Acoustic and optical phonon scattering in a single In(Ga)As quantum dot

Abstract

Coupling of acoustic and optical phonons to excitons in single InGaAs/GaAs quantum dots is investigated in detail experimentally and theoretically as a function of temperature. For the theoretical description of the luminescence spectrum, including acoustic and optical phonon scattering, we used the exactly solvable independent boson model. Surprisingly, only GaAs bulk-type longitudinal-optical (LO) phonons are detected in experiment. A quantitatively correct theoretical description of the optical-phonon replica is obtained by including a limited lifetime of the phonons and the dispersion of the LO phonon energy. Similarly, a numerically correct description of the acoustic phonon wings is again based on GaAs bulk material parameters for the phonon dispersion and deformation coupling. In addition, the line shape of the calculated spectra agrees with experiment only when realistic wave functions (e.g., based on eight-band k·p theory) are used for the electron-phonon coupling matrix elements. Gaussian wave functions describing the ground state of a harmonic oscillator fail to describe high-energy tails. Thus, fundamental insights of importance for the correct prediction of properties of nonclassical light sources, based on semiconductor nanostructures, are obtained.

Full text

RAPID COMMUNICATIONS
PHYSICAL REVIEW B 83, 041304(R) (2011)
Acoustic and optical phonon scattering in a single In(Ga)As quantum dot
Erik Stock,1,*Matthias-Rene Dachner,2Till Warming,1Andrei Schliwa,1Anatol Lochmann,1Axel Hoffmann,1
Aleksandr I. Toropov,3Askhat K. Bakarov,3Ilya A. Derebezov,3Marten Richter,2,4 Vladimir A. Haisler,3
Andreas Knorr,2and Dieter Bimberg1
1Institut f¨
ur Festk¨
orperphysik, Technische Universit¨
at Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
2Institut f¨
ur theoretische Physik, Technische Universit¨
at Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
3Institute of Semiconductor Physics, Lavrenteva av 13, Novosibirsk 630090, Russia
4Department of Chemistry, University of California, Irvine, California 92697-2025, USA
(Received 13 September 2010; published 19 January 2011)
Coupling of acoustic and optical phonons to excitons in single InGaAs/GaAs quantum dots is investigated
in detail experimentally and theoretically as a function of temperature. For the theoretical description of
the luminescence spectrum, including acoustic and optical phonon scattering, we used the exactly solvable
independent boson model. Surprisingly, only GaAs bulk-type longitudinal-optical (LO) phonons are detected in
experiment. A quantitatively correct theoretical description of the optical-phonon replica is obtained by including
a limited lifetime of the phonons and the dispersion of the LO phonon energy. Similarly, a numerically correct
description of the acoustic phonon wings is again based on GaAs bulk material parameters for the phonon
dispersion and deformation coupling. In addition, the line shape of the calculated spectra agrees with experiment
only when realistic wave functions (e.g., based on eight-band k·ptheory) are used for the electron-phonon
coupling matrix elements. Gaussian wave functions describing the ground state of a harmonic oscillator fail to
describe high-energy tails. Thus, fundamental insights of importance for the correct prediction of properties of
nonclassical light sources, based on semiconductor nanostructures, are obtained.
DOI: 10.1103/PhysRevB.83.041304 PACS number(s): 78.67.Hc, 73.63.Kv, 78.67.Lt
Single self-organized semiconductor quantum dots (QDs)
(Ref. 1) are most promising for the on-demand generation
of single polarized photons as q-bits2–4or entangled photon
pairs.5–9Since QDs can be easily embedded in electri-
cally pumped p-njunctions, compact nanophotonic10 and
nanoelectronic11 devices can be and have been developed. In
contrast to single photon emitters, based on atomic systems,12
the coupling of a QD to its semiconductor matrix drastically
influences the properties of the photons. In particular, the
excitonic recombination process is accompanied by a lattice
distortion and the generation of polarons.13,14 This scattering
process between the exciton and the phonon leads to a
dephasing that results in a reduction of the coherence length of
the photons. However, most quantum optic experiments, such
as those on indistinguishable photons or quantum cryptogra-
phy systems using double Mach-Zehnder interferometers,15
require long coherence length. Also, theory predicts that
exciton-phonon scattering may be the limiting factor for
generation of entangled photon pairs.16,17
On the other hand, luminescence broadening by phonon
scattering is useful in experiments on the strong light-matter
coupling: The phonon-assisted luminescence can be amplified
by the cavity and can be observed even in micropillars
containing only a single QD.18 Hence, to study the strong light-
matter coupling theoretically,19,20 a complete luminescence
spectrum, including phonon scattering, must be used.
Phonon scattering has been mostly studied until now on
polar CdSe QDs, in which the phonon coupling is much
stronger than, for example, in the nonpolar InGaAs QD system.
The latter one, however, was most successful for device
applications in the past decade, such as for the generation
of electrically driven single photons and entangled photon
pairs.3,4,9Photoluminescence measurements on ensembles of
InGaAs QDs lead to the conclusion that coupling to optical
phonons might be enhanced in QDs as compared to bulk
material.21 The line shape of a single longitudinal-optical (LO)
phonon replica has, however, never been studied experimen-
tally. In this respect, electrically pumped single QDs allow
us to study the exciton-phonon interaction on the fundamental
level: A single electron-hole pair recombines under generation
of a single photon and a phonon.
A number of theoretical approaches exist to describe, at
least in part, exciton- and electron-LO phonon interaction. One
of them suggested the importance of a second-order elastic
interaction between QD charge carriers and LO phonons.22
The resulting Gaussian shape of the LO replica was, however,
found to be an artifact.23,24 Another strong-coupling model
suggested a coupling of LO phonons to excited electronic
states in the QD or the wetting layer,25,26 leading to a more
triangular shape of the LO phonon replica. Only recently,
an exact solution of the electron-phonon interaction in the
quantum optical regime was presented.27 None of the models
has yet been compared with experimental data on single QDs
so far.
In this Rapid Communication we report on a comprehensive
study of both exciton-acoustic and optical-phonon interaction
in a highly efficient single-photon source, based on an
electrically driven single InGaAs/GaAs QD. By comparing
experimental data with calculated spectra, we demonstrate
that phonon coupling strongly depends on a correct and not
simplified description of the charge-carrier wave function,
phonon dispersion, and the lifetime of the phonons.
Our devices consist of InGaAs/GaAs QDs grown by
molecular beam epitaxy with a density of 5 ×108cm−2
embedded in a pin diode structure. Electrical pumping of
efficiently only one single QD is achieved by restricting the
041304-1
1098-0121/2011/83(4)/041304(4) ©2011 American Physical Society

RAPID COMMUNICATIONS
ERIK STOCK et al. PHYSICAL REVIEW B 83, 041304(R) (2011)
FIG. 1. (Color online) Semilogarithmic plot of the EL intensity
of a single QD. (a) With increasing temperature the line broadening
increases due to stronger acoustic phonon interaction. (b) Comparison
of the measured line shape with two calculated spectra, using
alternatively a simple Gaussian (red dashed curve) or a realistic
eight-band k·pwave function (green dotted curve).
injection current through an AlOXaperture. In order to increase
the photon out-coupling efficiency, a microcavity consisting
of 12 and 3 distributed Bragg mirror layers on the bottom
and top of the device, respectively, was grown. The resulting
cavity has a center energy at 1.292 eV with a Qfactor of
140. A detailed description of our device and single-photon
emission at a pump repetition rate of up to 1 GHz can be
found in Ref. 28. The electroluminescence (EL) of our device
is measured in a microluminescence setup, consisting of a
microscope objective [numerical aperture (NA) =0.8], a triple
monochromator with a spectral resolution of 40 μeV, and
a liquid-nitrogen-cooled charged couple device detector. In
order to obtain spectra with a sufficient large intensity dynamic
range, we sum up a series of spectra with 10 s integration
time each. The electroluminescence from only one QD can be
studied in detail, due to the strong current confinement in our
devices.
At a bias of 1.41 V and a current of 5 nA, we sum up
a series of 300 spectra of the exciton emission line for each
given temperature. In a semilogarithmic plot (Fig. 1), a clear
broadening of the emission is visible. The features at −7, −5,
and −3.5 meV from the dominating zero phonon line (ZPL)
can be attributed to the recombination of other complexes
in the QD or to weak luminescence from other QDs. These
features vary from device to device and are therefore not related
to phonon scattering. These features influence the phonon
sidebands and mainly contribute to the difference between
theory and experimental data in Fig. 1(b). The temperature
dependence of the exciton peak position has been removed
from Fig. 1(a) by plotting the spectra on a relative energy
scale. In our experiment the linewidth of the ZPL is limited by
the spectral resolution of our setup.
At low temperatures (15 and 35 K), the broad sidebands of
the ZPL are asymmetric due to spontaneous phonon emission,
with larger intensity at the lower-energy side. For increasing
temperatures (50 and 77 K) the broadening increases and be-
comes symmetric. Such properties of the sidebands are typical
for acoustic phonon scattering: By emission or absorption of
a phonon, the energy of the emitted photon will be reduced
or increased, respectively, by the phonon energy. At lower
temperatures, the phonon density nph is low and therefore, due
to spontaneous processes, phonon emission [∼(nph +1)] of an
(acoustic) phonon has a higher probability than the absorption
(∼nph). Hence, the resulting spectrum is asymmetric. With
increasing temperature the phonon density increases (nph 1)
and the probabilities for emission and absorption become
equal, resulting in a symmetric spectrum.
For the theoretical description of the luminescence spec-
trum, including acoustic phonon scattering, we applied the
exactly solvable independent boson model:13,29,30 This model
describes the lowest optically active quantum dot transition
as a two-level system coupled to phonons by band diagonal
interaction. For the numerical evaluation, we used well-known
GaAs bulk material parameters for the phonon dispersion and
the deformation coupling.31,32 Therefore, the line shape of
the calculated spectra depends only on the electron-phonon
coupling matrix elements that are determined essentially by
the wave functions of the electrons and holes in the QD. To
gain insight into the microscopic interaction and its sensitivity
to the character of the electronic wave functions, we compared
Gaussian wave functions (i.e., ground state of a harmonic
oscillator)33 and wave functions calculated for a realistic
QD using eight-band k·ptheory,34 respectively. Figure 1(b)
shows the measured spectrum at 15 K in comparison to the
theoretical line shape for both wave-function types. For a
Gaussian wave function, often used in theoretical studies, the
phonon scattering is strongly underestimated for any energy.
In contrast to this, k·pwave functions reproduce the measured
spectra very well. The main difference between the two
wave-function models is their real space behavior, transferred
to the decay of the coupling matrix elements in momentum
space: In contrast to Gaussian functions, the more realistic
eight-band k·pwave functions allow for an interaction at larger
wave numbers. Therefore, larger phonon energies have more
impact on the acoustic phonon sidebands. The large difference
between the two calculated spectra clearly demonstrates
the importance of realistic wave functions already for a
qualitatively correct description of electron-phonon scattering.
The measured spectra can be reproduced with our model by
using realistic eight-band k·pwave functions for temperatures
of up to 60 K. Another important result of the modeling of
the measured spectra is that they are perfectly reproduced by
using GaAs bulk phonon parameters only.
The influence of the wave function is also visible in
Fig. 2, where we show the spectral density (containing
coupling strength, interaction matrix elements, and phonon
dispersion) of the electron-phonon interaction for the two
different wave functions. The spectral density at small energies
(<10 meV) decays for a Gaussian wave function much faster
than for the k·pwave function. This result in energy space
can be directly translated to momentum space via the linear
dispersion. Even a change of the width of the Gaussian
curve never reproduces the measured spectra as well as the
eight-band k·pwave function.
Now, in order to study scattering processes of optical
phonons experimentally, we sum up a series of 1450 spectra
of the LO-phonon replica, resulting in a five times longer
integration time than for the acoustic phonon. Figure 3(a)
compares two spectra of the same QD under the same bias
condition: The black dotted curve shows the exciton emission
line around 1.3035 eV. For the second spectrum, the energy
scale [on the top of Fig. 3(a)] is shifted by 36.5 meV compared
to the bottom scale. Three pronounced peaks are visible in this
041304-2

RAPID COMMUNICATIONS
ACOUSTIC AND OPTICAL PHONON SCATTERING IN A ... PHYSICAL REVIEW B 83, 041304(R) (2011)
FIG. 2. (Color online) Comparison of the spectral phonon
density for two different wave functions. For acoustic phonons
(energy <10 meV) the Gaussian wave function (black curves)
decreases too quickly, leading to an underestimation of the phonon
scattering.
spectrum, and each of them has a “counterpart” in the black
spectrum. For other QDs, we also found for almost every
intense luminescence line a “counterpart” separated by around
−36 meV.
We attribute these corresponding lines to the recombination
of an exciton accompanied by the emission of an LO phonon.
For different QDs, the phonon energy varies between −35.7
and −36.6 meV with no systematic trend, as compared to the
ZPL energy. These energies are comparable to the LO phonon
energy of strain-free GaAs bulk material of 36.59 meV.31
The variation of the phonon energy can be explained by
differences in the strain of the GaAs material induced by size
and composition of the actual QD.35 At the energy of the InAs
LO phonon replica (30.3 meV), within our intensity dynamic
of five orders of magnitude, no luminescence is observed.
This is in agreement with the results for acoustic phonon
scattering. Consequently, the electron and holes in the QDs
interact predominantly with phonons from the surrounding
GaAs. The InAs phonon interaction is at least one order of
magnitude less. No emission caused by localized InAs-type
phonons in the QD is observed.
FIG. 3. (Color online) Optical-phonon replica of a single QD.
(a) The black dotted curve shows three ZPL lines, each one having
a replica, separated by 36.5 meV (red spectra, top energy scale).
(b) Comparison of the LO line shape between experimentally
measured (black) and calculated spectra with (red dotted curve) or
without (green dashed curve) taking into account a finite lifetime of
the phonon of 10 ps. A finite lifetime of the phonon provides a better
description of the broad line shape of the LO phonon replica.
Next, we focus on the coupling strength of the exciton-LO
phonon interaction. The coupling strength is usually described
by the Huang-Rhys factor, given by the intensity ratio of the
ZPL IZPL and the nth phonon replica In(at zero temperature) by
In/IZPL =Sne−S/n!.36,37 In Fig. 3the intensity is normalized
with respect to the ZPL. Both emission lines are located outside
the cavity resonance in the stop band, so their intensities can be
compared. The intensity of the LO phonon replica is about four
orders of magnitude lower than the ZPL intensity, resulting in
a Huang-Rhys factor of S≈10−4. This value is one order
of magnitude less than for GaAs bulk phonons,38 indicating
a reduced phonon coupling in our electrically pumped single
QD. We additionally derived the Huang-Rhys factor from the
calculated spectral density (Fig. 2).37 Our calculations yield
factors of S≈10−4, confirming the experimental result.
The line shape of the phonon replicas enables us to access
the coupling mechanism between the exciton and the LO
phonon: The recombination of an electron-hole pair in a QD
has a discrete energy and the linewidth of the ZPL is only
limited by the lifetime of the exciton to a few μeV.39 Therefore,
any broadening of the luminescence upon generation of an LO
phonon is due to the intrinsic phonon properties or coupling
of the phonons to other modes. Figure 3shows the LO phonon
replica at larger spectral resolution. The resulting full width at
half maximum of 700 μeV is well above the spectral resolution
and about one hundred times larger than for the ZPL. Hence,
LO phonons of varying energies are generated during the
recombination processes in the same single QD. Also, the line
shape is again slightly asymmetric, showing a faster decay on
the low-energy side than on the high-energy side. This specific
line shape is consistent with a broadening mechanism that
results from a realistic wave-number-dependent LO phonon
dispersion ωLO(q) beyond the Einstein model ωLO =const:
In such a dispersion relation, a higher energy corresponds
to a smaller momentum. Since the electron-phonon cou-
pling favors low-momentum phonons, the generation of
higher-energy phonons has a larger probability than that of
lower-energy phonons, and thus yields an asymmetry of the
line shape.
For a quantitative understanding of this coupling mech-
anism, we compare the experimental results with a model
(Fig. 3), where LO phonon assisted luminescence spectra
are calculated using the independent boson model and ap-
proximating the real optical-phonon dispersion by a cosine.
This approximation agrees very well with the experimentally
observed dispersion.31 For the coupling matrix elements, we
again compared Gaussian wave functions and eight-band
k·pwave functions, respectively. In contrast to the acoustic
phonon coupling, the influence of different wave functions
is less pronounced. In Fig. 2it can be seen that the LO
phonon part of the spectral density shows a much faster
decay than the acoustic phonon part. Hence the LO phonon
replicas for Gaussian and eight-band k·pwave functions
differ only slightly. For a complete description of the LO
phonon-scattering process, we also included a finite lifetime
of 10 ps (Refs. 40 and 41) of the LO phonons. The resulting
LO replica in Fig. 3(b) reproduces very well the measured line
shape, whereas a line shape with neglected phonon lifetime is
too narrow. A perfect fit to the experimental data results in a
phonon lifetime of 5 ps.
041304-3

RAPID COMMUNICATIONS
ERIK STOCK et al. PHYSICAL REVIEW B 83, 041304(R) (2011)
Additional phonon replica broadening, due to the influence
of excited and wetting layer states, was predicted for the more
enhanced electron-phonon coupling in CdSe systems.25 We
could not observe a significant influence in the GaAs material
system in our calculations.
In conclusion, we studied the acoustic and optical phonon
scattering on the most fundamental level, at which a single
photon and a phonon are generated by the recombination of a
single exciton in a single electrically driven InGaAs/GaAs
QD. By comparing experimental and calculated spectra,
we conclude that (i) GaAs bulk material phonon modes
represent the dominant broadening mechanism; (ii) local-
ized InAs-type phonons are not observed; (iii) a large LO-
replica broadening of ≈700 μeV is induced by interaction
with large wave-number phonons, having a finite lifetime;
(iv) k·pwave functions are essential for the description of
the excitons; and (v) the independent boson model using
these wave functions provides the proper wave-vector depen-
dence for a quantitative understanding of the experimental
results.
We acknowledge fruitful discussions with A. Baumgartner
and funding by the SFB 787 of the Deutsche Forschungsge-
meinschaft (DFG).
*erik@sol.physik.tu-berlin.de
1D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot
Heterostructures (Wiley, Chichester, 1998).
2P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff,
L. Zhang, E. Hu, and A. Imamoglu, Science 290, 2282 (2000).
3A. J. Shields, Nat. Photonics 1, 215 (2007).
4D. Bimberg et al.,IEEE Photon. J. 1, 58 (2009).
5N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron,
D. Gershoni, B. D. Gerardot, and P. M. Petroff, Phys. Rev. Lett.
96, 130501 (2006).
6R. M. Stevenson, R. J. Young, P.Atkinson, K. Cooper, D. A. Ritchie,
andA.J.Shields,Nature (London) 439, 179 (2006).
7A. Schliwa, M. Winkelnkemper, A. Lochmann, E. Stock, and
D. Bimberg, Phys. Rev. B 80, 161307(R) (2009).
8C. Kindel, S. Kako, T. Kawano, H. Oishi, Y. Arakawa, G. H¨
onig,
M. Winkelnkemper, A. Schliwa, A. Hoffmann, and D. Bimberg,
Phys.Rev.B81, 241309(R) (2010).
9C. L. Salter, R. M. Stevenson, I. Farrer, C. A. Nicoll, D. A. Ritchie,
andA.J.Shields,Nature (London) 465, 594 (2010).
10N. Kirstaedter et al.,Electron. Lett. 30, 1416 (1994).
11A. Marent, T. Nowozin, J. Gelze, F. Luckert, and D. Bimberg, Appl.
Phys. Lett. 95, 242114 (2009).
12H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691
(1977).
13B. Krummheuer, V. M. Axt, and T. Kuhn, Phys.Rev.B65, 195313
(2002).
14J. Forstner, C. Weber, J. Danckwerts, and A. Knorr, Phys. Rev. Lett.
91, 127401 (2003).
15N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys.
74, 145 (2002).
16U. Hohenester, G. Pfanner, and M. Seliger, Phys.Rev.Lett.99,
047402 (2007).
17A. Carmele, F. Milde, M.-R. Dachner, M. B. Harouni,
R. Roknizadeh, M. Richter, and A. Knorr, Phys. Rev. B 81, 195319
(2010).
18J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn,
S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke,
and A. Forchel, Nature (London) 432, 197 (2004).
19F. P. Laussy, E. del Valle, and C. Tejedor, Phys. Rev. Lett. 101,
083601 (2008).
20G. Tarel and V. Savona, Phys.Rev.B81, 075305 (2010).
21R. Heitz, I. Mukhametzhanov, O. Stier, A. Madhukar, and
D. Bimberg, Phys.Rev.Lett.83, 4654 (1999).
22A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mork, and R. Lang, Phys.
Rev. Lett. 85, 1516 (2000).
23E. A. Muljarov and R. Zimmermann, Phys. Rev. Lett. 96, 019703
(2006).
24A. V. Uskov, A. P. Jauho, B. Tromborg, J. Mork, and R. Lang, Phys.
Rev. Lett. 96, 019704 (2006).
25E. A. Muljarov and R. Zimmermann, Phys. Rev. Lett. 98, 187401
(2007).
26R. Z. E. A. Muljarov, Phys. Status Solidi 245, 1106
(2008).
27A. Carmele, M. Richter, W. W. Chow, and A. Knorr, Phys. Rev.
Lett. 104, 156801 (2010).
28A. Lochmann, E. Stock, J. A. Tofflinger, W. Unrau, A. Toropov,
A. Bakarov, V. Haisler, and D. Bimberg, Electron. Lett. 45, 566
(2009).
29R. Zimmermann and E. Runge, in Proceedings of the 26th ICPS
Edinburgh, edited by J. H. Davies (IOP Publishing, Bristol, UK,
2002), Vol. 171, paper M 3.1.
30J. Forstner, C. Weber, J. Danckwerts, and A. Knorr, Phys. Status
Solidi B 238, 419 (2003).
31Landolt-B ¨
ornstein—Semiconductors: Intrinsic Properties of Group
IV Elements and II-V, II-VI and I-VII Compounds (Springer, Berlin,
1987), Vol. III/22a.
32M. R. Dachner et al.,Phys. Status Solidi B 247, 809 (2010).
33A. Wojs, P. Hawrylak, S. Fafard, and L. Jacak, Phys. Rev. B 54,
5604 (1996).
34A. Schliwa, M. Winkelnkemper, and D. Bimberg, Phys.Rev.B76,
205324 (2007).
35R. Heitz et al.,Appl. Phys. Lett. 68, 361 (1996).
36K. Huang and A. Rhys, Proc.R.Soc.A204, 406 (1950).
37V. M a y a n d O . K ¨
uhn, Charge and Energy Transfer Dynamics in
Molecular Systems (Wiley-VCH Verlag, Berlin, 2000).
38S. Nomura and T. Kobayashi, Phys. Rev. B 45, 1305 (1992).
39P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin,
D. Ouyang, and D. Bimberg, Phys. Rev. Lett. 87, 157401 (2001).
40S. Rudin, T. L. Reinecke, and M. Bayer, Phys.Rev.B74, 161305(R)
(2006).
41A. R. Bhatt, K. W. Kim, and M. A. Stroscio, J. Appl. Phys. 76, 3905
(1994).
041304-4