Exciton

A characteristic transport length before exciton decay, commonly

L_{D} = D_{X} τ_{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.

Electron-hole Coulomb interaction ( $V (r) = - e^{2} / (4 π ϵ_{0} ϵ_{r} r)$ )

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.

Exciton reduced mass ( $μ^{*}$ )

The reduced band mass for relative electron–hole motion,

μ^{*} = m_{e}^{*} m_{h}^{*} / (m_{e}^{*} + m_{h}^{*})

. It differs from the total mass that controls centre-of-mass motion.

Biexciton

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} = 2 E_{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.

Exciton binding energy ( $E_{B} = E_{cont} - E_{X} > 0$ )

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_{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.

Thermal ionization of excitons

Thermal energy comparable with an exciton binding energy promotes dissociation and often washes out narrow bound-exciton resonances. Comparing

k_{B} T

with binding energy is a scale estimate, not an exact disappearance criterion: densities, screening, oscillator strength and linewidths also matter.

Wannier-Mott exciton ( $a_{B}^{}, R^{}$ )

A Wannier-Mott exciton has an internal wave function extending over many unit cells. Its simplest model assumes a three-dimensional bulk crystal, approximately parabolic valence band and conduction band, positive isotropic scalar effective masses

m_{e}^{*}, m_{h}^{*}

, a constant scalar relative permittivity

ϵ_{r}

, and a dilute pair population. Replace microscopic crystal forces by those effective masses and dielectric screening, and neglect exchange corrections and coupling to phonons. These assumptions must hold on the spatial and energy scales of the particular exciton.

With relative coordinate

r = r_{e} - r_{h}

, the internal Hamiltonian operator is the three-dimensional hydrogen atom problem with a reduced mass and a screened Coulomb potential:

H_{rel} = - \frac{ℏ ^{2}}{2 μ} \nabla_{r}^{2} - \frac{e ^{2}}{4 π ϵ _{0} ϵ _{r} r}, μ = \frac{m _{e}^{*} m _{h}^{*}}{m _{e}^{*} + m _{h}^{*}}, M = m_{e}^{*} + m_{h}^{*} .

Here

e

is the elementary charge,

ϵ_{0}

the vacuum permittivity, and

ℏ

the reduced Planck constant. The internal energy eigenvalues, radius scale, and exciton binding energies are

a_{B}^{*} = \frac{4 π ϵ _{0} ϵ _{r} ℏ ^{2}}{μ e ^{2}}, R^{*} = \frac{μ e ^{4}}{2 ( 4 π ϵ _{0} ϵ _{r} ) ^{2} ℏ ^{2}}, E_{rel, n} = - \frac{R ^{*}}{n ^{2}}, E_{B, n} = \frac{R ^{*}}{n ^{2}} .

(2)

The principal quantum number is

n = 1, 2, \dots

. This reduction and these scales are derived in Excitons in Semiconductor Quantum Dots by Matjaž Humar (§3, printed pp. 5–6, equations (13)–(26)).

To check the scales by dimensional analysis, write

C = e^{2} / (4 π ϵ_{0} ϵ_{r})

, which has units of energy times length. Then

a_{B}^{*} = ℏ^{2} / (μ C)

is a length and

R^{*} = μ C^{2} / (2 ℏ^{2})

is an energy. Rescaling

r = a_{B}^{*} ρ

makes

H_{rel} / R^{*} = - \nabla_{ρ}^{2} - 2/ ρ

, the dimensionless hydrogen atom operator. The Bohr radius

a_{B}^{*}

is a wave function scale, not the radius of a classical orbit; for this model's lowest internal bound state, the mean separation is

3 a_{B}^{*} /2

For the diagrams' hypothetical inputs

m_{e}^{*} = 0.10 m_{0}

m_{h}^{*} = 0.40 m_{0}

ϵ_{r} = 10

and

E_{g} = 1.500 eV

, where

m_{0}

is the free electron mass, independent substitution gives

μ = 0.080 m_{0}

a_{B}^{*} ≃ 6.615 nm

and

R^{*} ≃ 10.885 meV

. Here electronvolts express energy. The exciton binding energies for

n = 1, 2, 3

are approximately

10.885

2.721

and

1.209 meV

. These are model calculations, not measured material parameters. The large-radius approximation additionally requires

a_{B}^{*}

to be much larger than the crystal's unit cell length scale. In a thin layer, with anisotropic effective masses, or with spatially varying dielectric screening, this three-dimensional formula is not generally valid.

Finite-width quantum-well exciton ( $V (ρ) = - \frac{e ^{2}}{4 π ϵ _{0} ϵ _{r}} \int \frac{∣ χ _{e} ( z _{e} ) ∣ ^{2} ∣ χ _{h} ( z _{h} ) ∣ ^{2}}{ρ ^{2} + ( z _{e} - z _{h} ) ^{2}} d z_{e} d z_{h}$ )

In the envelope approximation, a quantum well confines electron and hole motion perpendicular to the plane. Averaging their screened attraction over the confinement envelopes gives the displayed effective in-plane potential. Its narrow-well limit approaches the ideal two-dimensional exciton; finite width softens the short-distance attraction. Dielectric mismatch, nonparabolic bands and strong electron–hole confinement mixing require more than this single-subband model.

Two-dimensional exciton

In the ideal screened-Coulomb model confined to a plane, the relative-motion energies are

- R^{*} / (ν - 1/2)^{2}

for

ν = 1, 2, \dots

, where

R^{*} = (μ / m_{e}) 13.6 eV / ε_{r}^{2}

. The ground-state binding is

4 R^{*}

rather than the three-dimensional value

R^{*}

. Finite-width wells need not be exactly planar.

Exciton dimensionality from peak spacings

The first two optically active hydrogenic levels have separation

32 R^{*} /9

in the planar model, versus

3 R^{*} /4

in the bulk model. The separation cancels an uncertain common continuum threshold, making it useful when rounded band parameters do not reproduce every absolute peak energy.

Exciton Bohr radius ( $a_{X} = 4 π ϵ_{0} ϵ_{r} ℏ^{2} / (μ^{*} e^{2})$ )

The characteristic relative-motion length of a hydrogenic Wannier-Mott exciton, with dielectric screening and exciton reduced mass

μ^{*}

. Comparing a quantum dot's dimensions with this scale helps distinguish strong carrier confinement from confinement of a comparatively intact bound pair.

Exciton Rydberg series ( $E_{n} = E_{g} - R^{*} / n^{2}$ )

The hydrogenic sequence of bound relative electron–hole energies

E_{n} = E_{g} - R^{*} / n^{2}

below the free-pair threshold, where

R^{*} = (μ^{*} / m_{e}) R_{H} / ε_{r}^{2}

. The effective-mass, screened-Coulomb approximation and optical selection rules limit the model.

Relative permittivity ( $ϵ_{r} = ϵ / ϵ_{0}$ )

The dimensionless ratio of a medium's permittivity to vacuum permittivity. The constant scalar value used in a Wannier-Mott exciton is an approximation to the relevant material response.

Dielectric function ( $ε (q, ω)$ )

The generally complex, frequency- and wavevector-dependent relative permittivity describing linear polarization response. Its imaginary part represents dissipative response with the sign fixed by the time convention; optical prefactors connect it to an absorption coefficient.

Static Lindhard dielectric function ( $ϵ (q, 0) = 1 + κ^{2} F (q /2 k_{F}) / q^{2}$ )

The static independent-electron density response of a uniform three-dimensional, two-spin Fermi gas, combined with self-consistent Poisson equation screening, gives the displayed dielectric function. For bare electron mass and vacuum background,

κ^{2} = m_{e} e^{2} k_{F} / (ϵ_{0} π^{2} ℏ^{2})

and

F (x) = \frac{1}{2} + \frac{1 - x ^{2}}{4 x} ln \frac{1 + x}{1 - x} .

Here

F (0) = 1

F (1) = 1/2

by continuous extension, and

F (x) \sim 1/ (3 x^{2})

at large

x

. A host dielectric background and band effective mass require replacing the vacuum parameters appropriately. This nonlocal response contains Friedel oscillations, which its long-wavelength Thomas-Fermi screening limit misses.

Kohn singularity of static Lindhard screening ( $F (1 + u) = \frac{1}{2} + \frac{u}{2} ln ∣ u ∣ + O (u)$ )

The three-dimensional static screening function is finite at

q = 2 k_{F}

, but its derivative diverges logarithmically. Expanding the explicit static Lindhard dielectric function gives the title's nonanalytic term. The singularity comes from particle–hole excitations connecting opposite parts of the Fermi surface. It produces a Friedel density tail from a derivative singularity; analogous electronic response can cause a Kohn anomaly in a coupled lattice vibration.

Dielectric screening

Modification of the interaction between electric charges by a medium's electrical response. In the simplest Wannier-Mott exciton model, the Coulomb potential is reduced by a constant relative permittivity; actual screening can depend on spatial and temporal scales.

Radial induced-charge screening kernel ( $g (q) = q [1 - ϵ (q)^{- 1}] / (2 π^{2})$ )

For a point source

Z e

and the Fourier transform convention

ρ (q) = \int e^{- i q \cdot r} ρ (r) d^{3} r

, self-consistent screening gives

ρ_{ind} (q) = Z e [ϵ (q)^{- 1} - 1]

. Angular inversion yields

ρ_{ind} (r) = - (Z e / r) \int_{0}^{\infty} g (q) sin (q r) d q

. The factor

1/ (2 π^{2})

includes the inverse-transform normalization and spherical angular integral. Perfect long-wavelength screening implies total induced charge

- Z e

Friedel oscillations

The sharp Fermi surface gives a nonlocal oscillatory electronic response around an impurity. In a three-dimensional isotropic metal its far-field oscillations have characteristic wavevector

2 k_{F}

and algebraically decaying amplitude. They are absent from the purely exponential Thomas-Fermi screening potential. The local screening formula is a useful long-wavelength model, not an exact asymptotic expression at every distance.

Friedel oscillation wavelength from carrier density ( $λ = π / (3 π^{2} n)^{1/3}$ )

A single isotropic three-dimensional valley with two equal spin populations has

n = k_{F}^{3} / (3 π^{2})

. A density oscillation at wavevector

2 k_{F}

therefore has the wavelength in the title. This depends on carrier density, independently of effective mass and dielectric screening strength. Extra valley degeneracy, spin polarization or reduced dimensionality changes the state-counting relation.

Friedel density tail from a derivative singularity ( $g^{''} (q) \sim C / (q - Q) \Rightarrow ρ_{ind} (r) \sim Z e C π cos (Q r) / r^{3}$ )

For the radial induced-charge screening kernel, integrate the sine transform twice by parts, with

g (0) = g (\infty) = 0

and vanishing remaining endpoint terms. This gives

ρ_{ind} = Z e \int g^{''} (q) sin (q r) d q / r^{3}

. A Cauchy principal value singularity at

Q > 0

contributes

C π cos (Q r)

to that integral: set

s = q - Q

and use

\int_{- \infty}^{\infty} sin (sr) d s / s = π

for

r > 0

. Smoothly localizing around

Q

justifies the large-

r

extension of the limits. In a three-dimensional isotropic Fermi gas,

Q = 2 k_{F}

Thomas-Fermi screening

In the long-wavelength static response of a degenerate conductor, a small electric potential

ϕ

changes the electron density by

δ n = e g (E_{F}) ϕ

, where

g (E_{F})

is the spin-summed density of states per volume per energy. The induced charge is therefore

- e^{2} g (E_{F}) ϕ

. With background permittivity

ε_{b}

, Poisson equation becomes

(\nabla^{2} - k_{TF}^{2}) ϕ = - ρ_{ext} / ε_{b}, k_{TF}^{2} = e^{2} g (E_{F}) / ε_{b} .

A point charge

Q

then has

ϕ (r) = Q e^{- k_{TF} r} / (4 π ε_{b} r)

. Larger electronic density of states gives stronger screening. This local response approximation does not capture the microscopic oscillatory tail of Friedel oscillations.^[ref]

Thomas-Fermi screening length ( $λ_{TF} = ε_{b} / [e^{2} g (E_{F})]$ )

The Thomas-Fermi screening length is the exponential decay scale of a statically screened potential in the local degenerate-electron approximation. For a three-dimensional parabolic free electron model,

g (E_{F}) = 3 n / (2 E_{F})

, so

λ_{TF}^{2} = 2 ε_{b} E_{F} / (3 n e^{2})

. At fixed mass and background permittivity it scales as

n^{- 1/6}

, rather than the classical Debye density power at fixed temperature.

Frenkel exciton

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]

Exciton optical transition ( $E_{opt} ≃ E_{g} - E_{B}$ )

For the ideal Wannier-Mott exciton in energy bands whose edges occur at the same crystal momentum, the pair excitation energy, measured from the unexcited crystal, is

E_{X, n} (K) = E_{g} - \frac{R ^{*}}{n ^{2}} + \frac{ℏ ^{2} K ^{2}}{2 M} .

Here

ℏ K

is the pair's total crystal momentum; the last term is center of mass kinetic energy. For a permitted photon absorption transition with negligible recoil energy,

ℏ ω ≃ E_{g} - E_{B, n}

. An electric-dipole selection rule and a nonzero transition matrix element are still needed: a bound state need not produce an observable spectral line. If the relevant energy band edges have different values of crystal momentum, a phonon can also be needed to conserve crystal momentum; subtracting a binding scale from a band gap alone does not specify the optical threshold.

Figure 1.
Exciton binding energy and hypothetical exciton optical transition positions
. Original three-dimensional Wannier-Mott exciton model with the inputs given above. Left: internal energy eigenvalues relative to the unbound continuum. Right: corresponding optical energy positions for hypothetical permitted transitions, with recoil neglected. Equal stick heights encode positions only, not predicted strengths or widths. The continuum threshold is $1.500 eV$ ; these sticks are not an empirical optical spectrum and are not the copper(I) oxide experiment below. **Source**. **Details**.

For example, the lowest internal bound state in this hypothetical model costs

1.489115 eV

to create optically, yet needs only

0.010885 eV

more to dissociate. The large creation energy and the small exciton binding energy answer different physical questions. An electron-hole recombination event can instead release excitation energy by photon emission or to other degrees of freedom; an emitted photon need not reproduce the original absorption energy if relaxation intervened.

Exciton absorption experiment in copper(I) oxide