Elliott formula

The Elliott formula describes analytically, or with few adjustable parameters such as the dephasing constant, the light absorption or emission spectra of solids.

It was originally derived by Roger James Elliott to describe linear absorption based on properties of a single electron–hole pair.

[1] The analysis can be extended to a many-body investigation with full predictive powers when all parameters are computed microscopically using, e.g., the semiconductor Bloch equations (abbreviated as SBEs) or the semiconductor luminescence equations (abbreviated as SLEs).

One of the most accurate theories of semiconductor absorption and photoluminescence is provided by the SBEs and SLEs, respectively.

All relevant many-body effects can be systematically included by using various techniques such as the cluster-expansion approach.

This homogeneous part yields an eigenvalue problem that can be expressed through the generalized Wannier equation that can be solved analytically in special cases.

In particular, the low-density Wannier equation is analogous to bound solutions of the hydrogen problem of quantum mechanics.

These are often referred to as exciton solutions and they formally describe Coulombic binding by oppositely charged electrons and holes.

The actual physical meaning of excitonic states is discussed further in connection with the SBEs and SLEs.

These exciton eigenstates provide valuable insight to SBEs and SLEs, especially, when one analyses the linear semiconductor absorption spectrum or photoluminescence at steady-state conditions.

[2] Under the steady-state conditions, the resulting equations can be solved analytically when one further approximates dephasing due to higher-order many-body effects.

After the exciton states are obtained, one can eventually express the linear absorption and steady-state photoluminescence analytically.

The same approach can be applied to compute absorption spectrum for fields that are in the terahertz (abbreviated as THz) range[3] of electromagnetic radiation.

Technically, the THz investigations are an extension of the ordinary SBEs and/or involve solving the dynamics of two-particle correlations explicitly.

[4] Like for the optical absorption and emission problem, one can diagonalize the homogeneous parts that emerge analytically with the help of the exciton eigenstates.

Each of the exciton resonances can produce a peak to the absorption spectrum when the photon energy matches with

For direct-gap semiconductors, the oscillator strength is proportional to the product of dipole-matrix element squared and

Therefore, a typical semiconductor's low-density absorption spectrum shows a series of exciton resonances and then a continuum-absorption tail.

increases more rapidly than the exciton-state spacing so that one typically resolves only few lowest exciton resonances in actual experiments.

The concentration of charge carriers influence the shape of the absorption spectrum considerably.

energies correspond to continuum states and some of the oscillators strengths may become negative-valued due to the Pauli-blocking effect.

Therefore, the corresponding electronic states can produce only photon emission that is seen as negative absorption, i.e., gain that is the prerequisite to realizing semiconductor lasers.

Even though one can understand the principal behavior of semiconductor absorption on the basis of the Elliott formula, detailed predictions of the exact

A further discussion of the relative weight and nature of plasma vs. exciton sources[7] is presented in connection with the SLEs.

This demonstrates that semiconductors are often subjects to massive Coulomb-induced renormalizations even when the system appears to have only electron–hole plasma states as emission resonances.

To make an accurate prediction of the exact position and shape at elevated carrier densities, one must resort to the full SLEs.

As discussed above, it is often meaningful to tune the electromagnetic field to be resonant with the transitions between two many-body states.

By starting from a steady-state configuration of electron–hole correlations, the diagonalization of THz-induced dynamics yields a THz absorption spectrum[4]

build up spontaneously and they describe correlated electron–hole plasma that is a state where electrons and holes move with respect to each other without forming bound pairs.

is broader than energetic spacing of n-p and (n+1)-p states making 1s-to-n-p and 1s-to-(n+1)p resonances merge into one asymmetric tail.

Characteristic linear absorption spectrum of bulk GaAs using two-band SBEs. The decay of polarization is approximated with a decay constant and is computed as function of the pump field's photon energy . The energy is shifted with respect to the band-gap energy and the semiconductor is initially unexcited. Due to the small dephasing constant used, several excitonic resonances appear (vertical lines) well below the bandgap energy. The magnitude of high-energy resonances are multiplied by 5 for visibility.
Photoluminescence intensity computed via the Elliott formula. The population of s-like exciton states follow a Boltzmann distribution at 35 Kelvin, where the 1 s population is suppressed to four percent and the dephasing constant is . The vertical lines indicate the position of s -like excitonic resonances, i.e., 1 s , 2 s , 3 s , etc. The bandgap energy is denoted by and 'arb. u.' means arbitrary units.
Terahertz absorption spectrum in bulk GaAs computed using the THz Elliott formula. The vertical lines indicate the n p -1 s transition energies of which the first one (2 p -1 s transition) is dominant. The 1 s -band-gap-transition lies slightly above 4meV, whereas the dephasing constant is chosen to be .