Resonant Rayleigh scattering of exciton-polaritons in multiple quantum wells.

Abstract

A theoretical concept of resonant Rayleigh scattering (RRS) of exciton-polaritons in multiple quantum wells (QWs) is presented. The optical coupling between excitons in different QWs can strongly affect the RRS dynamics, giving rise to characteristic temporal oscillations on a picosecond scale. Bragg and anti-Bragg arranged QW structures with the same excitonic parameters are predicted to have drastically different RRS spectra. Experimental data on the RRS from multiple QWs show the predicted strong temporal oscillations at small scattering angles, which are well explained by the presented theory.

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VOLUME 85, NUMBER 3PHYSICAL REVIEW LETTERS17 JULY 2000
Resonant Rayleigh Scattering of Exciton-Polaritons in Multiple Quantum Wells
Guillaume Malpuech and Alexey Kavokin
Laboratoire des Sciences et Matériaux pour l’Electronique, et d’Automatique, UMR 6602 du CNRS,
Université Blaise Pascal-Clermont-Ferrand II, 63177 Aubière Cedex, France
Wolfgang Langbein
Lehrstuhl für Experimentelle Physik EIIb, Universität Dortmund, Otto-Hahn Strasse 4, 44227 Dortmund, Germany
Jørn M. Hvam
Research Center COM, Technical University of Denmark, Building 349, DK-2800 Lyngby, Denmark
(Received 25 October 1999)
A theoretical concept of resonant Rayleigh scattering (RRS) of exciton-polaritons in multiple quantum
wells (QWs) is presented. The optical coupling between excitons in different QWs can strongly affect
the RRS dynamics, giving rise to characteristic temporal oscillations on a picosecond scale. Bragg and
anti-Bragg arranged QW structures with the same excitonic parameters are predicted to have drastically
different RRS spectra. Experimental data on the RRS from multiple QWs show the predicted strong
temporal oscillations at small scattering angles, which are well explained by the presented theory.
PACS numbers: 78.35.+c, 78.47.+p, 78.66.–w
The essential difference between the resonant Rayleigh
scattering [1] (RRS) and the photoluminescence (PL) is
that the optical coherence is kept in the former case and
broken in the latter case, which is not easily distinguished
experimentally. Recently, RRS by excitons in semicon-
ductor quantum wells (QWs) has been reported [2,3]. The
spectroscopic data have been interpreted in terms of scat-
tering of excitons by a disorder potential in the plane of
the QWs [4–6]. Strictly speaking, this interpretation con-
tains a controversy, since the bare excitons [7] themselves
are not coherent with the incident light. Actually, they
are formed as a result of dissociation of exciton-polaritons
with a loss of coherence. That is why the bare excitons
govern the PL spectra, while to describe the RRS spectra
one should speak in terms of the exciton-polaritons. This
difference is extremely important in multiple quantum well
(MQW) structures where the bare excitons are localized in
the individual QWs, but the exciton-polaritons occupy the
entire structure. Recently, there was a series of papers that
aimed to reveal the role of exciton-polaritons in cw and
time-resolved reflection of MQWs [8]. Moreover, Rabi
oscillations in the RRS spectra of microcavities have been
reported recently [9], which is a purely polaritonic effect.
This Letter aims to show that the RRS spectra of MQWs
are a result of exciton-polariton scattering involving all the
QWs. In particular, the RRS signal is sensitive to the pe-
riod of the MQW structure and the number of QWs. Es-
pecially Bragg- and anti-Bragg-arranged MQWs exhibit
drastically different RRS spectra. Experimental data on
MQWs with small interwell spacing are presented. Strong
temporal oscillations of the RRS are observed for small
scattering angles in a very good agreement with the pre-
sented theory.
The previous studies of the RRS by MQWs can be
briefly summarized as follows.The data reported in
Ref. [2] exhibit typically a RRS signal that initially rises
quadratically with time and then decays nonexponentially
on the time scale of the inverse inhomogeneous broaden-
ing of the exciton peak. This fast decay is followed by
a slower exponential decay. The rise and the fast initial
decay have been explained by a theoretical model [5]
which assumes an ensemble of classical oscillators moving
within a random potential with a finite correlation length.
Recent experiments on different MQW structures have
demonstrated a complex nonmonotonic decay sometimes
showing oscillations with a period of a few picoseconds
[4]. These oscillations have been associated in Ref. [4] to
the interference between light waves scattered by different
localized exciton states in a QW plane.
on single QWs [3] do not show such features and are
well explained using a correlation length shorter than
the exciton radius.Reference [10] has demonstrated a
possibility of the secondary emission (SE) from QWs due
to the quantum fluctuations of the light field. This theory
written for an ideal single QW has no direct relation to
the discussed experiments performed in the linear regime
on the a priori disordered MQW samples. An attempt to
take into account the polariton effects in the RRS has been
done by Citrin [11]; however, all presented numerical
calculations ignored the polariton effect, so that their role,
especially in the RRS of MQWs, remained unclear.
The observed discrepancies between the RRS from
SQWs and MQWs point out the need for an exciton-
polariton theory, which requires a revision of the physical
concept of the RRS in QWs, namely, introducing the
concept of scattering of coherent exciton-polariton states.
In our approach, the dielectric response of QWs con-
taining inhomogeneously broadened exciton resonances
is described in the framework of the nonlocal model
[12]. The parameters used for the individual QWs are the
Experiments
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VOLUME 85, NUMBER 3PHYSICAL REVIEW LETTERS17 JULY 2000
exciton radiative damping rate G0, the nonradiative homo-
geneous broadening g, and the Gaussian inhomogeneous
broadening D. For a MQW structure, we index by zjthe
coordinate of each QW and calculate the electric fields
Eg?zj,v? ? g?v?E?zj,v? (with j varying from 1 to N),
where E?z,v? is the stationary solution of the Maxwell
equations corresponding to a plane light-wave incident
on the structure from the left, and g?v? is the spectral
function of the incident pulse. In all further calculations
we assume the incident pulse of light to be 500 fs long
with a center frequency at the exciton resonance.
The probability to create an exciton having a frequency
v and a wave vector kk? 0 in the jth QW is proportional
to the electric field intensity in this well. The amplitude
of the electric field of the scattered exciton-polariton state
with kkfi 0 can then be found using Eq. (28) of Ref. [6]:
As?zj,v? ? Ps?v?Eg?zj,v?,(1)
where
Ps?v? ?
Z 1`
2`
3 exp?2?t?
µZ
?exp??t?t0?2fR? 2 1?dR
p
2t0?2?exp?ivt?dt ,
∂1?2
(2)
with the rise time t0?p2 ¯ h?D and fRbeing a potential
correlation function chosen in form:
fR? exp?2R2?2j2?,(3)
where j is the potential correlation length further assumed
to be j ? 5 nm. Note that Eq. (2) is written in the clas-
sical limit. The time-dependent amplitude of the scattered
light in the right part of Eq. (2) is real, while its Fourier
transform Ps?v? is a complex function. No random phase
fluctuations are taken into account.
The scattered amplitudes can be considered as light
emitted by each well. Using the generalized scattering-
states method [13], we calculate their propagation in the
scattering direction. Let us consider an emitter of light
situated in the center of the jth QW. We neglect the spa-
tial distribution of the emitter, which is a usual assumption
in the nonlocal model. We take into account the polari-
ton effect within each single QW, i.e., the possibility of
emission and absorption of photon by the same exciton.
Using the transfer matrix method for a given wave vector
of light kk, in the p polarization we calculate the reflection
and transmission coefficients of the right part of the struc-
ture (between z ? zjto z ? d, where d is the total length
of the MQW structure), labeled rj
as well as those of the left part of the structure (between
z ? zjto z ? 0), labeled rj
Formally, we consider our structure as a Fabry-Pérot
resonator with an infinitely thin central layer, where the
emitter is placed. Taking into account all multiple reflec-
tions in the resonator, we can write the electric field just
to the right from the source as
rand tj
r, respectively,
land tj
l, respectively.
˜E
zj,1
1 ?v,zj? ?
1
1 2 rj
rrj
l
,(4)
and the field just to the left from the emitter as
˜Ezj,1
2 ?v,zj? ?
rj
r
1 2 rj
rrj
l
.(5)
We calculate the electric field at the end and at the
beginning of the entire structure as
˜E
zj,1
1 ?v,d? ?
tj
r
1 2 rj
rj
rrj
l
,(6)
˜Ezj,1
2 ?v,0? ?
rtj
l
1 2 rj
rrj
l
,(7)
respectively.
In
˜E
the case of a d-pulse emitted form z ? zjin the negative
direction. Here the notation is completely analogous to
that used in Eqs. (4)–(7).
The amplitude of light emitted by the jth QW with a
spectral function As?zj,v? at the surface of the structure is
˜E
Thus, theamplitudeoflightscatteredinagivendirection
by the system containing N QWs is
N X
j?1
The time-resolved RRS signal is given by
Ç1
2p
2`
In summary, we take into account all the coherent
reflection-absorption processes for this scattered light
in the structure.We ignore the secondary scattering,
however.
Let us first compare the results of the present theory with
experimental data on the RRS from single quantum wells
(SQWs) and MQWs. In the following we discuss experi-
ments performed on GaAs?Al0.3Ga0.7As QWs grown by
molecular-beam epitaxy without growth interruption. The
MQWs consist of ten wells and have barrier thicknesses of
15 nm. They are placed in a helium cryostat at 5 K tem-
perature. The fundamental hh1-e1 1s exciton resonance
is excited by optical pulses from a mode-locked Ti:sap-
phire laser of about 500 fs Fourier-limited pulse duration.
The SE is passed through an analyzer parallel to the linear
excitation polarization (p polarization), a monochromator,
and is detected time and angular resolved by a synchroscan
streak camera with a time resolution of 3 ps. The angu-
lar resolution was adjusted close to a single speckle, i.e.,
to the diffraction limit of the excited area on the sample.
The spectral resolution of about 1 meV rejects nonreso-
nant emission but does not deteriorate the temporal reso-
lution. The excited exciton density of around 108cm22
per well was close to the low density limit. Using the
thesame
˜E
waywecalculate
zj,2
2 ?v,0?,
the
˜E
amplitudes
zj,2
1 ?v,d?, in
zj,2
1 ?v,zj?,
zj,2
2 ?v,zj?,
˜E
zj
A?v,0? ? As?zj,v??˜Ezj,1
2 ?v,0? 1˜Ezj,2
2 ?v,0??. (8)
rs?v? ?
˜E
zj
A?v,0?.(9)
Rs?t? ?
Z 1`
rs?v?exp?2ivt?dv
Ç2
.(10)
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VOLUME 85, NUMBER 3 PHYSICAL REVIEW LETTERS17 JULY 2000
speckle analysis technique [3], we deduce from the tem-
porally and directionally resolved emission intensity I?t, ? q?
the average emission intensity I?t? and the average coher-
ence c ? Icoh?I, where the average is taken over the scat-
tering directions ? q at fixed time, and Icohis the SE intensity
which is coherent to the excitation.
In Fig. 1 I?t? for a 16 nm MQW and a 15 nm SQW
sample are shown on the left for two different scattering
angles. Since the wells in both samples have nearly equal
thicknesses, the inhomogeneous broadening of the indi-
vidual wells due to the interface roughness is comparable
(just slightly more in the SQW structure having a thinner
QW).The SE linewidth (FWHM) for large scattering
angles is 170 meV for the SQW and 340 meV for the
MQW. The increased linewidth of the MQW is in good
agreement with the calculation and is due to the radiative
coupling between the wells, giving rise to ten radiative
modes of different radiative broadenings and renormal-
izations. The nonradiative homogeneous broadening g
is about 6 meV in both samples, as deduced from the
decay of the SE coherence decay for long times. While
for large scattering angles (bottom), the dynamics is quali-
tatively comparable in both structures, at small scattering
angles (top) the MQW shows strong modulations of the
RRS within the first 40 ps, which are not observed for the
SQW. On the right side of Fig. 1 the calculated RRS cor-
responding to the measurements on the left for the exact
structures of the SQW and MQW are shown, using the
G0? 0.02 meV, g ? 0.006 meV, and D ? 0.082 meV
for the MQW and D ? 0.10 meV for the SQW (note
that the FWHM of the Gaussian distribution is 2pln2 ? D
and that all parameters are determined by experiments
independent of the SE dynamics). For the small scattering
FIG. 1.
tensity (right) versus time for a 10 3 16 nm MQW (solid line)
and a 15 nm SQW (dotted line). Top: small scattering angle
(?6±); bottom: large scattering angle (?70±).
Measured SE intensity (left) and calculated RRS in-
angle, the calculations are in good agreement with the ex-
perimental data, confirming that the observed modulations
in the MQW structure are indeed due to the polaritonic
coupling and that the presented model is suitable to model
the RRS of exciton-polaritons in MQWs.
For the large angle scattering, the present model pre-
dicts only a slight reduction of the modulation in the
RRS, while in the experiment the modulations are strongly
suppressed. This discrepancy may be associated to the
higher-order scattering or to the angle dependence of the
inhomogeneous broadening parameter D. Note also that
the use of a classical limit in Eq. (2) is an approximation.
An increase of the inhomogeneous broadening is ex-
pected to reduce the importance of these modulations, as
in time-resolved reflection spectra [8]. We investigate this
using MQWs of different well thicknesses L, using the
fact that the inhomogeneous broadening is roughly propor-
tional to L23. In Fig. 2 we show I?t? at small scattering
angles from MQWs of 13, 10, and 8 nm well thick-
nesses, showing SE linewidths (FWHM) of 0.42, 0.7,
and 1.1 meV, respectively.
modulation is present for all structures, the duration
of this enhanced emission decreases with an increase
of the inhomogeneous broadening.
are shown for comparison (dotted line). The D used in
the calculations is determined using the experimental
linewidth of the 8 nm MQW sample, which is dominated
by the inhomogeneous broadening, and the L23scaling
as D ? 0.15, 0.34, and 0.66 meV, respectively. This is
compatible with the value of 0.1 meV used for the 15 nm
SQW in Fig. 1. G0was taken to be 22, 26, and 28 meV,
respectively, from [14]. We also used g ? 0.006 meV.
The initial part (up to 20 ps) of all the spectra is well
described by the theory (note, in particular, the double-
peak structure for the 13 nm MQWs). This confirms that
also the influence of the inhomogeneous broadening is
modeled correctly in the calculation.
the discrepancy between theory and experiment appears
which is likely due to secondary scattering.
While a strong initial RRS
Model calculations
At longer times,
FIG. 2.
intensity (dotted line) versus time for a 10 3 13 nm MQW
(left), 10 3 10 nm MQW (middle), and a 10 3 8 nm MQW
(right) for a small scattering angle (?6±).
Measured SE intensity (solid line) and calculated RRS
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VOLUME 85, NUMBER 3PHYSICAL REVIEW LETTERS17 JULY 2000
FIG. 3.
arranged (a) and 25 anti-Bragg-arranged (b) MQWs.
Time-resolved RRS spectra calculated for 25 Bragg-
The specific period of a MQW structure is strongly
effecting the RRS spectra. In order to demonstrate this,
let us compare the Bragg- and anti-Bragg-arranged MQW
structures with the same number of QWs. In the Bragg-
arranged MQWs the spacing between wells is equal to half
a wavelength of light at the exciton resonance frequency.
In this configuration, only one super-radiant exciton-
polariton mode is active [15]. In the anti-Bragg-arranged
QWs (spacing between wells equal to a quarter-wavelength
of light at the exciton frequency) the eigenmodes are sym-
metrically distributed around the bare exciton-resonance
frequency [16] thus allowing various quantum oscillation
effects.
Figure 3 shows the RRS calculation for 25 Bragg-
arranged and 25 anti-Bragg-arranged QWs. We assumed
a GaAs?Al0.3Ga0.7As structure embedded in Al0.3Ga0.7As
barriers and the following QW exciton parameters: G0?
0.028 meV (that corresponds to a QW of 8 nm), g ?
0.006 meV, and D ? 0.5 meV. The cap-layer thickness
for both structures is 15 nm. The incidence angle is 0±; the
scattering at small angles (within 5±) is considered. One
can see that there are no oscillations in the RRS spectra
from the Bragg-arranged MQW structure. On the contrary,
the RRS from the anti-Bragg-arranged MQWs exhibit
pronounced irregular oscillations. These oscillations arise
due to the interference between different bright polaritonic
modes in the MQWs. The drastic difference between RRS
spectra of Bragg and anti-Bragg-arranged QWs evidences
the dominant role of exciton-polaritons in the RRS.
In conclusion, RRS from multiple quantum well
structures is governed by the dynamics of the exciton-
polaritons, which are extended mixed exciton-photon
states occupying the entire system.
classical model takes into account the exciton-polariton
effect in MQWs and describes correctly the experimental
RRS dynamics from GaAs MQWs for small scattering
angles, which are dominated by the exciton-polariton
dynamics. Inhomogeneous broadening is found to limit
the time range on which exciton-polariton dynamics can
be dominating, both in experiment and simulation. Drastic
differences in the time-resolved RRS of Bragg- and anti-
Bragg-arranged MQWs are predicted.
The present semi-
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