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J. Chem. Phys. 151, 174710 (2019); https://doi.org/10.1063/1.5124913 151, 174710
© 2019 Author(s).
Contribution of electron-phonon coupling
to the luminescence spectra of single
colloidal quantum dots
Cite as: J. Chem. Phys. 151, 174710 (2019); https://doi.org/10.1063/1.5124913
Submitted: 20 August 2019 . Accepted: 21 October 2019 . Published Online: 07 November 2019
Eduard A. Podshivaylov, Maria A. Kniazeva, Aleksei A. Gorshelev, Ivan Yu. Eremchev, Andrei V.
Naumov, and Pavel A. Frantsuzov
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
Contribution of electron-phonon coupling
to the luminescence spectra of single colloidal
quantum dots
Cite as: J. Chem. Phys. 151, 174710 (2019); doi: 10.1063/1.5124913
Submitted: 20 August 2019 •Accepted: 21 October 2019 •
Published Online: 7 November 2019
Eduard A. Podshivaylov,1Maria A. Kniazeva,1Aleksei A. Gorshelev,2Ivan Yu. Eremchev,2,a) Andrei V. Naumov,2,3
and Pavel A. Frantsuzov1,4,b)
AFFILIATIONS
1Lomonosov Moscow State University, 119991 Moscow, Russia
2Institute of Spectroscopy RAS, 108840 Moscow, Russia
3Moscow State Pedagogical University, 119991 Moscow, Russia
4Voevodsky Institute of Chemical Kinetics and Combustion SB RAS, 630090 Novosibirsk, Russia
Note: This paper is part of the JCP Special Topic on Colloidal Quantum Dots.
a)eremchev@isan.troitsk.ru
b)pavel.frantsuzov@gmail.com
ABSTRACT
Luminescence spectroscopy experiments were realized for single colloidal quantum dots CdSe/ZnS in a broad temperature range above
room temperature in a nitrogen atmosphere. Broadening and shifts of spectra due to the temperature change as well as due to spectral
diffusion processes were detected and analyzed. A linear correlation between the positions of maxima and the squared linewidths of the
spectra was found. This dependence was explained by a model that takes into account the slow variation of the electron-phonon coupling
strength.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5124913
., s
I. INTRODUCTION
Colloidal semiconductor quantum dots (QDs) are very interest-
ing objects because of their unique optical properties such as a wide
absorption spectrum, a narrow emission line, a size-tunable emis-
sion wavelength, high photostability, and high fluorescence quan-
tum yield. The very first spectroscopic measurements of single CdSe
quantum dot photoluminescence revealed interesting phenomena
such as long-term fluctuations of the emission intensity (blinking)1
and very slow spectral diffusion (SD)2–4 with characteristic time
scales of up to hundreds of seconds. It was shown that at cryo-
genic temperatures, the observed emission spectrum linewidth of
single QDs depends on the signal accumulation time due to spectral
shifts,2,5 while the linewidths and the peak positions of the spectra
are correlated.6
Spectral diffusion at higher temperatures was observed by
Muller et al.7,8 in single CdSe QDs capped by a CdS rodlike shell.
The linewidth and the peak position of the emission spectrum are
found to be correlated at 5 K, 50 K, and room temperature. These
correlations at all temperatures were explained7,8 by the motion of
the net surface charge, which induces a Stark shift of the emis-
sion energy depending on the distance to the CdSe core, while
the spatial jitter of the charge density causes spectral line broad-
ening. Gomez et al.9noted that this hypothesis does not apply to
the spherically symmetric QDs. Besides, it should lead to varia-
tions of the linewidth with a change in the dielectric properties
of the medium. A series of spectroscopic experiments were per-
formed on single spherical QDs spin-coated on top of thin films
of various polymer matrices at room temperature. It was shown9
that there is a correlation between the linewidth and the peak posi-
tion of the emission spectrum in these particles without a signifi-
cant dependence on the dielectric permittivity of the matrix. Based
on this, it was concluded in Ref. 9that the mechanism respon-
sible for the correlated broadening and the peak position shift
J. Chem. Phys. 151, 174710 (2019); doi: 10.1063/1.5124913 151, 174710-1
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of the emission spectra in the PL has to be intrinsic to the QD
core.
Note that the broadenings of single QD emission spectra at
5 K and at room temperature are different in nature. At 5 K,
the zero-phonon line is observed and its width is much smaller
than the longitudinal optical (LO) phonon energy.10 The linewidth
at room temperature becomes greater than the energy of the LO
phonons, which means that the multiphonon nature of the broad-
ening should be taken into account.11,12 While the electron-phonon
coupling and spectral diffusion contributions to the spectra of chro-
mophore molecules in solid matrices have been studied in detail,13–15
the same contributions in QDs are still of much interest. Here, we
present an in-depth experimental and theoretical study of the dis-
cussed spectral characteristics of single colloidal semiconductor QDs
CdSe/ZnS in the context of their feasible relation to electron-phonon
coupling.
II. EXPERIMENT
We performed a set of spectroscopic experiments with single
QDs, including measurements with slow heating and cooling of a
sample.
Fluorescence images and spectra of single quantum dots were
recorded using a home-built fluorescence microscope equipped
with a prism spectrometer.13,16 Two optical schemes—a wide-field
scheme and a scanning confocal one (see Fig. 1)—were combined in
the microscope in order to simplify the procedure of single quan-
tum dot preliminary searching (by using fluorescence image pro-
cessing and antibunching identification) and to perform sequential
measurements of the fluorescence spectra of the selected QD. Quan-
tum dots (CdSe/ZnS from Sigma-Aldrich with the fluorescence peak
at 620 nm) were dispersed in a toluene solution of polyisobuty-
lene of low concentration and then spin coated onto a cover glass.
The thickness of the polymer films with single quantum dots var-
ied within the range of several tens of nanometers. The sample was
placed onto the piezo-driven stage (NanoScan Technology), which
allowed one to move the selected QD to the laser spot position
with high (nanometer) precision. Between the sample and the piezo-
driven stage, a thermoinsulating (fluoroplastic) substrate a few mil-
limeters thick was placed, with a hole in the center allowing the
microscope objective to approach the plane of the sample at the
required distance. The thermoinsulating substrate contained a tem-
perature sensor that had good thermal contact with the sample.
On top of the sample, a three-stage thermoelectric module was
pressed, which was used to heat or cool the sample. This opti-
cal scheme (including the piezo-driven stage with the sample, the
microscope objective, and the thermoelectric module) was mounted
inside a special home-built chamber, allowing measurements both
in a vacuum and in a gas nitrogen/helium atmosphere. In this
particular case, the measurements were performed in a nitrogen
atmosphere. The sample temperature was controlled by using a
LakeShore temperature controller. A tunable dye laser (Coherent
CR599) or solid state laser Coherent Verdi was used to excite quan-
tum dots at the wavelength of 580 nm (near the quantum dot
absorption band edge) or at 532 nm, respectively. The excitation
laser intensity (∼100 W/cm2in a focused spot) was attenuated by
neutral spectral density filters (Standa) and controlled by using a
Newport power meter. A set of interference filters (Semrock and
Thorlabs) was used for the separation of the QD fluorescence sig-
nal from the scattered laser radiation. Two highly sensitive cooled
electron multiplying charge-coupled device (EMCCD) cameras were
utilized to record single quantum dot images (Andor Luca) and
spectra (Andor Ixon Ultra). The Hanbury Brown and Twiss scheme
with broadband 50% splitter (Thorlabs) and two identical single-
photon avalanche diode (SPAD) detectors (EG&G SPCM-200PQ,
time resolution 1.3 ns, dead time 200 ns, QE 65%) was used to
measure the autocorrelation function for QD fluorescence intensity.
Each fluorescence spectrum from a single quantum dot was mea-
sured with an exposure time of 200 ms and a spectral resolution
of 0.7 nm, which was sufficient to achieve a good signal-to-noise
ratio.
FIG. 1. A schematic picture of the experimental setup.
J. Chem. Phys. 151, 174710 (2019); doi: 10.1063/1.5124913 151, 174710-2
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FIG. 2. Spectral traces (left panel) where the peak position is shown by the red line, time dependencies of the normalized PL intensity (central panel), and the linewidth (right
panel) for two different single CdSe/ZnS quantum dots (a) and (b) measured at room temperature.
J. Chem. Phys. 151, 174710 (2019); doi: 10.1063/1.5124913 151, 174710-3
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III. RESULTS
In the experiments at room temperature for each studied sin-
gle QD, we registered 2500–3000 emission spectra with 200 ms
accumulation time. The presence of both blinking and spectral dif-
fusion processes can be clearly seen. Spectral traces for two QDs are
shown in Fig. 2.
Each spectrum was fitted with a Gaussian function
G(ϵ)=G0
√2πσ exp−(ϵ−ϵ0)2
2σ2+b,
whose four parameters were peak emission photon energy ϵ0,
linewidth σ, amplitude G0, and background level b. An example of a
typical spectrum fitting is shown in Fig. 3.
The correlation between the peak energy and the linewidth
was found for all studied QDs. As seen in Fig. 4, the peak energy
dependence of the linewidth squared can be fitted by the linear
function
σ2=αkT(E0−ϵ0), (1)
where Tis the absolute temperature, kis the Boltzmann constant,
E0is the energy gap for particular QD, and the parameter αis
the linear dependence coefficient between the squared line width
and peak energy in the units of kT. The values of αare found
to be in the range from 0.48 to 0.63 for varied QDs at room
temperature.
In order to characterize the shift of the spectra, we found a
squared peak energy displacement of a typical single QD emission
spectrum
FIG. 3. The emission spectrum of a single CdSe/ZnS QD (blue line) and its fit
with a Gaussian (dashed line). The parameters of the fit are ϵ0= 2.029 eV and
σ= 18.3 meV.
D2(τ)=(ϵ0(t)−ϵ0(t+τ))2
as a function of time τfollowing Ref. 17. As can be seen in Fig. 5, the
averaged spectral shift squared D2is less than the σ2of a typical sin-
gle QD spectrum for all delay times, but more importantly, it is much
smaller than linewidth squared when τis equal to the signal accu-
mulation time τ= 200 ms. Thus, we can conclude that the observed
linewidth is not related to the spectral shifts during this time
period.
FIG. 4. The peak energy vs the linewidth
squared for different single QDs: (a)–(d)
at room temperature (black points) and
the linear fit Eq. (1) with T= 300 K (red
lines). The statistical error in the value of
alpha in each fit is less than 0.014.
J. Chem. Phys. 151, 174710 (2019); doi: 10.1063/1.5124913 151, 174710-4
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FIG. 5. Time dependence of D2function for a single QD at room temperature (black
diamonds). The red line is ∼τβ. The value of βis 0.603.
IV. THEORY AND DISCUSSION
Such a large value of the linewidth can be explained by multi-
phonon excitation.11,12 Let us consider the following Hamiltonian of
the QD electronic system interacting with Nphonon modes:
ˆ
H=ˆ
H0+A
N
∑
i=1
ˆ
qi(aiee+bigg), (2)
where
ˆ
H0=N
∑
i=1
ˆ
p2
i
2+ω2
i
2ˆ
q2
i+E0ee, (3)
E0is the energy gap, and |gand |eare the ground state and
excited electronic state of the QD, respectively. The parameter A
characterizes the electron-phonon interaction strength. ˆ
qiand ˆ
pi
are the coordinate and momentum operators of ith phonon mode,
characterized by the frequency ωi. Both the excited state and the
ground state are connected with the photon modes in the model.
The interaction of the excited state and the ground state with the
ith phonon is described by the dimensionless coefficients aiand bi,
correspondingly.
The emission spectrum at a given value of Ahas a Gaussian
form with the following parameters (see details of the derivation in
the Appendix):
ϵ0=E0−A2S
∑
i=1
ai(ai−bi)
ω2
i
, (4)
σ2=kTA2S
∑
i=1
(ai−bi)2
ω2
i
. (5)
Fluctuations of the linewidth within the model are explained by
a slow variation of the parameter A. Such variations of the electron-
phonon interaction were observed experimentally in single colloidal
QDs6,8 as well as in single chromoprotein molecules.18,19
Variations of the parameter Awith time lead to shifts in the
position of the maximum (spectral diffusion) correlated with the
linewidth. Equations (4) and (5) give the linear dependence of Eq. (1)
where
α=N
∑
i=1
ai(ai−bi)
ω2
i−1N
∑
i=1
(ai−bi)2
ω2
i
.
As seen in Fig. 4, the parameter αvaries from one QD to another,
as well as E0. The model predictions are consistent with the exper-
imental results of Gomez et al.9since the electron-phonon inter-
action in QDs is not related to the dielectric properties of the
environment.
In order to check the theory at various temperatures, the spec-
troscopic experiment on one quantum dot was performed with
heating and cooling of the sample. Twenty spectra were measured
sequentially at each selected temperature in the range from 305.5 K
to 353.6 K. It was found that all the data can be well fitted by Eq. (1),
provided that the energy gap E0depends on temperature and αkeeps
constant, as seen in Fig. 6. The E0value proves to decrease with
temperature rise. The changes in the effective band gap presum-
ably occur due to thermal expansion. Importantly, this result does
not depend on the process which led the system to that temperature
(heating or cooling). Note that this dependence of the “pure” energy
gap E0on temperature is not due to the electron-phonon interaction
as is usually considered.20–22
Therefore, the suggested model explains the fluctuations of the
linewidth of a single QD emission at temperatures 300 K and above,
but at the same time, it predicts the spectral shifts correlated with the
linewidth.
What is the mechanism of the variations in the magnitude of
the electron-phonon interaction? The estimate shows that at a given
excitation intensity of 100 W/cm2, the average time between the
absorption of photons of one QD is about 1 μs. Thus, the effect of
multiexciton states can be excluded from consideration. An increase
in the temperature of a single QD after absorption of photons
does not exceed 2 K, as estimated by Kuno et al.,23 while thermal
relaxation is of the order of 10 ps. This means that local heating
can also be excluded from the possible causes of the phenomenon.
FIG. 6. The peak energy vs the linewidth squared for a single QD at various tem-
peratures (points). The solid lines indicate the theoretical prediction at four different
temperatures. Parameter α= 0.63.
J. Chem. Phys. 151, 174710 (2019); doi: 10.1063/1.5124913 151, 174710-5
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Plakhotnik et al.17 showed that the squared energy displacement
of a single QD emission at cryogenic temperatures has an anoma-
lous (sublinear) behavior at short times D2∼τβ, where β<1. It
was explained by introducing a number of stochastic two-level sys-
tems (TLSs) having a wide distribution of flipping rates. The squared
energy displacement calculated with the use of our experimental
data at room temperature also shows a similar sublinear time depen-
dence (Fig. 5). This means that the TLS based model can be used to
describe the fluctuations of the electron-phonon interaction value at
high temperatures as well. A possible microscopic origin of the con-
formation change in the TLS could be due to the jumps of the surface
or interface atom between two quasistable positions.24 Note the gen-
eral interest regarding the microscopic nature of the SD processes
that were observed for most single quantum emitters: single organic
dye molecules,25–27 single light harvesting complexes and proteins,28
color centers in diamonds,29 and single rare-earth ions in crystals.30
In many cases, SD has been attributed to the tunneling processes in
an emitter and/or its local surroundings. At the same time, the rela-
tion between SD and phonon-assisted optical dephasing was always
under discussion.
Empedocles and Bawendi6observed changes in the electron-
phonon interaction parameter upon application of an external elec-
tric field. It can be assumed that the shift of the peak of the spectrum
in an external electric field is partially determined by a change in the
electron phonon interaction. To verify this assumption, additional
experiments could be performed.
Single QD blinking can also be explained within a TLS based
model of Ref. 31. The Multiple Recombination Center (MRC) model
suggested by Frantsuzov et al.31 reproduces the key properties of
single QD blinking, such as the ON and OFF time distribution func-
tions,31 the power spectral density,32 and the long-term correlations
between subsequent blinking times.33 The similarity in temporal
fluctuations of the spectrum and the emission intensity of a single
QD allows one to make the assumption that both phenomena can be
explained by a unified mechanism.34
In conclusion, our experiments show a linear correlation
between the position of the maximum and the linewidth squared of
a single QD emission spectrum at room temperature and above. In
order to explain the experimental results, we consider a model of QD
emission spectrum linewidth fluctuations based on a slow variation
of the electron-phonon interaction. The model was tested using the
data of a unique single QD spectroscopy experiment under heating
and cooling conditions.
ACKNOWLEDGMENTS
This study was supported by the Russian Foundation for Basic
Research, Project No. 16-02-00713. The measurements were car-
ried out under the State Contract of the Institute of Spectroscopy
RAS. The luminescence microscopy technique with detection of
single quantum dots with nanometer spatial resolution is devel-
oped under support of the Russian Science Foundation (Project No.
17-72-20266, head I. Yu. Eremchev).
APPENDIX: DERIVATION OF EQUATIONS (4) AND (5)
Potential energy of the excited electronic state is given by the
following formula:
Ue(q)=N
∑
i=1ω2
i
2q2
i+Aqiai.
In the classical limit
hωi≪kT, the probability distribution function
of the coordinates is given by the Boltzmann distribution,
P(q)=1
Zexp−1
kT
N
∑
i=1ω2
i
2q2
i+Aqiai, (A1)
where Zis the partition function,
Z=∫dq1∫dq2⋯∫dqNexp−1
kT
N
∑
i=1ω2
i
2q2
i+Aqiai.
It is assumed that the thermal relaxation is much faster than the vari-
ations of the parameter A. The energy of the emitted photon at given
values of the phonon coordinates is equal to the difference between
the energies of the excited and ground states,
ϵ=E0+A
N
∑
i=1(ai−bi)qi. (A2)
Equation (A1) is a multidimension Gaussian distribution, and
Eq. (A2) is a linear function of the coordinates qi. Consequently, the
distribution of ϵis also Gaussian,
p(ϵ)=1
√2πσ exp−(ϵ−ϵ0)2
2σ2,
where the parameters ϵ0and σ2can be found by averaging over the
distribution (A1),
ϵ0=¯
ϵ=E0+A
N
∑
i=1(ai−bi)qi, (A3)
σ2=(ϵ−¯
ϵ)2=A2N
∑
i=1(ai−bi)2(qi−¯
qi)2. (A4)
The mean values for the coordinates can be easily found by integra-
tion over distribution (A1)
¯
qi=−Aai
ω2
i
(qi−¯
qi)2=kT
ω2
i
.
Substituting these expressions into Eqs. (A3) and (A4) gives
Eqs. (4) and (5).
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