^{1}, C. M. Hess

^{1}, P. J. Whitham

^{1}and P. J. Reid

^{1,a)}

### Abstract

The photoluminescence intermittency (PI) exhibited by single emitters has been studied for over a decade. To date, the vast majority of PI analyses involve parsing the data into emissive and non-emissive events, constructing histograms of event durations, and fitting these histograms to either exponential or power law probability distributions functions (PDFs). Here, a new method for analyzing PI data is presented where the data are used directly to construct a cumulative distribution function(CDF), and maximum-likelihood estimation techniques are used to determine the best fit of a model PDF to the CDF. Statistical tests are then employed to quantitatively evaluate the hypothesis that the CDF (data) is represented by the model PDF. The analysis method is outlined and applied to PI exhibited by single CdSe/CdS core-shell nanocrystals and the organic chromophore violamine R isolated in single crystals of potassium-acid phthalate. Contrary to previous studies, the analysis presented here demonstrates that the PI exhibited by these systems is not described by a power law. The analysis developed here is also used to quantify heterogeneity within PI data obtained from a collection of CdSe/CdS nanocrytals, and for the determination of statistically significant changes in PI accompanying perturbation of the emitter. In summary, the analysis methodology presented here provides a more statistically robust approach for analyzing PI data.

This work was supported by the National Science Foundation (NSF) (Grant No. DMR 1005819). Part of this work was conducted at the University of Washington NanoTech User Facility, a member of the NSF National Nanotechnology Infrastructure Network (NNIN).

I. INTRODUCTION

II. EXPERIMENTAL METHODS

A. Sample preparation

B. Single-molecule microscopy

III. RESULTS AND DISCUSSION

A. Current analysis and maximum likelihood estimation

B. Cumulative distribution functions

C. An alternative method for analyzing PI data

1. Estimating PDF best-fit parameters

2. Statistically evaluating PDFs

3. Extension to other PDFs

D. Comparing PI data sets without PDFs

IV. CONCLUSION

### Key Topics

- Quantum dots
- 42.0
- Cumulative distribution functions
- 25.0
- Data analysis
- 25.0
- II-VI semiconductors
- 15.0
- Statistical analysis
- 11.0

##### B82B1/00

## Figures

*On*- and *off*-event duration histograms for 220 CdSe/CdS quantum QDs in PMMA. (a-b). Raw PI histograms presented on a log-log plot. The tail of the data is fanned, representing sparse events. (c-d). Each point in the raw histogram is divided by the average time to the nearest neighbors, resulting in a continuous probability density. The data in the right are fitted with linear least squares regression to obtain the power-law exponent. (e-f) CDFs generated from the data in (a) and (b), the CDF is a naturally decaying distribution that can be fitted and differentiated to generate the PDF without the need to smooth the data. The curvature of the CDF indicates that the data are not consistent with power-law.

*On*- and *off*-event duration histograms for 220 CdSe/CdS quantum QDs in PMMA. (a-b). Raw PI histograms presented on a log-log plot. The tail of the data is fanned, representing sparse events. (c-d). Each point in the raw histogram is divided by the average time to the nearest neighbors, resulting in a continuous probability density. The data in the right are fitted with linear least squares regression to obtain the power-law exponent. (e-f) CDFs generated from the data in (a) and (b), the CDF is a naturally decaying distribution that can be fitted and differentiated to generate the PDF without the need to smooth the data. The curvature of the CDF indicates that the data are not consistent with power-law.

An illustration of the process by which simulated data are created. Simulated data sets for the power-law are generated by a uniform sampling of the experimental data below *t* _{ min } with probability 1-*n* _{ tail }/*n* (shaded region), and randomly from a power-law (or other PDF) employing best fit parameters determined using maximum likelihood estimation with sampling probability *n* _{ tail }/*n*.

An illustration of the process by which simulated data are created. Simulated data sets for the power-law are generated by a uniform sampling of the experimental data below *t* _{ min } with probability 1-*n* _{ tail }/*n* (shaded region), and randomly from a power-law (or other PDF) employing best fit parameters determined using maximum likelihood estimation with sampling probability *n* _{ tail }/*n*.

(Left) Best fits of the 0.5 μW data to power-law(-·-), lognormal(-), and Weibull(·) PDFs for 220 CdSe/CdS quantum dots in PMMA for *on*-times (a) and *off* times (b). None of the PDFs represent the data, with all fits rejected having p-values <0.0001 (derived from 10 000 simulated data sets as described in the text). For power-law best fit *t* _{ min } was 0.01 for both *on* and *off* with α = 1.55 (*off*) and 1.67 (*on*). (Right) Identical analysis applied to the PI data derived from a single CdSe/CdS quantum dot *on*-times (c) and *off*-times (d). The single QD data also fails the statistical test for all three distributions to 0.01. This was performed for all QDs at 0.5 μW, not a single QD (*on* or *off*) was represented by these functions.

(Left) Best fits of the 0.5 μW data to power-law(-·-), lognormal(-), and Weibull(·) PDFs for 220 CdSe/CdS quantum dots in PMMA for *on*-times (a) and *off* times (b). None of the PDFs represent the data, with all fits rejected having p-values <0.0001 (derived from 10 000 simulated data sets as described in the text). For power-law best fit *t* _{ min } was 0.01 for both *on* and *off* with α = 1.55 (*off*) and 1.67 (*on*). (Right) Identical analysis applied to the PI data derived from a single CdSe/CdS quantum dot *on*-times (c) and *off*-times (d). The single QD data also fails the statistical test for all three distributions to 0.01. This was performed for all QDs at 0.5 μW, not a single QD (*on* or *off*) was represented by these functions.

Complimentary CDFs for the *on* (a) and *off* (b) times observed VR-KAP at room temperature (black) and 60 °C (gray). Best fits to lognormal functions are overlaid on the data (dashed lines). On-interval data become strongly lognormal at 60 °C (p-value increases from 0.024 to 0.118), but the room temperature tail diverges from the fit. Off interval data are not well fit to lognormal at either temperature (p-values of 10^{−4} and 0.035 for room temperature and 60 °C, respectively).

Complimentary CDFs for the *on* (a) and *off* (b) times observed VR-KAP at room temperature (black) and 60 °C (gray). Best fits to lognormal functions are overlaid on the data (dashed lines). On-interval data become strongly lognormal at 60 °C (p-value increases from 0.024 to 0.118), but the room temperature tail diverges from the fit. Off interval data are not well fit to lognormal at either temperature (p-values of 10^{−4} and 0.035 for room temperature and 60 °C, respectively).

(Left) Complimentary CDF's on log-log axes for CdSe/CdS QDs *on*- (a) and *off*- (b) times for data taken using 0.5 μW (solid line) and 8.2 μW (dashed line). 110 QDs are used at each power. *D* values are 0.0397 (on times) and 0.073 (off times). (Right) Control study for determining the distribution of observed *D* values for CdSe/CdS QDs in PMMA with 0.5 μW excitation at 532 nm. The data from 220 CdSe/CdS quantum dots were randomly parsed into two sets and the CDF's for the *on*- (c) and *off*- (d) times were determined and *D* value computed. The above histogram represents 1000 random trials. The shaded box indicates *D* values corresponding to p-values that *accept* the hypothesis that the two distributions are the same. Notice that 60%–65% of the time this hypothesis fails. The dashed line represents a lognormal fit to the data. Integrating the tail of the normalized lognormal PDF greater than *D* obtained from the two power comparison gives 0.36% (on-times) and 0.001% (off times) probability that they could be the same, demonstrating a power-dependence for both on and off times.

(Left) Complimentary CDF's on log-log axes for CdSe/CdS QDs *on*- (a) and *off*- (b) times for data taken using 0.5 μW (solid line) and 8.2 μW (dashed line). 110 QDs are used at each power. *D* values are 0.0397 (on times) and 0.073 (off times). (Right) Control study for determining the distribution of observed *D* values for CdSe/CdS QDs in PMMA with 0.5 μW excitation at 532 nm. The data from 220 CdSe/CdS quantum dots were randomly parsed into two sets and the CDF's for the *on*- (c) and *off*- (d) times were determined and *D* value computed. The above histogram represents 1000 random trials. The shaded box indicates *D* values corresponding to p-values that *accept* the hypothesis that the two distributions are the same. Notice that 60%–65% of the time this hypothesis fails. The dashed line represents a lognormal fit to the data. Integrating the tail of the normalized lognormal PDF greater than *D* obtained from the two power comparison gives 0.36% (on-times) and 0.001% (off times) probability that they could be the same, demonstrating a power-dependence for both on and off times.

0.5 μW QD CDF's for different thresholds of 3 (solid), 6 (dashed), and 9 (dotted) standard deviations above RMS noise depicted in (a). *On*-times (b) show a faster fall off in probability at long-times with increased threshold. *D* value of 0.0153 between the thresholds of 3 and 9. The opposite is observed in the *off* times (c), where *D* is 0.0378 between the thresholds of 3 and 9 standard deviations.

0.5 μW QD CDF's for different thresholds of 3 (solid), 6 (dashed), and 9 (dotted) standard deviations above RMS noise depicted in (a). *On*-times (b) show a faster fall off in probability at long-times with increased threshold. *D* value of 0.0153 between the thresholds of 3 and 9. The opposite is observed in the *off* times (c), where *D* is 0.0378 between the thresholds of 3 and 9 standard deviations.

*D* values observed at 0.5 and 8.2 μW between individual quantum QDs within their own ensembles. The *on*- (a, b) and *off*- (c, d) times become more homogeneous with increasing power (the distributions of *D* shift toward 0). These histograms correspond to the p-value being accepted 41% (*off*, c) and 33% (*on*, a) for 0.5 μW, and 29% (*off*, d) 22% (*on*, b) at 8.2 μW, consitent with a decrease in the *off*-times resulting in more switches.

*D* values observed at 0.5 and 8.2 μW between individual quantum QDs within their own ensembles. The *on*- (a, b) and *off*- (c, d) times become more homogeneous with increasing power (the distributions of *D* shift toward 0). These histograms correspond to the p-value being accepted 41% (*off*, c) and 33% (*on*, a) for 0.5 μW, and 29% (*off*, d) 22% (*on*, b) at 8.2 μW, consitent with a decrease in the *off*-times resulting in more switches.

The mean and standard deviations of the *on* (a, b)- and *off* (c, d) intervals computed for the 0.5 μW data. These data illustrate just how difficult it is to judge how similar the quantum QDs are. When the KS test is used on either of these distributions in a simulation like that used on the ensemble CDF's the distributions are considered the same 93% of the time. This illustrates that for the case of distributed data the mean and standard deviation are not reliable methods for capturing the degree of heterogeneity in the ensemble.

The mean and standard deviations of the *on* (a, b)- and *off* (c, d) intervals computed for the 0.5 μW data. These data illustrate just how difficult it is to judge how similar the quantum QDs are. When the KS test is used on either of these distributions in a simulation like that used on the ensemble CDF's the distributions are considered the same 93% of the time. This illustrates that for the case of distributed data the mean and standard deviation are not reliable methods for capturing the degree of heterogeneity in the ensemble.

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