Kinetics of Photoluminescence Decay of Colloidal Quantum Dots

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C: Physical Processes in Nanomaterials and Nanostructures

Kinetics of Photoluminescence Decay of Colloidal Quantum Dots: Non-Exponential Behavior and Detrapping of Charge Carriers Evgeny N. Bodunov, and Ana Luisa Simões Gamboa J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02779 • Publication Date (Web): 23 Apr 2018 Downloaded from http://pubs.acs.org on April 23, 2018

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The Journal of Physical Chemistry

Kinetics of Photoluminescence Decay of Colloidal Quantum Dots: Non-Exponential Behavior and Detrapping of Charge Carriers E. N. Bodunov1,* and A. L. Simões Gamboa2,* 1

Department of Physics, Emperor Alexander I St. Petersburg State Transport University, St.

Petersburg, 190031 Russia 2

International Research and Education Centre for Physics of Nanostructures, ITMO University, St.

Petersburg, 197101 Russia

Corresponding Authors *E-mail: [email protected], [email protected]

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ABSTRACT: Understanding the physics behind the kinetics of photoluminescence decay of colloidal quantum dots is of critical importance for the characterization and potential applications of these materials. The non-exponential decays typically observed present a challenge due to a lack of models for their description that are simultaneously physically meaningful and practical to use. In this work, a new function for the description of the whole photoluminescence decay curves of drop-cast films of CdSe/ZnS quantum dots at room temperature is proposed whose parameters have a straightforward physical meaning, accounting for the long-time tails of the decays and highlighting the role of the detrapping of charge carriers. This function and the interpretation of the photoluminescence decay it provides represent an alternative to the widespread assumption that Förster Resonance Energy Transfer takes place in systems of densely packed nearly monodisperse colloidal quantum dots.


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Introduction Pioneer work on semiconductor nanocrystals dispersed in glass1,2 and in a liquid phase3,4 has led to intense research in the resulting new field of colloidal quantum dots (QDs) over the past thirty-five years, as the optical properties due to size quantization effects in these nanocrystals5-7 make them potentially interesting materials for applications in sensing,8,9 imaging,10,11 light harvesting and photovoltaics,12,13 lasing,12,14 and optoelectronics.8,13,15,16 The relaxation processes that shape the decay of band-edge excitons in QDs, and namely in systems of densely packed QDs, are the critical processes that determine the suitability of these materials for a potential application;6,7 understanding the kinetics of photoluminescence (PL) decay of QDs, and in particular of systems of densely packed QDs, is therefore of critical importance for their characterization and synthetic development. However, one of the challenges posed by the behavior of QDs to experimenters and theorists alike is precisely the interpretation of the decay kinetics of these nanocrystals, which is typically non-exponential as reported for a variety of systems: colloidal solutions (nearly monodisperse QDs,17-19 electrostatically bound QD structures,20 doped QDs,21 QDs with various molecular species attached to the nanocrystal surface22-24), QDs dispersed in a gelatin matrix,25 drop-cast films (nearly monodisperse QDs,19,26 mixtures of QDs,26 QDs encapsulated in a polymer matrix27), Langmuir-Blodgett28,29 and Langmuir-Schaefer28 QD films on various substrates, QDs embedded in photonic crystals,30 in a porous medium,31 and incorporated into hollow polymer microspheres,16 for example. The analysis of these non-exponential decays is challenging due to a lack of models for their description that are simultaneously physically meaningful and practical to use. Time-resolved luminescence measurements are typically analyzed resolving the luminescence decay profiles in terms of discrete exponential components with distinct lifetimes, and QD systems are no exception. Sums of exponentials, however, have physical meaning only in the simplest cases and, otherwise, the choice of the number of exponentials is to a great extent arbitrary.32,33 3 ACS Paragon Plus Environment


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In QD systems, the complex PL decay profiles resulting from interactions between the nanocrystals and with their environment may be the signature of an infinite (continuous or discrete) distribution of rate constants. It has been attempted to recover the underlying probability distribution function of rate constants from the corresponding integral equation for QD PL decays.34 However, this task is tricky because it is an ill-posed problem.33 Another approach consists in choosing a known mathematical function as the probability distribution function of rate constants. Care must be exercised in this case as well since, as in the case of the multiple exponentials mentioned above, the chosen function may approximate the QD PL decay well but have no physical significance for the system (log-normal distribution,30 for example). The QD PL decay kinetics have also been fitted using a stochastic kinetic model that considers a Poisson distribution of quenchers23,35 on the surface of each QD (unidentified surface traps, dyes or metal nanoparticles attached to the nanocrystal surface)36-38 based on an analogy with the kinetic model of luminescence quenching in micellar systems originally proposed by Infelta and Grätzel in 1974.39-41 A function that has often been used42-45 as a convenient mathematical description of QD PL decays is the Kohlrausch or stretched exponential function

( )

Φ (t ) = exp  − t τ 

β

,  

(1)

where 0 < β ≤ 1 and τ has the dimensions of time.33 The stretched exponential function has firm grounds in the description of the luminescence intensity decay I(t) in condensed matter46 in the following form, firstly derived by Förster from realistic models of luminescence quenching,47,48 β  t t   t I (t ) = exp  − − a    = exp  −  τ  τ    τ 

  I rel ( t ) , 


(2)

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where the term exp(-t/τ) describes the exponential part of the kinetics, determined by the intrinsic radiative and non-radiative transitions in the species under study, and the second term,

  t β  I rel ( t ) = exp  −a    ,  τ    

(3)

accounts for the relaxation part, shaped by Förster Resonance Energy Transfer (FRET) or FRETtype mechanisms. In eq 2, τ is the lifetime of the excited state of the donor in the absence of the quencher and the parameter a is proportional to the concentration of the acceptor and depends on the Förster radius. Recently, it has been shown that data that can be satisfactorily described by eq 1 may be equally well described by eq 2 within the precision limits of the measurements,49 with smaller values of the parameter β. The use of eq 2 for fitting the experimental non-exponential PL decay of QD systems at room temperature allows resolving the decay into the exponential component exp(-t/τ), characterizing the QD intrinsic radiative and non-radiative processes, and the non-exponential component, due to additional channels of energy relaxation. Furthermore, it can give valuable indication of the mechanism of PL quenching in the QD system through the value of the stretching parameter β: long-range resonance energy transfer to energy-acceptor species in the environment of the QDs which act as energy donors (β ≤ 0.5) or contact quenching mechanisms (FRET with Förster radius smaller than or equal to the minimal distance between donors and acceptors, exchange energy transfer, or charge transfer) by quenchers distributed according to Poisson statistics on the surface of the dots (β > 0.5). For β > 0.5 it has also been shown that a decay that is well approximated by the fit using eq 2 can be equally well described by a stochastic kinetic model that considers a Poisson distribution of identical quenchers on the surface of a given QD.49 According to the well-established description of the exciton structure in the most widely studied CdSe QDs,50-52 the PL decay of an ensemble of nearly monodisperse QDs at room temperature should be single-exponential, exp(-t/τ), with τ equal to twice the radiative lifetime of the QD bright 5 ACS Paragon Plus Environment


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state.12,50 However, this is seldom observed experimentally. The data are typically approximated by a sum of exponentials26 or fit to a single-exponential53 ignoring the long-time tails of the decays. On the other hand, recently, the room temperature PL decay of trioctylphosphine oxide (TOPO)-capped CdSe/ZnS QDs in solution has been approximated using eq 2 with β = 0.5 and this value interpreted as the result of FRET from the QDs (energy donors) to high-frequency anharmonic vibrations (energy acceptors) of the randomly distributed -CH groups of the TOPO molecules.54 In the present work, the interpretation of the room temperature PL decay of a densely packed ensemble of QDs has been revisited. The experimental PL decay of nearly monodisperse TOPOcapped CdSe/ZnS QDs in drop-cast films has been analyzed with the purpose of finding a function that is able to express the physical meaning of the shape of the whole decay curves (that is, without ignoring the long-time tails), with a minimum number of free parameters.

Experimental Section TOPO-capped CdSe/ZnS QDs (Research Institute for Physicochemical Problems, Belarusian State University10) of mean core diameter dQD ≈ 2.5 nm were drop-cast on glass from colloidal solutions in toluene. The QDs in the drop-cast films were characterized by absorption and steady-state luminescence spectroscopy using a Shimadzu UV Probe 3600 spectrophotometer and a Zeiss confocal laser scanning microscope LSM 710 operating at a wavelength of 405 nm, respectively (Figure 1). The optical density of the QD films was less than 0.3 at the first excitonic peak in order to minimize reabsorption and the emission peak was at 2.31 eV with a FWHM of 6% (0.14 eV). Room temperature PL decays were measured by time-correlated single photon counting (TCSPC) using a MicroTime100 PicoQuant spectrometer with a pulsed diode laser operating at a wavelength of 405 nm, average power 1 mW, repetition rate of 40 MHz, and pulse width of 70 ps. The detection energy was varied in the range 2.25-2.38 eV (520-550 nm).


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0.3

0.2

I (a. u.)

Absorbance (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.1

0.0 2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

Energy (eV)

Figure 1. Room temperature absorption (blue line) and emission (red line) spectra from a dense film of nearly monodisperse (6% size dispersion) TOPO-capped CdSe/ZnS QDs (mean core diameter dQD ≈ 2.5 nm) drop-cast from toluene.

Results and Discussion The spectrally resolved PL decays (Figure 2) show a progressively faster decay with increasing detection energy, which is considered to be a fingerprint of the quenching of the PL of the smaller QDs (larger band gap) by the larger QDs (smaller band gap) in the ensemble. This behavior of densely packed nearly monodisperse QDs has been equated in the literature with electronic energy transfer from the QDs acting as energy donors to the neighboring resonant QDs acting as energy acceptors, but evidence supported by a detailed analysis of the kinetics of the PL decay based on accepted models of luminescence quenching has not been provided.26,28,55 The experimental decays in this work can be well approximated by a fit using a sum of three exponentials (with time constants of the order of 100 ns, 10 ns, and 2 ns). However, as discussed previously, this function is not supported by a physical model. Then, it was attempted to fit the experimental data to the theoretical models corresponding to the luminescence quenching 7 ACS Paragon Plus Environment


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mechanisms mentioned above in the Introduction (FRET and contact quenching). If the decays reflect electronic energy transfer within the sample inhomogeneous distribution, a successful fit to eq 2 with β ≤ 0.5 should be obtained. However, this was not verified. Further, it was found that the fit to a stochastic kinetic model that considers a Poisson distribution of identical quenchers on the surface of a given QD,49  t  t    I (t ) = exp  − − N  1 − exp  − K    , τ      τ

(4)

where N is the average number of quenchers per QD and K the dimensionless quenching rate of the luminescence of a QD by one quencher, is not satisfactory either. These findings show that FRET or contact quenching mechanisms in the ensemble of densely packed QDs cannot explain the shape of the PL decays obtained. In order to analyze the data, a theoretical model that takes trapping and detrapping of excited charges into account is proposed here. The long-time tails of QD PL decays at room temperature have been associated in the literature with charge detrapping processes which are also assumed to be related to the PL intermittency (blinking) observed in constant wave excitation over significantly long times.56,57 However, these processes have not been translated into a function for analyzing the decays that is physically meaningful and practical to use for the experimenter.


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1

I (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.1

2.25 eV

0.01 2.38 eV

0

10

20

30

40

50

60

70

80

90

t (ns)

Figure 2. Spectrally resolved room temperature photoluminescence dynamics of a dense film of TOPO-capped CdSe/ZnS QDs drop-cast from toluene. The intensity I is normalized to the maximum number of counts. The detection energy is varied in the range 2.25 - 2.38 eV. The experimental decays (open circles) were approximated using eq 13 (solid lines) with τ = 2τB = 30 ns.

In the model presented here for an ensemble of densely packed QDs, a QD at room temperature is regarded as a two-level system (a ground state and an excited state, the latter corresponding to the QD bright state) in which the excited state decays radiatively with rate 1/τ (with τ equal to twice the intrinsic radiative lifetime of the bright state, τB).12,50 Let the QD be surrounded by N identical traps (electron or hole traps), each of which can trap an electron (or hole) with a finite trapping rate, k1, considered to be the same for all traps. This assumption is made to limit the number of variables in the model. Let it be assumed that the total trapping rate is large, that is, Nk1 >> 1/τ, and that each trap can release the trapped electron (or hole) back to the QD with rate k2 (k1>k2). Let it also be assumed that the decay rate of a trapped electron (or hole) within the trap, 1/τtrap, is negligible when compared to k2, that is, 1/ τtrap > 1/τ, the coefficients A and B practically do not depend on the number of traps per QD and are approximately given by A≈

k1 k2 . ,B≈ k1 + k 2 k1 + k 2

(9)

Besides, for Nk1 >> 1/τ, in eq 7 the first term in the first exponential and the second exponential also do not depend on the number of traps per QD. Let it be assumed that the number of traps, N, is different for each QD in the ensemble and that the distribution of traps is governed by a Poisson distribution,

NN f (N ) = exp ( − N ) , N!

(10)

where N is the average number of traps per QD in the ensemble. Averaging the decay kinetics of the excited state of the QDs, given by eq 7, with the distribution function given by eq 10 yields the normalized kinetics of PL decay of the QD ensemble 10 ACS Paragon Plus Environment

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I (t ) =

 t k1 k1 exp  − − N 1 − exp ( − ( k1 + k2 ) t ) k1 + k2  τ k1 + k2

(



) + k k+ k 

2

1

 t k2  exp  − . 2  τ k1 + k2 

(11)

In order to approximate the experimental PL decay kinetics it is convenient to work with the dimensionless rates K1 = k1τ and K2 = k2τ. Then eq 11 can be rewritten as

I (t ) =

 t K1  t K2   K1 t   K2  − N 1 − exp  − ( K1 + K 2 )    + exp  − exp  − . K1 + K 2 τ    K1 + K 2    τ K1 + K 2   τ K1 + K 2

(12)

The first term in eq 12 describes the nonexponential component of the PL decay, which is determined by the trapping of electrons (or holes) and the concomitant establishment of equilibrium between the populations of the electron (or hole) state and the trap state. This term decays fast, with a rate of the order of magnitude of N (K1 + K2)/τ. The second term describes the exponential component of the PL decay, governed by the detrapping of the electrons (or holes). This is a slow process, with a rate less than 1/τ, as K2 < K1. It is convenient to use τ, N , K0 = K1 + K2, and K2 as fitting parameters. For these parameters, eq 12 takes the form

I (t ) =

 t K0 − K2  t K2  K0 − K2 t   K   − N 1 − exp  − K 0    + 2 exp  − exp  − . K0 τ   K0    τ K0   τ K0

(13)

Figure 2 shows that the experimental PL decays are well described by eq 13 with τ = 30 ns (which gives a radiative lifetime of the QD bright state τB = 15 ns, in good agreement with the values predicted theoretically for CdSe QDs50 and determined experimentally27,58) and N and K1 increasing with increasing detection energy (Figure 3). The model proposed is therefore consistent with the experimental data.



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3.6

4.2

3.2

4.0

2.8

N

3.8 2.4 3.6

K1

2.0

3.4

1.6 2.24

2.28

2.32

2.36

2.40

PL detection energy (eV)

Figure 3. Dependence of the average number of traps per QD in the ensemble, N (black squares) and the dimensionless trapping rate of an electron (or hole), K1 (open circles) on the PL detection energy.

Concerning the nature of the traps, these might be traps "intrinsic" to a given QD, due to structural defects and dangling bonds.6 The number of traps intrinsic to a given QD should increase with decreasing QD size. This is due to the concomitant increase of the surface-to-volume ratio, which leads to (i) less effective packing of the ligands on the QD surface (and therefore less effective passivation of this surface) and (ii) more defects in the CdSe/ZnS core/shell boundary, due to lattice mismatch. It can be hypothesized that the higher density of trap states in the "matrix" of densely packed QDs in the drop-cast films might act to enhance the trapping probability, leading to the slightly increase of N with detection energy experimentally observed. Alternatively, the traps might be the QDs of smaller band gap (larger size) surrounding a given QD of larger band gap (smaller size) in the inhomogeneous ensemble (besides size, the ensemble may be also inhomogeneous in respect to crystal shape51). This is also in agreement with the observation that N slightly increases with detection energy, consistent with a picture where the 12 ACS Paragon Plus Environment

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number of nearest neighbors of smaller band gap surrounding a given QD is slightly larger for the QDs of larger band gap. An excited electron (or hole) may be trapped by a nearest neighbor QD of smaller band gap, giving rise to a long-lived charged state. The possible reason for this might be preferential interactions developing within the environment created by the densely packed QDs and leading to electron tunneling within this QD "matrix". It can also be assumed that the most probable pathway for electron-hole recombination is for the electron (or hole) to return, thanks to thermal excitation, to the original QD of larger band gap. The slower components of the PL decays are then expected to be controlled by thermal detrapping of charge carriers from the QD "matrix" surrounding a particular QD. Since the model presupposes that thermal equilibrium is attained between the populations of the electron (or hole) state and the trap state, it is possible to estimate the energy difference between the electron (or hole) and trap state levels, ∆Etrap, by the relation

 ∆Etrap K2 = exp  − K1  k BT

 , 

(14)

where kB is the Boltzmann constant and T is the temperature, using the parameters K1 and K2 from the fit of the PL decays to eq 13. This gives ∆Etrap ≈ 0.12-0.15 eV, increasing with PL detection energy. These values are comparable to the FWHM of the PL spectrum of the QD ensemble due to the size distribution in the ensemble, 0.14 eV. This means that, for all the four detection energies investigated, the QDs have as nearest neighbors other QDs of smaller band gap (larger size) that can serve as charge traps for the former.

Conclusions To conclude, in this work, the interpretation of QD PL decay in a densely packed ensemble has been revisited. Attempts to approximate the whole PL decay curves of drop-cast films of CdSe/ZnS QDs at room temperature to the kinetics describing FRET-type or contact quenching mechanisms have 13 ACS Paragon Plus Environment


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not proved successful. A new function for approximating the PL decay curves is introduced whose parameters have a straightforward physical meaning, accounting for the long-time tails of the decays and highlighting the role of the detrapping of charge carriers. The function describes the timedependence of the whole PL decay, with four adjustable parameters, one of which (τ) may be estimated theoretically and determined experimentally independently.

Acknowledgement This work was partially supported by the Ministry of Education and Science of the Russian Federation (Grant No. 14.Y26.31.0028).

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Kinetics of Photoluminescence Decay of Colloidal Quantum Dots

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