Chinese Journal of Chemical Physics  2018, Vol. 31 Issue (4): 595-598

#### The article information

Yong-lei Sun, Jing Zhao

Average Arrival Time: an Alternative Approach for Studying Fluorescent Behavior of Single Quantum Dots

Chinese Journal of Chemical Physics, 2018, 31(4): 595-598

http://dx.doi.org/10.1063/1674-0068/31/cjcp1806149

### Article history

Accepted on: July 23, 2018
Average Arrival Time: an Alternative Approach for Studying Fluorescent Behavior of Single Quantum Dots
Yong-lei Suna, Jing Zhaoa,b
Dated: Received on June 21, 2018; Accepted on July 23, 2018
a. Institute of Materials Science, University of Connecticut, Storrs, Connecticut 06269, USA;
b. Department of Chemistry, University of Connecticut, Storrs, Connecticut 06269, USA
*Author to whom correspondence should be addressed. Jing Zhao, E-mail:jing.zhao@uconn.edu
Abstract: Due to photoluminescence intermittency of single colloidal quantum dots (QDs), the traditional exponential fluorescence lifetime analysis is not perfect to characterize QDs' fluorescent emission behavior. In this work we used the time-tagged time-resolved (TTTR) mode to record the fluorescent photons from single QDs. We showed that this method is compatible with the traditional lifetime analysis. In addition, by constructing the trajectory over time and the distribution of average arrival time (AAT) of the fluorescent photons, more details about the emission behavior of QDs were revealed.
Key words: Quantum dots    Fluorescence lifetime    Time-tagged time-resolved mode    Average arrival time
Ⅰ. INTRODUCTION

Fluorescence lifetime is a critical parameter to characterize quantum dots (QDs) as well as other fluorophores on the ensemble or single-emitter level. The value of lifetime comes from single-exponential fitting of the time-dependent fluorescence decay curve. This ideal situation happens when QDs have singular emission state; however, in real experiments QDs have been reported to have multiple emission states, as revealed by single particle level experiments [1]. The bright and dark emission states are indicated by high and low fluorescence intensity segments in single QD's intensity trajectory over time, a phenomenon also known as fluorescence intermittency or "blinking" [2]. The fluorescence decay kinetics is different for the bright and dark emission states. As a result, the decay curve deviates from perfect single exponential form. In principle, all decay kinetics curves can be fitted multi-exponentially within a certain margin of error. However, this pure mathematical treatment has two disadvantages: (ⅰ) physical meanings are usually difficult to be assigned to the lifetime components from multi-exponential fitting; (ⅱ) no information about the change of emission states over time can be obtained.

Another approach to interpreting the decay kinetics that deviates from single exponentiality is the lifetime distribution analysis [3]. In this approach, instead of finding discrete lifetime components, a Laplace transform is involved to determine the distribution or spectrum of lifetime:

 $\begin{eqnarray} I(t)=\int_0^\infty \sigma(\tau) \textrm{e}^{-t/\tau} \textrm{d}\tau \end{eqnarray}$ (1)

where $I$($t$) denotes the decay curve. Currently, the determination of lifetime distribution $\sigma$($\tau$) by deconvoluting $I$($t$), or in other words, inverting Laplace transform, is usually realized by applying the CONTIN program [4], which consumes significant time and computing resources. Moreover, this method does not reveal the change of emission states during acquisition either. In order to achieve that, the lifetime trajectory over time similar to fluorescence intensity trajectory is constructed. In a lifetime trajectory, the bin time is selected to ensure the singular emission state during it, i.e., the decay kinetics in a bin time is single exponential. However, the dwelling time of bright emission states in colloidal QDs can be as short as several milliseconds [5]. During an acquisition bin time on millisecond level, the number of detected photons is often so small that it is insufficient for exponential fitting, or it leads to fallacious fit lifetime [6]. One could collect more photons in a bin time by increasing the excitation power. But apart from the concomitant rising of background signal, other interfering factors can also occur like the multi-exciton emission. To summarize, the short bin time and sufficient detected photons for exponential fitting during the bin time can hardly be simultaneously satisfied in real experiments. This paradox comes from the pre-assumption that lifetime is determined by exponential fitting.

To solve the aforementioned problem, in this work we used the average arrival time (AAT) of fluorescent photons in each bin time to characterize the emission behavior of colloidal QDs, instead of the traditional exponential fitting method. To obtain the AAT of fluorescent photons, time-tagged time-resolved (TTTR) mode was used in the experiments, which is readily available in many of the time-correlated single photon counting modules. Without the pre-assumption of exponential lifetime, we can construct the trajectory and distribution of the AAT easily. And though different by definitions, we show the AAT is compatible with the traditional exponential lifetime in analysis. Therefore albeit still being statistical, our method could possibly reveal subtle changes of the emission behavior of single QDs.

Ⅱ. METHOD A. Theoretical

Suppose the bin time is chosen so that the decay kinetics has a single exponential form $A\textrm{e}^{-t/\tau}$ during it, which can be readily achieved by our method. Then the number of collected photons $N_i$ of the $i$-th channel during a bin time in time-resolved fluorescence spectrometer is proportional to $A\textrm{e}^{-t_i/\tau} \Delta t$, where $\Delta t$ is the time resolution, and $t_i$=$i\Delta t$. The AAT $\tau'$, i.e., the mean value of the arrival time of the photons within this bin time is:

 $\begin{eqnarray} \tau'&=&\left(\sum\limits_i^\infty t_i N_i\right)\Big/\left(\sum\limits_i^\infty N_i \right)\nonumber\\ &=&\left(\sum\limits_i^\infty t_i A\textrm{e}^{-t_i/\tau} \Delta t\right)\Big/\left(\sum\limits_i^\infty A\textrm{e}^{-t_i/\tau} \Delta t\right)\nonumber \\ &=&\left(\sum\limits_i^\infty t_i \textrm{e}^{-t_i/\tau} \Delta t\right)\Big/\left(\sum\limits_i^\infty \textrm{e}^{-t_i/\tau} \Delta t\right) \end{eqnarray}$ (2)

The time resolution $\Delta t$ is usually on the picosecond scale, much smaller than the bin time. It is reasonable to treat it as $\Delta t$$\rightarrow$0:

 $\begin{array}{l} \mathop {\lim }\limits_{\Delta t \to 0} \tau ' \to \left( {\int_0^\infty t {{\rm{e}}^{ - t/\tau }}{\rm{d}}t} \right)/\left( {\int_0^\infty {{{\rm{e}}^{ - t/\tau }}} {\rm{d}}t} \right)\\ = \frac{{{\tau ^2}}}{\tau } = \tau \end{array}$ (3)

Therefore we prove the AAT is compatible with the traditional exponential lifetime in analysis as long as the bin time is much bigger than the time resolution.

B. Experimental

To test the AAT method, single colloidal CdSe/CdS QDs were studied. The QDs were synthesized using previously published method [7]. A highly diluted solution of the QDs in hexane was spun cast onto a glass coverslip and then mounted onto a home-built confocal fluorescence microscope. The QDs were excited by a pulsed laser (PDL 800-B, PicoQuant) at 450 nm with a repetition rate of 2.5 MHz. The pumping power of the laser was kept low to avoid significant multi-exciton emission of QDs. The fluorescence signal of single QDs was collected through a 100$\times$ oil-immersion objective (Nikon, N.A.=1.3) and then detected by a single photon detector ($\tau$-SPAD, PicoQuant). TTTR data of photons were recorded by a single photon counting system (PicoHarp 300, PicoQuant). All the optical measurements were performed at room temperature.

Ⅲ. RESULTS AND DISCUSSION

For each individual photon, a photon counting system operated in TTTR mode records not only the photon's arrival time after the laser synchronized pulse, but also its macroscopic arrival time with respect to the beginning of measurement, hence offering more flexibility in analysis. FIG. 1 shows the intensity trajectory over time and fluorescence decay dynamics of a single QD recovered from TTTR data. The various high and low intensity states are clearly seen in FIG. 1(a), proving the signal was from a single QD. The switch between bright and dark emission states affected the decay kinetics, as the decay curve in FIG. 1(b) cannot be fitted by a single exponential decay indicated by the dashed curve. The solid curve in FIG. 1(b) denotes a double exponential fitting: $A_0$+$A_1 \textrm{e}^{-(t-t_0)/\tau_1 }$+$A_2 \textrm{e}^{-(t-t_0)/\tau_2 }$, where $\tau_i$ ($i$=1, 2) are the two lifetime components. In previous studies a decay curve of a single QD could be fitted single exponentially, albeit not perfectly [8]. The difference between our results and theirs comes from the advance of the synthesis method of QDs. The QDs used in previous studies have distinct binary bright and dark emission states indicated by the intensity trajectories, while blinking is significantly suppressed in the QD we studied. In the former case, the majority of the photoluminescence signal from the single QD comes from the bright state, whereas the dark state contributes very little to the signal. The time-dependent photoluminescence measurement essentially measures the decay of the bright state, which follows a single exponential decay function. In the latter case, the reduced dark state and more intermediate states can be found in the intensity trajectory in FIG. 1(a) (see FIG. S1 in supplementary materials). These states contribute considerably to the photoluminescence signal, and they do not share the same decay kinetics. Therefore, the decay curve deviates from a single exponential decay function due to the nature of the photoluminescence signal, which originates from multiple emission states.

 FIG. 1 (a) Fluorescence intensity trajectory of a single QD. (b) Fluorescence decay kinetics of the same QD. The dashed curve represents the single exponential fitting curve, while the solid curve represents the double exponential fitting curve.

The deviation from single exponential decay curve of this single QD was further investigated by constructing the AAT trajectory and distribution shown in FIG. 2. The time resolution of PicoHarp 300 is 4 ps, much smaller than our choice of bin time 50 ms, therefore the requirement in Section Ⅱ is obviously satisfied. FIG. 2(a) shows that the AAT trajectory over time that has high and low value segments, is similar to the intensity trajectory. Actually the bins with higher intensities are likely to have larger AAT values. The correlation coefficient between the intensity and AAT corr($I, \tau'$) gives a value of 0.47, verifying this positive correlation (see FIG. S2 in supplementary materials).

 FIG. 2 (a) Fluorescence AAT trajectory of the same QD in FIG. 1. (b) The distribution of AAT. The two solid lines denote the values of the exponential lifetime components and relative magnitudes of the corresponding pre-factors of the double exponential fitting curve in FIG. 1(b).

FIG. 2(b) shows the AAT distribution which features a peak and its tail at the small end, corresponding to the high and low value segments of the AAT trajectory. The broad peak and its tail clearly demonstrate the single QD's multiple emission states, thus explain for the deviation from single exponential decay curve. The positions of the solid lines in FIG. 2(b) represent the values of the lifetime components $\tau_i$ ($i$=1, 2) from the double exponential fitting in FIG. 1(b), while the lengths of the lines are proportional to the corresponding pre-factors $A_i$ ($i$=1, 2). The positions and lengths of the solid lines in general match the AAT distribution, but these lines are not exactly in the center of the peaks of the AAT distribution. This is because the multi-exponential fitting is macroscopic, while the AAT of each bin time is determined by the nature of the emission behavior of the single QD during the short time, as well as stochasticity. Theoretically a peak in a distribution is indistinguishable with a single component at certain significant level [9]. However, considering the physical and chemical properties of the real system, the peaks of AAT distribution of a single QD cannot be simply treated as the broadening of the lifetime components from multi-exponential fitting. Similar observations were made on other QDs as well (see FIGs. S3 and S4 in supplementary materials for representative data). It has been reported that a single QD has multiple bright emission states, and these states are claimed to be continuous [10]. The fluctuations of the excitation condition and chemical environment also contribute to the broad distribution of AAT. As a result, the analysis gives more insight into the complicated emission behavior of the single QD.

Ⅳ. CONCLUSION

We have demonstrated that the AAT method is not only compatible with the traditional exponential lifetime analysis, but also able to present small changes in temporal and spectral domains while the latter cannot. The AAT method is thus a more exquisite approach to characterizing a single QD, suitable for many research topics in studying the QD fluorescence. Examples are identifying single QDs, detecting the changes of emission behaviors when changing excitation condition or chemical environment, and so on. Finally we would like to point out this method is general, as it can be applied to other fluorophores like dye molecules and fluorescent proteins.

Supplementary materials: Intensity distribution, intensity-AAT correlation, additional AAT trajectory and distribution are available in the supplementary materials.

Ⅴ. ACKNOWLEDGMENTS

This work was supported by the National Science Foundation CAREER award (CHE-1554800).

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a. 美国康涅狄格大学材料科学研究院, 康涅狄格州, 斯托斯 06269;
b. 美国康涅狄格大学化学系, 康涅狄格州, 斯托斯 06269