Coherent and incoherent dynamics of a dilute exciton system

PERGAMON

Solid State Communications 116 (2000) 253±257

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Coherent and incoherent dynamics of a dilute exciton system D.B. Tran Thoai 1, H. Haug* Institut fuÈr Theoretische Physik, J.W. Goethe UniversitaÈt Frankfurt, Robert-Mayer-Str. 8, D-60054 Frankfurt a.M., Germany Received 6 July 2000; received in revised form 25 July 2000; accepted 31 July 2000 by J. Kuhl

Abstract The kinetic equations for the polarization, the exciton distribution function and the anomalous correlation function of a system of coherently excited weakly interacting excitons are studied including the mean-®eld terms and their corresponding collision terms. These kinetic equations are solved by an adiabatic elimination of the anomalous correlation function. Our numerical results show besides the existence of the Bose±Einstein condensation (BEC) above the critical density also the slowing-down of the dephasing, when the critical density of the BEC is approached. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Semiconductors; B. Optical properties; E. Time-resolved optical spectroscopies PACS: 71.35.Lk

1. Introduction Bose±Einstein condensation has found renewed interest due to recent experimental observations in atomic trapped gases [1,2] and in excitonic systems [3]. For optically excited semiconductors, many authors have treated excitons as weakly interacting bosons, either in the purely coherent limit [4,5], or in a totally incoherent regime [6,7]. Only recently, Schmitt et al. [8] have included both aspects on a microscopic level. The kinetics of the exciton dephasing after coherent pulse excitation have been studied by introducing the anomalous correlation function and its collision terms. The anomalous correlation function, which plays a crucial role in the dynamics of the system for several reasons [5,9], has been often neglected in the derivation of the mean-®eld equations. If one only considers the exciton± exciton scattering without taking into account the anomalous correlation function, the coherent amplitude of the exciton system would stay coherent forever and no dephasing could take place. But taking into account the anomalous correlation, Schmitt et al. [8] have solved the kinetic

* Corresponding author. Tel.: 49-69-798-22-334; fax: 49-69-79828-354. E-mail address: [email protected] (H. Haug). 1 Permanent address: Institute of Physics, Mac Dinh Chi 1, Ho Chi Minh City, Vietnam.

equation for ZnSe and found a pronounced slowing-down of the dephasing near the critical density. However, above the critical density the system did not reach a stable condensed phase. Note that the inverse process of dephasing of a coherent amplitude, namely the build-up of a coherent amplitude out of an originally incoherent system has been studied by many authors in the partially Bose-condensed trapped gas [10±14] and also in the problem of Bose star formation from the dark matter in the Universe [15]. Stoof [10,13] used the Schwinger±Keldysh formalism to describe the dynamics of the atomic BEC, his approach is similar to the treament of the exciton system by Schmitt et al. [8]. In spite of many similarities, the excitons differ in one respect essentially from the trapped atomic gases. In the considered exciton system, the gauge symmetry has been broken initially by the coherent light pulse and at least a small phase remains until the condensation sets in. Therefore, stochastic ¯uctuations are not required to be included as in the case of atomic systems [10,13,16]. In this work, we focus our attention on the kinetics of the coherently excited dilute exciton system with na3B p 1 where aB is the exciton Bohr radius. If the boson system is suf®ciently dilute, the anomalous correlation function will evolve rapidly [11]. Therefore, we shall try to ®nd the BEC solution of the microscopically derived kinetic equations by using a formal adiabatic elimination of the anomalous correlation function.

0038-1098/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(00)00320-3

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2. Kinetic equations Let us consider a system of excitons generated by a coherent light pulse via optically allowed dipole transitions. The Hamiltonian of the system is given by: X X 1 1 1 Hˆ W 1k b1 b b b b k bk 1 4V k ;k ;q k1 k2 k2 2q k1 1q k 1

2

p 2 V …dE…t†b1 0 1 h:c:†;

…1†

where b1 k and bk are the creation and annihilation operator of a 1s-exciton with the momentum k, respectively,

2p pW 2 X ˆ p ‰n n …1 1 nk † 2 …1 1 nq † 2r col "V 2 k ;q q k2q 1 £ …1 1 nk2q †nk Š £ d…1q 1 1k2q 2 1k 2 10 †; 2nk ˆ C22 …k† 1 C12 …k†; 2t col C22 …k† ˆ

pW 2 X p ‰nk1q nk1 2q …1 1 nk1 †…1 1 nk † "V 2 k1 ;q

£ d…1k1q 1 1k1 2q 2 1k 2 1k1 †;

is the kinetic energy of the exciton, d the dipole exciton matrix element, E(t) the classical light ®eld and V is the volume of the crystal. The exciton±exciton interaction potential W is assumed to be a contact potential, which can be evaluated analytically in the small momentum approximation [17]: 26 3 paB ER 3

where ER is the exciton binding energy. Note that in Eq. (1), the coherent light pulse excites the ground state exciton with zero center-of-mass momentum, because we neglect for simplicity the ®nite photon momentum. p The kinetic equations of the polarization p ˆ 1= V kb0 l; the anomalous correlation function Fk ˆ kbk b2k l and the exciton distribution function nk ˆ kb1 k bk l can be derived by the equation-of-motion method. The set of the resulting equations is a time-dependent version of Hartree±Fock± Bogoliubov equations, which had been studied in equilibrium many years ago [18,19]. The resulting equations are: 2 0 1 3 2p 14 @1 1 X A ˆ 2i W n 1 n 1 10 5p 2t " 2 0 V q q W 1 X 1 2p Fq pp 1 i dE0 …t† 1 ; 2" V q 2" 2t col

…2†

20 1 3 2nk W 4@ 2 1 X A p 5 2nk Fq Fk 1 ; Im p 1 ˆ " V q 2t 2t col

…3†

2i

…6†

2 …1 1 nk1q †…1 1 nk1 2q †nk1 nk Š

"2 k2 1k ˆ 2m

Wˆ

(5)

2 0 1 3 2Fk 24 @ 1 X A ˆi W n0 1 n 1 1k 5Fk 2t " V q q 0 1 X W 1 2Fk 2 2 i …1 1 2nk †@p 1 Fq A 1 ; 2" V q 2t col where 2 n0 ˆ kb1 0 bl ˆ upu

…4†

C12 …k† ˆ

2pW 2 X p n0 {‰nk1q …1 1 nq †…1 1 nk † "V 2 k1 ;q 2 …1 1 nk1q †nq nk Šd…1k1q 1 10 2 1q 2 1k † 1

1 ‰n n …1 1 nk † 2 …1 1 nk2q †…1 1 nq †nk Š 2 k2q q

 d…1k2q 1 1q 2 1k 2 10 †}; 2 3 2Fk pW 2 4 1 X ˆ B F 1 Ck n0 5: Ak Fk 1 V q k;q k1q 2t col "

…7†

Our kinetic equations consist of the fully coherent mean ®eld terms (Eqs. (2)±(4)) and the incoherent collision terms (Eqs. (5)±(7)). The collision integral C22 describes the collisions between non-condensate excitons while C12 describes collisions which involve condensate excitons. Many studies on exciton population dynamics only took into account the collision integral C22 [6,7]. Expressions for Ak, Bk,q and Ck are lengthy and have been given in Ref. [8]. The kinetic equations derived in Ref. [8] using the Green-function tech2nk nique are identical to ours. All collision terms 2p 2t ucol ; 2t ucol ; 2Fk 2 2t ucol are proportional to W (®rst Born approximation). For a dilute Bose system like excitons in semiconductors, the high-order anomalous correlation function Fk varies faster compared to the polarization and the distribution function. Therefore, we could formally eliminate the anomalous correlation function Fk from the equation of motion by integrating over its effects during collisions. In a recent work, Proukakis et al. [11] have derived a time-dependent generalized nonlinear SchroÈdinger equation for partially Bose-condensed trapped atoms by introducing a formal adiabatic elimination of the anomalous correlation function for the set of the Hartree±Fock±Bogoliubov equations of the trapped Bose gas. We shall limit ourselves to the second order in the interacting potential W. Note that in the equations of the polarization and of the distribution function, Fk only appears in the combination WFk. We formally solve the equation of the anomalous correlation function to ®rst order in the

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255

Fig. 1. Density n and n0 for a subcritical density ntotal ˆ n 1 n0 ˆ 2:6 £ 1016 cm23 : Exciting pulse has been also plotted.

Fig. 3. Density n and n0 for a supercritical density ntotal ˆ 1:76 £ 1017 cm23 :

interaction potential W and obtain: " " ! W Z 2i 1 X dt1 exp 2 W n0 1 nq Fk ˆ 2i 2" " V # # ! 1 X 2 1 1k …t 2 t1 † …2nk 1 1† p 1 Fq V

any excitons in k ± 0 states. The coherent excitation would stay coherent forever. 3. Numerical results and discussion …8†

The sum of Fq in the solution (8) is also disregarded because its contribution in the equation of p and nk will be of the order of W 3, if we iteratively replace Fq by its ®rst order value. Therefore, we get " " ! W Z 2i 1 X Fk ˆ 2i nq dt1 exp 2 W n0 1 2" " V # # 1 1k …t 2 t1 † …2nk 1 1†p2 :

…9†

This equation together with the equations of the polarization and distribution function will be numerically solved in the next section. In two recent investigations of Grif®n et al. [12,14], the two-¯uid hydrodynamic equations of a trapped Bose gas have been studied by neglecting the anomalous correlation function Fk totally. A total neglect of Fk in the kinetic equations of the coherently excited exciton gas would not scatter

Fig. 2. Decay of the polarization for a subcritical density ntotal ˆ 2:6 £ 1016 cm23 :

In this section, we carry out numerical calculations for ZnSe using the same parameters as Schmitt et al. [8] with an exciton Rydberg ER ˆ 19 meV and an exciton Bohr radius aB ˆ 6 nm: The laser pulse-width (FWHM) is 700 fs and the laser is tuned to the exciton ground state …10 ˆP0†: Fig. 1 shows the noncondensate density n ˆ nk and condensate density n0 ˆ upu2 versus time for ntotal ˆ n 1 n0 ˆ 2:6 £ 1016 cm23 ; which is below the critical density. The condensate density is growing during the pulse and decaying after the pulse as expected. Fig. 2 shows the corresponding decay of the polarization at the same ntotal ˆ 2:6 £ 1016 cm23 : Fig. 3 shows the kinetics above the critical density. One sees a weak beating shortly after the pulse due to the condensed and non-condensed excitons. Our most striking result is the existence of a Bose±Einstein condensation. It means that only a proper treatment of the equation of the

Fig. 4. Asymptotic exciton distribution function (points) versus exciton energy for a supercritical density ntotal ˆ 1:76 £ 1017 cm23 : Solid line: Bose distribution function with chemical potential m ˆ 0 and temperature T ˆ 9:5 K: Inset: Same curves in semilogarithmic plot.

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retrieve the interesting results obtained by Schmitt et al. [8], namely the critical slowing-down of the dephasing rate when the critical density is approached. 4. Conclusions

Fig. 5. Asymptotic sum of the anomalous correlation uFk u versus excitation density.

anomalous correlation function, which has been recently discussed by Proukakis et al. [11] in the context of the trapped Bose gas, can lead to the existence of the Bose± Einstein condensation. In a recent paper on the exciton± phonon kinetics including both the mean-®eld terms and their corresponding collisions terms [20], we have used the same method to adiabatically eliminate the faster anomalous correlation function. We have found that any improper k treatment of the collision terms 2F 2t ucol would destroy the BEC even for excitation densities far above the critical density. These ®ndings hold both for exciton±exciton kinetics and for exciton±phonon kinetics [20]. In Fig. 4, we plot the corresponding asymptotic exciton distribution function (points) versus exciton energy for the supercritical density ntotal ˆ 1:76 £ 107 cm23 …na3B ˆ 0:036 p 1† reached for t . 2 ps: This distribution function can be ®tted quite well with a Bose distribution function with chemical potentialP m ˆ 0 and temperature T ˆ 9:5 K: In Fig. 5, we display k uFk …t ˆ 1†u versus the excitation density. This sum shows a pronounced threshold behavior as expected in the phase transition, also it does not vanish strictly below threshold. In Fig. 6, we plot various quantities versus excitation density: the dephasing time T2 and the chemical potential m of the asymptotic distribution function. One can observe that the chemical potential approaches zero from below. We

In conclusion, we have numerically solved the kinetic equations for a weakly interacting exciton system on a microscopic level by adiabatically eliminating the anomalous correlation function. To the best of our knowledge, no explicit numerical calculations have been carried out before for microscopically derived kinetic equations of atomic trapped gases or of related systems. A numeric study on kinetics of BEC in Ref. [15] is based on generalized Boltzmann scattering rates in Eqs. (3) and (4) (without anomalous correlation function). In this framework, the kinetics due to particle±particle scattering toward a stable BE-condensation phase has been obtained. We hope that our results will stimulate corresponding experimental and theoretical investigations of the kinetics of the excitonic BEC. The best candidate for the observation of the excitonic BEC is probably Cu2O. However, the excitation of the excitons in this material is dipole-forbidden, therefore the theory would have to be modi®ed accordingly. For the sake of simplicity, Schmitt et al. [8] and we have used the contact potential for the exciton±exciton interacting potential. A further improvement of our study would be the replacement of the contact potential by the two-body (or many-body) T-matrix as discussed in atomic trapped gases [10,11,13]. Acknowledgements The authors appreciate a fruitful cooperation with L. BaÂnyai, P. Gartner and particularly with O. Schmitt on the exciton±exciton kinetics. This work has been supported by the Deutsche Forschungsgemeinschaft in the framework of the research initiative: Quantum Coherence in Semiconductors. References

Fig. 6. Dephasing time T2 and chemical potential m versus the excitation density.

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