer excitons sit between these two limits: the electron and hole reside on adjacent molecules or atomic sites. These are particularly relevant in organic donor-acceptor systems.

Surface excitons form at interfaces between different materials or at a material's surface, where the abrupt change in dielectric environment modifies the binding.

Binding Energy

The exciton binding energy $E_b$ is the energy needed to dissociate the electron-hole pair into free carriers. It's calculated using a hydrogen-like model:

$E_b = \frac{\mu e^4}{2\hbar^2 \epsilon^2}$

• $\mu$ is the reduced mass of the electron-hole pair: $\frac{1}{\mu} = \frac{1}{m_e^*} + \frac{1}{m_h^*}$
• $\epsilon$ is the dielectric constant of the material
• $e$ is the elementary charge

The binding energy scales inversely with $\epsilon^2$, which is why high-dielectric semiconductors produce weakly bound Wannier-Mott excitons while low-dielectric organics produce strongly bound Frenkel excitons. Typical values range from a few meV in bulk GaAs (~4.2 meV) to several eV in molecular crystals.

Exciton Properties

Optical Characteristics

Excitons absorb and emit light at energies slightly below the band gap, since the binding energy lowers the total energy of the electron-hole pair relative to free carriers. This produces sharp excitonic absorption peaks just below the band edge in optical spectra.

When an exciton recombines radiatively, the electron drops back into the hole and a photon is emitted. This photoluminescence has a characteristic wavelength set by the exciton energy. In systems where excitons couple strongly to cavity photons, hybrid quasiparticles called exciton-polaritons form, with dramatically modified dispersion and optical response.

Spatial Extent

The exciton Bohr radius $a_B$ quantifies how spread out the exciton is:

• Wannier-Mott excitons: $a_B$ is typically 10–100 times the lattice constant (e.g., ~13 nm in GaAs)
• Frenkel excitons: $a_B$ is comparable to atomic dimensions, confined within a single unit cell

The spatial extent matters because it determines how strongly the exciton interacts with defects, impurities, phonons, and other excitons.

Lifetime and Decay

Exciton lifetimes span a wide range depending on the material and conditions:

• Radiative decay: the electron and hole recombine, emitting a photon. This is the desired channel in light-emitting applications.
• Non-radiative decay: energy is lost to Auger recombination (energy transferred to a third carrier) or phonon-assisted processes (energy dissipated as heat).
• Thermal dissociation: at elevated temperatures, thermal energy can ionize the exciton into free carriers.

The exciton diffusion length $L_D$ is set by $L_D = \sqrt{D \tau}$, where $D$ is the diffusion coefficient and $\tau$ is the lifetime. This length scale is critical for device design, since excitons must reach an interface or recombination site before decaying.

Excitons in Semiconductors

Band Structure Effects

The electronic band structure strongly shapes exciton behavior. In direct band gap semiconductors like GaAs, the conduction band minimum and valence band maximum sit at the same crystal momentum. Excitons form and recombine efficiently because no phonon is needed to conserve momentum.

In indirect band gap materials like silicon, the band extrema occur at different momenta. Exciton formation and radiative recombination require phonon assistance, making both processes much slower and less efficient.

Valence band degeneracy introduces additional complexity. The heavy-hole and light-hole bands produce distinct exciton series with different reduced masses and binding energies, visible as separate features in high-resolution optical spectra.

Wannier-Mott vs. Frenkel Excitons

Exciton-Polaritons

When excitons are placed inside a semiconductor microcavity and the exciton-photon coupling strength exceeds the decay rates, the system enters the strong coupling regime. The resulting exciton-polaritons are hybrid light-matter quasiparticles with two branches in their dispersion relation: an upper polariton branch and a lower polariton branch, separated by the vacuum Rabi splitting.

Because polaritons inherit a very light effective mass from their photonic component (roughly $10^{-5} m_e$), they can undergo Bose-Einstein condensation at temperatures far above what's needed for bare excitons. This enables phenomena like polariton lasing, superfluidity, and quantum simulation in accessible experimental conditions.

Experimental Techniques

Photoluminescence Spectroscopy

This technique excites a sample optically (usually with a laser above the band gap) and collects the emitted light as excitons recombine. The emission spectrum reveals exciton energy levels and binding energies. Temperature-dependent measurements track how excitons thermally dissociate into free carriers, while time-resolved photoluminescence (using pulsed excitation and fast detectors) directly measures exciton lifetimes and relaxation pathways.

Absorption Spectroscopy

By measuring how much light a sample absorbs as a function of photon energy, you can identify excitonic absorption peaks below the band edge. The position and strength of these peaks yield the exciton binding energy and oscillator strength. Applying a magnetic field (magneto-absorption) splits and shifts excitonic levels, providing additional information about effective masses and the exciton's internal structure through Landau level spectroscopy.

Time-Resolved Measurements

Ultrafast techniques probe exciton dynamics on femtosecond to nanosecond timescales:

• Pump-probe spectroscopy: a pump pulse creates excitons, and a delayed probe pulse measures how the absorption changes as excitons evolve
• Transient absorption: closely related to pump-probe, maps out excited-state dynamics
• Two-dimensional spectroscopy: correlates excitation and detection frequencies to reveal energy transfer pathways between excitonic states
• Terahertz spectroscopy: probes transitions within the exciton (between its internal energy levels) and tracks exciton transport

Applications of Excitons

Light-Emitting Diodes

Exciton recombination is the fundamental light-emission mechanism in organic LEDs (OLEDs) and quantum dot LEDs. Electrical injection creates electrons and holes that form excitons, which then recombine radiatively.

A key challenge is spin statistics: electrical injection produces 25% singlet and 75% triplet excitons, but only singlets emit light efficiently in fluorescent materials. Two strategies address this:

• Phosphorescent emitters (using heavy-metal complexes like Ir or Pt) enable triplet emission through strong spin-orbit coupling, reaching nearly 100% internal quantum efficiency
• Thermally activated delayed fluorescence (TADF) materials have a small singlet-triplet energy gap, allowing reverse intersystem crossing to convert triplets back to emissive singlets

Solar Cells

In organic photovoltaics, absorbed photons create excitons that must diffuse to a donor-acceptor interface and dissociate into free charges before they can contribute to photocurrent. The exciton diffusion length (typically 5–20 nm in organics) sets a fundamental constraint on active layer morphology.

Two advanced concepts push beyond conventional efficiency limits:

• Singlet fission: certain materials (like pentacene) convert one singlet exciton into two triplet excitons, potentially generating two electron-hole pairs per absorbed photon and exceeding the Shockley-Queisser limit
• Hot carrier extraction: collecting carriers from high-energy excitons before they thermalize to the band edge could recover energy normally lost as heat

Quantum Information

Excitons confined in quantum dots can serve as qubits, with the exciton's presence or absence encoding quantum information. Entangled photon pairs generated through the biexciton cascade in quantum dots are used in quantum cryptography protocols. Exciton-polariton condensates in microcavities have been proposed as platforms for quantum simulation of interacting many-body systems.

Excitons in Low-Dimensional Systems

Reducing dimensionality enhances Coulomb interactions by restricting the spatial extent of wavefunctions and reducing dielectric screening. This systematically increases exciton binding energies and oscillator strengths.

Quantum Wells

In a quantum well (a thin semiconductor layer sandwiched between wider-gap barriers), excitons are confined in two dimensions. The binding energy increases roughly by a factor of 4 in the ideal 2D limit compared to bulk. The quantum-confined Stark effect allows an applied electric field to tune the exciton energy, which is the basis for electro-absorption modulators.

In coupled quantum wells, spatially indirect excitons form with the electron and hole in different layers. These have long lifetimes (because the spatial separation reduces recombination overlap) and can be manipulated with electric fields.

Quantum Wires

One-dimensional confinement (as in semiconductor nanowires or carbon nanotubes) further enhances binding energies and oscillator strengths. Excitons propagate along the wire axis with reduced scattering compared to bulk. Carbon nanotubes are a prototypical 1D system where excitons dominate the optical response due to the strong, poorly screened Coulomb interaction.

Quantum Dots

In quantum dots, confinement in all three dimensions produces discrete, atom-like energy levels. The exciton emission wavelength is tunable through the dot size: smaller dots emit at higher energies (bluer), larger dots at lower energies (redder). Binding energies can reach hundreds of meV due to strong confinement.

At higher excitation levels, multi-exciton complexes form:

• Biexcitons: two excitons bound together, analogous to a hydrogen molecule
• Trions: a neutral exciton plus an extra electron (negative trion) or hole (positive trion)

These complexes produce distinct spectral signatures and are important for quantum dot lasers and single-photon sources.

Many-Body Effects

Exciton-Exciton Interactions

At low densities, excitons behave as independent quasiparticles. As density increases, interactions become important:

• Dipole-dipole interactions shift and broaden spectral lines
• Exciton-exciton annihilation (one exciton transfers its energy to another, which then dissociates) limits the achievable exciton density
• Attractive interactions can bind two excitons into a biexciton
• At very high densities, repulsive interactions drive the exciton Mott transition, where bound excitons ionize into an unbound electron-hole plasma

Biexcitons and Trions

Biexcitons are four-particle complexes (two electrons, two holes) bound by correlations beyond what holds each individual exciton together. Their binding energy is typically 10–20% of the single exciton binding energy.

Trions (charged excitons) form when a neutral exciton captures an additional free carrier. They're especially prominent in doped semiconductors and low-dimensional systems where carrier densities are significant. Trions modify both optical spectra and transport properties.

Bose-Einstein Condensation

Excitons are composite bosons (integer total spin), so at sufficiently low temperatures and high densities they can undergo Bose-Einstein condensation (BEC). In practice, achieving exciton BEC is difficult because excitons have finite lifetimes and must reach thermal equilibrium before decaying.

Exciton-polaritons offer a workaround: their extremely light effective mass (from the photon component) raises the critical temperature for condensation dramatically. Polariton condensates have been observed at temperatures up to ~room temperature in some systems, displaying macroscopic coherence and superfluid-like behavior.

Theoretical Models

Effective Mass Approximation

This approach replaces the full crystal potential with effective masses for the electron and hole, derived from the curvature of the energy bands:

$m^* = \hbar^2 \left(\frac{d^2E}{dk^2}\right)^{-1}$

The electron and hole then interact through a screened Coulomb potential, reducing the exciton problem to a hydrogen-like two-body problem. This works well for Wannier-Mott excitons in materials with parabolic bands near the band extrema, but breaks down when bands are strongly non-parabolic or when electron-hole correlations are strong.

Hydrogen-Like Model

Building on the effective mass approximation, the exciton energy levels follow a Rydberg series:

$E_n = E_g - \frac{\mu e^4}{2\hbar^2 \epsilon^2 n^2}$

• $E_g$ is the band gap energy
• $n = 1, 2, 3, \ldots$ is the principal quantum number
• The $n = 1$ state has the largest binding energy; higher states converge toward the band edge

This predicts a series of discrete absorption lines below the band gap, merging into the continuum at $E_g$. The model works quantitatively for Wannier-Mott excitons (the $n = 1, 2, 3$ lines are clearly resolved in GaAs at low temperature) but fails for Frenkel excitons, where the electron-hole separation is too small for the continuum dielectric screening assumption to hold.

Computational Approaches

For quantitative predictions beyond the hydrogen model, several computational methods are used:

• GW + Bethe-Salpeter equation (BSE): the current gold standard for exciton calculations. The GW approximation gives accurate quasiparticle band structures, and the BSE describes the electron-hole interaction. Computationally expensive but reliable.
• Time-dependent DFT (TDDFT): less expensive than GW-BSE but often less accurate for exciton binding energies, depending on the exchange-correlation functional used.
• Quantum Monte Carlo: handles strong correlations well but is computationally demanding and harder to apply to large systems.
• Tight-binding and $\mathbf{k \cdot p}$ models: efficient semi-empirical approaches for calculating exciton properties in nanostructures where full ab initio methods are too costly.

Excitons in Novel Materials

Transition Metal Dichalcogenides

Monolayer TMDs like $\text{MoS}_2$ and $\text{WSe}_2$ are direct-gap semiconductors with remarkably strong excitonic effects. Reduced dimensionality and weak dielectric screening produce exciton binding energies of 300–700 meV, meaning excitons are stable well above room temperature.

These materials also have valley-specific optical selection rules: left- and right-circularly polarized light selectively excite excitons in the K and K' valleys of the Brillouin zone. This valley degree of freedom can be used to encode and process information (valleytronics).

In van der Waals heterostructures (stacked layers of different TMDs), interlayer excitons form with the electron and hole in separate layers. These have long lifetimes and large permanent dipole moments, making them attractive for studying collective excitonic phenomena.

Perovskites

Hybrid organic-inorganic perovskites like $\text{CH}_3\text{NH}_3\text{PbI}_3$ have achieved remarkable solar cell efficiencies (>25%). Their exciton binding energies are moderate (~10–50 meV in 3D perovskites), placing them near the boundary between excitonic and free-carrier behavior at room temperature.

Reducing dimensionality to 2D layered perovskites dramatically increases the binding energy (to ~200–400 meV) due to both quantum confinement and the dielectric contrast between organic and inorganic layers. The interplay between excitons, polarons (carriers dressed by lattice distortions), and dynamic structural disorder makes the photophysics of perovskites unusually rich and still actively debated.

Carbon Nanotubes

Single-walled carbon nanotubes are quasi-1D systems where excitons dominate the optical response. The combination of 1D confinement and weak dielectric screening produces binding energies of 300–500 meV, even larger than the thermal energy at room temperature.

Each nanotube chirality (defined by its diameter and wrapping angle) has distinct electronic and excitonic properties. Both bright (optically active) and dark (optically inactive) exciton states exist, with the dark states often lying lower in energy and influencing relaxation dynamics. These chirality-dependent properties enable applications in near-infrared light emission, photodetection, and biological sensing.