Intermittency of CsPbBr_3 perovskite quantum dots analyzed by an unbiased statistical analysis
Isabelle M. Palstra, Ilse Maillette de Buy Wenniger, Biplab K. Patra, Erik C. Garnett, A. Femius Koenderink
IIntermittency of CsPbBr perovskite quantumdots analyzed by an unbiased statistical analysis Isabelle M. Palstra, † , ‡ Ilse Maillette de Buy Wenniger, ‡ Biplab K. Patra, ‡ Erik C.Garnett, ‡ and A. Femius Koenderink ∗ , ‡ † Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam,The Netherlands ‡ Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG, Amsterdam TheNetherlands
E-mail: [email protected]
Abstract
We analyze intermittency in intensity and fluorescence lifetime of CsPbBr per-ovskite quantum dots by applying unbiased Bayesian inference analysis methods. Weapply changepoint analysis (CPA) and a Bayesian state clustering algorithm to deter-mine the timing of switching events and the number of states between which switchingoccurs in a statistically unbiased manner, which we have benchmarked particularly toapply to highly multistate emitters. We conclude that perovskite quantum dots displaya plethora of gray states in which brightness broadly speaking correlates inversely withdecay rate, confirming the multiple recombination centers model. We leverage the CPApartitioning analysis to examine aging and memory effects. We find that dots tend toreturn to the bright state before jumping to a dim state, and that when choosing adim state they tend to explore the entire set of states available. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b ntroduction Cesium lead halide perovskite nanocrystals, introduced in a seminal paper by Protesescuet al. have emerged as highly attractive quantum dots, with advantageous properties incomparison to traditional colloidal II-VI semiconductor quantum dots. These include verylarge photon absorption cross sections, a wide degree of tunability by both size and halide(Br, I, Cl) composition, and reportedly a very high luminescence quantum yield without theneed of protecting the nanodot core with epitaxial shells, as is required for CdSe quantumdots. Furthermore, inorganic halide perovskite materials generally show an exceptionallyhigh tolerance to defects. Owing to these properties perovskite nanocrystals are intensivelypursued as solar cell materials, as emitters for LEDs, display technologies and lasers, and could be interesting as single photon sources. For the purpose of single photon sources,emitters need to satisfy a variety of requirements beyond brightness, tuneability and highquantum efficiency, which includes single-photon purity, tight constraints on inhomogeneousspectral broadening, and stability in spectrum, decay rate and intensity. Perovskite nanocrystals unfortunately follow the almost universally valid rule that solid-state single emitters at room temperature show intermittency.
In the field of II-VIquantum dots, intermittency has been studied for over two decades, with the aim of iden-tifying the nature of the usually two or three distinct bright, dark, and gray states, andthe mechanism by which switching occurs, by analysis of the apparently discrete switchingevents between dark and bright states, and concomitant jumps in spectrum and life-time. For instance, for II-VI quantum dots a popular model (reviewed in Ref. ) is thecharging/discharging model whereby quantum dots turn from bright to dark upon acquiringa single charge, with typically power-law distributed residence times for on and off statesdetermined by the mechanism by which charges are exchanged with the environment. For perovskite nanocrystals several groups studied intermittency and found quite dif-ferent physics. A set of works observe that perovskite quantum dots do not show bimodalbehavior, like II-VI quantum dots do, but instead a continuous distribution of states between2hich they switch.
These observations are consistent with the so-called ‘multiple re-combination centers (MRC) model’ proposed for CsPbBr dots by Li et al., wherein asingle quantum dot does not immediately switch from bright to dark on introduction of asingle charge or defect, but instead intrinsically has a multitude of recombination centersthat may be activated to add nonradiative decay channels. Activation of individual recom-bination centers then shows as switching events, but with a wide distribution over intensityand rate. Other groups, in contrast, have analyzed intermittency on basis of changepointanalysis and cluster analysis, which are Bayesian inference tools for the unbiased estimate ofthe number of states. These reports claim that just of order 2-4 states are involved insteadof a continuum. Finally, a recent study points at memory effects in intermittency, visible inthat work as correlations between subsequent dwell times in the brightest state. Intermittency analysis is a field known to be fraught by statistical bias in analysis meth-ods, primarily due to binning of data prior to analysis. This is a recognized problemalready for interpreting data from bimodal dots. These artefacts may be even more severefor multilevel dots. In this work we report a study of cesium-lead-bromide nanocrystal inter-mittency, analyzing the photon statistics of a large number of dots using unbiased Bayesianstatistics analysis tools, tracing brightness and fluorescence lifetimes simultaneously, andscreening for memory effects. These Bayesian statistics methods were first developed byWatkins and Yang and have since been applied in a small set of papers to two/few-levelII-VI dots, and in one recent work to CsPbBr dots. Our implementation is through afreely available Python-based analysis toolbox, which we have specifically benchmarkedby Monte Carlo methods for application to highly multi-state, instead of bi-modal, systems.In this work, a first main purpose is to obtain statistically unbiased estimates, or at leastlower bounds, for the number of dark/gray states of perovskite quantum dots from a largenumber of single dot measurements. Our conclusions solidly support the MRC model with asingle well-defined bright state and a continuum — or at least over 10 — gray/darker states,in agreement with, but not Ref. Next, our purpose is to screen for memory effects in3esidence times, intensity levels and decay rate sequences in data that has been separated insegments by unbiased changepoint analysis, thereby extending Ref., which did not leveragethe benefit of CPA analysis. We find no evidence for memory in residence times, but do findthat a substantial fraction of dots tend to switch back and forth repeatedly between thequite uniquely defined bright state and the band of gray states, instead of jumping throughall states in an uncorrelated random fashion.Figure 1: Properties of a CsPbBr quantum dot. (A) a SEM image showing a cluster ofCsPbBr quantum dots. The scale bar is 100 nm. A time trace of the (B) intensity and(C) lifetime of a typical qdot when split into bins of 10 ms (green). We find a single peakin both the intensities and lifetimes around 60 counts/ms and 0.05 ns − , respectively. Forvisualization purposes, we also show the photon events binned into 0.5 ms bins (purple).(D) The g ( τ ) of this qdot. The dots used in this analysis were selected for having g (0) < . · g (100ns). (E) The spectrum of this qdot. We find a peak in the emission at 505 nm.(F) The decay trace of all the photon events combined. We have excluded an electronicartefact between 20 and 30 ns. We have a reasonable fit to a bi-exponential decay with ratesof γ , γ = 0 . , .
03 ns − , respectively. (G) The FDID diagram of this dot. We see a mainpeak at I, γ = 0 . × cts/s and 0.09 ns − xperimental methods To introduce our measurement protocol and the photophysics of the CsPbBr dots at handwe first present in Figure 1 the typical behavior of a CsPbBr quantum dot, as analyzedwith the standard approach of plotting time binned data. Preparation of cesium oleate.
We load 0.814 g of Cs CO into a 100 mL 3-neck flaskalong with 40 ml of octadecene (ODE) and 2.5 ml of oleic acid (OA) and dry this for 1hour at 120 ◦ C. This is then heated under an N atmosphere to 150 ◦ C until all Cs CO hasreacted with OA. To prepare for the next step, we preheat the resulting cesium oleate to 100 ◦ C before injection. This is necessary as it precipitates out from ODE at room temperature.
Synthesis of CsPbBr nanocubes. We load 0.188 mmol of PbBr in 5 ml of ODE, 0.5ml of oleylamine and 0.5 ml of OA into a three-neck round bottom flask and dry this undervacuum at 120 ◦ C for an hour, after which the reaction atmosphere is made inert by flushingthe flask with N . After complete solubilization of PbBr , the temperature is raised to 200 ◦ C and 0.4 ml of the preheated cesium oleate is injected into the three-neck flask. Afterthe injection, the color of the solution turns from colorless to greenish yellow indicating theformation of perovskite cubes. Then we lower the temperature to 160 ◦ C and anneal thesolution at that temperature for 10 min to get uniform size dispersion of the cubes. Afterthat we cool down the solution using ice water bath for further use.
Isolation and purification of CsPbBr3 cubes.
After the synthesis, we centrifugeour solution twice to collect the cubes. First, we take 1 ml from the stock solution justafter the synthesis and centrifuge at 8000 rpm for 20 min to collect all CsPbBr3 particlesfrom the solution. We discard the supernatant, gently wash the inner wall of the tube usingtissue paper and add 2 ml of toluene to disperse the CsPbBr3 solid. The second step ofcentrifugation is run at 2000 rpm for 5 min to get rid of all the particles that are too large.In the supernatant, we have 2 ml of toluene containing CsPbBr nanocubes having a sizedistribution around 10-15 nm. As the scanning electron micrograph in Figure 1(A) showsour quantum dots are essentially cubic in shape.5efore the measurement, about 400 µ L of the solution is spin coated at 1000 rpm onglass coverslips that had been cleaned in a base piranha solution. In order to protect thequantum dots from moisture in the air, the quantum dots were covered by a layer of PMMAA8, by spincoating for 60 s at 4000 rpm.
Single emitter microscope.
For optical characterisation and measurements, we use aninverted optical microscope to confocally pump the dots at 450 nm (LDH-P-C-450B pulsedlaser, PicoQuant) at 10 MHz repetition rate of <
70 ps pulses, with 90 nW inserted into themicroscope. An oil objective (Nikon Plan APO VC, NA=1.4) focuses the pump laser onto thesample and collects the fluorescence. The excitation provides similar pulse energy density asin at the lowest energy density (cid:104) N (cid:105) (cid:28) < Measurement protocol.
Using the camera and wide-field pump illumination, we selectan emitter that appears to be diffraction-limited. After driving it to the laser spot, we doa time-correlated single-photon counting (TCSPC) measurement to collect photon arrivaltimes. To calculate the photon correlations we use a home-built TCSPC toolkit that utilizesthe algorithm developed by Wahl et al. to calculate g ( τ ) and the lifetimes for the differentemitters and for the individual CPA segments. From g ( τ ) we select the emitters with astrong anti-bunching signal (normalized g ( τ ) < . Initial characterization.
Figure 1 presents initial characterization of an exemplarysingle dot on basis of standard timebinned analysis, where the data is sliced in 10 ms longsegments, to each of which intensity and decay rate is fitted. Throughout this work weconsider photon counting data, in which absolute time-stamps are collected with 0.165 nsresolution for all collected photons and concomitant excitation laser pulses, on two avalanchephotodiodes (APDs) in a Hanbury-Brown and Twiss configuration. This allows to construct a posteriori from one single data set the intensity, fluorescence decay rate, and the g (2) ( τ )photon-photon correlation. In our optical measurements we post select all single nanopar-ticles on basis of photon antibunching ( g (2) ( τ = 0) < . g (2) ( τ ) that is constructed from the full photon record. The quantum dot in Fig. 1(E)shows a time-averaged emission spectrum that peaks at around 5 nm and has a spectralFWHM of 20 nm, which is consistent with reports by Protesescu et al., and together withthe antibunching photon statistics points at quantum confinement. The time-integrated flu-orescence decay trace (Fig. 1(F)) is markedly non-single exponential. Fitted to a doubleexponential decay we find decay rates of γ , γ = 0 . , .
03 ns − . We must note, however,that a double exponential is often not sufficient to fit these emitters, and typical decayrates for our dots range from 0 .
05 to 0 . − . At these decay rates, the fastest decay ratecomponent of the quantum dots generally span at least 10 timing card bin widths.Figure 1(B, C) shows just a fraction of the intensity and decay rate time trace, plottedaccording to the common practice of partitioning the single photon data stream in bins. Thefluorescence decay rate for each bin is obtained by fitting data within each 10 ms bin toa single exponential decay law employing a maximum likelyhood estimator method that isappropriate for Poissonian statistics. As expected from prior reports on single pervoskitenanocrystal blinking, the intensity and decay rate time trace show clear evidence for7ntermittency. The intensity varies from essentially zero to 150 counts per ms. Figure 1(B,right panel) shows a histogram of intensities, binned over the entire time trace (for all dotsin this work, 120 s, or till bleaching occurred). The histogram shows a broad distributionof intensities with most frequent intensities around 60 cts/ms. This is in contrast with thetypical bi- or trimodal physics of II-VI quantum dots, which usually show distinct brightand dark states.
However, the width of the peak well exceeds the Poisson vari-ance expected at these count rates, suggesting that there are many intensity levels. Thedecay rate histogram also displays intermittent behavior, in step with the intensity blinking.The most frequent decay rate is around 0.07 ns − . Fig. 1(G) displays a Fluorescence DecayRate Intensity Diagram [FDID], a 2D histogram displaying the frequency of occurrence ofintensity-decay rate combinations. This type of visualization was first introduced by toidentify correlations between intensity and fluorescence decay rate (FLIDs in those works,using lifetime instead of decay rate). For II-VI quantum dots, FDID diagrams typically sep-arate out bright and slowly decaying states from dark, quickly decaying states.
Instead,for the perovskite quantum dot at hand, the FDID diagram presents a broad distributionwith a long tail towards dim states with a fast decay.The picture that emerges from Fig. 1 is consistent with recent observations of severalgroups, and qualitatively supports the notion of a continuous distribution of dark,gray state, consistent with the MRC model. This should be contrasted to typical II-VIquantum dot behavior, and also the recent report by on very similar CsPbBr dots, buttaken under very low repetition rate excitation conditions (fs pulses at very low repetitionrate, as opposed to picoseond pulses at ≥
10 MHz — at similar (cid:104) N (cid:105) < . Computational methods
Since extreme caution is warranted when scrutinizing photon counting statistics to deter-mine quantitative intermittency metrics due to artefacts of binning, we proceed to8nalyze the data of a large number of dots with state-of-the-art bias-free statistical analysisto determine a lower bound to the number of involved states, and the switching dynamicsand memory effects therein. We apply tools of Bayesian statistics, specifically, changepointanalysis (CPA) to partition the data in segments separated by switching events, and level-clustering to determine (a lower bound to) the number of states, as a rigorous and bias-freeapproach to investigate the intermittency of quantum dots. These tools were first proposedby Watkins and Yang, and later also used and extended in the context of quantum dot in-termittency by. We refer to Ref. for our freely available implementation anda detailed description of benchmarking of this tool set. Here we summarize just the salientoutcomes relevant for this work, obtained by extensive Monte Carlo based benchmarking todetermine the performance of CPA and clustering for highly multilevel emitters.CPA performs segmentation of the time record of single photon counting events intointervals within which the count rate is most likely a constant value, delineated by switchingevents or ‘changepoints’ at which the count rate changes, in as far as can be judged giventhe shot noise in the data. Since CPA works on a full time series with many jumps byfinding a single jump at a time, and successively subdividing the time stream until segmentswith no further jumps are found, the ultimate performance is ultimately set by how wellCPA can pinpoint in the last stage of the subdivision single jumps in short fragments of thephoton stream. For significant intensity contrasts CPA detects changepoint in very shortfragments (e.g., to accurately resolve a jump with a 5-fold count rate contrast, a record ofjust 200 photons suffice), with single-photon event accuracy. Smaller jumps are missed unlessfragments are longer (e.g., factors 1.5 contrast jumps require fragments of ca. 10 photonsfor near sure ( > practical count rates of 10 cts/s this meansthat switching events further apart than 10 ms are accurately identified as long as jumpcontrasts exceed a factor 1.5 (ca. 100 ms for contrasts as small as 1.2). Switching eventsthat are even closer in time are missed by CPA. This is intrinsic to the photon budget, i.e.,the ultimate information content in the discrete event time stream fundamentally does not9llow pinpointing even more closely spaced switching evens.After dividing the time trace into segments spaced by changepoints, one is left withsequences containing the residence times T q for each segment, photon counts N q and instan-taneous segment intensities ( I q = N q /T q ), as well as decay rates γ q , obtained by maximumlikelihood fitting of the decay trace from each segment to a single exponential decay. Thequestion how many actual intensity levels most likely underlie the measured noisy sequence I m r can be determined using Watkins & Yang’s clustering algorithm. While Watkins andYang considered Poisson distributed noise, as in this work, we recommend also the work ofLi and Yang as a very clear explanation of the method, though applied to Gaussian dis-tributed noise. The idea is that expectation-maximization is used to group the most similarsegments together into n G intensity levels, where n G = 1 , , , . . . . After this, the most likelynumber of levels, n G = m r , required to describe the data, given that photon counts are Pois-son distributed, can be determined by a so-called Bayesian Information Criterion (BIC). We have extensively verified by Monte Carlo simulations the performance of CPA and levelclustering for dots with many assumed discrete intensity levels in a separate work.
Inbrief, at small photon budgets in a total time series, only few levels can be detected, butconversely at the total photon budgets in this work, exceeding 5 · events, clustering hasa >
95% success rate in pinpointing the exact number of levels in dots with at least 10assumed intensity levels. Moreover, for photon budgets that are too small to pinpoint alllevels exactly (e.g., at 10 counts in a total measurement record, only up to 4 levels canbe accurately discerned), clustering always returns a lower bound for the actual number ofintensity states. 10igure 2: (A) An example of the intensity time trace of a measured quantum dot (purple,binned in 0.5 ms bins for visualisation purposes), and the intensity segments found by CPA(green). In the lower panel the lifetime for the found CPA segments is shown. On theright are histograms of the occurrences of the intensities for both treatments with segmentsweighted by their duration. (B-D) Three FDID plots weighting each CPA segment (B)equally, (C) by their number of counts, (D) by their duration. The choice of weights putsemphasis on different parts of the intensity-decay rate diagram, as they report on differentlydefined probability density functions. 11 esults and discussion Changepoint analysis and FDID diagrams
We have applied the unbiased CPA analysis and Bayesian inference tools to data from40 single CsPbBr quantum dots. We first discuss an exemplary single dot as example,and then discuss statistics over many single dots. The example dot is identical to the oneconsidered in Figure 1 and refer to the supporting information for results on all dots. InFigure 2A we see that CPA is able to accurately follow the intensity trace of a typicalCsPbBr quantum dot. We show only a section of the total measurement for clarity, andstrictly for plotting purposes only, binned the photon arrival times in 0.5 ms intervals. Notethat this binning is only for visualization, and does not enter the CPA algorithm. Figure 2Bdisplays the fitted decay rates for the same selected time interval, obtained by fitting eachof the identified segments. The right-hand panels of Figures 2A and B show histogramsof intensity and lifetime as accumulated over the full time trace. It should be noted thatthese histograms are intrinsically different from those in Figure 1 for two reasons. First,binned data has entries from bins containing jumps, leading to a smearing of the histogram.Second, since histogramming of segment values I q is agnostic to segment duration , events aredifferently weighted. Thus the histogram of intensities now shows a bimodal distribution.The histogram of the decay rates still exhibits only a single peak at ca. 0.05 ns − .Next we construct correlation diagrams of fluorescent decay rate versus intensity (FDIDs)from CPA data. Customarily FDIDs are 2D histograms of intensity and decay rate asextracted from equally long time bins in binned data. As the length of segments found byCPA can vary over many orders of magnitude, an important questions is with what weighta given segment should contribute to a CPA-derived FDID. A first approach is to give allsegments an equal contribution to the FDID, which emphasizes the probability for a dotto jump to a given intensity-decay rate combination. Alternatively, one could weight thecontribution of each segment to the histogram by the amount of counts it contributes. This12istogram hence emphasizes those entries that contribute the most to the time-integratedobserved photon flux. Lastly, if one uses the segment durations as weights for contributionof segments to the FDID one obtains an FDID closest in interpretation to the conventionalFDID diagram, which presents the probability density for being in a certain state at a giventime. Figures 2D, E and F provide all three visualizations. The data shows variations inintensity levels over approximately a factor 10, with concomitant decay rates also varying overan order of magnitude. Overall, all diagrams suggest an inverse dependence qualitativelyconsistent with the notion that the dots experience a fixed radiative rate, yet a dynamicvariation in the number of available non-radiative decay channels, that make the dot bothdarker and faster emitting. The unweighted and photon count weighted FDIDs show a peakat similar intensity and decay rate at γ, I = 0.06 ns − , 12 × s − , indicative of the mostfrequently occuring intensity/rate combination that is simultaneously the apparent brightstate. The different FDID weightings emphasize different aspect of the data. For instance,weighting by counts highlights mainly the emissive states and underrepresents the long tailof darker state, with respect to the other weighting approaches. This qualitative differencecan result in a quantitative difference for extracted parameters, such as the apparently mostfrequently occurring combination of intensity and decay rate.FDIDs for essentially all dots (see supporting information) are much like the exampleshown in Figure 2, showing a slow decaying bright state with a long tail towards bothlower intensity and faster decay. In fact, we can collapse the FDIDs of all 40 dots ontoeach other by summing histograms (no weighting by, e.g., segment duration) of normalized intensity I/ (cid:104) I (cid:105) versus γ , which further underlines this generic behavior, see Figure 3). Anappealing explanation for the observed dynamics is if the perovskite dots are characterizedby always emitting from one unique bright state that is efficient and has a slow rate of decay γ r [labelled as radiative decay rate], while suffering fluctuations in both brightness and ratethrough abrupt jumps in a nonradiative rate γ nr , as in the MRC model. In this picture, onewould expect the FDID feature to be parametrizable as I ∝ B + I γ r / ( γ r + γ nr ). The feature13igure 3: FDID of all 40 single dots, obtained by summing single dot FDIDs for which thesegment intensities were normalized to the mean intensity. A simple histogramming was used(no specific weighting of entries by duration or counts). Overplotted is a parametric curveof the form ( γ r + γ nr , B + I γ r / ( γ r + γ nr )) with as input a fixed value γ r , and a background B = 0 . I , with I adjusted to match the peak in the FDID, and γ nr scanned.14n the collapsed FDID plot can indeed be reasonably parametrized as such a hyperbola. Thisparametrization is consistent with Ref. in which a linear relation between intensity andfluorescence lifetime was reported. The required radiative decay rate for the bright stateis γ r ∼ .
075 ns − , while the parametrization requires a residual background B = 0 . I .This residual background is not attributable to set up background or substrate fluorescence,suggesting a weak, slow luminescence component from the dots themselves. Moreover, wenote that the FDID feature clearly has a somewhat stronger curvature then the hyperbolicparametrization (steepness of feature at γ < . − , and I/ (cid:104) I (cid:105) > . Clustering analysis
The FDID diagrams at hand qualitatively support the continuous distribution of states hy-pothesized in the MRC model, and observed by Refs.
As quantification of the numberof states involved we perform clustering analysis to estimate the most likely numberof intensity states describing the data on basis of Bayesian inference. A plot of the BayesianInformation Criterion as function of the number of levels n G allowed for describing the dataof the specific example dot at hand is shown in Figure 4A. Strikingly, the BIC does notexhibit any maximum in the range n G = 1 . . .
5, but at n G = 13. Recalling that the BICcriterion in clustering analysis for multistate dots at finite budget generally report a lowerbound, this finding indicates that the data for this dot requires at least as many levels to beaccurately described, if a discrete level model is at all appropriate.Similar conclusions can be drawn from Figure 4B. We have found in Monte Carlo simula-tions that if one allows the level clustering algorithm to find the best description of intensitytraces in n G levels for dots that in fact have just m < n G levels, then the returned descriptionof the data utilizes just m levels, with the remaining levels having zero occupancy in the bestdescription of the data returned by the algorithm. Figure 4B shows the occupancy assignedby the clustering algorithm for our measured quantum dot as function of the number ofstates offered to the algorithm for describing the segmented intensity trace. Each additional15igure 4: (a) th BIC of a single CsPbBr quantum dot. We see that the BIC of this dotpeaks at n G = 13. (b) The occupancy diagram of the same quantum dot. The numberof occupied states keeps growing with the number of available states, somewhat saturatingaround n G = 15. (c) A histogram showing the durations of the found CPA segments ofthe CsPbBr quantum dots. For this dot, we find a powerlaw tail with an exponent of α = 2 . quantum dot, when the intensityis calculated over 1 ms bins. A truncated powerlaw is fitted to the data. We find an exponentof C = 0 . .
39. Figures demonstrating the switching times, the long-term autocorrelations,the m-state likelihood, and the occupancy of a single dot.16tate offered to the clustering algorithm is in fact used by the algorithm, whereas MonteCarlo simulations have shown that at the photon budgets involved (5 . × photons) theclustering algorithm generally does not assign occupancy to more than m levels to simu-lated m -levels dots. The occupancy diagram hence confirms the conclusion from the BICcriterion that the dot at hand requires many levels, or even a continuous set of levels, to bedescribed.For all 40 dots we extracted wavelength, brightness, and performed the same CPA andclustering analysis as for the example dot. Moreover, we examined segment duration statis-tics for power law exponents. The supporting information contains a detailed graphicaloverview of the CPA results for each of the 40 dots, while summarized results are shown inFigure 5. Figure 5A shows that the dots have a low dispersion in peak emission wavelength,with emission between 500 and 510 nm. All considered dots offered between 2 and 8 × photon events (Figure 5B) for analysis (120 seconds collection time, or until photobleaching).The mean intensity per measured dot (histogram Figure 5C) is typically in the range from15 × and 80 × cts/s, with one single dot as bright as 110 × cts/s. According to theMonte Carlo analysis in the total collected photon count for all dots therefore provides asufficient photon budget to differentiate with high certainty at least up to 10 states. We canthus with confidence exclude that intermittency in these perovskite quantum dots involvesswitching between just two or three states as in usual quantum dots. Instead any physicalpicture that invokes a set of m discrete levels requires a description in upwards of m = 10levels. In how far further distinctions between >
10 discrete levels, or instead a continuousband can be made on basis of data is fundamentally limited by the finite photon budgetthat can be extracted from a single emitter. This quantification matches the observationin Ref. (based on examining time-binned FDID diagrams), but is at variance withGibson et al., who used CPA and clustering on similar CsPbBr quantum dots, but arriveat an estimate m r = 2 .
6. This difference may be attributable to the different excitationconditions that are unique to Gibson et al. relative to all other works.17 esidence times In Figure 4C we show a histogram of the segment lengths found by CPA. In other works,on-states and off-states are often separated explicitly by thresholding following which on-times and off-times are separately analyzed, for instance to ascertain the almost universallyobserved power-law dependencies and their exponents. In the case of our CsPbBr quantumdots a level assignment in on and off states is not obvious. Therefore we simply combine all segment lengths irrespective of intensity level in a single histogram. These switching timesare power-law distributed, at least from minimum time durations of 10 ms onwards. Theshort-time roll off is consistent with the limitations of the information content of the discretephoton event data stream: for segments shorter than 50 photons or so, even if physicallythere would be a jump, the photon number would not suffice to resolve it. Thus the roll-offdoes not exclude that power-law behavior also occurs for shorter times, but instead signifiesthat the testability of such a hypothesis is fundamentally limited. Fitting the power law t − α for time >
10 ms indicates a power-law exponent of α = 2 . ± . α ∼ . α = 2 . Also, thesevalues are significantly higher than the exponents reported for on-times of CsPbBr dotsextracted from intensity-thresholded time-binned data. We note that one can (somewhatarbitrarily) threshold CPA-segmented data in an attempt to isolate ’on-times’ for the brightstate from the ‘residence times’ associated with the long dark/gray tail of states. Doing sowith thresholds I/ (cid:104) I (cid:105) > . I/ (cid:104) I (cid:105) < . We notethat apart from the methodological difference of not working with binned thresholded databut with CPA analysis, also the selection of dots reported on may matter. In this workwe report on all dots identified as single photon emitters by their g (2) (0). Instead in Ref. dots are reported to have been selected as those for which inspection of binned time tracessuggested the most apparent contrast between bright and dim states, qualitatively appearingclosest to bimodal behavior. According to our analysis of FDIDs and in light of the MRCmodel, this post-selection may not single out the most representative dots.An alternative approach to quantifying blinking statistics and power law exponents thatrequires neither thresholding binned data nor CPA is to simply determine intensity auto-correlation functions g (2) for time scales from ms to seconds, as proposed by Houel et al. According to Houel et al. the normalized autocorrelation minus 1 may be fit with the equa-tion At − C exp( − Bt ). Figure 4(D) show such an analysis for the exemplary dot at hand,for which we find a reasonable fit with C = 0 .
39. As Figure 5F shows, across our collectionof dots we generally fit exponents C in the range 0.10 to 0.75 to intensity autocorrelationtraces. We note that the relation α = 2 − C put forward by Houel et al. is only expectedto hold for two-state quantum dot, and C does not relate directly to α for quantum dots inwhich more than two states are at play.Figure 5: Summary of the behavior of the 40 measured single quantum dots. We show thedistribution of found (A) peak wavelengths, (B) total photon count, (C) intensities, (D)most likely number of states, (E) powerlaw exponents of the switching time α , and (F) thepower-law exponent of the autocorrelation C .19 emory effects, aging and correlations in CPA sequences , V $ γ Q V − % W L P H V l og ( T n ) & log ( T n ) , F W V V ' log ( T n ) γ Q V − ( ρ Q R U P Figure 6: Analysis of (absence of) aging during photocycling of a single perovskite quantumdots, histogramming intensity (A), decay rate (B) and segment duration (C) in slices of0.9 sec for a total measurement of 90 sec. (D, E) correlation histograms of intensity ver-sus segment duration, and decay rate versus segment duration, evidencing that these areuncorrelated quantities.Finally we examine the dots for aging and memory effects, leveraging the fact that CPAgives an unbiased data segmentation into segments n = 1 . . . N that are classified by seg-ment duration T , T , . . . , intensity in counts/sec I , I , . . . , and decay rate γ , γ , . . . that isestablished without any distorting temporal binning. We present results again for the sameexample dot as in Fig. 1 in Figures 6 and 7. With regard to aging, one can ask if over the fullmeasurement time in which a dot undergoes of order 10 excitation cycles, the distributionof segment duration, intensity and decay rate show any sign of change. To this end, we sub-divide the total measurement period (e.g., Fig.6(A-C), total measurement time 60 s for thisdot) in 100 slices that are equal length in terms of wall-clock time, and examine the evolu-tion of histograms of I n , γ n and T n for these short measurement intervals as function of theiroccurrence in the measurement time. As the residence times are very widely distributed, weplot histograms of log T q , with q the index of the segments. There is no evidence that anyof these observables change their statistical distribution over the time of the measurement.While Fig.6(A-C) shows an example for just one dot, this conclusion holds for all dots in our20easurement sets, with the caveat that for some dots drifts in microscope focus caused asmall gradual downward drift in intensity. We observed no photobrightening of dots duringthe experiment.Clustering allows us to ask questions that are not accessible with simple binning of data,as we can examine the datasets for correlations between parameters and between subsequentsegments. In terms of cross-correlating different observables, beyond FDIDs that correlateintensity and decay rate, one can also examine correlations between intensity and segmentduration, and between decay rate and segment duration. Histogramming the clustered datato screen for such correlations (Fig.6(D,E)) show that both the distribution of intensities andof decay rates are uncorrelated, or only very weakly correlated, with the segment duration.In other words, we find no evidence that within the distribution of states between which thedot switches, some states have different residence time distributions than others.Memory effects should appear as correlations in the values for any given observablein subsequent segments, i.e. in conditional probabilities that quantify what the probability P ∆ n ( A | B ) is that a chosen observable to obtain a value A is given that it had a value B in theprevious segment (∆ n = 1), or generally counting ∆ n events further back into the history ofprevious segments. Figure 7 shows such conditional probabilities for ∆ n = 1 (panels A-C)and ∆ n = 2 (panels D-F) , for intensity (panels A,D), decay rate (panels B,E) and segmentduration (panels C,F). These diagrams are obtained by applying a simple 2D histogrammingapproach, listing the value of B as x − axis, the value of A as y − axis, and normalizing thesum of each of the columns to obtain a conditional probability. We note that this approachmeans that at the extremes of the histograms (far left, and far right), there are few eventsto normalize to, leading to large uncertainty. When screening for memory in intensities, it’simportant to consider that the CPA algorithm itself selects for intensity jumps. Due to thisthe intensity after one jump (∆ n = 1) is a priori very unlikely to achieve a similar value,which leads to a near-zero conditional probability at the diagonal of Figure 7(a). Nonetheless,the distinct features in the diagram at ∆ n = 2 (Fig. 7(b)) do suggest that the dot generally21 I n + F W V V $ γ n + Q V − % l og ( T n + ( s )) & I n F W V V I n + F W V V ' γ n Q V − γ n + Q V − ( log ( T n (s)) l og ( T n + ( s )) ) U H O D W L Y H V H J P H Q W Q U D X W R F R U U I * U H O D W L Y H V H J P H Q W Q U D X W R F R U U γ + U H O D W L Y H V H J P H Q W Q U D X W R F R U U 7 , ρ Q R U P Figure 7: (A-F) Conditional probability of observing a value for intensity (A, D), rate (B,E) or segment duration (C, F), given the value of the same obersvable one or two stepsearlier, respectively. (G,H,I) Normalized autocorrelation (difference from 1) of the sequenceof intensity, decay rate and segment durations. This data is for the same single dot asconsidered in Fig. 6. 22lternates repeatedly back and forth between a bright and a dark state. More telling thandiagrams for intensity are those for decay rate. They show that if, in a given step the decayrate is low (slow, bright feature in FDID at < . − ), then in the subsequent step thedecay rate is usually fast, yet widely distributed from 0.1 to 1 ns − , and vice versa from anyof the fast decaying states, the dot is likely to jump to the quite narrowly defined slow rate ofthe bright state. If one considers the conditional rate at ∆ n = 2, the conclusion is that if thedot is in the bright state with its slow decay rate at a given step, then likely after two jumpsit comes back to this bright, slowly decaying state. If, however, the rate was fast anywherein the interval from 0.1 to 1 ns − , then after an excursion to the slow rate at ∆ n = 1, thedot likely in the second step again takes on a fast rate in the interval from 0.1 to 1 ns − butwithout a particularly clear preference for any value in that wide interval. Finally we notethat there is no indication in our data that subsequent residence times ( P ( T m +∆ n | T m ) showany memory (Fig. 7(c,f), showing result for log T m ). Similarly we found no memory effectsfor the sequence of on-times, as selected from CPA by thresholding at I/ (cid:104) I (cid:105) > . I q , γ q and T q .We plot normalized autocorrelation traces G (∆ q ) − H ∈ I q , γ q , T q ,one defines G (∆ q ) = (cid:104) H ( q ) H ( q + ∆ q ) (cid:105) / (cid:104) H ( q ) (cid:105) (where (cid:104) . (cid:105) denotes the mean over q are all segment indices and ∆ q is 1 , , . . . ), so that atlong times G m − G (∆ q ) − without any regard for their time duration . For a conventional two-leveldot, the autocorrelation trace I q would oscillate with large contrast up to very large q . In-stead, we find that the dot at hand shows an oscillation with a distinct contrast in the23ntensity segment autocorrelation contrast for up to 5—10 cycles. In the normalized auto-correlation for decay rates the memory is far less evident. We attribute this not to a lack ofmemory, but note that if a dot switches between a state of well defined slow rate, and anarray of states with highly distributed fast rates, then upon averaging the wide distributionof fast rates washes out any autocorrelation signature. Finally, the residence times, whichwe already found to be uncorrelated between subsequent jumps, show no autocorrelationsignature for z (cid:54) = 0. A similar behavior to that shown in Figure 7(G) was observed for circa30% of the dots studied, with other dots showing no clear intensity autocorrelation. Conclusion
To conclude, we have reported on intermittency properties of a large number of CsPbBr quantum dots on basis of a Bayesian inference data analysis. This approach works with raw,unbinned, photon counting data streams and thereby avoids artefacts commonly associatedwith the analysis of timebinned data. Our data generally confirms the multiple recombina-tion centers model. In particular, we find that dots have in addition to their bright emissivestate a tail of gray states that qualitatively appears continuous in FDID diagrams, and thataccording to clustering analysis requires at least 10 to 20 levels to describe, if a discrete-leveldescription would be appropriate. Thereby our work provides a confirmation of claims inearlier works under similar excitation conditions, with the distinction that we do notuse time binned data but rigorously exploit all the information in the data stream to thelevel that its intrinsic noise allows. We note that the same type of dots have displayed adifferent behavior, indicative of 2 to 3 levels, in Ref. Since that work uses almost identicalBayesian inference methods, we conclude that this distinction is really due to the differentphysical realization. While differences in sample preparation can not be excluded, we notethat Ref. stands out from all other reports due to its very different excitation conditions.As in the first report proposing the MRC model for perovskite dots, we qualitatively find24hat the tail of gray states display an inverse correlation between intensity and rate, sug-gesting that the dots have a unique bright state with a given decay rate, to which randomactiviation of recombination centers add nonradiative decay channels. However, we note thatthis observation merits further refinement of models: while plotting intensity versus lifetimemay point at strict proportionality, plotting rate instead of lifetime accentuates deviations,noteably a deviation in curvature of our data relative to inverse proportional dependence.Finally, we have analyzed correlations in the measured CPA-segmented sequences ofintensity-levels, decay rates and segment lengths. We find no evidence for aging, i.e., gradualshifts in e.g., decay rate or blinking dynamics during photocycling of dots through 10 to10 detected photons (i.e., well over 10 cycles). Also, our data indicate that residencetimes are not correlated to the state that a dot is in. The residence times can be fiduciallyextracted for a limited time dynamic range from ca. 5-15 ms, limited intrisically by countrate, to ca. 10 s, limited by the length of the photon record. We note that in residencetime histograms determined by CPA, according to Monte Carlo simulations at long timesthe analysis fiducially reports on power laws without introducing artefacts, such as apparentlong-time roll offs. The exponents that we find are in the range from 1.5 to 3.0, which appearshigh compared to the near-universal value of 1.5 observed for II-VI single photon sources.In the domain of CsPbBr dots, reports have appeared of even lower exponents (down to1.2) with exponential roll offs at times ∼ . T q ) reported by Hou et al. for on-times.In our view, this rich data set will stimulate further theory development in the domainof inorganic quantum dot intermittency. The CPA-segmented data for all quantum dots ismade available in digital form for all quantum dot single photon emitters that we report onin this work. Supporting Information Available
PDF containing CPA-based intermittency analysis report for 40 single dots - appended tothis preprint. For select dots, data is posted with the Python analysis code of Ref. atgithub.
Acknowledgement
This work is part of the research program of the Netherlands Organization for ScientificResearch (NWO). We would like to express our gratitude to Tom Gregorkiewicz who passedaway in 2019, and whose encouragement and guidance in the early stages of the project wereinvaluable. 26 eferences (1) Protesescu, L.; Yakunin, S.; Bodnarchuk, M. I.; Krieg, F.; Caputo, R.; Hendon, C. H.;Yang, R. X.; Walsh, A.; Kovalenko, M. V. Nanocrystals of Cesium Lead Halide Per-ovskites (CsPbX 3 , X = Cl, Br, and I): Novel Optoelectronic Materials Showing BrightEmission with Wide Color Gamut.
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