PACS: 71.35.+z, 71.36.+c, 71.38.+i 1. Introduction scattering, which leads to a high rate of the exciton-topolariton transitions . The case of an intermediate It is well known that the exciton–photon coupling leads to level of optical excitation where the energy distribution the creation of exciton polaritons, which are new elementary of the exciton polaritons (and therefore, the spectrum of excitations of the crystalline semiconductor [1,2]. Under the polariton radiation) is determined by elastic exciton– non-resonant excitation the exciton polaritons are generated exciton collisions has not been experimentally studied yet.
with a high momentum (k) on the lower polariton branch According to theoretical predictions by Bisti  an elastic (LPB) of the dispersion curve . Subsequently the collision of two excitons results in a relaxation of one of polaritons relax in energy and momentum due to their them into a polariton and a transition of the second exciton interaction with phonons and finally escape from the crystal.
into the high-momentum region of the LPB. This should lead Already the first studies of exciton polaritons have shown to the appearance of an additional polariton luminescence that the spectrum of the polariton radiation is determined line that is red-shifted from the free exciton transition.
by their spatial and energy distribution [4–6]. It was Up to now such a line has not been observed experimentally.
established, that the decrease of the radiative lifetime The main obstacle for an experimental observation of this and the exciton–phonon scattering rate with decreasing line is its masking by the impurity-bound exciton lines, polariton momentum results in a bottleneck effect, which which are usually present in the near-bandgap luminescence has been theoretically predicted by Toyozawa  and has of semiconductor crystals.
been experimentally observed later by many groups [7–12].
Recently, we reported on the growth of ultra-high The bottleneck in the relaxation process leads to a strong quality Alx Ga1-xAs layers by molecular beam epitaxy .
nonequilibrium energy distribution of polaritons so that the The thin X line of the free exciton radiation dominates the concentration of particles with a momentum k < k0 (where luminescence spectra of these layers, whereas the bound k0 is the momentum value at the crossover between excitons exciton lines are absent. Therefore, this layers was the and photons on the dispersion curve) is smaller than the good object to search the new line of LPB predicted by concentration of the particles with a momentum k > k0.
Bisti . It is necessary to note that in spite of the theory In the following, we refer to polaritons as particles with of bulk polaritons is based on models of uniform media or k < k0, and to excitons as particles with k > k0.
crystals, but one could argue that neither uniformity, nor By now, experimental and theoretical studies of the crystal structure of a substance are necessary for polaritons formation and evolution of the energy distribution of to occur. As an example one can refer to excistence exciton polaritons at low and high levels of non-resonant of the polaritons associated with quasi-2D excitons in optical excitation have been performed [4–14]. At low quantum-size heterostructures. As another example, in excitation levels the energy distribution of exciton polaritons spite of opinion that excitonic polariton does not occur in is determined by collisions (both elastic and inelastic) solid solution we demonstrated recently excitonic polariton of excitons with impurities and phonons . Due to controlled low temperature the optical absorption in our the small rate of the exciton-to-polariton transitions the AlxGa1-x As layers up to temperatures 155 K .
polariton concentration is very low in this case. At high In this paper, we report on the experimental observation excitation levels, the energy distribution of the exciton of a new polaritonic luminescence line, which is due to elaspolaritons is determined by the inelastic exciton–exciton tic exciton–exciton collisions in ultra-high quality AlGaAs ¶ layers. It is shown that the rate of the exciton–polariton E-mail: firstname.lastname@example.org Fax: 8-383-3332771 transitions caused by elastic exciton–exciton collisions is Exciton–polariton transition induced by elastic exciton–exciton collisions in ultra-high quality AlGaAs alloys determined not only by the density of the exciton gas shown in Fig. 3. Fig. 4, a demonstrates the PL kinetics of but also by its temperature, in accordance to theoretical the X and Y lines after the pulsed laser excitation with a predictions by Bisti .
pulse energy of 2 nJ. The intensities of both lines increase within the initial 200 ps after the excitation pulse. This initial rise time is attributed to the exciton creation and 2. Experimental The studied layers of ultra-high quality Alx Ga1-xAs with AlAs fractions of x = 0.21 and x = 0.26 were obtained by molecular beam epitaxy. The details of the layer growth and the equipment used for the recording of continuous wave (cw) photoluminescence (PL) were described elsewhere . The cw PL was excited using an Ar-ion laser with a wavelength of 488 nm. The excitation power density was changed from 3 · 10-4 to 400 W/cm2 using neutral density filters. The time-resolved PL was excited using a dye laser synchronously pumped by a mode-locked Ar-ion laser. This resulted in about 20 ps duration pulses at a wavelength of 580 nm. The intrinsic repetition rate of 80 MHz set by the Ar-ion laser cavity length was reduced to 4 MHz using a cavity damper. The PL was analyzed using a CROMEX 250IS spectrometer and detected by a Hamamatsu C4334 Streakscope camera. The time resolution of the system was better than 50 ps. Samples were placed in a closed-cycle CTI-Cryogenics cryostat.
The laser spot focused on the sample had a Gaussian shape with a half-width of 250 µm during both the cw and timeresolved measurements. The image of the excited spot was projected onto the entrance slit of the monochromator with Figure 1. Low-temperature (T = 4.2K) photoluminescence a unity magnification. Since the monochromator slit widths spectra of a pure AlxGa1-x As layer with the AlAs fraction of 10-20 µm were used, the data were recorded with a x = 0.21 measured at different excitation powers, which are quasi-uniform excitation.
labeled in the figure.
3. Experimental results Fig. 1 shows the 4.2 K photoluminescence spectra of the AlxGa1-x As layer with an AlAs fraction of x = 0.measured at different excitation levels. The line labeled X dominates in all spectra. Recently, we have shown that the PL maximum of the X line coincides with the transmission minimum  allowing to attribute this line to the lower polariton branch radiation . The line which is related to the radiation of the upper polariton branch is absent in the spectra, which evidences the ultra-low concentration of shallow donors in the layer . A new line labeled Y appears on the low-energy tail of the X line when the excitation power exceeds 0.5 mW . This Y line shifts towards lower energy when the excitation power is increased. The variation of the relative position of the Y line with respect to the X line as a function of the laser power is depicted in Fig. 2 for layers with AlAs fractions x = 0.21 and x = 0.26. The dependence of the Y line position on the laser power shows a nonlinear behavior with a maximum energy shift of 9 meV. The relative peak Figure 2. The red shift of the Y line as a function of excitation intensity of the Y line remains nearly constant when the power in AlxGa1-x As layers with AlAs fractions of x = 0.21 and excitation power is increased from 0.5 to 10 mW, whereas x = 0.26. The calculated energy shift of the maximum position of it decreases when the excitation power exceeds 10 mW as the Y line is given as a solid line.
Физика и техника полупроводников, 2006, том 40, вып. 544 T.S. Shamirzaev, A.I. Toropov, A.K. Bakarov, K.S. Zhuravlev, A.Yu. Kobitski, H.P. Wagner, D.R.T. Zahn energy relaxation [21–24]. Thereby after the end of the excitation pulse the Y line intensity increases more rapidly than that of the X line, while the decay of both lines starts simultaneously. In addition oscillations are observed at the beginning of the decay curves. The amplitude of these oscillations diminishes when the excitation power is reduced and the oscillations dissappear when the pulse energy is lower than 0.2 nJ as shown in Fig. 4, b. The origin of these oscillations which is not yet understood might be attributed to a spatial variation of the excitonic gas density . The decay curve of the X emission is expressed by an exponential function with a decay time of 1090 ps.
The decay of the Y line shows a biexponential behavior where the fast initial dynamics has a decay time of 170 ps followed by a slower decay rate with a time constant of 510 ps. When the excitation density is decreased the kinetics of the Y line changes. At excitation densities lower than 0.2 nJ the decay of the Y line becomes single exponential with a decay time of 510 ps.
4. Discussion 4.1. Identification of the Y line There are several possibilities of explain an emission line on the low energy side of the free exciton transition. Such a line can have its origin (i) due to the recombination of bound excitons, (ii) due to phonon replicas of free exciton transitions , (iii) or to inelastic exciton–exciton collisions, which result in the so-called P-line  and further can be caused (iv) by the recombination of biand multi-excitonic molecules, which corresponds to the socalled M-line [14,27]. The nonlinear shift of the Y line with encreasing excitation power unambiguously clarifies that this line is not related to impurity bound excitons or Figure 4. The kinetics of the X and Y line intensity after pulse excitation. The pulse energy was 2 nJ (a) and 0.2 nJ (b). The thick dashed line is a calculated curve for the additional Y line caused by elastic exciton–exciton collisions. In order to clarify the figure the calculated curve is shifted relative to the experimental data.
The time dependence of the exciton gas temperature is shown by the dotted line.
to a phonon replica of the excitonic PL, since the spectral position of these lines is independent on the excitation lavel. As the energy difference of a P-line to the free exciton transition has a minimum value of 3/4 of the free exciton finding energy being 6 meV in AlxGa1-x As with x = 0.21  this interpretation is also ruled out. Finally the value of the low energy shift of the Y line is higher Figure 3. The relative intensity of the Y line as a function of than the calculated biexciton binding energy in AlGaAs excitation power in AlxGa1-x As layers with the AlAs fractions being 1–2meV [29,30] at moderate intensities. Moreover, of x = 0.21 and x = 0.26. The dashed line gives a fitting of -5/in contrast to the P- and M-lines, the relative intensity the relative Y line intensity by the expression IY /IX n0.1Tex.
of the Y line decreases with increasing excitation power.
The exciton gas temperature Tex as a function of excitation power is expressed by stars. The solid line is given as a guide for the eyes. At the same time, the direction and the nonlinear shift Физика и техника полупроводников, 2006, том 40, вып. Exciton–polariton transition induced by elastic exciton–exciton collisions in ultra-high quality AlGaAs alloys of the Y line with increasing excitation density indicates that this line can be related to a new polariton line, which appears due to elastic exciton–exciton collisions of a portion of excited excitons during their lifetime1 as predicted by Bisti . The polaritonic picture is suitable in alloys when the probability of polariton decay determined by scattering on impurities and fluctuations of alloy composition is less than the probability of exciton–photon coupling. Since the band gap fluctuations resulting from fluctuations in the alloy composition are the main reason for broadening of the free exciton line , we estimated the magnitude of band gap fluctuations from the width of the X line. In contrast to AIIBIV or AIIIBV nitride ternary alloys with a high magnitude of alloy band gap fluctuations  the magnitude of alloy fluctuations in the AlGaAs layers is about 1 meV only.
The low values of impurity concentration  and alloy potential fluctuations responsible for polariton scattering allow us to suggest that the appearance of the Y line can be described in terms of the polaritonic model. Recently, we had a direct experimental evidence of the existence of polaritons in our AlGaAs samples . We have demonstrated that the integrated optical absorption of the free excitons decreases with temperature decrease, which Figure 5. Polariton dispersion curve E(K) for AlGaAs layer with indicates a transition form the excitonic to the polaritonic the AlAs fraction of x = 0.21. Population of excitonic polaritons as regime at the lowest temperature. a function of polariton energy calculated according to the models proposed in Ref. 4 (1) and by Bisti (2) at an exciton concentration of 5 · 1014 cm-3.
4.2. Bisti’s theory It is well known that the spectrum of a LPB radiation I(E) which is emitted normal to sample’s surface is given result of the bottleneck in the exciton relaxation process by the equation [4,5]:
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