and of the decrease of the polariton escape time in this range. A polaritonic spectrum calculated from Eg. (1) using I(E) =TS(E)F(E, z )|z =0 (E)v(E), (1) such an exciton–polariton population exhibits a single line where TS(E) is the transmittance of the surface, only. In completion to [4,5] Bisti solved the Boltzmann F(E, z )|z =0 is the distribution function of exciton–polaritons equation for intermediate excitation densities where the at the surface, (E) and v(E) are the density of ex- exciton scattering is dominated by elastic exciton–exciton citonic polaritons and their group velocity, respectively.
collisions. For simplicity exciton–impurity and exciton– Consequently the exciton–polariton spectrum is determined phonon scattering have been neglected and a uniform spatial by its energy and spatial distribution F(E, z ) taken at exciton distribution was assumed in these calculations.
the crystal’s surface. In general the distribution function The calculated exciton–polariton population following Bisti’s F(E, z ) is determined by solving the Boltzmann equation.
theory is shown in Fig. 5 as solid curve 2. In the energy For low exciton concentration where the exciton scattering range above the crossover point the population obtained by is dominated by collisions with impurities and phonons Bisti’s approach coincides with that calculated previously, the Boltzmann equation was solved by many groups .
but on the low energy side an additional line appears It was shown that most of the excitons are concentrated that is shifted to lower energies compared to the free above the crossing point of the excitonic and photonic exciton transition energy. It is important to note that dispersion curves. Fig. 5 depicts the calculated polariton the exciton–exciton collisions become dominant when the population which is a product of the distribution function probability of these collisions ( ) exceeds the probabilities ee and density of states (E)F(E, z )|z =0 as a function of their of the exciton–acoustic phonon ( ), and exciton–impurity ep energy using the model proposed in Ref. 4. The low scattering processes ( ). A comparison of the values of ei energy truncation of the exciton–polariton population is a the and in GaAs was calculated by Bisti revealing ep ee At intermediate temperatures when both excitons and free electrons that exceeds if the concentration of excitons exceeds ee ep are present due to the thermal dissociation of a part of the excitons, a 1013 cm-3. In order to compare with we can use the ee ei scattering of excitons by electrons should be also considerable. However, results of Ziljaev et al. . For pure GaAs it was shown we suppose that at lower temperatures, when excitons are not thermally dissociated, the exciton–exciton scattering dominates. that the value of exceeds that of by more than an ei ep 3 Ôèçèêà è òåõíèêà ïîëóïðîâîäíèêîâ, 2006, òîì 40, âûï. 546 T.S. Shamirzaev, A.I. Toropov, A.K. Bakarov, K.S. Zhuravlev, A.Yu. Kobitski, H.P. Wagner, D.R.T. Zahn order of magnitude even when the impurity concentration is lower than 1014 cm-3. Therefore, the energy distribution of polaritons predicted by the Bisti’s theory can be realized experimentally only in very pure semiconductor crystals, which might be the reason for the lack of any experimental observation of that additional polariton line so far in GaAs.
Another possible reason may be connected with exciton screening at high excitation densities. The influence of the exciton screening is greater at greater exciton radii. It is possible that in GaAs with the exciton radiums greater than that in AlGaAs the screening at high excitation densities is more important than the exciton–exciton collision process.
4.3. Comparison of the Bisti’s theory with Figure 6. Low temperature photoluminescence spectra: a — experiment, cw PL spectra experimentally observed and b — calculated at different exciton Using Bisti’s theory we calculated the LPB radiation concentrations: 1 — 2 · 1013 cm-3, 2 — 1014 cm-3, 3 — 7 · 1014 cm-3. The calculated spectrum at an exciton concentration I(E) determining the energetic position and intensity of the of 2 · 1013 cm-3 is also shown in the left figure as dashed line.
additional polariton line as functions of the density (n) and temperature (Tex) of the exciton gas. The values of the permittivity and effective masses of the charge carriers in AlGaAs alloys used in the calculations were taken from the experimental data. The best fit of the radio of the Y line Ref. 28. The probability of the nonradiative recombination to X line intensity versus the exciton concentration is given of excitons, which is a parameter in the calculation, was -5/by IY /IX n0.1Tex. Since the intensity of the X line is taken to be 3 · 108 s-1 to reach the best agreement between proportional to the exciton concentration , the intensity the experimental data and calculation. The energy value -5/of the Y line is described by IY n1.1Tex. We suppose at the crossover between exciton and photon dispersion that the difference between the experimental and calculated curves as a function of the AlAs fraction was calculated values of the Y line intensity can be explained by two factors:
using the AlGaAs band-gap and free exciton binding energy (i) a deviation of the spatial polariton distribution from the taken from Ref. 34 and Ref. 35, respectively. In order uniform distribution assumed by Bisti; and (ii) a deviation to compare the experimental spectra with calculated data of the energy distribution of polaritons from that postulated we estimated the concentration of nonequilibrium carriers by Bisti due to the scattering of excitons caused by alloy using the method described in Ref. 28. For this estimation, disorder. In fact, the intensity of the Y line decreases we took a charge carrier lifetime of 1 ns (evaluated from when the crystal thickness decreases . Actually, the our time-resolved data) and a carrier diffusion length of effective thickness of the cristal is lower than the sample 1 µm (from Ref. 28). In order to determine the value thickness taken in our calculation because the realistic of Tex, which raises with increasing excitation power, we spatial polariton distribution is a decaying function of depth, fitted the high-energy tail of the X line by a temperatureas shown by Rossin . In addition the extra scattering dependent exponential function f ( ) =a exp(- /kTex), by alloy disorder, which competes with exciton–exciton where a is a constant, is the photon energy, and k scattering, leads to a decrease of the exciton-to-polariton is the Boltzmann’s constant . As shown in Fig. 3 the transition rate.
temperature of the exciton gas increases from 6 to 20 K within the applied excitation power range.
The experimentally observed and calculated photolumi4.4. Kinetics of the polariton emission nescence spectra are given in Fig. 6, a and Fig. 6, b, respecThe most intriguing result of the time-resolved PL tively, showing good accordance. Indeed, the novel Y line experiment is that the decay curve of the Y line shows appears in the calculated spectra at an exciton concentration a biexponential behavior at high excitation powers while higher than 1013 cm-3 in agreement to the experimental it reveals a single exponential decay at the low excitation data. The width of the Y line in the experimental spectra is power. In the latter case the characteristic decay time is slightly broader compared to the calculated spectra which is half the time constant of the X emission, which is typical attributed to alloy disorder neglected by the applied theory.
for a radiation generated by exciton–exciton collisions .
The calculated energy position of the Y line as a function We will show that the biexponential behavior at high of the exciton concentration n is given by the logarithmic excitation powers is due to a cooling of the exciton gas dependence kT ln n, which fits the experimental data of Fig. 2 well. Note, however, that the calculated depen- after the excitation pulse.
dence of the Y line intensity on the exciton concentration, If the exciton-to-polariton transitions are due to collisions -5/given by an expression IY n1.3Tex, slightly differs from of excitons, then the kinetics of the polariton concentration Ôèçèêà è òåõíèêà ïîëóïðîâîäíèêîâ, 2006, òîì 40, âûï. Exciton–polariton transition induced by elastic exciton–exciton collisions in ultra-high quality AlGaAs alloys is described by the high excitation level coincides with the cooling of the exciton gas. When the exciton gas temperature comes close dnp np to the lattice temperature, the slope of the Y line decay = - + B(t), (2) dt p curve approaches half of the decay time of the X emission.
where np and p are the concentration and the radiative lifetime of polaritons, respectively, and B(t) is the rate of the 5. Summary exciton-to-polariton transitions due to the exciton–exciton In conclusion, we have observed a new polariton radicollisions.2 The low polariton state density and the low ation line in the photoluminescence spectra of ultra-high value of the polariton lifetime  allow us to neglect the quality AlGaAs layers. We have shown that this line appears inverse polariton–exciton transitions in expression (2). Since as a result of the exciton-to-polariton transition caused by the exciton gas has a Maxwellian energy distribution, which the elastic exciton-exciton collisions. The rate of the elastic is evidenced by the exponential shape of the high-energy exciton-to-polariton scattering is determined not only by the tail of the X line in both the cw and time-resolved PL density of the excitonic gas, but also by its temperature.
spectra , the rate of the exciton-to-polariton transitions due to the elastic exciton–exciton collisions may be written following Bisti  asAcknowledgments n2 (t) E E ex We acknowledge the support of this work by the Russian B(t) J exp -, (3) kTex(t) kTex(t) kTex(t) Foundation for Basic Research (N 04-02-16774) and by the Graduiertenkolleg Thin films and non-crystalline materials“ where nex is the concentration of excitons, E is the differ” of the Technische Universitt Chemnitz.
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