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. do]
10.1103/PhysRevLett.79.5170
/ref>[Kira, M.; Koch, S. W. (2011). ''Semiconductor Quantum Optics''. Cambridge University Press. .] describe luminescence
Luminescence is spontaneous emission of light by a substance not resulting from heat; or "cold light".
It is thus a form of cold-body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions or stress on a crysta ...
of semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
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 to physics. For transport ...
of spontaneously emitted light. This description established the first step toward semiconductor quantum optics
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
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 force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. Th ...
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" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
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 of the atom.
General theory
Typically the photon contai ...
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 popul ...
with coherent 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" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
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
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
.
Starting point
The derivation of the SLEs starts from a system 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. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
, 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 odd half-integer s ...
creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
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 particle, massless ...
-number operator.
When the photon coherences, here the expectation value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
, vanish and the system becomes quasistationary, semiconductors emit incoherent light spontaneously, commonly referred to as luminescence
Luminescence is spontaneous emission of light by a substance not resulting from heat; or "cold light".
It is thus a form of cold-body radiation. It can be caused by chemical reactions, electrical energy, subatomic motions or stress on a crysta ...
(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 (cor ...
.) 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 to physics. For transport ...
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. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
, 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 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 classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
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 one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structura ...
. 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
10.1103/PhysRevB.75.155315
/ref>[Berstermann, T.; Auer, T.; Kurtze, H.; Schwab, M.; Yakovlev, D.; Bayer, M.; Wiersig, J.; Gies, C.; Jahnke, F.; Reuter, D.; Wieck, A. (2007). "Systematic study of carrier correlations in the electron–hole recombination dynamics of quantum dots". ''Physical Review B'' 76 (16). do]
10.1103/PhysRevB.76.165318
/ref>[Shuvayev, V.; Kuskovsky, I.; Deych, L.; Gu, Y.; Gong, Y.; Neumark, G.; Tamargo, M.; Lisyansky, A. (2009). "Dynamics of the radiative recombination in cylindrical nanostructures with type-II band alignment". ''Physical Review B'' 79 (11). do]
10.1103/PhysRevB.79.115307
/ref>
destroys as many electron–hole pairs as it creates photons because due to the general conservation law
\frac \sum_\omega \langle \hat^\dagger_ \hat_ \rangle = -\frac \sum_\mathbf f^e_\mathbf.
Besides the terms already described above, the photon-assisted polarization dynamics contains a spontaneous-emission source
\Omega^\mathrm_ = \mathrm \mathcal_ \Bigl(f^e_ f^h_ + \sum_ c_\mathrm^ \Bigr)\,.
Intuitively, f^e_\mathbf \, f^h_\mathbf describes the probability to find electron and hole with same \mathbf when electrons and holes are uncorrelated, i.e., plasma. Such form is to be expected for a probability of two uncorrelated events to occur simultaneously at a desired \mathbf value. The possibility to have truly correlated electron–hole pairs is defined by a two-particle correlation c_\mathrm^; the corresponding probability is directly proportional to the correlation. In practice, c_\mathrm^ becomes large when electron–hole pairs are bound as excitons
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. Th ...
via their mutual Coulomb attraction. Nevertheless, both the presence of electron–hole plasma and excitons can equivalently induce the spontaneous-emission source.
As the semiconductor emits light spontaneously, the luminescence is further altered by a stimulated contribution
\Delta\Omega_^\mathrm = \mathrm \sum_ \mathcal_ \,\Delta \langle \hat^\dagger_ \hat_ \rangle
that is particularly important when describing spontaneous emission in semiconductor microcavities and lasers
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" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
because then spontaneously emitted light can return to the emitter (i.e., the semiconductor), either stimulating or inhibiting further spontaneous-emission processes. This term is also responsible for the Purcell effect
Henry Purcell (, rare: September 1659 – 21 November 1695) was an English composer.
Purcell's style of Baroque music was uniquely English, although it incorporated Italian and French elements. Generally considered among the greatest E ...
.
To complete the SLEs, one must additionally solve the quantum dynamics of exciton correlations
\begin
\mathrm\hbar\frac c_\mathrm^ = & \left(
\tilde_ - \tilde_ \right)\, c_\mathrm^
+ S_\mathrm^
\\
&+ \Bigl( 1-f^e_-f^h_ \Bigr) \sum_ V_ \,c_\mathrm^
- \Bigl( 1-f^e_-f^h_ \Bigr) \sum_ V_ \,c_\mathrm^
\\
&+ D_\mathrm^+ T_\mathrm^\,.
\end
The first line contains the Coulomb-renormalized kinetic energy of electron–hole pairs and the second line defines a source that results from a Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermod ...
-type in- and out-scattering of two electrons and two holes due to the Coulomb interaction. The second line contains the main Coulomb sums that correlate electron–hole pairs into exciton
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. Th ...
s whenever the excitation conditions are suitable. The remaining two- and three-particle correlations are presented symbolically by D_\mathrm^ and T_\mathrm^, respectively.[Kira, M.; Koch, S.W. (2006). "Many-body correlations and excitonic effects in semiconductor spectroscopy". ''Progress in Quantum Electronics'' 30 (5): 155–296. do]
10.1016/j.pquantelec.2006.12.002
/ref>
Interpretation and consequences
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. As a result, \Omega^\mathrm_ is finite and it drives 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 particle, massless ...
-assisted processes for all those \mathbf values that correspond to the excited states. This means that \Pi_ is simultaneously generated for many \mathbf values. Since the Coulomb interaction couples \Pi_ with all \mathbf values, the characteristic transition energy follows from the exciton energy, not the bare kinetic energy of an electron–hole pair. More mathematically, the homogeneous part of the \Pi_ dynamics has eigenenergies that are defined by the generalized Wannier equation The Wannier equation describes a quantum mechanical eigenvalue problem in solids where an electron in a conduction band and an electronic vacancy (i.e. hole) within a valence band attract each other via the Coulomb interaction. For one electron and ...
not the free-carrier energies. For low electron–hole densities, the Wannier equation produces a set of bound eigenstates which define the exciton
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and some liquids. Th ...
resonances.
Therefore, \Pi_ shows a discrete set of exciton resonances regardless which many-body state initiated the emission through the spontaneous-emission source. These resonances are directly transferred to excitonic peaks in the luminescence itself. This yields an unexpected consequence; the excitonic resonance can equally well originate from an electron–hole plasma or the presence of excitons.[Kira, M.; Jahnke, F.; Koch, S. (1998). "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence". ''Physical Review Letters'' 81 (15): 3263–3266. do]
10.1103/PhysRevLett.81.3263
/ref> At first, this consequence of SLEs seems counterintuitive because in few-particle picture an unbound electron–hole pair cannot recombine and release energy corresponding to the exciton resonance because that energy is well below the energy an unbound electron–hole pair possesses.
However, the excitonic plasma luminescence is a genuine many-body effect where plasma emits ''collectively'' to the exciton resonance. Namely, when a high number of electronic states participate in the emission of a single photon, one can always distribute the energy of initial many-body state between the one photon at exciton energy and remaining many-body state (with one electron–hole pair removed) without violating the energy conservation. The Coulomb interaction mediates such energy rearrangements very efficiently. A thorough analysis of energy and many-body state rearrangement is given in Ref.[
In general, excitonic plasma luminescence explains many nonequilibrium emission properties observed in present-day semiconductor luminescence experiments. In fact, the dominance of excitonic plasma luminescence has been measured in both quantum-well][Chatterjee, S.; Ell, C.; Mosor, S.; Khitrova, G.; Gibbs, H.; Hoyer, W.; Kira, M.; Koch, S.; Prineas, J.; Stolz, H. (2004). "Excitonic Photoluminescence in Semiconductor Quantum Wells: Plasma versus Excitons". ''Physical Review Letters'' 92 (6). do]
10.1103/PhysRevLett.92.067402
/ref> and quantum-dot systems.[Schwab, M.; Kurtze, H.; Auer, T.; Berstermann, T.; Bayer, M.; Wiersig, J.; Baer, N.; Gies, C.; Jahnke, F.; Reithmaier, J.; Forchel, A.; Benyoucef, M.; Michler, P. (2006). "Radiative emission dynamics of quantum dots in a single cavity micropillar". ''Physical Review B'' 74 (4). do]
10.1103/PhysRevB.74.045323
/ref> Only when excitons are present abundantly, the role of excitonic plasma luminescence can be ignored.
Connections and generalizations
Structurally, the SLEs resemble the semiconductor Bloch equations (SBEs) if the \Pi_ are compared with the microscopic polarization within the SBEs. As the main difference, \Pi_ also has a photon index \omega, its dynamics is driven spontaneously, and it is directly coupled to three-particle correlations. Technically, the SLEs are more difficult to solve numerically than the SBEs due to the additional \omega degree of freedom. 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[Hader, J.; Moloney, J. V.; Koch, S. W. (2006). "Influence of internal fields on gain and spontaneous emission in InGaN quantum wells". ''Applied Physics Letters'' 89 (17): 171120. do]
10.1063/1.2372443
/ref>[Hader, J.; Hardesty, G.; Wang, T.; Yarborough, M. J.; Kaneda, Y.; Moloney, J. V.; Kunert, B.; Stolz, W. et al. (2010). "Predictive Microscopic Modeling of VECSELs". ''IEEE J. Quantum Electron.'' 46: 810. do]
10.1109/JQE.2009.2035714
/ref> and disorder studies.[Rubel, O.; Baranovskii, S. D.; Hantke, K.; Heber, J. D.; Koch, J.; Thomas, P. V.; Marshall, J. M.; Stolz, W. et al. (2005). "On the theoretical description of luminescence in disordered quantum structures". ''J. Optoelectron. Adv. M.'' 7 (1): 115.]
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. Many explicit examples are given in Refs.[Imhof, S.; Bückers, C.; Thränhardt, A.; Hader, J.; Moloney, J. V.; Koch, S. W. (2008). "Microscopic theory of the optical properties of Ga(AsBi)/GaAs quantum wells". ''Semicond. Sci. Technol.'' 23 (12): 125009.] for quantum-well and quantum-wire systems, and in Refs.[Feldtmann, T.; Schneebeli, L.; Kira, M.; Koch, S. (2006). "Quantum theory of light emission from a semiconductor quantum dot". ''Physical Review B'' 73 (15). do]
10.1103/PhysRevB.73.155319
/ref>[Baer, N.; Gies, C.; Wiersig, J.; Jahnke, F. (2006). "Luminescence of a semiconductor quantum dot system". ''The European Physical Journal B'' 50 (3): 411–418. do]
10.1140/epjb/e2006-00164-3
/ref> for quantum-dot systems.
Semiconductors also can show several resonances well below the fundamental exciton resonance when phonon-assisted electron–hole recombination takes place. These processes are describable by three-particle correlations (or higher) where photon, electron–hole pair, and a lattice vibration, i.e., a phonon, become correlated. The dynamics of phonon-assisted correlations are similar to the phonon-free SLEs. Like for the excitonic luminescence, also excitonic phonon sidebands can equally well be initiated by either electron–hole plasma or excitons.[Böttge, C. N.; Kira, M.; Koch, S. W. (2012). "Enhancement of the phonon-sideband luminescence in semiconductor microcavities". ''Physical Review B'' 85 (9). do]
10.1103/PhysRevB.85.094301
/ref>
The SLEs can also be used as a systematic starting point for semiconductor quantum optics
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
.[Gies, Christopher; Wiersig, Jan; Jahnke, Frank (2008). "Output Characteristics of Pulsed and Continuous-Wave-Excited Quantum-Dot Microcavity Lasers". ''Physical Review Letters'' 101 (6). do]
10.1103/PhysRevLett.101.067401
/ref>[Aßmann, M.; Veit, F.; Bayer, M.; Gies, C.; Jahnke, F.; Reitzenstein, S.; Höfling, S.; Worschech, L. et al. (2010). "Ultrafast tracking of second-order photon correlations in the emission of quantum-dot microresonator lasers". ''Physical Review B'' 81 (16). do]
10.1103/PhysRevB.81.165314
/ref> As a first step, one also includes two-photon absorption correlations, \Delta \langle \hat_\omega \hat_ \rangle, and then continues toward higher-order photon-correlation effects. This approach can be applied to analyze the 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 of the atom.
General theory
Typically the photon contai ...
effects and to realize and understand the quantum-optical spectroscopy.
See also
* Coherent effects in semiconductor optics
* 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 ...
* Photoluminescence
Photoluminescence (abbreviated as PL) is light emission from any form of matter after the absorption of photons (electromagnetic radiation). It is one of many forms of luminescence (light emission) and is initiated by photoexcitation (i.e. pho ...
* 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 ...
* 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 bas ...
* Semiconductor laser theory
Semiconductor lasers or laser diodes play an important part in our everyday lives by providing cheap and compact-size lasers. They consist of complex multi-layer structures requiring nanometer scale accuracy and an elaborate design. Their theor ...
References
Further reading
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* {{cite book, last1=Kalt, first1=H., last2=Hetterich, first2=M., title=Optics of Semiconductors and Their Nanostructures, year=2004, publisher=Springer, isbn=978-3540383451
Semiconductor analysis
Quantum optics