Editing Photoluminescence (section)

=== Ideal quantum-well structures ===

An ideal, defect-free semiconductor [[quantum well]] structure is a useful  model system to illustrate the fundamental processes in typical PL experiments. The discussion is based on results published in Klingshirn (2012)<ref>Klingshirn, Claus F. (2012). ''Semiconductor Optics.'' Springer. {{ISBN|978-3-642-28361-1}} {{OCLC|905285603}}.</ref> and Balkan (1998).<ref>Balkan, Naci (1998). ''Hot Electrons in Semiconductors: Physics and Devices.'' Oxford University Press. {{ISBN|0198500580}}.</ref>

The fictive model structure for this discussion has two confined quantized electronic and two hole [[subband]]s, e<sub>1</sub>, e<sub>2</sub> and h<sub>1</sub>, h<sub>2</sub>, respectively.
The linear [[absorption spectrum]] of such a structure shows the [[exciton]] resonances of the first (e1h1) and the second quantum well subbands (e<sub>2</sub>, h<sub>2</sub>), as well as the absorption from the corresponding continuum states and from the barrier.

==== Photoexcitation ====

In general, three different excitation conditions are distinguished: resonant, quasi-resonant, and non-resonant. For the resonant excitation, the central energy of the laser corresponds to the lowest [[exciton]] resonance of the [[quantum well]]. No, or only a negligible amount of the excess, energy is injected to the carrier system. For these conditions, coherent processes contribute significantly to the spontaneous emission.<ref name="Kira1999" /><ref name="KiraJahnke1999">Kira, M.; Jahnke, F.; Hoyer, W.; Koch, S. W. (1999). "Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures". ''Progress in Quantum Electronics'' '''23''' (6): 189–279. [https://dx.doi.org/10.1016%2FS0079-6727%2899%2900008-7 doi:10.1016/S0079-6727(99)00008-7.]</ref> The decay of polarization creates excitons directly. The detection of  PL is challenging for resonant excitation as it is difficult to discriminate contributions from the excitation, i.e., stray-light and diffuse scattering from surface roughness. Thus, [[speckle pattern|speckle]] and resonant [[Rayleigh scattering|Rayleigh-scattering]] are always superimposed to the [[Coherence (physics)|incoherent]] emission.

In case of the non-resonant excitation, the structure is excited with some excess energy. This is the typical situation used in most PL experiments as the excitation energy can be discriminated using a [[spectrometer]] or an [[optical filter]]. One has to distinguish between quasi-resonant excitation and barrier excitation.

For quasi-resonant conditions, the energy of the excitation is tuned above the ground state but still below the [[Potential barrier|barrier]] [[absorption edge]], for example, into the continuum of the first subband. The polarization decay for these conditions is much faster than for resonant excitation and coherent contributions to the quantum well emission are negligible. The initial temperature of the carrier system is significantly higher than the lattice temperature due to the surplus energy of the injected carriers. Finally, only the electron-hole plasma is initially created. It is then followed by the formation of excitons.<ref name="KaindlCarnahan2003">Kaindl, R. A.; Carnahan, M. A.; Hägele, D.; Lövenich, R.; Chemla, D. S. (2003). "Ultrafast terahertz probes of transient conducting and insulating phases in an electron–hole gas". ''Nature'' '''423''' (6941): 734–738. [https://dx.doi.org/10.1038%2Fnature01676 doi:10.1038/nature01676.]</ref><ref name="ChatterjeeEll2004">Chatterjee, S.; Ell, C.; Mosor, S.; [[Galina Khitrova|Khitrova, G.]]; Gibbs, H.; Hoyer, W.; Kira, M.; Koch, S. W.; Prineas, J.; Stolz, H. (2004). "Excitonic Photoluminescence in Semiconductor Quantum Wells: Plasma versus Excitons". ''Physical Review Letters'' '''92''' (6). [https://dx.doi.org/10.1103%2FPhysRevLett.92.067402 doi:10.1103/PhysRevLett.92.067402.]</ref>

In case of barrier excitation, the initial carrier distribution in the quantum well strongly depends on the carrier scattering between barrier and the well.

==== Relaxation ====

Initially, the laser light induces coherent polarization in the sample, i.e., the transitions between electron and hole states oscillate with the laser frequency and a fixed phase. The polarization dephases typically on a sub-100 fs time-scale in case of nonresonant excitation due to  ultra-fast Coulomb- and phonon-scattering.<ref name="ArltSiegner1999">Arlt, S.; Siegner, U.; Kunde, J.; Morier-Genoud, F.; Keller, U. (1999). "Ultrafast dephasing of continuum transitions in bulk semiconductors". ''Physical Review B'' '''59''' (23): 14860–14863. [https://dx.doi.org/10.1103%2FPhysRevB.59.14860 doi:10.1103/PhysRevB.59.14860.]</ref>

The dephasing of the polarization leads to creation of populations of electrons and holes in the conduction and the valence bands, respectively. The lifetime of the carrier populations is rather long, limited by radiative and non-radiative recombination such as [[Auger recombination]]. During this lifetime a fraction of electrons and holes may form excitons, this topic is still controversially discussed in the literature. The formation rate depends on the experimental conditions such as lattice temperature, excitation density, as well as on the general material parameters, e.g., the strength of the Coulomb-interaction or the exciton binding energy.

The characteristic time-scales are in the range of hundreds of [[picosecond]]s in GaAs;<ref name="KaindlCarnahan2003" /> they appear to be much shorter in [[wide-bandgap semiconductors|wide-gap semiconductors]].<ref name="UmlauffHoffmann1998">Umlauff, M.; Hoffmann, J.; Kalt, H.; Langbein, W.; Hvam, J.; Scholl, M.; Söllner, J.; Heuken, M.; Jobst, B.; Hommel, D. (1998). "Direct observation of free-exciton thermalization in quantum-well structures". ''Physical Review B'' '''57''' (3): 1390–1393. [https://dx.doi.org/10.1103%2FPhysRevB.57.1390 doi:10.1103/PhysRevB.57.1390].</ref>

Directly after the excitation with short (femtosecond) pulses and the quasi-instantaneous decay of the polarization, the carrier distribution is mainly determined by the spectral width of the excitation, e.g., a [[laser]] pulse. The distribution is thus highly non-thermal and resembles a [[Gaussian distribution]], centered at a finite momentum. In the first hundreds of [[femtosecond]]s, the carriers are scattered by phonons, or  at elevated carrier densities via Coulomb-interaction. The carrier system successively relaxes to the [[Fermi–Dirac distribution]] typically within the first picosecond. Finally, the carrier system cools down under the emission of phonons. This can take up to several [[nanoseconds]], depending on the material system, the lattice temperature, and the excitation conditions such as the surplus energy.

Initially, the carrier temperature decreases fast via emission of [[Phonon#Acoustic and optical phonons|optical phonons]]. This is quite efficient due to the comparatively large energy associated with optical phonons, (36meV or 420K in GaAs) and their rather flat dispersion, allowing for a wide range of scattering processes under conservation of energy and momentum. Once the carrier temperature decreases below the value corresponding to the optical phonon energy, [[Phonon#Acoustic and optical phonons|acoustic phonons]] dominate the relaxation. Here, cooling is less efficient due their [[Acoustic dispersion|dispersion]] and small energies and the temperature decreases much slower beyond the first tens of picoseconds.<ref name="KashShah1984">Kash, Kathleen; Shah, Jagdeep (1984). "Carrier energy relaxation in In0.53Ga0.47As determined from picosecond luminescence studies". ''Applied Physics Letters'' '''45''' (4): 401. [https://dx.doi.org/10.1063%2F1.95235 doi:10.1063/1.95235.]</ref><ref name="PollandRühle1987">Polland, H.; Rühle, W.; Kuhl, J.; Ploog, K.; Fujiwara, K.; Nakayama, T. (1987). "Nonequilibrium cooling of thermalized electrons and holes in GaAs/Al_{x}Ga_{1-x}As quantum wells". ''Physical Review B'' '''35''' (15): 8273–8276. [https://dx.doi.org/10.1103%2FPhysRevB.35.8273 doi:10.1103/PhysRevB.35.8273.]</ref> At elevated excitation densities, the carrier cooling is further inhibited by the so-called [[hot-phonon effect]].<ref name="ShahLeite1970">Shah, Jagdeep; Leite, R.C.C.; Scott, J.F. (1970). "Photoexcited hot LO phonons in GaAs". ''Solid State Communications'' '''8''' (14): 1089–1093. [https://dx.doi.org/10.1016%2F0038-1098%2870%2990002-5 doi:10.1016/0038-1098(70)90002-5.]</ref> The relaxation of a large number of  hot carriers leads to a high generation rate of optical phonons which exceeds the decay rate into acoustic phonons. This creates a non-equilibrium "over-population" of optical phonons and thus causes their increased reabsorption by the charge-carriers significantly suppressing any cooling. Thus, a system cools slower, the higher the carrier density is.

==== Radiative recombination ====
The emission directly after the excitation is spectrally very broad, yet still centered in the vicinity of the strongest exciton resonance.  As the carrier distribution relaxes and cools, the width of the PL peak decreases and the emission energy shifts to match the ground state of the exciton (such as an electron) for ideal samples without disorder. The PL spectrum  approaches its quasi-steady-state shape defined by the distribution of electrons and holes. Increasing the excitation density will change the emission spectra. They are dominated by the excitonic ground state for low densities. Additional peaks from higher subband transitions appear as the carrier density or lattice temperature are increased as these states get more and more populated. Also, the width of the main PL peak increases significantly with rising excitation due to excitation-induced dephasing<ref name="WangFerrio1993">Wang, Hailin; Ferrio, Kyle; Steel, Duncan; Hu, Y.; Binder, R.; Koch, S. W. (1993). "Transient nonlinear optical response from excitation induced dephasing in GaAs". ''Physical Review Letters'' '''71''' (8): 1261–1264. [https://dx.doi.org/10.1103%2FPhysRevLett.71.1261 doi:10.1103/PhysRevLett.71.1261.]</ref> and the emission peak experiences a small shift in energy due to the Coulomb-renormalization and phase-filling.<ref name="Haug2009" />

In general, both exciton populations and plasma, uncorrelated electrons and holes, can act as sources for photoluminescence as described in the [[semiconductor-luminescence equations]]. Both yield very similar spectral features  which are difficult to distinguish; their emission dynamics, however, vary significantly. The decay of excitons yields a single-exponential decay function since the probability of their radiative recombination does not depend on the carrier density. The probability of spontaneous emission for uncorrelated electrons and holes, is approximately proportional to the product of electron and hole populations eventually leading to a non-single-exponential decay described by a [[hyperbolic function]].