Semiconductor luminescence equations
The semiconductor luminescence equations (SLEs)[1][2] describe luminescence of semiconductors resulting from spontaneous recombination of electronic excitations, producing a flux of spontaneously emitted light.
Due to randomness of the vacuum-field fluctuations, semiconductor luminescence is incoherent whereas the extensions of the SLEs include[2] the possibility to study resonance fluorescence resulting from optical pumping with coherent laser light.
At this level, one is often interested to control and access higher-order photon-correlation effects, distinct many-body states, as well as light–semiconductor entanglement.
Such investigations are the basis of realizing and developing the field of quantum-optical spectroscopy which is a branch of quantum optics.
, vanish and the system becomes quasistationary, semiconductors emit incoherent light spontaneously, commonly referred to as luminescence (L).
The corresponding luminescence flux is proportional to the temporal change in photon number,[2]
determines the corresponding electron–hole recombination operator defining also the microscopic polarization within semiconductor.
contains the dipole-matrix element for interband transition, light-mode's mode function, and vacuum-field amplitude.
In general, the SLEs includes all single- and two-particle correlations needed to compute the luminescence spectrum self-consistently.
More specifically, a systematic derivation produces a set of equations involving photon-number-like correlations
The Coulomb renormalization are identical to those that appear in the semiconductor Bloch equations (SBEs), showing that all photon-assisted polarizations are coupled with each other via the unscreened Coulomb-interaction
The excitation level of a semiconductor is characterized by electron and hole occupations,
[2] It is straight forward to verify that the photon-assisted recombination[3][4][5] destroys as many electron–hole pairs as it creates photons because due to the general conservation law
Such form is to be expected for a probability of two uncorrelated events to occur simultaneously at a desired
becomes large when electron–hole pairs are bound as excitons via their mutual Coulomb attraction.
As the semiconductor emits light spontaneously, the luminescence is further altered by a stimulated contribution
that is particularly important when describing spontaneous emission in semiconductor microcavities and lasers because then spontaneously emitted light can return to the emitter (i.e., the semiconductor), either stimulating or inhibiting further spontaneous-emission processes.
To complete the SLEs, one must additionally solve the quantum dynamics of exciton correlations
The second line contains the main Coulomb sums that correlate electron–hole pairs into excitons whenever the excitation conditions are suitable.
[2][6] Microscopically, the luminescence processes are initiated whenever the semiconductor is excited because at least the electron and hole distributions, that enter the spontaneous-emission source, are nonvanishing.
dynamics has eigenenergies that are defined by the generalized Wannier equation not the free-carrier energies.
For low electron–hole densities, the Wannier equation produces a set of bound eigenstates which define the exciton resonances.
shows a discrete set of exciton resonances regardless which many-body state initiated the emission through the spontaneous-emission source.
In fact, the dominance of excitonic plasma luminescence has been measured in both quantum-well[8] and quantum-dot systems.
Technically, the SLEs are more difficult to solve numerically than the SBEs due to the additional
However, the SLEs often are the only (at low carrier densities) or more convenient (lasing regime) to compute luminescence accurately.
Furthermore, the SLEs not only yield a full predictability without the need for phenomenological approximations but they also can be used as a systematic starting point for more general investigations such as laser design[10][11] and disorder studies.
[12] The presented SLEs discussion does not specify the dimensionality or the band structure of the system studied.
As one analyses a specified system, one often has to explicitly include the electronic bands involved, the dimensionality of wave vectors, photon, and exciton center-of-mass momentum.
This approach can be applied to analyze the resonance fluorescence effects and to realize and understand the quantum-optical spectroscopy.
