Screening effect on the photoluminescence spectra in quantum-well wires

PERGAMON

Solid State Communications 118 (2001) 211±214

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Screening effect on the photoluminescence spectra in quantum-well wires H.T. Duc, D.B. Tran Thoai* Ho Chi Minh City Institute of Physics, Vietnam Centre for Natural Science and Technology, 1 Mac Dinh Chi, Ho Chi Minh City, Viet Nam Received 4 December 2000; accepted 17 January 2001 by P. Burlet

Abstract Photoluminescence spectra of quantum-well wires have been studied within the statically screened approximation. In agreement with recent experimental results in GaAs quantum wires, we found that the exciton peak shows almost no shift. q 2001 Elsevier Science Ltd. All rights reserved. PACS: 71.35.2y; 71.35.Ee; 73.23.Ps; 78.55.2m Keywords: A. Semiconductor; A. Quantum-well wires; B. Electron±electron interactions; B. Optical properties

1. Introduction Many-body effects in dense electron±hole plasmas in semiconductors and their heterostructures have found a growing interest in recent years. Under strong laser excitation, exchange-correlation effects and screening of Coulomb interaction cause band-gap renormalization (BGR). The band-gap decrease with increasing electron±hole pair densities affects the excitation process and leads to optical nonlinearities [1±3]. The BGR and absorption spectra have been studied intensely in bulk and quantum well semiconductors [1±6]. Recently, there have been some studies on the BGR, absorption and photoluminescence (PL) spectra in semiconductor quantum-well wires (QWWs) [7±10]. In such quasi-onedimensional structures, the motion is free along the wire axis but is quantized in the two dimensions perpendicular to it. While most authors have studied the BGR and absorption spectra for one subband, GreÂus et al. [9] were the ®rst, who calculated the PL for several subbands but only in the Hartree±Fock approximation, thereby neglecting the effects of the screening of Coulomb interaction. Within their theoretical model, GreÂus et al. [9] found a systematic difference concerning the shift of the PL lines compared to their experimental results in InGaAs, which showed a constancy of the ®rst peak vs. the electron±hole density. * Corresponding author. E-mail address: [email protected] (D.B. Tran Thoai).

One may raise the question whether the neglect of the screening in the interaction has caused the shift of the calculated PL lines or not. In this work, we calculate the PL spectra by solving the Bethe±Salpeter equation (BSE) of the optical susceptibility for a multi-subband QWW within the statically screened approximation. We shall critically compare our results with those in Hartree±Fock approximation. 2. Theory We start from a 2D quantum well in which the electrons and holes are con®ned laterally by a harmonic oscillator potential with a total subband spacing of V . The Fourier transformation along the z-direction of the 2D bare Coulomb interaction between the charge carriers is given by the average over the eigenfunctions of the con®ned oscillator potential: V nrsm …q† ˆ

2e2 Z Z 0 p dz dz c n …z†c rp …z 0 †c s …z 0 †c m …z†K0 e0  …uq…z 2 z 0 †u†;

…1†

where K0(x) is the zeroth-order modi®ed Bessel function and 0 2s 3 s1 12   1 m V m V mV 5 A exp 2 z 2 Hn 4 z ; …2† c n …z† ˆ @ n 2 n! p" 2" "

0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00061-8

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Fig. 1. PL spectra of GaAs QWW for T ˆ 50 K and various densities in the unscreened Hartree±Fock approximation.

where m is the reduced electron±hole mass and Hn(x) the Hermite polynomial. The luminescence spectrum R(v ) is obtained from the imaginary part of the total optical susceptibility: R…v† , v 3

Imx…v† eb…"v2m† 2 1

screened Coulomb potential. For simplicity, we have incorporated broadening effects by a phenomenological damping constant g . We calculate the self-energy within the leading order screening approximation: j X

2

ud u 1 Im , v eh V eb…"v2m† 2 1

X

…3†

l

where b is the inverse thermal plasma energy, m the total chemical potential of the electron±hole pair, deh the interband dipole transition element. The optical susceptibility x k,l(v ) obeys an integral equation, which corresponds to the BSE of the interband polarization. The BSE for the interband polarization (with only one subband) in bulk and quantum well semiconductors has been solved within the statically screened approximation [1,4]. Within this approximation the integral equation for the optical susceptibility x k,l(v ) reads: X xk;l …v† ˆ x0k;l …v† 1 x0k;l …v† Vs;ll 0 ll 0 …k 2 k 0 †xk 0 ;l 0 ; …4†

;

3

k;l

xk;l …v†;

k 0 ;l 0

x0k;l …v†

is the free-carrier susceptibility. In order to ensure that the crossover from gain to absorption occurs at the total chemical potential "v ˆ m even in the presence of broadening, we take the free-carrier susceptibility in the spectral representation [1,3,9]:

x0k;l …v†

Z dv 0 2"g ˆ 2p …eek;l 1 ehk;l 2 "v 0 †2 1 "2 g2 1 2 f e …"v 0 2 ehk;l † 2 f h …ehk;l †  ; v 0 2 v 2 id

j X l;HF

where Pf is the0 Fermi distribution P function, ejk;l ˆ Eg j j j j "2 k2 1 1 …k† 1 1 …l 1 † V ; e 1 is the l k;l l …k† 2mj 2 2 renormalized single-particle energy, VS is the statically

k 0 ;z 0 ;l 0

j X

…k† 1

0 0 VS;ll 0 ;ll 0 …k 2 k 0 ; z 2 z 0 †G0j l 0 l 0 …k ; z †

…k; z†;

…6†

…7†

l;corr

where j X

…k† ˆ 2

l;HF

X k 0 ;l 0

Vll 0 ll 0 …k 2 k 0 †f j …ejk 0 ;l 0 †

…8†

P is the unscreening Hartree±Fock self-energy and jl;corr is the correlation self-energy. For practical purposes, the RPA dielectric function can be replaced by the plasmon-pole approximation [1±5,7,10]. Within this simpler approximation the correlation self-energy is given by [10]: j X l;corr

…k† ˆ

X k 0 ;l 0

Vll 0 ll 0 …k 2 k 0 †

" 

1 …5†

X

…k; z† ˆ 2 b1

v2pl;ll 0 ll 0 2vk2k 0 ;ll 0 ll 0

1 1 g…vk2k 0 ;ll 0 ll 0 † 2 f j …ejk 0 ;l 0 †

ejk;l 2 ejk 0 ;l 0 2 vk2k 0 ;ll 0 ll 0

g…vk2k 0 ;ll 0 ll 0 † 1 f j …ejk 0 ;l 0 †

ejk;l 2 ejk 0 ;l 0 1 vk2k 0 ;ll 0 ll 0

# ;

and the statically screened potential VS;ll 0 ll 0 …q† reads ! v2pl;ll 0 ll 0 …q† ; VS;ll 0 ll 0 …q† ˆ Vll 0 ll 0 …q† 1 2 v2q;ll 0 ll 0

…9†

…10†

H.T. Duc, D.B. Tran Thoai / Solid State Communications 118 (2001) 211±214

213

Fig. 2. (a) PL spectra of GaAs QWW for T ˆ 50 K and various densities in the statically screened approximation with correlation self-energy in the dynamical approximation. (b) PL spectra of GaAs QWW for T ˆ 50 K and various densities in the statically screened approximation with correlation self-energy in the quasi-static approximation.

where

v2q;ll 0 ll 0 ˆ v2pl;ll 0 ll 0 1 1

v2pl;ll 0 ll 0 …q† ˆ

q x2ll 0 ll 0

! 1

tion and also in the quasi-static approximation. If one neglects the correlation self-energy and uses the unscreened potential V instead of the statically screening potential VS in Eq. (4) one retrieves the BSE treated by GreÂus et al. [9].

4

q ; 4m2

X Vll20 l 00 l 0 …q† nj 00 q2 l ; 0 ll 0 …q† V mj 00 ll l ;j

X

x2ll 0 ll 0 ˆ 2b

2

l 00 ;j;k

3. Numerical results and discussions

Vll200 l 00 l 0 …q† f j …ejl 00 †…1 2 f j …ejl 00 †† X l 00 ;j

Vll200 l 00 l 0 …q†…njl 00 =mj †

v2pl;ll 0 ll 0 ;

njl is the particle density in the subband l, g…v† ˆ …ebv 2 1†21 is Bose distribution function. In solving the BSE (for one band) most authors [1,3±5,7] have determined the correlation self-energy in quasi-static approximation by neglecting the recoil energies in the denominator of the Eq. (9). In the following, we shall solve the BSE with the correlation self-energy calculated in the dynamical approxima-

In this section we carry out numerical calculations for GaAs QWW with three subbands using the following parameters: me ˆ 0:067 m0 ; mh ˆ 0:46 m0 ; e0 ˆ 13:1; Eg ˆ 1:6 eV; intersubband spacing V ˆ 5 E 0 . 23 meV; damping constant g ˆ 1 E0 . 4:6 meV; where E0 ˆ 4:6 meV is the 3D exciton Rydberg. For comparison we display the PL spectra for T ˆ 50 K and for various electron±hole densities (from n . 8:3 £ 104 cm21 to n . 1:0 £ 106 cm21 † in the Hartree± Fock approximation and in the statically screened approximation. Fig. 1 corresponds to the Hartree±Fock approximation. In that approximation, the exciton peak shifts strongly

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with increasing density. Besides that, the relative intensities of the spectra show irregular values. The spectra intensities increase suddenly when the total chemical potential coincides with excitonic resonance energies. Fig. 2a corresponds to the correlation self-energy in the dynamical approximation and shows the exciton peak at 1.585 eV. Fig. 2b corresponds to the correlation self-energy in the quasi-static approximation and shows the exciton peak at 1.582 eV. The difference of the peak position between Fig. 2a and b is due to the overestimate of the BGR in the quasistatic approximation compared to the dynamical approximation. It is obviously from Fig. 2 that the most striking effect of the screening is: the exciton peak does not shift much with increasing density (maximum shift is 1.1 meV). These calculated results agree rather well with recent measurements performed on V-groove GaAs quantum wire samples at the estimated temperature 50±60 K by R. Ambigapathy et al. [11]. They found no signi®cant shift in the exciton peak with increasing density in agreement with measurements in InGaAs QWW [9]. In summary, we have calculated the PL spectra of photo excited QWWs and have found a large difference between the screened approximation and the unscreened Hartree± Fock approximation. While the screened approximation agrees rather well with recent experimental ®ndings [11], the unscreened Hartree±Fock approximation fails to-tally to describe the experiments.

Acknowledgements We gratefully acknowledge the ®nancial support of the National Program for Basic Research. References [1] H. Haug, S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scienti®c, Singapore, 1993. [2] R. Zimmermann, Many Particle Theory of Highly Excited Semiconductors, Teubner, Liepzig, 1988. [3] H. Haug, S. Schmitt-Rink, Prog. Quant. Electron. 9 (1984) 3. [4] J.P. LoÈwenau, S. Schmitt-Rink, H. Haug, Phys. Rev. Lett. 49 (1982) 1511. [5] C. Ell, R. Blank, S.W. Koch, H.E. Schmidt, H. Haug, Solid State Commun. 52 (1984) 123. [6] S. Das Sarma, R. Jalabert, S.R.E. Yang, Phys. Rev. B39 (1989) 5516. [7] S. Benner, H. Haug, Europhys. Lett. 16 (6) (1991) 579. [8] B. Hu, S. Das Sarma, Phys. Rev. Lett. 68 (1992) 1750. [9] Ch. GreÂus, A. Forchel, R. Spiegel, F. Faller, S. Benner, H. Haug, Europhys. Lett. 34 (1996) 213. [10] D.B. Tran Thoai, H. Thien Cao, Solid State Commun. 111 (1999) 67. [11] R. Ambigapathy, I. Bar-Joseph, D.Y. Oberli, S. Haacke, M.J. Brasil, F. Reinhardt, E. Kapon, B. Deveaud, Phys. Rev. Lett. 78 (1997) 3579.