Elliott formula

Linear optical absorption

Linear absorption of broadband weak optical probe can then be expressed as

Linear Elliott formula

${\displaystyle \alpha (\omega )=\mathrm{I}\mathrm{m}\left[\sum _{\lambda }\frac{F_{\lambda }}{E_{\lambda }-\hslash \omega -\mathrm{i}{\gamma }_{\lambda }(\omega )}\right]}$

where ${\displaystyle \hslash \omega }$ is the probe-photon energy, ${\displaystyle F_{\lambda }}$ is the oscillator strength of the exciton state ${\displaystyle \lambda }$, and ${\displaystyle {\gamma }_{\lambda }}$ is the dephasing constant associated with the exciton state ${\displaystyle \lambda }$. For a phenomenological description, ${\displaystyle {\gamma }_{\lambda }}$ can be used as a single fit parameter, i.e., ${\displaystyle {\gamma }_{\lambda }\to \gamma }$. However, a full microscopic computation generally produces ${\displaystyle {\gamma }_{\lambda }(\omega )}$ that depends on both exciton index ${\displaystyle \lambda }$ and photon frequency. As a general tendency, ${\displaystyle {\gamma }_{\lambda }(\omega )}$ increases for elevated ${\displaystyle E_{\lambda }}$ while the ${\displaystyle \omega }$ dependence is often weak.

Each of the exciton resonances can produce a peak to the absorption spectrum when the photon energy matches with ${\displaystyle E_{\lambda }}$. For direct-gap semiconductors, the oscillator strength is proportional to the product of dipole-matrix element squared and ${\displaystyle |\sum _{\mathbf{k}}{\varphi }_{\lambda }(\mathbf{k}){|}^2}$ that vanishes for all states except for those that are spherically symmetric. In other words, ${\displaystyle F_{\lambda }}$ is nonvanishing only for the ${\displaystyle s}$-like states, following the quantum-number convention of the hydrogen problem. Therefore, optical spectrum of direct-gap semiconductors produces an absorption resonance only for the ${\displaystyle s}$-like state. The width of the resonance is determined by the corresponding dephasing constant.

In general, the exciton eigen energies consist of a series of bound states that emerge energetically well below the fundamental bandgap energy and a continuum of unbound states that appear for energies above the bandgap. Therefore, a typical semiconductor's low-density absorption spectrum shows a series of exciton resonances and then a continuum-absorption tail. For realistic situations, ${\displaystyle {\gamma }_{\lambda }}$ 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. For high enough densities, all ${\displaystyle E_{\lambda }}$ energies correspond to continuum states and some of the oscillators strengths may become negative-valued due to the Pauli-blocking effect. Physically, this can be understood as the elementary property of Fermions; if a given electronic state is already excited it cannot be excited a second time due to the Pauli exclusion among Fermions. 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 ${\displaystyle E_{\lambda }}$, ${\displaystyle F_{\lambda }}$, and ${\displaystyle {\gamma }_{\lambda }(\omega )}$ requires a full many-body computation already for moderate carrier densities.

Photoluminescence Elliott formula

After the semiconductor becomes electronically excited, the carrier system relaxes into a quasiequilibrium. At the same time, vacuum-field fluctuations trigger spontaneous recombination of electrons and holes (electronic vacancies) via spontaneous emission of photons. At quasiequilibrium, this yields a steady-state photon flux emitted by the semiconductor. By starting from the SLEs, the steady-state photoluminescence (abbreviated as PL) can be cast into the form

Photoluminescence Elliott formula

${\displaystyle \mathrm{P}\mathrm{L}(\omega )=\mathrm{I}\mathrm{m}\left[\sum _{\lambda }\frac{F_{\lambda }{S}_{\lambda }}{E_{\lambda }-\hslash \omega -\mathrm{i}{\gamma }_{\lambda }(\omega )}\right],}$

that is very similar to the Elliott formula for the optical absorption. As a major difference, the numerator has a new contribution – the spontaneous-emission source

${\displaystyle S_{\lambda }=\sum _{\mathbf{k}}|{\varphi }_{\lambda }(\mathbf{k}){|}^2{f}_{\mathbf{k}}^e{f}_{\mathbf{k}}^h+\mathrm{\Delta }{N}_{\lambda }}$

that contains electron and hole distributions ${\displaystyle f_{\mathbf{k}}^e}$ and ${\displaystyle f_{\mathbf{k}}^h}$, respectively, where ${\displaystyle \hslash \mathbf{k}}$ is the carrier momentum. Additionally, ${\displaystyle S_{\lambda }}$ contains also a direct contribution from exciton populations ${\displaystyle \mathrm{\Delta }{N}_{\lambda }}$ that describes truly bound electron–hole pairs.

The ${\displaystyle f_{\mathbf{k}}^e{f}_{\mathbf{k}}^h}$ term defines the probability to find an electron and a hole with same ${\displaystyle \mathbf{k}}$. Such a form is expected for a probability of two uncorrelated events to occur simultaneously at a desired ${\displaystyle \mathbf{k}}$ value. Therefore, ${\displaystyle f_{\mathbf{k}}^e{f}_{\mathbf{k}}^h}$ is the spontaneous-emission source originating from uncorrelated electron–hole plasma. The possibility to have truly correlated electron–hole pairs is defined by a two-particle exciton correlation ${\displaystyle \mathrm{\Delta }{N}_{\lambda }}$; the corresponding probability is directly proportional to the correlation. Nevertheless, both the presence of electron–hole plasma and excitons can equivalently induce the spontaneous emission. A further discussion of the relative weight and nature of plasma vs. exciton sources is presented in connection with the SLEs.

Like for the absorption, a direct-gap semiconductor emits light only at the resonances corresponding to the ${\displaystyle s}$-like states. As a typical trend, a quasiequilibrium emission is strongly peaked around the 1s resonance because ${\displaystyle S_{\lambda }}$ is usually largest for the ${\displaystyle \lambda =1s}$ ground state. This emission peak often remains well below the fundamental bandgap energy even at the high excitations where all states are continuum states. 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.

Terahertz Elliott formula

As discussed above, it is often meaningful to tune the electromagnetic field to be resonant with the transitions between two many-body states. For example, one can follow how a bound exciton is excited from its 1s ground state to a 2p state. In several semiconductor systems, one needs THz fields to induce such transitions. By starting from a steady-state configuration of electron–hole correlations, the diagonalization of THz-induced dynamics yields a THz absorption spectrum

THz Elliott formula

${\displaystyle {\alpha }_{\mathrm{T}\mathrm{H}\mathrm{z}}(\omega )=\mathrm{I}\mathrm{m}\left[\frac{\sum _{\nu ,\lambda }{S}^{\nu ,\lambda }(\omega )\mathrm{\Delta }{N}_{\nu ,\lambda }-{\left[S^{\nu ,\lambda }(-\omega )\mathrm{\Delta }{N}_{\nu ,\lambda }\right]}^{\star }}{\omega (\hslash \omega +\mathrm{i}\gamma (\omega ))}\right].}$

In this notation, the diagonal contributions ${\displaystyle \mathrm{\Delta }{N}_{\lambda ,\lambda }}$ determine the population of ${\displaystyle \lambda }$ excitons. The off-diagonal ${\displaystyle \mathrm{\Delta }{N}_{\nu ,\lambda \ne \nu }}$ elements formally determine transition amplitudes between two exciton states ${\displaystyle \nu }$ and ${\displaystyle \lambda \ne \nu }$. For elevated densities, ${\displaystyle \mathrm{\Delta }{N}_{\nu ,\lambda \ne \nu }}$ 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.

In contrast to optical absorption and photoluminescence, THz absorption may involve all exciton states. This can be seen from the spectral response function

${\displaystyle S^{\nu ,\lambda }(\omega )=\sum _{\beta }\frac{(E_{\beta }-E_{\nu })J_{\nu \beta }{J}_{\beta \lambda }}{E_{\beta }-E_{\nu }-\hslash \omega -\mathrm{i}{\gamma }_{\lambda ,\nu }(\omega )}}$

that contains the current-matrix elements ${\displaystyle J_{\nu \beta }\propto \sum _{\mathbf{k}}{\varphi }_{\nu }^{\star }(\mathbf{k})\mathbf{k}\cdot {\mathbf{k}}_{\mathrm{T}\mathrm{H}\mathrm{z}}{\varphi }_{\beta }(\mathbf{k})}$ between two exciton states. The unit vector ${\displaystyle {\mathbf{k}}_{\mathrm{T}\mathrm{H}\mathrm{z}}}$ is determined by the direction of the THz field. This leads to dipole selection rules among exciton states, in full analog to the atomic dipole selection rules. Each allowed transition produces a resonance in ${\displaystyle S^{\nu ,\lambda }(\omega )}$ and the resonance width is determined by a dephasing constant ${\displaystyle {\gamma }_{\lambda ,\nu }(\omega )}$ that generally depends on exciton states involved and the THz frequency ${\displaystyle \omega }$. The THz response also contains ${\displaystyle \gamma (\omega )}$ that stems from the decay constant of macroscopic THz currents.

In contrast to optical and photoluminescence spectroscopy, THz absorption can directly measure the presence of exciton populations in full analogy to atomic spectroscopy. For example, the presence of a pronounced 1s-to-2p resonance in THz absorption uniquely identifies the presence of excitons as detected experimentally in Ref. As a major difference to atomic spectroscopy, semiconductor resonances contain a strong excitation-induced dephasing that produces much broader resonances than in atomic spectroscopy. In fact, one typically can resolve only one 1s-to-2p resonance because the dephasing constant ${\displaystyle {\gamma }_{\nu ,\lambda }}$ 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.