The semiconductor luminescence equations (SLEs)
[Kira, M.; Jahnke, F.; Koch, S.; Berger, J.; Wick, D.; Nelson, T.; Khitrova, G.; Gibbs, H. (1997). "Quantum Theory of Nonlinear Semiconductor Microcavity Luminescence Explaining "Boser" Experiments". ''Physical Review Letters'' 79 (25): 5170–5173. doi:10.1103/PhysRevLett.79.5170][Kira, M.; Koch, S. W. (2011). ''Semiconductor Quantum Optics''. Cambridge University Press. .] describe
luminescence
Luminescence is a spontaneous emission of radiation from an electronically or vibrationally excited species not in thermal equilibrium with its environment. A luminescent object emits ''cold light'' in contrast to incandescence, where an obje ...
of
semiconductor
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
s resulting from spontaneous
recombination of electronic excitations, producing a
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...
of
spontaneously emitted light. This description established the first step toward
semiconductor quantum optics because the SLEs simultaneously includes the quantized light–matter interaction and the
Coulomb-interaction coupling among electronic excitations within a semiconductor. The SLEs are one of the most accurate methods to describe light emission in semiconductors and they are suited for a systematic modeling of semiconductor emission ranging from
exciton
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb's law, Coulomb force resulting from their opposite charges. It is an electrically neutral quasiparticle regarded as ...
ic luminescence to
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
s.
Due to randomness of the
vacuum-field fluctuations, semiconductor luminescence is
incoherent whereas the extensions of the SLEs include
the possibility to study
resonance fluorescence
Resonance fluorescence is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency
Natural frequency, measured in terms of '' eigenfreque ...
resulting from
optical pumping
Optical pumping is a process in which light is used to raise (or "pump") electrons from a lower energy level in an atom or molecule to a higher one. It is commonly used in laser construction to pump the active laser medium so as to achieve popu ...
with
coherent
Coherence is, in general, a state or situation in which all the parts or ideas fit together well so that they form a united whole.
More specifically, coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics ...
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
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
Quantum-optical spectroscopyKira, M.; Koch, S. (2006).
"Quantum-optical spectroscopy of semiconductors". ''Physical Review A'' 73 (1).
doibr>10.1103/PhysRevA.73.013813 .Koch, S. W.; Kira, M.; Khitrova, G.; Gibbs, H. M. (2006). "Semiconductor exc ...
which is a branch of
quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction ...
.
Starting point
The derivation of the SLEs starts from a system
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
that fully includes many-body interactions, quantized light field, and quantized light–matter interaction. Like almost always in
many-body physics
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Terminology
''Microscopic'' here implies that quantum mechanics has to be ...
, it is most convenient to apply the
second-quantization formalism. For example, a light field corresponding to frequency
is then described through
Boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
creation and annihilation operators
Creation operators and annihilation operators are Operator (mathematics), mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilatio ...
and
, respectively, where the "hat" over
signifies the operator nature of the quantity. The operator-combination
determines the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
-number operator.
When the photon coherences, here the
expectation value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected va ...
, vanish and the system becomes
quasistationary, semiconductors emit
incoherent light spontaneously, commonly referred to as
luminescence
Luminescence is a spontaneous emission of radiation from an electronically or vibrationally excited species not in thermal equilibrium with its environment. A luminescent object emits ''cold light'' in contrast to incandescence, where an obje ...
(L). (This is the underlying principle behind
light-emitting diodes
A light-emitting diode (LED) is a semiconductor device that emits light when current flows through it. Electrons in the semiconductor recombine with electron holes, releasing energy in the form of photons. The color of the light (corresp ...
.) The corresponding luminescence
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...
is proportional to the temporal change in photon number,
As a result, the luminescence becomes directly generated by a photon-assisted electron–hole recombination,
that describes a correlated emission of a photon
when an electron with wave vector
recombines with a
hole
A hole is an opening in or through a particular medium, usually a solid Body (physics), body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in m ...
, i.e., an electronic vacancy. Here,
determines the corresponding electron–hole recombination operator defining also the microscopic polarization within semiconductor. Therefore,
can also be viewed as photon-assisted polarization.
Many electron–hole pairs contribute to the photon emission at frequency
; the explicit
notation within
denotes that the correlated part of the expectation value
is constructed using the
cluster-expansion approach
The cluster-expansion approach is a technique in quantum mechanics that systematically truncates the BBGKY hierarchy problem that arises when quantum dynamics of interacting systems is solved. This method is well suited for producing a closed set ...
. The quantity
contains the
dipole-matrix element for
interband transition, light-mode's
mode function, and
vacuum-field amplitude.
Principal structure of SLEs
In general, the SLEs includes all
single- and two-particle correlations needed to compute the luminescence spectrum
self-consistent
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of ...
ly. More specifically, a systematic derivation produces a set of equations involving photon-number-like correlations
whose diagonal form reduces to the luminescence formula above. The dynamics of photon-assisted correlations follows from
where the first contribution,
, contains the
Coulomb-renormalized single-particle energy that is determined by the
bandstructure of the
solid
Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
. 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 three-particle correlations that appear are indicated symbolically via the