Photoluminescence Spectroscopy of One-Dimensional Resonant Photonic Crystals

ARTICLE IN PRESS

Journal of Luminescence 125 (2007) 112–117 www.elsevier.com/locate/jlumin

Photoluminescence spectroscopy of one-dimensional resonant photonic crystals

M.M. Voronova, E.L. Ivchenkoa,Ã, M.V. Erementchoukb, L.I. Deychb, A.A. Lisyanskyb

aA.F. Ioffe Physico-Technical Institute, Russian Academy of Science, 194021 St. Petersburg, Russia bPhysics Department, Queens College, City University of New York, Flushing, New York 11367, USA

Available online 12 September 2006

Abstract

We report on a theoretical study of photoluminescence (PL) of the resonant one-dimensional photonic crystals, or the near-Bragg quantum-well structures. The PL spectral intensity is found by including random sources in the equations for the exciton dielectric polarization and introducing the discrete Green’s function. The structures without and with the dielectric contrast are considered. The positions of peaks in the calculated PL spectra are in agreement with the real parts of the exciton–polariton eigenfrequencies. A relation between the PL and absorption spectra is discussed. r 2006 Elsevier B.V. All rights reserved.

Keywords: Exciton polariton; Photoluminescence; Multiple quantum wells; Resonant photonic crystal

1. Introduction SQW structure. In the simplest kinetic theory, the intensity Iq of electromagnetic wave emitted by a SQW and The physics of photonic crystals, i.e. structures with characterized by the wave vector q can be written by using periodically modulated dielectric function allowing for Fermi’s golden rule: Bragg diffraction of light, is a rapidly developing field. 2p 2 c Among one-dimensional (1D) photonic crystals, of parti- I q ¼ _oqwq; wq ¼ f jMq j d _ q Eq . (1) _ qjj jj jj cular interest are the so-called resonant Bragg struc- nb tures with the period d satisfying the Bragg condition pffiffiffiffi Here oq ¼ cq/nb is the wave frequency, nb ¼ b, eb is d/l(o0) ¼ 0.5 at the exciton resonance frequency o0, see the dielectric constant of the barrier material assumed to the book [1] and references therein. Experimentally, coincide with the background dielectric constant ea of the photoluminescence (PL) spectra of the resonant Bragg QW (the general case of structures with the dielectric and near-Bragg quantum well (QW) structures were contrast ea6¼eb is considered in Section 3), wq is the studied in Refs. [2–5]. In this work we propose a theory probability rate for the photon emission per unit in-plane of secondary radiation of exciton–polaritons in multiple area, qJ the in-plane component of q, Ek ¼ _o0 þ QW (MQW) structures, derive equations for the PL ð_2k2=2MÞ is the two-dimensional (2D) exciton excitation spectral intensity, present the results of numerical calcula- energy as a function of the 2D wave vector k, o0 and M tion and compare the PL and absorption spectra. are, respectively, the exciton resonance frequency and in- plane translational effective mass, fk is the distribution 2. PL of a single QW (SQW) function defined inside the circle kpq0,withq0 being the light wave vector at the exciton resonance frequency q0 ¼ In order to derive a general equation for the spectral nbo0=c and Mk is the matrix element of emission of a intensity of PL from MQWs we first present that for a photon by an exciton with the wave vector k ¼ qJ. Eqs. (1) take into account that only excitons with koq0 can decay Ã Corresponding author. Fax: +812 247 1017. radiatively while excitons with k4q0 are dark, they induce E-mail address: [email protected] (E.L. Ivchenko). in the adjusting barriers evanescent electromagnetic field

0022-2313/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2006.08.015 ARTICLE IN PRESS M.M. Voronov et al. / Journal of Luminescence 125 (2007) 112–117 113 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 which exponentially decays as expð k q0jzjÞ along the In the following we focus on normally emitted light normal z to the interface plane. ðqjj ¼ 0Þ and fix its particular transverse polarization. This Neglecting exciton scattering within the circle kpq0 we will allow us to use the scalar amplitudes P0 Pk¼0 and can write S0 Sk¼0 instead of the vectors Pk and Sk and to present the electric field of the outgoing light wave in the barrier as 1 iqjzj f k ¼ tkGk ; ¼ 2ðG0k þ GkÞ , (2) E0e where the coordinate z is referred to the QW center. tk The amplitude E0(o) and the exciton polarization P0(o) where G0k and Gk are the exciton radiative and nonradia- are related by [1] tive damping rates, G is the rate of acoustic-phonon- k i induced transitions from the exciton states with the wave E ðoÞ¼ P ðoÞ; x ¼ b , 0 x 0 2pq a vectors k0 lying outside the circle of radiative states into the 0 radiative state k, namely, where a is the QW thickness assumed to be small as 1 X compared to q0 . Then for the intensity of normally 2p 2 G ¼ f 0 jV j ½ðN þ 1Þdð_o E 0 þ _O Þd 0 emitted light we obtain k _ k Q Q q k Q k ;kþQjj 0 k Q 2 cnb 2 cnb hjS0ðoÞj i _ 0 _ 0 IðoÞ¼ hjE ðoÞj i¼ , (7) þ NQdð oq Ek OQÞdk ;kQjj . ð3Þ 0 2 2 2 2p 2px ðo0 oÞ þðG0 þ GÞ Here Q and OQ are the acoustic phonon wave vector and where G0 ¼ G0;k¼0 and G ¼ Gk¼0. Comparison between frequency, respectively, NQ is the phonon occupation Eq. (5) and Eq. (7) allows us to relate the spectral density of number, VQ the matrix element of the exciton–phonon 2 the phenomenological source hjS0ðoÞj i to microscopic interaction, and both the phonon emission and absorption characteristics of the system: processes are taken into account. In order to go beyond validity of the kinetic equation hjS ðoÞj2i¼ b jM j2G . (8) 0 _ 2 0 0 and take into account the effect of finite exciton lifetime on 2 q0a the PL spectral width we can replace the exact d-function in Eq. (1) by the smoothed function 3. Exciton–polariton luminescence in finite near-Bragg c 1 G þ G MQW structures D _ q E ¼ 0k k , n qjj _p 2 2 b ½cq=nb oðqjjÞ þðG0k þ GkÞ Now we can derive the general equation for the intensity (4) of PL from a structure containing N equidistant QWs between semi-infinite barrier layers. In the following the where oðkÞ¼Ek=_. Then the radiation intensity can be rewritten as QW width, the width of a barrier separating the nearest QWs and the period are denoted by a, b and d ¼ a+b, 2 p jMq j Gq respectively, and the QWs are labeled by the index I ¼ _o jj jj . (5) q q _2 2 2 n ¼ 1,y,N. Taking into account that the random sources ½cq=nb oðqjjÞ þðG0qjj þ Gqjj Þ Sn in different QWs are uncorrelated we can write the PL 2 We assume the typical kinetic energy _2k0 =2M to exceed intensity of the normally emitted light as a sum: the exciton damping rate. This allows one to retain the X cnb ðnÞ 2 G IðoÞ¼ hjE j i . (9) exact d-function for the phonon-induced transition rate k 2p in Eq. (3). n Alternatively, Eq. (5) can be obtained with the help of a Here E(n) is the electric field of the secondary radiation phenomenological approach, in which noncoherent polar- outgoing from the nth QW, i.e., with the radiation source ization responsible for luminescence is modeled by a in the nth QW, and coming out into one of the semi-infinite random source term, SkðoÞ, in the equation of motion of barriers, say, the field at the plane shifted by d/2 from the the exciton-related polarization: center of the leftmost QW. One can show that it is given by iqd=2 ½oðkÞo iðG0k þ GkÞPkðoÞ¼SkðoÞ. (6) t r t i S e EðnÞ ¼ n1 þ Nn n n , This equation takes into account the interaction between 1 rn1rNnþ1 1 rnrNn x o0 o iðG þ G0Þ excitons and the emitted radiation by means of the term (10) G , which represents exciton radiative decay rate. We, 0k where rm and tm are the amplitude reflection and however, neglect here a radiative shift of the exciton transmission coefficients, respectively, for a substructure frequency. From Eq. (6) one immediately obtains comprized of m QWs and bounded by the planes shifted by 7 hjS ðoÞj2i d/2 from the centers of the leftmost and rightmost QWs. hj ð Þj2i¼ k Pk o 2 2 ; For the particular case m ¼ 0, i.e., when the bounding ½oðkÞo þðG0k þ GkÞ planes do not enclose any QW, rm and tm are equal to 0 where the angular brackets denote the averaging over and 1, respectively. Using properties of reflection and realizations of the random source. transmission coefficients determined by their places in a ARTICLE IN PRESS 114 M.M. Voronov et al. / Journal of Luminescence 125 (2007) 112–117 transfer matrix describing propagation of the field through results in the structure, this expression for E(n) can be presented in an ( b_ðoo0Þ alternative form e j1 for o o0 d; JðoÞ¼ (14) j for o o d; ÀÁiqd=2 2 0 ðnÞ tN iqd i Sne E ¼ 1 þ rNne , (11) tNn x o0 o iG where j1, j2 are slowly varying functions of o. where tN is the transmission coefficient for the entire For simplicity, in the following we present the PL spectra structure, and the resonant denominator describing ex- calculated neglecting the frequency dependence of the citons dressed by the radiative field (term Sn=½o0 o generation rate G0(o; n). One should bear in mind that this iðG þ G0Þ) is replaced by the resonant term corresponding dependence can be taken into account by multiplying the to initial purely mechanical excitons ðSn=½o0 o iG Þ. spectra by a factor (Eq. (14)) or by a more complicated Eq. (11) emphasizes the distinct role of two factors approximation of the o-dependent function defined determining the luminescent properties of the structures according to Eq. (3). Simultaneously, a frequency depen- under consideration: transmission coefficient through the dence of G has to be taken into consideration according to structure tN reflects the role of the normal photonic modes X 2p 2 2GðoÞ¼ jV j ½N dð_o E 0 þ _O Þd 0 of the MQW structrure, while initial exciton susceptibility _ Q Q k Q k ;Qjj determines the efficiency of the initial excitation of the k0Q þð þ Þ ð_ 0 _ Þ 0 excitons by the noncoherent source. NQ 1 d o Ek OQ dk ;Qjj . In structures without the dielectric contrast one can use a The electric field E(n) can be related to the source S by simpler approach based on a set of coupled linear n the discrete Green’s function G 0 of the system as follows equations n n X ðnÞ ðo0 o iGÞG1n d1n E ¼ Sn . ðo0 o iGÞPn0 þ Ln0n00 Pn00 ¼ Sn dn0n, xG0 n00 iqdjn0n00j The Green’s function satisfies the equation set Ln0n00 ¼iG0e , ð12Þ X ðo0 o iGÞGn0n þ Ln0n00 Gn00n ¼ dn0n, (15) 0 for the dielectric polarizations Pn0 (n ¼ 1, 2, y, N) with the n00 random source located in the nth QW. Similar to Eq. (8) and can be presented in the form one has for the mean square ðnÞ iKdn0 ðnÞ iKdn0 G 0 ¼ P e þ P e n n þ ( b 0 hjS ðoÞj2i¼ jM j2G ðo; nÞ, (13) P0 for n ¼ n; n _ 2 0 0 2 q0a þ 0 PeiKdjn nj for n0an: with G0(o; n) given by Eq. (3) where f k0 should be replaced Here K is the wave vector of an exciton polariton in an ðnÞ infinite QW structure, it satisfies the dispersion equation [1] by the 2D-exciton distribution function f k0 in the nth QW. In order to analyze the role of frequency dependence of the G0 generation rate G (o; n) we consider a sum of two cos Kd ¼ cos qd þ sin qd. (16) 0 o o þ iG dimensionless integrals, 0 P , P determine the Green’s function for an infinite QW Z 0 1 structure and are given by JðoÞ¼b dEebE½NðE þ _o _oÞþ1 0 iG sin qd Eþ 0 Z P ¼ 2 , E ðo0 o iGÞ sin Kd bE þ b dEe Nð_o E _o0Þ, 0 1 P0 ¼ P þ , o0 o iG related to the terms in Eq. (3) due to acoustic-phonon PðnÞ emission and absorption, respectively. The integral J(o) result from the internal reflection of exciton polaritons from the outermost barriers, can be used for the analysis of G0(o; n) assuming the matrix iF iKdð2Nn2Þ element of exciton-phonon scattering, VQ, to be indepen- e þ re PðnÞ ¼ rP , dent of the phonon wave vector. However one has to bear 1 r2e2iKdðN1Þ in mind that this matrix element vanishes at small Q and where the occupation number NðE þ _o0 _oÞ diverges as E tends to _ðo o Þ. These two complications can approxi- 1 eiðqKÞd 0 r ¼ , mately be avoided by introducing in the above integrals the 1 eiðqþKÞd limits Eþ ¼ maxf0; _ðo o0Þþdg and E ¼ maxf0; _ðo ¼ ð Þ ¼ ð Þ o0Þdg, where d is a fixed positive energy. The integration Fþ Kd n 2 ; F Kd 2N n . ARTICLE IN PRESS M.M. Voronov et al. / Journal of Luminescence 125 (2007) 112–117 115

For structures with the dielectric contrast one can use the 0.5 equations [6]

r sinðmKdÞ 0.4 r ¼ 1 , m sinðmKdÞt sin½ðm 1ÞKd 1 1 t1 sinðKdÞ tm ¼ . ð17Þ 0.3 sinðmKdÞt1 sin½ðm 1ÞKd Here K is a function of o satisfying the dispersion 0.2 equation for exciton polaritons in an inﬁnite MQW

structure, reflection and transmission coefficients from a PL intensity (arb.un.) 2 single QW, r and t , can be found in Ref. [6].Forthe 0.1 1 1 3 structure containing N QWs, a semi-infinite back barrier and a front barrier of the thickness b0 between the first-QW left interface and vacuum, the expression for E(n) is multiplied 0.0 if -0.002 -0.001 0.000 0.001 0.002 by the factor t=ð1 e rrN Þ,wheret ¼ 2nb=ðnb þ 1Þ, r ¼ ω ω ω 0 ( - 0)/ 0 ðnb 1Þ=ðnb þ 1Þ and f ¼ 2½b ðb=2Þonb=c. Fig. 1 illustrates the PL spectra calculated for a MQW Fig. 2. Effect of the dielectric contrast on the PL spectral shape from the structure without the dielectric contrast. The structure near-Bragg MQW structure with 100 wells and the cap layer of thickness parameters are indicated in the caption. For simplicity, we b0 ¼ d(a/2). Curves 1–3 are calculated for the QW background refractive 2 index given by n /n ¼ 0.9; 1 and 1.1, respectively. Other parameters set hjSnðoÞj i¼1. The arrows indicate positions of real a b parts of eigenfrequencies for four exciton–polariton modes coincide with those indicated in the caption to Fig. 1. in the N-QW structure. The labels j ¼ N2, N3 and N4 correspond to three polaritons with Re {o} satisfying 0.5 Eq. (16) with Kj ¼ðpj=NdÞ. Fig. 2 shows the effect of the dielectric contrast mismatch on the PL spectral shape. 0.4 Effects of the detuning from the Bragg condition and the increasing number of QWs are demonstrated in Figs. 3 and 3 4, respectively. 0.3 1

4. Relation between absorption and PL spectra 0.2 PL intensity (arb.un.) We deﬁne the absorbance A in an N-QW structure with 0.1 semi-inﬁnite left- and rightmost barriers as 1 jr j2 jt j2. N N 2 Using the representation of rN, tN in the form of Eqs. (15) 0.0 0.6 -0.002 -0.001 0.000 0.001 0.002 N-3 ω ω ω ( - 0)/ 0 0.5 N-4 Fig. 3. Variation of the PL spectra with the period of the MQW structure

containing 100 wells. Curves 1–3 are calculated for d/l(o0) ¼ 0.495, 0.5, 0.4 and 0.505, respectively, and the cap layer of the thickness b0 ¼ d(a/2). N-2 Other parameters coincide with those indicated in the caption to Fig. 1. 0.3

0.2 and (18) in Ref. [7] we derive the following expression for

PL intensity (arb.un.) the absorbance: 2 0.1 Y 2 2 1 ðo ojÞ þðGj GÞ 1 AðoÞ¼1 , (18) 2 2 2 0.0 j ðo ojÞ þðGj þ GÞ -0.002 -0.001 0.000 0.001 0.002 (ω-ω )/ω where oj iðGj þ GÞ are the eigenfrequencies of exciton 0 0 polaritons in the N-QW structure (j ¼ 1, 2, y, N). In the Fig. 1. Calculated PL spectra for a near-Bragg MQW structure (1) with limit of small nonradiative damping the exact Eq. (18) semi-infinite barriers and (2) with one semi-infinite barrier and a finite cap reduces to layer of thickness b0 ¼ d ða=2Þ. The structure parameters chosen are as _ X follows: thepffiffiffiffiffi number of QWs N ¼ 100, d/l(o0) ¼ 0.495, o0 ¼ 1.5 eV, 2GGj _ AðoÞ¼ . (19) na ¼ nb ¼ 13, G0 ¼ 27 meV, and G ¼ G0. Arrows indicate real parts of ðo o Þ2 þ G2 four exciton–polariton eigenfrequencies. j j j ARTICLE IN PRESS 116 M.M. Voronov et al. / Journal of Luminescence 125 (2007) 112–117

of the packet. The ﬂux of photons transmitted from the 0.075 right-hand medium into vacuum via the multi-layered system equals dq dq dq _ c _ 0 x y z o Nð oÞT ðoÞ 3 0.050 nb ð2pÞ dq dq ¼ _oNð_oÞT 0ðoÞdo x y , 1 ð2pÞ3 0.025 2 0

PL intensity (arb.un.) where T (o) is the transmittance, i.e., the fraction of photon 3 flux that passes through the system from the right to the left, 4 and the photon dispersion o ¼ cq=nb in the medium of the 0.000 refractive index nb is taken into account. The PL spectral -0.002 -0.001 0.000 0.001 0.002 intensity, I0(o), emitted by the multi-layered system into ω ω ω vacuum in the state of equilibrium is defined as ( - 0)/ 0 dq dq Fig. 4. Variation of the PL spectra with the well number N in the MQW x y I 0ðoÞdo 3 . structure. Solid curves are calculated for N ¼ 10 (curve 1), 30 (2), 50 (3), ð2pÞ and 100 (4). The calculation is performed for the structure with 0 Since in equilibrium the photon flux should be zero we d/l(o0) ¼ 0.5, na/nb ¼ 1.1, b ¼ d(a/2), and other parameters are indicated in the caption of Fig. 1. The dotted curve shows a trajectory obtain of the PL maximum in the IPLo plane with increasing QW number. 0 _oNð_oÞ½1 RðoÞT ðoÞ ¼ I 0ðoÞ .(21) Due to the time inversion symmetry T0(o)mustcoincide with the system transmittance from the left to the right. In real systems the damping rate G usually exceeds G0. Therefore, Eq. (21) can be rewritten in the form However, Eq. (19) can be useful for testing the numerical _ _ _ computation of A(o). If the MQW structure is bounded I 0ðoÞ¼ oNð oÞAð oÞ,(22) from one side, say, from the left, by vacuum the relating the absorbance A(o) with the emission spectral absorbance is defined by intensity I0(o) in accordance with Kirchhoff’s law of thermal radiation. If the PL arises due to the indirect phonon-assisted AðoÞ¼1 RðoÞTðoÞ; RðoÞ¼jrvacðoÞj2, N emission of 2D excitons and the energy distribution of vac 2 TðoÞ¼nbjtN ðoÞj , ð20Þ nonequilibrium excitons is characterized by the phonon vac vac temperature T, i.e. if f k ¼ exp½bðm EkÞ,wherem is the where rN ðoÞ and tN ðoÞ are the amplitude reflection and transmission coefficients from the N-QW structure under exciton chemical potential, one can write instead of Eq. (22) normal incidence of light from vacuum. IðoÞ¼ebm_oNð_oÞAð_oÞ,(23) It is instructive to relate the PL and absorbance spectra where I(o) is the spectral intensity of spontaneous emission for planar structures similarly as is done for absorption and of photoexcited multi-layered system. In equilibrium m ¼ 0 emission spectra of 3D crystals [8]. Let us consider a planar and Eq. (23) reduces to Eq. (22). If the frequency interval multi-layered system of the thickness D neighboring under consideration is narrow as compared with the thermal vacuum on the left and semi-infinite homogeneous medium energy k T one can set IðoÞ/Að_oÞ. with the refraction index n on the right. For photons in B b Note that an equation relating the PL spectrum with the vacuum and the homogeneous medium we use the bases pffiffiffiffiffiffiffiffiffi absorption spectrum multiplied by a Boltzmann factor was expðiqrÞ= V where V and V are two arbitrary 1;2 1 2 obtained in Ref. [9] for exciton–polaritons in a quantum normalizing volumes and q is the photon wave vector. At microcavity by introducing the polariton density of states thermal equilibrium the flux of normally incident photons and considering the PL as a direct transmission of cavity of frequency o ¼ cq upon the multi-layered system from polaritons into vacuum. However, this kind of derivation vacuum is given by presents difficulties for exciton–polaritons in finite close-to- dq dq dq Bragg MQWs, that are open systems characterized by short _ocNð_oÞ½1 RðoÞ x y z ð2pÞ3 polariton radiative lifetimes, in which case the definition of the polariton density of states is problematic [10]. On the dqxdqy ¼ _oNð_oÞ½1 RðoÞdo , other hand, Eq. (22) derived here by using the requirement ð Þ3 2p of zero for the photon flux at thermal equilibrium where c is the light velocity in vacuum, N(_o) the photon presupposes no additional assumptions and does not occupation number ½expðb_oÞ11 expðb_oÞ, require a definition of polariton density of states. More- 1 b ¼ðkBTÞ , kB is the Boltzmann’s constant, T the absolute over, Eq. (23) is derived for indirect exciton emission under temperature, R(o) the reflectance, and the differentials dqx, the single assumption of Boltzmann distribution of dqy,dqz are small compared to the average wave vector, qJz, photoexcited excitons outside the radiative circle. ARTICLE IN PRESS M.M. Voronov et al. / Journal of Luminescence 125 (2007) 112–117 117

The general relation between absorption and emission sensitive to the geometrical parameters of the structure, can be explicitly derived for MQW structure under a namely, the period d, the number of QWs and the cap-layer simplifying assumption of thin QWs. In this case using thickness, as well as the ratio of the exciton radiative and definition of absorbance given by Eq. (20) one can derive nonradiative damping rates. A quantitative comparison an expression for A(o) in the following form: with experiment [4,5] needs a thorough fitting of these parameters and, probably, allowance for inhomogeneous 2G G XN Að Þ¼ 0 jjE 2 broadening. o 2 2 n , (24) ðo o0Þ þ G n¼1 where |E |2 is the magnitude of the field at the center of nth n Acknowledgments QW calculated with scattering boundary conditions. This field, therefore, is different from the field entering Eq. (9) Work done at Ioffe Institute was financially supported for the emission intensity. In order to relate this expression by RFBR and the program of Russian Ministry of Science to the luminescence spectra we have to rewrite it in terms of and Education, while the work done at Queens College was the reflection and transmission coefficients. It can be shown supported by AFOSR Grant # F49620-02-1-0305 and by that if the field behind the right boundary of the structure is PSC-CUNY Grants. tN, then one has at the right boundary of nth QW tN iqd=2 iqd=2 En ¼ e þ rNne . tNn References Substituting this expression into Eq. (24) and comparing the results with Eq. (11) we finally find the following [1] E.L. Ivchenko, Optical Spectroscopy of Semiconductor Nanostruc- tures, Alpha Science International, Harrow, UK, 2005. relationship between absorption and emission spectra [2] M. Hu¨bner, J. Prineas, C. Ell, P. Brick, E.S. Lee, G. Khitrova, 2 H.M. Gibbs, S.W. Koch, Phys. Rev. Lett. 83 (1999) 2841. cnb hjSnj i [3] Ch. Spiegelberg, H.M. Gibbs, J.P. Prineas, C. Ell, P. Brick, E.S. Lee, IðoÞ¼ 2 AðoÞ, (25) 2px 2G0G G. Khitrova, V.S. Zapasski, M. Hu¨bner, S.W. Koch, Phys. Status 2 Solidi B 221 (2000) 85. where hjSnj i is assumed to be independent of n. For a [4] A.V. Mintsev, L.V. Butov, C. Ell, S. Mosor, G. Khitrova, structure with a cap layer of finite thickness, light emission H.M. Gibbs, JETP Lett. 76 (2002) 637. into vacuum and absorption under incidence from vacuum, [5] A.V. Mintsev, In: Proceedings of the 11th International Symposium the relation coefficient in Eq. (25) is multiplied by the on Nanostructures: Physics and Technology, Ioffe Institute, St. Petersburg, Russia, 2003, p. 195. refractive index nb. The calculation of the emission spectra 2 [6] E.L. Ivchenko, V.P. Kochereshko, A.V. Platonov, D.R. Yakovlev, for frequency-independent hjSnj i,orG (o; n), and the 0 A. Waag, W. Ossau, G. Landwehr, Fiz. Tverd. Tela 39 (1997) 2072 absorbance spectra for frequency-independent G comple- Phys. Solid State 39 (1997) 1852. tely confirms this relation. [7] E.L. Ivchenko, A.I. Nesvizhskii, S. Jorda, Fiz. Tverd. Tela 36 (1994) 2118 Sov. Phys. Solid State 36 (1994) 1156. 5. Conclusion [8] W. van Roosbroeck, W. Schockly, Phys. Rev. 94 (1954) 1558. [9] R.P. Stanley, R. Houdre´, C. Weisbuch, U. Oesterle, M. Ilegems, Phys. Rev. B 53 (1996) 10995. In conclusion, we have developed a theory of PL for [10] A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, multiple QWs and established relation between their PL G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and absorbance spectra. The PL spectral shape is quite C.M. Bowden, Phys. Rev. E 68 (2003) 26614.