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1 Pergamon Solid State Communications, Vol. 18, No. 4, pp , Published by Elsevier Science Ltd /98 $ - see front matter PII: S38 198(98)35- THERMODYNAMIC EQUILIBRIUM OF SCREENED EXCITON SYSTEM BY ELECTRON HOLE PLASMA IN THE TWO-DIMENSIONAL STRUCTURE N. Ben Brahim Aouani, a L. Mandhour, a R. Bennaceur, a, * S. Jaziri, a T. Amand b and X. Marie b a Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis-16-Tunisie b Laboratoire de Physique de la Matière Condensée, CNRS-URA 74, INSA, 3177 Toulouse Cedex, France (Received 5 September 1997; accepted 2 July 1998 by G. Bastard) In this work, we present the results for the thermodynamical equilibrium of a two dimensional (2D) system composed of excitons and free electronhole pairs taking into account the screening of the Coulomb interaction by free carriers. We found that the 1S exciton binding energies are significantly decreased within the studied range of temperature and total pair density and that the equilibrium exciton electron hole plasma is displaced towards the excitonic ionisation when the total pair density increase if we take into account the screening effect. We show that screening must be taken into account for the determination of the excitonic energy in thin quantum wells at high temperature and high total pair densities Published by Elsevier Science Ltd Keywords: A. quantum wells, A. semiconductors, D. optical properties, D. thermodynamic properties. 1. INTRODUCTION In three-dimensional semiconductor systems (3D), the absolute excitonic line stays constant even at very high electron hole pair densities. With increasing free carrier concentration, screening gives rise to a red shift, whereas the weakening of attractive electron hole interaction due to the exclusion principle gives rise to a blue shift. So the lowest exciton line remains unchanged, due to the high degree of compensation between the two last effects [1]. However D. Hulin et al. [2, 3] observe a high energy exciton line shift in absorption spectrum of GaAs AlGaAs multiple-quantum well structures. They found that the magnitude of the blue shift depends predominantly upon the GaAs layer thickness and increases when the well size decreases. They attribute this effect to a strong reduction of long-range many body Coulomb interaction which occurs in two-dimensional system. These results are consistent with the theoretical analysis of Schmitt Rink [4]. Moreover, a blue shift and a splitting of the excitonic line has been observed [5 7] in a dense and highly polarised exciton system. * Corresponding author. raouf.bennaceur@fst. rnu.tn 199 Recently, the authors of [8] have studied by the technique of time-resolved photoluminescence spectroscopy, the dynamic of excitons and free carriers, at low temperature. In the case of exciton bimolecular formation process, they show that a 2D mass action law, describing the coexistence of the free carriers and excitons, explains the time evolution of the exciton photoluminescence line width. This effect confirm a theoretical result of Chemla [9] and experimental one of Colocci [1], that a thermodynamic equilibrium is established between excitons, electrons and holes, and that this equilibrium is maintained on a large temperature range. In this paper we are interested in the screening Coulomb interaction effect on the thermodynamic equilibrium. 2. BINDING ENERGY IN THE 2D SCREENED EXCITON SYSTEM BY FREE CARRIERS The Coulomb interaction in a bound electron hole pair is screened by the presence of free carriers. In a 2D system, the binding energy of the exciton is indeed reduced by the screening. But the electron hole pair remains bound even for fairly strong screening, due to the fact that screening parameter saturates for large

2 2 THERMODYNAMIC EQUILIBRIUM OF ELECTRON HOLE PLASMA Vol. 18, No. 4 density or small temperature q s ¼ 2 m e þ m h : a m However, in 3D case, where the screening can prevent the pair to bound at high free carrier densities, when the electron hole bound state is no longer stable. We propose in this work to study the evolution of exciton electron hole plasma system, at a given total pair density and temperature, if we take into account the screening effect particularly at high density and temperature. We neglect the screening due to the neutral exciton gas, which is less efficient than the one due to the electron hole plasma [4] and the band-gap renormalisation for densities less than 1 11 cm ¹2 and temperature less than 1 K [12]. The screened Coulomb potential in a 2D system is given by Stern and Howard [11], and is represented by the following expression: V S ðrþ ¼¹ e2 q dq J ðqrþ e q þ q s ¼ e2 er þ e2 q s dq J ðqrþ ð1þ e q þ q s where J is the Bessel function of the first kind, e is the dielectric static constant and q s is the screening 2D parameter in Debye model which depend on the free electron and hole plasma density resulting from the excitonic dissociation n p ¼ n e ¼ n h and T through [11]: q s ðn p ; TÞ ¼ 2 m e a m 1 ¹ exp ¹ p 2 n p m ek B T þ m h m 1 ¹ exp ¹ p 2 n p m h k ð2þ BT where a is the effective 3D exciton Bohr radius, m e, m h are respectively the effective electron and hole masses and m is the reduced exciton mass. The hamiltonian of the system within the effective mass approximation is given by: H ¼ p2 2m þ V sðrþ (3) The exciton energy is obtained by using the perturbative method up to second order. We add and subtract to the hamiltonian H a non screened 2D Coulomb potential, which allows us to write H as summation of two hamiltonians, where the solution of the first one H is known and is exact. And the second term H 1 can be considered as a perturbation. H 1 is small when q s and also when r, i.e. where the exciton S wave function is largest, so that one can expect its corrections H ¼ H þ H 1 (4) with H ¼ p2 2m ¹ e2 er : (5a) The eigenvalues and the eigenvectors (the hydrogenic states) of H are given by [9]: En ðþ ¼¹ R ðn ¹ 1 2 Þ2; F n;l ðr; vþ ¼a n;l e ilv r jlj e ¹ r=2 L 2jlj n ¹jlj¹1 ðrþ ð5b ð5cþ 2r r ¼ and ðn ¹ 1=2Þa r ( " # 2 ) 1=2 2 1 ðn ¹jlj¹1Þ! a n;l ¼ p ðn ¹ 1 2 Þa ðn þjlj¹1þ!ð2n ¹ 1Þ n ¼ 1; 2; ; l ¼ ; 1; 2; ; n ¹ 1; L p mðxþ are the associated Laguerre polynomes, R is 3D effective Rydberg. The corresponding fundamental wave function and energy have the following expressions: f 1S ðrþ ¼ 1 4 p e ¹ð2=aÞr ; E ðþ 1S 2p a ¼¹4R The hamiltonian H 1, considered as a perturbation, is written as: H 1 ¼¹ e2 f ðq s ; rþ¹ 1 (6) e r where f ðq s ; rþ ¼ q dq J ðqrþ q þ q s The energy correction at first and second order are given respectively by: E ð1þ 1S ¼ f 1SjH 1 jf 1S ¼ 2pa 2 1 and E ð2þ 1S ¼ q s¹a 1 a 2 1 n;l 1S; ¹ p e 2 e 1 a 2 1 þq2 s j f n;l jh 1 jf 1S j 2 E ðþ 1S ¹ E n q s a 1 ða 2 1 þ q2 s Þ p! q s ð a 2 1 log þq s¹q s Þ p a 1 ð a 2 1 þq s þa 1 Þ where f n;l jh 1 jf 1S ¼ 2pa n a 1 Aða n ; q s Þ with Aða n ; q s Þ¼ e2 e q s dr re ¹ nanr L n ¹ 1 ða n rþ dq q þ q s J ðqrþ and a n ¼ ð7aþ (7b) 4 ð2n ¹ 1Þa ð8þ

3 Vol. 18, No. 4 THERMODYNAMIC EQUILIBRIUM OF ELECTRON HOLE PLASMA We determine the excitonic binding associated to the fundamental state at each value of q s, so for a given plasma density and temperature ðe n ¼¹E 1S ¼ E B ðn p ; TÞÞ. We note that the second order of energy is much less than the first one ðe ð2þ 1S :127EðÞ 1S Þ and that the screening reduces the excitonic binding energy, as expected. 3. THERMODYNAMIC EQUILIBRIUM BETWEEN EXCITONS AND FREE CARRIERS The excitons play an important role in optical absorption, it is interesting to study their formation which occurs through a bimolecular process [8]. The 2D mass action law describes the coexistence of excitons and free carriers at a given temperature. On the ground of the mass action law, Chemla [9], explains the optical non-linearities in quantum well at room temperature. Colocci et al. [1] applied with success the 2D mass action law for the analysis of the thermal ionisation of excitons in GaAs GaAlAs quantum wells in photoluminescence and excitation photoluminescence experiments. Using the photoluminescence spectrum line shape, they have deduced the relative population of excitons and free carriers. Moreover they show that the thermodynamic equilibrium is achieved between these entities within a large temperature range. For weakly confined systems excitons are screened by free carriers with increasing density and temperature. When the confining effects are large, excitons become more stable. Under this condition they coexist with free carriers. The equilibrium between the two phases is established and it is described by a mass action law. At a given equilibrium temperature of the whole system and total pair density, we use the coupled system given by (9) with a non screened binding energy in order to determine the density of plasma which gives us the screening parameter (2). After we calculate the screened binding energy using a perturbative method and we evaluate the new plasma density. Thus until the system converge. The 2D mass action law is given by [8, 1]. If we take into account the screening effect, the 2D mass action law have the following expression: n e n h n x ¼ n2 p n x ¼ m 2p 2k BT exp ¹ E Bðn p ; TÞ ¼ Kðn p ; TÞ k B T n t ¼ n p þ n x 21 The equilibrium constant Kðn p ; TÞ correspond to ratio between excitonic ionisation coefficient a and bimolecular formation coefficient gðkðn p ; TÞ ¼a=gÞ. We set n p ¼ n e ¼ n h where n p is the plasma density and n t is the total pair density. We deduce from (1) the relative plasma density: s n p ¼ Kðn p; TÞ þ 2 þ Kðn p; TÞ Kðnp ; TÞ (1) n t 2n t 2n t 2n t (9) At a fixed total pair density and temperature, we solve self-consistently the equation (1) and the excitonic Schrödinger equation, using the effective parameter for GaAs: m e ¼ :67m and m h ¼ :45m, a ¼ 165 Å, R ¼ 3:48 mev. The relative plasma density is represented as a function of temperature at different total plasma densities, Fig. 1. Relative plasma density, as a function of temperature for total pair densities ranging from cm ¹2 to cm ¹2.

4 22 THERMODYNAMIC EQUILIBRIUM OF ELECTRON HOLE PLASMA Vol. 18, No. 4 Fig. 2. Relative plasma density, as a function of temperature for total pair densities ranging from cm ¹2 and cm ¹2 (¹ with screening and X without screening). on Fig. 1, taking account the screening of the Coulomb interaction. On Fig. 2 we compare the results with and without Coulomb interaction screening. These figures show that the equilibrium is displaced towards the plasma when the temperature increases and the total pair density decreases. However, the 2D mass action law displaces the equilibrium towards the exciton formation, when the total pair density increases, in spite of the screening effect which only slows down this behaviour particularly at high total density even at relatively low temperature. The exciton ionisation and formation processes are essentially dependent on the temperature and the total pair density. The introduction of screened Coulomb interaction by free carriers, implies an equilibrium constant which depends on both temperature and plasma pair density. These two parameters act effectively on the equilibrium plasmaexciton. We represent in Fig. 3 the equilibrium constant as a function of temperature at different total plasma densities. Figure 3 shows that the equilibrium constant increases with the total plasma density at a given temperature, but this growth is less rapid than the total plasma density. This behaviour imply that the equilibrium is displaced towards the exciton formation even at high screening. The equilibrium is displaced Fig. 3. The equilibrium constant (in units of total pair density), as a function of temperature for total pair densities ranging from cm ¹2 and cm ¹2.

5 Vol. 18, No. 4 THERMODYNAMIC EQUILIBRIUM OF ELECTRON HOLE PLASMA 23 Fig. 4. The screening parameter, as a function of temperature for total pair densities ranging from cm ¹2 to cm ¹2. towards the excitonic ionisation if the temperature increases, when T attend 5 K the composition of system is relatively stabilised. The screening parameter and the binding energy (obtained after the self-consistent calculation) are represented on Figs 4 and 5 respectively, as a function of temperature and for total pair densities ranging from cm ¹2 to cm ¹2. The Fig. 4 shows that the screening parameter increases with the temperature until T ¼ 5 K, where the majority of the pairs are ionised and then begins to decrease. The corresponding binding energy begins to increase starting from T ¼ 5 K. 4. CONCLUSION In this work, we have shown that the screening reduces considerably the 1S exciton binding energy. It acts on the displacement of the equilibrium excitonplasma for the studied system. We noted that the equilibrium continues to displace toward the formation of the excitons despite the considerable reduction of the binding energy under the effect of the screening. However, apart from the screening, other complementary effects, such as the phase space filling and exchange are present specially at higher plasma densities, where the exciton becomes unstable. It is interesting to analyse Fig. 5. The binding energy normalised to the two-dimensional exciton Rydberg (4R with R ¼ 3:48 mev), as a function of temperature for total pair densities ranging from cm ¹2 to cm ¹2.

6 24 THERMODYNAMIC EQUILIBRIUM OF ELECTRON HOLE PLASMA Vol. 18, No. 4 these effects in order to describe the experimental behaviour near the critical density. Acknowledgements We would like to thank the referee for interesting and useful advice. REFERENCES 1. Huang, H. and Schmitt-Rink, S., Progr. Quant. Electron., 9, 1984, Hulin, D., Mysyrowicz, A., Antonetti, A., Migus, A., Masselink, W.T., Morkx, H., Gibbs, H.M. and Peyghambarian, N., Phys. Rev., B33, 1986, Peyghambarian, N., Gibbs, H.M., Jewell, J.L., Antonetti, A., Migus, A., Hulin, D. and Mysyrowicz, A., Phys. Rev. Lett., 53, 1984, Schmitt-Rink, S., Chemla, D.S. and Miller, D.A.B., Phys. Rev., B32, 1985, Amand, T., Marie, X., Baylac, B., Dareys, B., Barrau, J., Brousseau, M., Planel, R. and Dunstan, D.J., Phys. Lett., A193, 1994, Robart, D., Amand, T., Marie, X., Brousseau, M., Barrau, J. and Baquet, G., J. Opt. Soc. Am., B13, 1996, Wang, H., Ferrio, K., Steel, D.G., Hu, Y.Z., Binder, R. and Koch, S.W., Phys. Rev. Lett., 71, 1993, Robart, D., Marie, X., Baylac, B., Amand, T., Brousseau, M., Baquet, G., Debart, G. and Planel, R., Solid State Commun., 95, 1995, Chemla, D.S., Helv. Phys. Acta., 56, 1983; J. Lumin., 3, 1985, Colocci, M., Gurioli, M. and Vinattierri, A., J. Appl. Phys., 68, 199, Stern, F. and Howard, W., Phys. Rev., 163, 1967, Guven, K. and Tanatar, B., Superlatt. and Microstruct., 2, 1996, 81.