Model of fluorescence intermittency of single colloidal semiconductor quantum dots using multiple recombination centers
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Model of fluorescence intermittency of single colloidal semiconductor quantum dotsusing multiple recombination centers
Pavel A. Frantsuzov, S´andor Volk´an-Kacs´o and Bolizs´ar Jank´o
Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA (Dated: November 29, 2018)We present a new physical model resolving a long-standing mystery of the power-law distributionsof the blinking times in single colloidal quantum dot fluorescence. The model considers the non-radiative relaxation of the exciton through multiple recombination centers. Each center is allowed toswitch between two quasi-stationary states. We point out that the conventional threshold analysismethod used to extract the exponents of the distributions for the on-times and off-times has a seriousflaw: The qualitative properties of the distributions strongly depend on the threshold value chosenfor separating the on and off states. Our new model explains naturally this threshold dependence,as well as other key experimental features of the single quantum dot fluorescence trajectories, suchas the power-law power spectrum (1/f noise).
Substantial progress has been made recently in thestudy of long range correlations in the fluctuations ofthe emission intensity (blinking) in single colloidal semi-conductor nanocrystals (QD) [1, 2, 3, 4, 5], nanorods[6], nanowires[7] and some organic molecules[8]. By in-troducing an intensity threshold level to separate bright(on) and dark (off) states, Kuno et al. [2] found that theon- and off-time distributions in QDs exhibit a spectac-ular power-law dependence over 5-6 orders of magnitudein time. p on/off( t ) ∼ t − m (1)As discovered later by Shimizu et al. [3], the power-law on-time distribution is cut off at times ranging froma few seconds to 100 s, depending on the dot and itsenvironment. During the past eight years or so thetruncated power-law form of the blinking on-time dis-tributions was confirmed by many experimental groups(see [4, 5, 9, 10] and references therein), but its micro-scopic origin remains a mystery. Remarkably, there areno ”set” values for the on-time and off-time exponents.They are scattered in the region from 1.2 to 2.0. Similaron- and off-time distributions were found recently for theother blinking systems mentioned above: semiconductornanorods(NRs)[6, 11], nanowires (NWs)[7] and organicdyes [8]. The generality of the phenomenon is ratherintriguing. We argued that there must be a commonunderlying mechanism responsible for the long time cor-related fluorescence intermittency detected in all thesesystems[12]. Most theoretical explanations of the QDblinking [2, 3, 13, 14, 15] are based on the Efros/Rosencharging mechanism [16]. The mechanism attributes on-and off- states to a neutral and a charged QD, respec-tively. The light-induced electronic excitation in thecharged QD is quenched by a fast Auger recombinationprocess. A number of experimental results indicate, how-ever, that there are no unique bright (on) and dark (off)states of the QD, but a continuous set of emission intensi-ties [17, 18, 19]. One can therefore suggest an alternativemechanism of the blinking, assuming slow fluctuations in the non-radiative recombination rate of the excited state[20, 21, 22].In order to gain further insight into the possible blink-ing mechanism, we performed an extensive analysis ofthe on- and off- distributions of actual single QD fluores-cence trajectories. Our procedure is different from theconventional ones, as we applied the Maximum Likeli-hood Estimator (MLE) method to find the best Gamma-distribution p ( t ) ∼ t − m exp( − t/T ) fit for the set of onand off durations. The MLE approach [23] gives an un-biased estimation for the parameters of the power-lawdistribution with minimal statistical error. These prop-erties are crucial and allow for the investigation of a sin-gle trajectory. Our approach, in contrast with procedureused by Hoogenboom et al. [23], allows us to find opti-mal values for not only for m , but for the truncation time T as well. The fluorescence trajectories we investigatedwere obtained by Protasenko and Kuno and have alreadybeen analyzed by others [24, 25].Our fitting procedure is performed repeatedly for anumber of threshold values for each trajectory. In all thecases the off-time distribution truncation time is foundtoo long to be detected. Also, the threshold depen-dence of the distributions was all but ignored until now.The only exception is the recent observation made onnanorods (not QDs) by Drndic and her coworkers [11]. Inany case, the fundamental nature of this dependence wasnot revealed until now. An example of the threshold de-pendency of the power law exponent (slope on log-scale)and on- truncation time for a singe QD trajectory is pre-sented in Fig 1. While we investigated a large number oftrajectories, we have deliberately chosen for this paperone with clearly visible telegraph noise-like features andwell-defined on and off maxima in the intensity histogram[see inset in Fig. 1(b)]. As it is evident from Fig 1, evenfor this apparently ideal case, the distribution parame-ters are strongly threshold dependent. While the ma-jority of the analyzed trajectories are not like telegraphnoise, we mention that all show similar threshold depen-dence. The on-time truncation time decreases monoton-ically with increasing of the threshold. This trend is thesame for most single QD fluorescence trajectories we an-alyzed. The scaling of the slope as a function of thethreshold is more complicated. The exponent of the off-time distribution shows several extrema, whereas the on-time exponent has a minimum as the threshold value isvaried. We wish to emphasize that dependence of on- andoff-time exponents on threshold can qualitatively changefrom one trajectory to another.
20 40 60 80 100 120
Threshold (counts / 10 ms) E xpon e n t a
20 40 60 80 100 120
Threshold (counts / 10 ms) T r un ca ti on ti m e ( s ) I n t e n s it y ( c oun t s / m s ) b FIG. 1: (color online) The threshold dependence of the on-time (red squares) and off-time (blue circles) distribution ex-ponents (a) and on-time distribution truncation time (b) ob-tained from the experimentally measured single QD fluores-cence trajectory. Error bars show the standard deviation val-ues. Insert b: a part of the trajectory and an intensity his-togram.
We interpret this strong dependence on threshold asa clear indication that the standard trajectory analysis -based on the separation between on and off events with asomewhat arbitrary threshold - is not quite adequate, andthe trajectories should be analyzed over the full range ofthreshold parameter. It also explains wide distributionof the exponents found by different groups. As shownbelow, one of the key results of this paper is to exploitthe threshold dependence of the trajectory parameters toretrieve important information about the physical mech-anism of the fluctuations.The power spectrum of the fluorescence trajectory of a single QD has a power law form [26, 27] S I ( ω ) ∼ ω − l where l is close to 1. Therefore, we can consider theQD blinking process as an example of single particle 1/f(flicker) noise. The generally accepted phenomenologicalmodel for the electrical 1/f noise generation in solids isthat of electrical transport in the presence of an environ-ment consisting of multiple stochastic two-level systems(TLS)[28, 29]. In the case of QD blinking we suggest asimilar physical model based on a TLS environment [30].In our model the non-radiative relaxation of the QDexcitation occurs via trapping of holes to one of the N quenching centers, followed by a non-radiative recom-bination with the remaining electron. Each of thesequenching centers could be dynamically switched be-tween inactive and active conformations. The two confor-mational states differ in their ability to trap holes: thehole trapping rate is much larger in the active confor-mation than it is in the inactive state. Recent studiesof trapping rates in the single QDs [32] showed that thenumber of hole traps on the surface and on the core/shellinterface is in order of 10. Interestingly, we find that weonly need a similar number of recombination centers inorder to reproduce the basic features of the fluorescencetrajectories. A possible microscopic origin of the con-formation change in the recombination center could bedue to the light-induced jumps of the surface or inter-face atom between two quasi-stable positions. The sur-face atoms in such a small object as colloidal QD can befound in a variety of local crystal configurations. Conse-quently, we can expect a wide distribution of switchingrates. The non-radiative trapping rate in our model cantherefore be expressed as k t ( t ) = N X i =1 k i σ i ( t ) + k . (2)For each TLS the stochastic variable σ i ( t ) randomlyjumps between two values 0 and 1, corresponding to inac-tive and active conformations, respectively. Furthermore, k i is the trapping rate in the active configuration, and k is the background non-radiative relaxation rate. Thetime distribution functions for the σ i = 0 → σ i = 1 → γ + i and γ − i , respectively. While in the simplest modelthe transition rates for the individual TLS are constants(non-interacting TLSs), we will show that a more generalcase of the interacting TLS systems must also be consid-ered. The power spectral density of the process (2) withinthe non-interacting TLS model is a sum of Lorentzians S k ( ω ) = 1 π N X i =1 γ + i γ − i γ + i + γ − i k i ω + ( γ + i + γ − i ) . (3)The number of parameters in the above expression canbe drastically reduced if the experimental constraint of1 /f noise spectrum is imposed. Indeed, after choosing k i = k and γ + i = γ − i = γ i ≡ γ a i , where a ≪
1, onecan effectively fit the spectrum in Eq. (3) with 1 /ω inthe frequency region γ N ≪ ω ≪ γ [28]. Assuming lowexcitation intensities and steady-state conditions for thefermionic degrees of freedom, the quantum yield Y ( t ) isgiven by[20]: Y ( t ) = k r k r + k t ( t ) , (4)where k r is the radiative relaxation rate.Let us now show that our suggested model of fluores-cence fluctuations exhibits strong threshold dependenceof the on- and off-time distribution parameters. Theproblem of finding these distributions for the stochasticprocess Y ( t ) with known properties and threshold value y is equivalent to the well-known crossing problem [33].There are only few cases when this problem can be solvedexactly [34]. Fortunately, our present model can be re-duced to such an exactly solvable case. The system at anymoment t could be completely described by the configu-ration Σ = { σ , . . . , σ N } . Clearly, there are 2 N differentconfigurations. A random walk in the given configurationspace is a Markovian stochastic process. The vector ~P containing probabilities of all configurations P Σ satisfiesthe Master equation ddt ~P ( t ) = ˆ W ~P ( t ) (5)where the transition matrix ˆ W contains the followingnonzero elements W Σ + i Σ − i = γ + i , W Σ − i Σ + i = γ − i , W ΣΣ = − X Σ ′ =Σ W Σ ′ Σ , where Σ − i = { σ , . . . , σ i = 0 , . . . , σ N } and Σ + i = { σ , . . . , σ i = 1 , . . . , σ N } for each given Σ. The non-radiative relaxation rate for given configuration Σ can beexpressed by Eq. (2), which allows us to find the corre-sponding emission intensity level Y Σ from Eq. (4). Let usintroduce a threshold value for the quantum yield y . Bydefinition, the QD is in the bright (on) state if Y ( t ) ≥ y and dark (off) state otherwise. For each threshold level y all configurations can be separated to a bright group, sat-isfying a condition Y Σ ≥ y and a dark group. The vectorof probabilities can be presented in the form ~P = (cid:18) ~P b ~P d (cid:19) ,where vectors ~P b and ~P d contain probabilities of brightand dark configurations, respectively. The transition ma-trix can be recast in block form ˆ W = (cid:18) ˆ W bb ˆ W bd ˆ W db ˆ W dd (cid:19) . Theexpressions for the normalized on-time and off-time dis-tribution functions in this notations are well-known [35] p on( t ) = D ~ , ˆ W db exp( ˆ W bb t ) ˆ W bd ~P e E D ~ , ˆ W bd ~P e E − p off( t ) = D ~ , ˆ W bd exp( ˆ W dd t ) ˆ W db ~P e E D ~ , ˆ W db ~P e E − (6) where h ~a,~b i denotes the scalar product, ~ ~P e is the equilibrium probabilities vector, sat-isfying a stationary condition ˆ W ~P e = 0. We found thatthe on-time and off-time distributions generated by Eqs.(6) can be fitted by a power law function (1) (see theinsert in Fig. 2a). Beyond a certain off/on time valuethe power law behavior sharply changes to exponentialasymptotic behavior exp( − t/T ). In our analysis, thisvalue of T is defined as a truncation time. We performedsimulations of on-time and off-time distributions for themodel system of non-interacting TLS. This relativelysimple model reproduces the general trend seen exper-imentally in the truncation times: The on-time trunca-tion decreases and the off-time truncation increases whenthe threshold value goes up.While this simple, non-interacting TLS model is use-ful in illustrating our procedure, it cannot reproduce thethreshold dependence of the exponents. The slope ofthe on-time distribution monotonically increases with thethreshold value, when the off-time exponent has an op-posite trend. In order to make our model more realistic,we introduce interaction between TLS in the simplestmean-field form (similar to Ref.[31]). The interaction ischaracterized by the parameter α , whereas the bias foran individual TLS is parameterized by β : γ ± i = γ i exp ± α N X i =1 ( σ i − / ± β ! (7)Fig. 2 provides the numerical calculation results forthe interacting model with the following parameters: N = 10, γ = 1, a = 10 − / , k r /k = 1, k = 0, α = 0 . β = − .
13. As seen from this figure, the thresh-old dependence of the truncation times keep the sametrend as for noninteracting case. In contrast, the slopesnow show a non-monotonic threshold dependence repro-ducing qualitatively the experimental behavior shown inFig.1a. The insert in Fig. 2b shows that the interactingTLS model is capable of generating the two-maximumintensity distribution seen in Fig 1b. The relative easewith which our simple phenomenological model capturedthe experimental trend gives us hope that the model canbe used to extract interaction parameters for the TLSenvironment. These parameters could provide useful ex-perimental constraints on future microscopic models forthe TLS environment of a variety of systems showingfluorescence intermittency. The model proposed herealso explains recent observations of the non-blinking dots.Furthermore, a similar model can be constructed for thefluorescence intermittency seen in quantum wires. Thedetails for these results will be published elsewhere.In conclusion, the phenomenological model we pro-posed in this paper succeeds in qualitatively explainingthe key experimental facts characterizing long-correlatedfluorescence intensity fluctuations of the single colloidalquantum dots: (1) the truncated power-law distributions
Threshold E xpon e n t Time -12 -10 -8 -6 -4 -2 O n / o ff ti m e d i s t r i bu ti on a onoff Threshold T r un ca ti on ti m e Quantum yield P D F b FIG. 2: (color online) The theoretical threshold dependence ofthe on-time (red squares) and off-time (blue circles) distribu-tion exponents (a) and truncation times (b) for the interact-ing TLS model (7). Insert a: the on- and off-time distributionfunctions at the threshold value y=0.25. Insert b: probabilitydistribution function (PDF) of the quantum yield. for on- and off-times obtained by the commonly usedthreshold procedure; (2) the strong threshold dependenceof the distribution parameters m and T and wide rangeof the the extracted exponents; (3) the 1/f noise form ofthe power spectrum of the intensity fluctuations; (4) thecontinuous distribution of emission intensities and exci-tation lifetimes; (5) the weak temperature dependenceof the fluorescence intermittency due to the light-drivencharacter of the TLS switching process.We would like to thank Dr. Vladimir Protashenko andespecially Professor Masaru Kuno for many useful con-versations and for providing us with high quality exper-imental data. We would also like to acknowledge thesupport of the Institute for Theoretical Sciences, the De-partment of Energy, Basic Energy Sciences, and the Na-tional Science Foundation via the NSF-NIRT grant No.ECS-0609249. [1] M. Nirmal et al. , Nature , 802 (1996). [2] M. Kuno et al. , J.Chem.Phys. , 3117 (2000); ,1028 (2001).[3] K.T. Shimizu et al. Phys.Rev. B , 205316 (2001).[4] D.E. Gomez, M. Califano, and P. Mulvaney, Phys. Chem.Chem. Phys., , 4989 (2006).[5] F. Cichos, C. von Borczyskowski, and M. Orrit, Cur.Opin. Col. Inter. Sci. , 272 (2007).[6] S. Wang et al. , J. Phys. Chem. B , 23221 (2006).[7] V.V. Protasenko, K.L. Hull, and M. Kuno, Adv. Mat. , 2942 (2005).[8] J.P. Hoogenboom et al. , ChemPhysChem. , 823 (2007).[9] F.D. Stefani, J.P. Hoogenboom, and E. Barkai, PhysicsToday , No. 2, 34 (2009).[10] J.J. Peterson and D.J. Nesbitt, Nano Lett. , 338 (2009).[11] S. Wang et al. , Nano Lett. , 4020 (2008).[12] P. Frantsuzov, M. Kuno, B. Janko, and R.A. Marcus,Nature Physics , 519 (2008).[13] R. Verberk, A.M. van Oijen, and M. Orrit, Phys.Rev. B , 233202 (2002).[14] G. Margolin and E. Barkai, J. Chem. Phys. , 1566(2004).[15] J. Tang and R.A. Marcus, Phys. Rev. Lett. , 107401(2005).[16] Al.L. Efros and M. Rosen, Phys. Rev. Lett. , 1110(1997); Al.L. Efros, Nature materials , 612 (2008).[17] G. Schlegel, J. Bohnenberger, I. Potapova, and A. Mews,Phys.Rev.Lett. , 137401 (2002).[18] B.R. Fisher et al. , J.Phys.Chem. B , 143 (2004).[19] K. Zhang et al. , Nano Lett. , 843 (2006).[20] P.A. Frantsuzov and R.A. Marcus, Phys. Rev. B ,155321 (2005).[21] N.I. Hammer et al. , J. Phys. Chem. B , 14167 (2006).[22] S.J. Park et al. , Chem. Phys. , 169 (2007).[23] J.P. Hoogenboom, V.K. den Otter, and H.L. Offerhaus,J. Chem. Phys. , 204713 (2006).[24] G. Margolin et al. , Adv. Chem. Phys. , 327 (2006).[25] S. Bianco, P. Grigolini, and P. Paradisi, J. Chem. Phys. , 174704 (2005).[26] M. Pelton, D.G. Grier, and P. Guyot-Sionnest, Appl.Phys. Lett. , 819 (2004).[27] M. Pelton, G. Smith, N.F. Scherer, and R.A. Marcus,Proc. Natl. Acad. Sci. , 14249 (2007)[28] S. Kogan, Electronic Noise and Fluctuations in Solids(Cambridge University Press, England) (1996).[29] M.B. Weissman, Rev. Mod. Phys. , 537 (1988).[30] Note, that another QD blinking model containing mul-tiple TLS was suggested recently in different context:Bianco et al. [31] considered a model of equivelent in-teracting TLS-s (clocks) without specifing a molecularmechanism. The model generates intermittent behaviorwith power-law distribution function ( m = 1 . et al. , Physica A , 1387 (2008).[32] M. Jones, S.S. Lo, and G.D. Scholes, Proc. Natl. Acad.Sci. U.S.A. , 3011 (2009).[33] R.L. Stratonovich, Topics in the theory of random noise,Vol 2. (Gordon and Beach, New York) 1967.[34] S. Majumdar, Curr. Sci. , 370 (1999).[35] D.R. Fredkin and J.A. Rice, J. Appl. Prob.23