PHYSICAL REVIEW 8 VOLUME 50, NUMBER 16 15 OCTOBER 1994-II Exciton thermalization in quantum-well structures Massimo Gurioli, Anna Vinattieri, Juan Martinez-Pastor, and Marcello Colocci European Laboratory for Non Linear Spectroscopy and Department of Physics, Uni Uersity of Florence, Largo Enrico Fermi 2, 50125 Firenze, Italy (Received 13 June 1994) We show that a unified picture, namely, thermalization within the inhomogeneously broadened exci- ton band, can be used to explain the excitonic photoluminescence (PL) spectra in quantum wells in- dependently of the presence, or lack thereof, of the low-energy shift of the PL line with respect to the ab- sorption peak (Stokes shift). The Stokes shift itself turns out to be a mere consequence of the thermal equilibrium with a quadratic dependence on the absorption linewidth and a linear dependence on the in- verse of the excitonic temperature. The predictions are found to be in excellent agreement with careful measurements in GaAs/A1„Ga& „As single-quantum-well structures. I. INTRODUCTION Excitons in low-dimensional semiconductor systems, in particular quantum-well (QW) structures, have attracted much interest in the past years due to their peculiar prop- erties, which, making them different from bulk excitons, have permitted the realization of a new generation of electro-optic devices. The optical techniques, and in par- ticular photoluminescence (PL), ' are widely recognized as the most powerful tools not only for studying the intrin- sic features of the confined excitons but also for charac- terizing the defect content and the sample quality. So far two different approaches have been used for the analysis of the QW excitonic optical spectra, mainly de- pending on the presence, or lack thereof, of a redshift of the PL emission with respect to the fundamental absorp- tion transition, usually referred to as the Stokes shift (SS). If the SS is negligible, a free-exciton picture is usually ap- plied to the recombination kinetics in QW's. The exciton center-of-mass wave function is supposed to be delocal- ized over the whole QW plane and, as a consequence of the polariton effect, a very short lifetime is predicted (often referred to as superradiance). The much longer PL decay times measured after nonresonant excitation are then explained in terms of the thermalization between ac- tive and silent excitonic states, which gives rise to a temperature-dependent lifetime. On the contrary, the presence of a sizable SS is commonly assumed " as a direct proof of exciton trapping at interface defects, thus implying both localization of the exciton center-of-mass wave function and lack of thermalization. This picture has strong consequences on the whole recombination kinetics; in fact, apart from the trapping dynamics itself, the radiative lifetime is predicted to increase by nearly one order of magnitude as a consequence of the reduction of the excitonic coherence length. We can therefore sum- marize the commonly accepted pictures as follows: the presence of a SS means exciton localization and lack of thermalization, while its absence is the signature of free excitons and, eventually, of a thermalized distribution. In this paper we show that a unified picture, namely, thermalization within the inhomogeneously broadened exciton band, can be applied for explaining the whole phenomenology of the QW exciton optical spectra. We suppose that the recombining carriers, after momentum and energy relaxation, populate the excitonic states with a quasiequilibrium distribution described by an effective temperature Tz. Then, if the inhomogeneous broadening is small compared to the thermal energy KTC, each exci- ton state has essentially the same thermal population: the PL reflects the absorption and the SS is not observ- able. On the contrary, if the excitonic linewidth exceeds the thermal energy, the low-energy side of the inhomo- geneous exciton band is enhanced by the thermal popula- tion and a SS appears. On the basis of such a simple model we are able to find the general relationship be- tween the SS, the photogenerated carrier temperature Tc, and the absorption linewidth h. In particular we predict that, for a given Tc, the SS is quadratic with 5 rather than linear, as recently claimed in Refs. 10 and 11. We have performed a detailed study of the PL line shape in a set of good quality GaAs/Alo 3Gao 7As single- quantum-well (SQW) structures, with a Stokes shift rang- ing between 0 and 4. 4 meV, by means of photolumines- cence and photoluminescence excitation (PLE) measure- ments under low-power continuous-wave (cw) excitation. The thermalization of the photogenerated carriers is ex- perimentally demonstrated, at all temperatures (1.8 — 120 K) and in all the QW's investigated independently of the SS values, by the presence in the PL spectra of free car- rier and light-hole exciton recombinations. The equilibri- um distribution, which necessarily follows the cw excita- tion, can then be described, with a very good approxima- tion, by a thermal distribution. The resulting carrier temperature Tz is directly obtained from a fit of the PL free-carrier slope; we find that T& is, in general, higher than the lattice temperature TL. Only for resonant exci- tation at the heavy-hole (HH) exciton energy and/or high lattice temperatures does Tc correspond to TL, in a11 oth- er cases T& & TL. As a consequence, care has to be used when assuming that the low-temperature condition KTL «5 implies exciton trapping or when using the nominal lattice temperature TI whenever comparing the 0163-1829/94/50(16)/11817(10)/$06. 00 50 11 817 1994 The American Physical Society