Anomalous exciton lifetime by electromagnetic coupling of self-assembled InAs/GaAs quantum dots

Anomalous exciton lifetime by electromagnetic coupling of

self-assembled InAs/GaAs quantum dots

Citation for published version (APA):

Bogaart, E. W., & Haverkort, J. E. M. (2010). Anomalous exciton lifetime by electromagnetic coupling of self-assembled InAs/GaAs quantum dots. Journal of Applied Physics, 107(6), 064313-1/5. [064313].

https://doi.org/10.1063/1.3354080

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10.1063/1.3354080 Document status and date: Published: 01/01/2010

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Anomalous exciton lifetime by electromagnetic coupling of self-assembled

InAs/GaAs quantum dots

E. W. Bogaarta兲and J. E. M. Haverkort

Department of Applied Physics, COBRA Inter-University Research Institute, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

共Received 1 November 2009; accepted 2 February 2010; published online 29 March 2010兲 We report on the experimental observation of a hitherto ignored long-range electromagnetic coupling between self-assembled InAs/GaAs quantum dots共QDs兲. A 12 times enhancement of the QD exciton lifetime is observed by means of time-resolved differential reflection spectroscopy. The enhancement is due to local field effects within the QD ensemble. The electromagnetic coupling of the QDs results in a collective polarizability, and is observed as a suppression of the emission rate. Our results reveal that the mutual coupling strength can be optically tuned by varying the pump excitation density. This enables us to optically tune the exciton lifetime. © 2010 American Institute

of Physics.关doi:10.1063/1.3354080兴

I. INTRODUCTION

Realization of nanoscaled semiconductor systems, e.g., quantum dots共QDs兲, has opened up the possibility of inves-tigating the effects of the local electrodynamic environment on the radiative emission of confined excitons. One of the consequences of a modified electromagnetic density-of-states 共DOS兲 is the enhancement or suppression of the spontaneous emission rate. Modification of the local density of electro-magnetic states in the vicinity of an emitter can be estab-lished by inserting the emitter into a nanocavity1,2 or by the mutual interaction between closely spaced emitters. The lat-ter can be described by Förster coupling,3–5 the Dicke-effect,6,7or local field effects.8The Dicke-effect is ob-served as an enhancement or suppression of the emission rate—known as superradiance and subradiance, respectively, and is sensitive to the distance between the interacting emit-ters. In order to observe superradiance or subradiance in solid state nanostructures, the system has to be engineered with high accuracy as established for a high-quality multiple quantum well structure.9 Unlike quantum wells, self-assembled QD nanostructures have a significant inhomoge-neously broadened DOS and a lateral distribution. As a re-sult, pure super- and subradiance tend to cancel each other.7 Experimentally, it is ignored that optical excitation of nanoscale objects, e.g., QDs, quantum rings, and carbon nanotubes, has a profound impact on the permittivity of the nanoscaled object and near its lattice site. For InAs/GaAs QDs this means that due to the Lorenzian-shaped permittiv-ity of optically excited QD ␧QD共␻兲, which is much larger

than the permittivity of the GaAs host medium ␧h,10–12 the

electromagnetic field pattern is strongly modified. Hereby, the radiative decay channels8,13 of the emitter are affected. Hence, the high contrast permittivity landscape formed by excited QDs provides a strong dipole scattering of electro-magnetic fields,8such that local electromagnetic fields acting on each QD are modified by the adjacent polarized QDs. In

this perspective, a QD ensemble can be described as an array of mutually interacting point sources, the strength of which is given by the self-dressed polarizability.

By plotting the QD-exciton lifetime versus the QD tran-sition energy an anomalous QD-exciton lifetime spectrum is revealed, as depicted in Fig. 1. The lifetime spectrum— deduced from transient differential reflection measurements—has a pronounced resonantlike behavior with respect to the optically excited QD DOS with a 12 times enhancement at the mean transition energy of the QD ground state.

In this paper, we report on a significant modification of the QD-exciton lifetime observed by means of time-resolved differential reflection spectroscopy 共TRDR兲,14,15 owing to mutual electromagnetic coupling between QD-excitons within the ensemble.16The profound modification of the QD-exciton lifetime is a direct consequence of the collective po-larizability of the nanostructure. To explain this collective effect, we utilize and significantly extend the electromagnetic response theory as presented by Ref.8for a two-dimensional 共2D兲 array of identical QDs. Our extended model describes the effective electromagnetic response of nanostructures, which takes into account the inhomogeneously broadened

a兲_{Electronic mail: erik.bogaart@dalsa.com.}

1.05 1.10 1.15 1.20 0 500 1000 1500 2000 0 250 500 750 1134 nm 1078 nm 1103 nm Reflectan ce (norm.) Time (ps) 5 K Energy (eV) Q D-exciton lif etime (ps ) 1st Excited state Ground state Photoluminescence

FIG. 1. 共Color online兲 QD-exciton lifetime vs the probe photon energy measured at 300 W/cm2 _{pump excitation, revealing anomalous lifetime} resonance for the QD ground and excited state. A PL spectrum obtained at high excitation density is added as a reference, to emphasize the QD first excited state. The inset shows three TRDR signals共normalized兲 obtained for different probe photon energies.

共2010兲

DOS as observed for more realistic QD ensembles. The sig-nificant lifetime enhancement illustrates a mutual interaction between resonant QD-excitons, which extends over distances considerably beyond the nearest neighbor separation.

II. MODEL

Let a single random inhomogeneous layer of QDs be placed in a homogeneous isotropic host medium. Our model incorporates two factors of the irregularity such as:共i兲 inho-mogeneous broadening of the exciton spectrum and共ii兲 ran-dom distribution of QDs in the layer. As a first step, we assume all QDs to be identical and eliminate thus the inho-mogeneous broadening. Then, the effective boundary condi-tions derived for regular arrays in Ref.8are extended to the case of randomly inhomogeneous 2D ensembles of QDs with homogeneous line broadening. In the strong confinement regime,8 the effective permittivity of a QD is written as a Lorentzian function17–19 with a single resonance frequency 共eigenfrequency兲␻0

␧QD共␻兲 = ␧h

冉

1 +

␻QD

␻−␻0+ i⌫0

冊

. 共1兲

Parameter ␻_QD is related to the QD oscillator strength f₀ through the spontaneous emission rate ⌫₀

␻QD= 2⌫₀ k₀3VQD , with ⌫₀=␧h 1/2_␻ 0 2 e2f0 6␲␧0mec3 . 共2兲

Here e and medenote the elementary charge and the effective

mass, respectively.⌫0governs the homogeneous linewidth of

a QD with volume V_QD=共4␲/3兲R_QD3 . The electromagnetic response utilizes the Clausius–Mossotti relation20,21to calcu-late the collective polarizability of the ensemble

␣= R_QD3 ␧QD−␧h ␧QD+ 2␧h

. 共3兲

Further analysis is based on general principles of the wave propagation in randomly inhomogeneous media.

The resulting local field effects are visualized by the optical response functions. We use the reflection coefficient—s-polarization—of a planar periodic array of identical QDs to determine the collective radiative decay rate. In analogy with Ref. 22, the plane wave QD reflection coefficient rQD is obtained by solving the jump boundary

conditions22–24 on the field components, such that

rQD共␻兲 =

i⌳

冑

␧h− i⌳

, with ⌳ =2␲␻0␧h␣

cd_QD2 . 共4兲

Note that, the reflection of the QD ensemble depends on the 2D periodic lattice spacing dQD. The emission rate of an

electromagnetically coupled QD array⌫coupled_{—at resonance}

ប␻0—is governed by the imaginary part of the frequency

pole of the reflection coefficient,8and can be written as ⌫coupled_=⌫isolated₋k0

冑

␧h␻QDVQD

2d_QD2 . 共5兲

Here ⌫isolated is a weighted sum of the dephasing and the radiative emission rate of an isolated QD-exciton, i.e., an

uncoupled QD-exciton.25–29 k0 denotes the vacuum wave

vector and ␻QD is the phenomenological parameter

propor-tional to the QD oscillator strength.8Equation共5兲shows that the lifetime 1/⌫coupled_{is governed by the lattice spacing d}

QD.

To apply our theoretical model to self-assembled QD nanostructures, the inhomogeneously broadened DOS and random spatial QD distribution must be introduced in the model. We emphasize, that the electromagnetic coupling manifests itself only between 共quasi兲 resonant QD-excitons and is governed by the spectral overlap of the homoge-neously broadened permittivity of the excited QDs, typically described by a Lorenzian profile. Lateral disorder means dis-tortion of the ideal lattice, and is seen as a fluctuation of distances between adjacent sites. Disorder effects of excitons in semiconductor nanostructures can be described within the usual single-site approximation, e.g., the coherent-potential approximation 共CPA兲.30–32 If Z different scatterers are ran-domly distributed on the lattice sites of a plane, the CPA can be used to treat the disordered system within a mean-field context. Hereby, the effective CPA medium consists of iden-tical scatterers at all sites of the plane, i.e., an isotropic QD array. By making use of the CPA formulism, the whole QD ensemble with inhomogeneously broadened DOS G共␻兲 can be divided into smaller subensembles, each with their own narrow DOS G共␻j兲 and unique ordering—a 2D array with

periodicity d_QDres共␻j兲, where each lattice site is occupied by a

QD-exciton with the effective polarizability.

The average distance between the QDs within each sub-ensemble depends on the location of these QDs within the overall DOS, 1/d_QDres共␻j兲=

冑

N共␻j兲=

冑

NQDG共␻j兲. Here N共␻j兲

and NQDare the area QD density of a subensemble with peak

energy ប␻jand of the whole ensemble, respectively. Taking

this into account for the whole ensemble, Eq.共5兲is rewritten as

⌫coupled_{共␻兲 = ⌫}isolated_{− 2}k0

冑

␧h␻QDVQD

2 NQDG

2_{共␻兲. 共6兲}

The additional factor two in the second term on the righthand side takes into account the QD spin degeneracy. Equation共6兲 reveals that the QD-exciton lifetime has a maximum at the center of the QD-size distribution, where the average dis-tance between radiatively coupled QDs within the suben-semble has statistically a minimum value. Equation 共6兲also predicts a resonantlike QD-exciton lifetime spectrum. The lifetime spectrum will have a narrower spectral width—a factor

冑

2 narrower—than the QD distribution G共␻兲, due to the quadratic dependence. This prediction is indeed observed experimentally, as will be shown below.

Besides the imaginary part, the real part of the frequency poles of rQDdetermines the depolarization shift, such that we

obtain

␻c= Re关rQD共␻−␻0兲兴 =

␻QD

3 . 共7兲

As is observed from Eq.共7兲, the shift does not depend on the periodicity of the QD array, but is solely governed by the parameter␻QD. This means that the oscillator strength of the

QD defines the depolarization shift in the ensemble.

III. SAMPLE GROWTH AND EXPERIMENTAL DETAILS

To test the result of the theoretical electromagnetic re-sponse model as given by Eq.共6兲, differential reflection mea-surements are performed on a five layer self-assembled InAs QD nanostructure grown by molecular beam epitaxy on a 共100兲 GaAs substrate. After the deposition of 295 nm GaAs buffer layer at 580 ° C, the temperature was lowered to 490 ° C for the growth of the multiple QD layers. A 30 nm GaAs layer was deposited before the growth of the five lay-ers of QDs. Each QD layer consists of 2.1 ML of InAs fol-lowed by a 30 nm GaAs spacer layer. Finally, the sample is capped by 137 nm GaAs at a temperature of 580 ° C. Atomic force microscopy images of uncovered dots show that the QDs are formed with a density of approximately 2.8 ⫻1010 _cm−2_.

The QD-exciton lifetime is investigated by means of pump-probe TRDR共Refs. 14,15, and33兲 at 5 K. The QDs

are nonresonantly excited using short laser pulses from a mode-locked Ti:sapphire laser. The pump photon energy is tuned above the band gap energy of the GaAs barrier mate-rial and creates free carriers within the GaAs host, which are subsequently captured into the QDs. The carrier-induced re-flection change⌬R/R0共␻兲 is monitored by tuning the photon

energy of the probe laser over the QD optical transition en-ergy within the ensemble. These resonant probe pulses are generated from an optical parametric oscillator, synchroni-cally pumped by the Ti:sapphire laser. Both lasers emit 2 ps pulses, corresponding to a spectral resolution of approxi-mately 1 meV. Thus, in the center of the size distribution, only 1.7% of all QDs are in resonance with the probe laser. This small fraction of resonant QDs forms an interactive coupled subensemble, which collectively responds to the electromagnetic probe field. Since the average dot-to-dot dis-tance in our sample is approximately 60 nm, the average separation between resonant dots extends to approximately 460 nm in the center of the size distribution.

IV. RESULTS AND DISCUSSION

Figure1depicts the QD-exciton lifetime as a function of the QD transition energy. We note that the QD-exciton life-time spectrum shows a pronounced resonantlike behavior with respect to the photoluminescence 共PL兲 spectrum. The lifetime spectrum reveals, that the QD-exciton lifetime at the mean transition energy of the QD ground state is 12 times larger than that of the excitons at the wings of the distribu-tion. In other words, Fig.1reveals a 12 times enhancement of the QD-exciton lifetime. Moreover, the lifetime spectrum is narrower than the PL spectrum which is in agreement with the predictions as deduced from Eq. 共6兲. Surprisingly, also the lifetime of the QD first excited state at 1.163 eV has an energy dependence. These preliminary observations already provide some qualitative experimental confirmations of our theoretical model. To make a more quantitative statement, additional investigations of the TRDR measurement results must be performed. Analysis of the lifetime spectrum by us-ing Gaussian fits, reveals two peaks that are ascribed to the QD ground and first excited state with peak energies of 1.104 and 1.161 eV, respectively. In addition, the spectral width of

the ground共first excited兲 state is 27 共23兲 meV. The PL spec-trum has a peak energy of 1.107 eV for the ground state and a spectral width of 44 meV. We note that the lifetime spec-trum is slightly shifted with respect to the PL specspec-trum.

At the side of the distribution the QD-exciton lifetime approaches 150 ps. These QDs can be regarded as uncoupled and enable us to determine⌫isolated. Our observation of a 150 ps QD-exciton lifetime is not surprising if we consider that our QDs have a total confinement energy of approximately 168 meV.14From the results as reported in Ref.29, a radia-tively limited lifetime of a few hundred picoseconds is ex-pected.

More experimental evidence of electromagnetic cou-pling between optically excited QDs is provided by the pump excitation density dependence of the QD-exciton lifetime en-hancement, as depicted in Fig.2. From the spectra shown in Figs.1and2, it is clear that the lifetime-spectrum amplitude exhibits a strong dependence on the pump-induced carrier density ␩, but that the shape remains unaffected. The in-creased pump-induced carrier density induces a higher QD population, resulting in a nonzero change in the QD-exciton emission rate,14,33 ⌬⌫=⳵⌫/⳵␩⫽0. Hence, the QD-exciton lifetime is governed by the density of occupied states, ⌫coupled_{共␻兲⬃G}

occupied

2 _{共␻兲. Thus, the number of optically}

ex-cited QDs determines the effective coupling distance and hereby the coupling strength.

In order to compare our theoretical model with the ex-perimental findings, Figs.1and2, the QD DOS G共␻兲 has to

be known. We utilize the differential reflection spectrum ob-tained by plotting the amplitude of the TRDR time-traces, as depicted in the inset of Fig.1, versus the probe photon en-ergy. The resultant reflectivity spectrum is depicted in Fig.3. Two peaks are observed for the QD ground state and for the QD first excited state. Both energy states have a local mini-mum near the peak of the ground and first excited state tran-sition energy. In other words, for both energy states a split-ting is observed in the differential reflection spectrum and appears as a double peak shifted ⫾ប␻c with respect to the

mean transition energy of each state. From the reflectivity spectrum, we deduce a splitting of 2ប␻c= 27共15兲 meV for

the QD ground共first excited兲 state.

The amplitude⌬R共␻兲 of the differential reflectivity, Fig.

3, is expressed as14,33 5 K 1.05 1.10 1.15 1.20 0 500 1000 1500 2000 Q D -Pol ari ton lif eti me (ps ) Energy (eV)

FIG. 2. 共Color online兲 QD-exciton lifetime vs the probe photon energy obtained for a pump excitation density of 200 共triangle兲 and 400 W/cm2 共square兲, corresponding to approximately one and two electron-hole pairs in a single QD, respectively.

⌬R共␻兲 = −

冕

H共␻−␻

⬘

兲G共␻

⬘

兲d␻

⬘

,

with H共␻兲 = ⌬⌫共␻兲关⌫共␻兲L

⬘

共␻兲 + L共␻兲兴, 共8兲 in which L共␻兲 is a Lorentzian line shape function of a single isolated QD modeling the homogeneous broadening, and

L

⬘

共␻兲 is its first derivative. Combining the measured

reflec-tivity spectrum ⌬R/R0共␻兲 with the pump excitation

depen-dence of the emission rate⌬⌫共␻兲, as deduced from Figs.1

and2, the energy distribution G共␻兲 is extracted by using Eq. 共8兲. The result is depicted in Fig.4.

From Eq.共6兲it is expected that the width of the lifetime spectrum is a factor

冑

2 narrower than the DOS. This means, we have to compare the lifetime spectra with the extracted distribution G共␻兲 and not with the PL spectrum. From our analysis we observe that the extracted distribution is de-scribed by a superposition of two Gaussian functions with mean energies of 1.103 and 1.154 eV, respectively. The main peak, ascribed to the QD ground state, has a spectral width of approximately 38 meV. The theoretical prediction is accu-rately verified since the 27 meV spectral width of⌫coupled_共␻兲

is found to be a factor 1.41 smaller than the 38 meV spectral width of G共␻兲. Thus, our theoretical model of electromag-neticly coupled QD-excitons is experimentally verified.

Now let us determine parameter␻QDfrom which we can

derive the mutual coupling distance d_QDres. We employ the

ex-perimentally observed splitting in the TRDR spectrum 共Fig.

3兲. This splitting can be described in terms of an

exciton-photon coupling—inducing a QD-polariton splitting—with energy ប␻c, and is a direct measure of the mixing between

the QD-exciton and photon states. This allows us to derive the longitudinal-transverse splitting ប␻LT in QDs,34,35 ␻c

=

冑

␻₀␻LT/2. From the polariton splitting of the ground state

transition, 2ប␻c= 27 meV, an effective LT-splitting of 0.33

meV is determined. Finally, we obtain the phenomenological parameter ␻_QD= 3␻c. We emphasize, the ratio ␻LT/␻0⬇3

⫻10−4 _{is in perfect agreement with the value typically}

ob-served for semiconductors.34

Applying Eq. 共6兲 to our experimental results, the 12 times enhancement of the exciton lifetime at the center of the QD-size distribution—deduced from the data depicted in Fig. 1—corresponds to an average distance of d_QDres = 490⫾20 nm between the radiatively coupled QD-excitons. We note, that the exciton lifetime at the wings of the distribution are thus governed by an even larger separa-tion of the mutually interacting QDs. These QDs have a weaker coupling strength and therefore a smaller lifetime enhancement.

Our observation of a long-range electromagnetic interac-tion between QD-excitons, implies that photonic lattices formed by QDs共Refs.34and36兲 are promising systems for

the development of future nanophotonic devices37 such as a quantum processor.38Because long-range interaction mecha-nisms in which the interaction is not limited to the nearest neighbors, are essential for building a scalable quantum computer.39 We expect that our results will open intriguing perspectives for the emerging fields of quantum logics, in which the photonic lattice can be regarded as a quantum register with each QD-exciton on a lattice site acting as a qubit.

V. SUMMARY

In summary, electromagnetic interaction between distant QDs is observed from transient differential reflectivity mea-surements. The QD-exciton lifetime is measured as a func-tion of the transifunc-tion energy and shows a strong resonantlike behavior with respect to the QD DOS. The obtained lifetime spectrum reveals a 12 times lifetime enhancement at the cen-ter of the ground state energy distribution, due to the collec-tive effect of electromagneticly coupled QDs.

ACKNOWLEDGMENTS

The authors thank R. Nötzel, G. Ya. Slepyan, and S. A. Maksimenko for fruitful discussions. This work is financially supported by the Stichting voor Fundamenteel Onderzoek der Materie共FOM兲.

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