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arXiv:cond-mat/0103370 · Journal of Experimental and Theoretical Physics Letters

Low-temperature kinetics of 2D exciton gas cooling in quantum well bilayer

S. V. Iordanski, A. Kashuba

Published May 1, 2001• 2 citations
arXivHTMLPDF

Abstract

We study the kinetics of 2D Bose gas cooling provided Bose particles interact with 3D phonons. At low temperatures, phonon emission is prohibited by energy and momentum conservation. We show that both particle—particle scattering and impurity scattering assist Bose gas cooling. The temporal relaxation of temperature follows the law % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jd9sq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcKbay-l% qa-Xgagaafaaaa!38AA! $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} $$ above the Berezinski-Kosterlitz-Thouless phase transition point and T∼1/t after a Bose-Einstein 2D quasi-condensate develops.

DOI
10.1134/1.1385662

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We study the kinetics of 2D Bose gas cooling provided Bose particles interact with 3D phonons. At low temperatures, phonon emission is prohibited by energy and momentum conservation. We show that both particle—particle scattering and impurity scattering assist Bose gas cooling. The temporal relaxation of temperature follows the law % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jd9sq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcKbay-l% qa-Xgagaafaaaa!38AA! l\lower0.5em⌣\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} l\lower0.5em⌣ above the Berezinski-Kosterlitz-Thouless phase transition point and T∼1/t after a Bose-Einstein 2D quasi-condensate develops.
The relaxation kinetics of an ideal Bose gas coupled to a phonon thermostat at temperature Tb below or equal to the critical temperature Tc consists of an initial transient and the following adiabatic stage. The transient depends on the initial distribution of bosons at t = 0 and lasts a few characteristic scattering times. The following nonexponential relaxation towards a quantum degenerate equilibrium state with a Bose-Einstein (BE) condensate is a slow adiabatic process. We find a generic solution of the phonon-assisted Boltzmann kinetics at the adiabatic stage. This solution is completely defined by the ratio Tc/Tb. Here, one has a unique example of the fundamental solution of Boltzmann kinetics that is insensitive to the initial distribution. For Tb → Tc from above, we develop a theory of the thermodynamic critical slowing-down which also leads to the nonexponential thermalization of bosons. We apply our results to the relaxation kinetics of para-excitons in Cu2O and reinterpret the recent reports on the observation of BE condensation of para-excitons.
Quasiequilibrium relaxational thermodynamics is developed to understand LA-phonon-assisted thermalization of Bose-Einstein distributed excitons in quantum wells. We study quantum-statistical effects in the relaxational dynamics of the effective temperature of excitons T=T(t).T=T(t).T=T(t). When T is less than the degeneracy temperature T0,{T}_{0},T0​, well-developed Bose-Einstein statistics of quantum-well excitons leads to nonexponential and density-dependent thermalization. At low bath temperatures ${T}_{b}\ensuremath{\rightarrow}0,$ the thermalization of quantum statistically degenerate excitons effectively slows down and $T(t)\ensuremath{\propto}1/\mathrm{ln}t.$ We also analyze the optical decay of Bose-Einstein distributed excitons in perfect quantum wells, and show how nonclassical statistics influences the effective lifetime ${\ensuremath{\tau}}_{\mathrm{opt}}.$ In particular, ${\ensuremath{\tau}}_{\mathrm{opt}}$ of a strongly degenerate gas of excitons is given by $2{\ensuremath{\tau}}_{R},$ where ${\ensuremath{\tau}}_{R}$ is the intrinsic radiative lifetime of quasi-two-dimensional excitons. Kinetics of resonant photoluminescence of quantum-well excitons during their thermalization is studied within the thermodynamic approach and taking into account Bose-Einstein statistics. We find density-dependent photoluminescence dynamics of statistically degenerate excitons. Numerical modeling of the thermalization and photoluminescence kinetics of quasi-two-dimensional excitons are given for ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ quantum wells.
Photo-excited excitons, i.e. closely bound electron-hole pairs, are in the low-density regime approximately bosons. For sufficiently low temperatures and sufficiently high densities of the exciton gas, a nonequilibrium phase transition in the form of a Bose-Einstein condensation (BEC) can be expected. Various levels of the description of the kinetics of relaxation and condensation due to the coupling of the excitons to a cold reservoir of acoustic phonons are discussed. For noninteracting excitons a rate equation approach with the exciton Boltzmann equation can be used. For interacting excitons the nonmarkovian quantum kinetics is required for the description of the BEC in order to incorporate the changes of the energy spectrum during the condensation self-consistently. Finally colored Langevin fluctuations can be used to calculate the phase diffusion of the condensate.
We study the kinetics of condensation in a trapped two-dimensional exciton gas using the quantum Boltzmann equation. We assume that the exciton gas interacts with a lattice that is kept at a constant temperature. Our simulations allow us to estimate and compare the condensation time for Bose gases with and without interactions. We find that anti-Stokes Raman-type two-phonon processes dominate the relaxation rate of low-energy states for a noninteracting gas. The presence of dissipation from the low-energy states results in a heating effect for the exciton gas which in turn increases the mean number of Bose particles required for the onset of condensation. Our results indicate that the condensation time scale in trapped two-dimensional excitons is determined by the phonon emission rate into the ground state. This time scale is orders of magnitude shorter than what is predicted for three-dimensional excitons. The steady state in a dissipative gas is a nonequilibrium condensate, or equivalently an exciton matter laser, where the exciton temperature remains above the lattice temperature.
Summary form only given. In the dilute limit excitons behave as weakly interacting Bose particles and are expected to undergo the Bose-Einstein condensation (BEC). Because the exciton mass is very small at experimentally accessible exciton densities the critical temperature for exciton BEC should reach several Kelvins, i.e., about six orders of magnitude higher than the T/sub c/ for atom BEC. To create degenerate Bose-gas of excitons we propose quasi-2D gas of indirect excitons in coupled quantum wells (CQWs) which have two primary advantages for realizing a quantum degenerate exciton gas and, ultimately, the BEC-like phase transition: (i) the long lifetime of indirect excitons and (ii) the short exciton cooling time.
The relaxation kinetics of an ideal Bose gas coupled to a phonon bath at temperatures below or equal to the critical temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ is given within a unique scenario. During the first transient stage, which lasts a few characteristic scattering times, the initial distribution disappears. The following nonexponential relaxation towards a quantum degenerate equilibrium state with a Bose-Einstein condensate is a slow adiabatic process. For this second kinetic stage we find an analytic generic solution of the Boltzmann equation. The generic solution is independent of the initial distribution and completely defined by the ratio Tc{\mathrm{T}}_{\mathrm{c}}Tc​/Tb{\mathrm{T}}_{\mathrm{b}}Tb​ of the critical and bath temperatures.