1. Introduction

The second-order photon cross correlation function, g⁽²⁾(τ), plays a critical role in characterizing the photon emission statistics of nanoscale light emitters including single molecules [1–3], quantum dots [4,5], carbon nanotubes [6], and diamond nanocrystals [7–9]. It is widely used to test the correlation of intensities on two different photon detectors in a Hanburry Brown and Twiss configuration [10]. In a pulsed laser experiment, it is well known that measuring a single quantum emitter will result in photon anti-bunching, which is signified by the disappearance of a peak at zero time delay. This arises from the fact that a single quantum emitter can only emit a single photon per excitation pulse, a property that is utilized in single molecule spectroscopy to prove the isolation of a single quantum emitter [1–13]. In fact, the area ratio between center peak and lateral or side peaks, R, has been used to determine the number of m identical emitters under illumination: R = (m-1)/m [10–12].

In some emitters, such as quantum dots, optical generation of multi-excitons can lead to the emission of multiple photons in a single excitation cycle via a quantum cascade process [14–16]. When the emission of these multi-excitons are not spectrally excluded in a g⁽²⁾ measurement, they contribute to the zero time delay peak. Thus, it has been shown that for a single, well isolated, quantum dot in the low pump intensity limit, R is directly proportional to the ratio between the bi-exciton quantum yield, Q_BX, and single exciton quantum yield, Q_X i.e. R = Q_BX/Q_X [13,17].

Recently, much effort has been devoted to the study of nanocrystal quantum dots (NQDs) with large bi-exciton quantum yields [17–19]. NQDs with high bi-exciton quantum yield are important for light amplification [20], multi-exciton generation [21,22], and photon pair source applications [23,24]. As Q_BX plays a critical role in understanding the relationship between NQD PL blinking [25,26] and the suppression of Auger recombination, determining Q_BX is an important part of this puzzle [17,18,27–29].

Importantly, g⁽²⁾ based Q_BX measurements hinge upon the complete isolation of individual NQDs. However, when the combined effects of both clustering and multi-exciton emission are considered together, the ratio R can be expressed as:

Here we assume that all m NQDs are identical: having equal excitation probabilities and identical Q_X and Q_BX values. Therefore, when measuring large Q_BX values, or rather, R > 0.5, an independent measure of NQD isolation becomes necessary [30]. Independent verification of NQD isolation might include high-resolution techniques such as AFM, SEM, or TEM. However, such techniques have limited utility in that they require additional specialized instrumentation, they are time consuming, and it is difficult to perform these high resolution microscopies and single dot spectroscopy on the same set of NQDs. Most importantly, these techniques are not appropriate for many types of samples such as NQDs embedded in polymer films or biological tissues. In this regard, an all-optical approach capable of disentangling the contributions of clustering and multi-exciton emission is essential to interpret g⁽²⁾ measurements without ambiguity.

We now demonstrate that the application of an appropriate time gate to a g⁽²⁾ measurement allows for the separation of various factors contributing to R. Specifically, this technique allows for the determination of the extent of clustering (m), as well as the Q_BX/Q_X ratio. Furthermore, by conducting a detailed analysis on the decay functionality of R with respect to the gate delay time (GDT), we determine general conditions required for effective utilization of this approach. Analyzing this decay functionality also yields a method for determining lifetimes of bi-exciton states (τ_BX). Existing approaches for measuring τ_BX involve fitting PL decays measured as a function of pump power with multi-exponential functions [31,32]. In contrast, our approach requires fewer fitting parameters, and more importantly, allows for the determination of τ_BX at low pump powers, where contributions from higher order multi-excitons are minimal. In this regard, this approach could serve as a valuable tool in investigating technologically important multi-exciton processes in NQDs, such as optical amplification and multiple exciton generation.

2. Time-gating principle

Our approach exploits the fact that the bi-exciton emission always precedes single exciton emission. We selectively detect the photon emissions of the single excitons through the use of a time gate (only photons arriving within the gate time are analyzed) where the beginning of the gate (the gate delay time – GDT) is set to be much longer than the lifetime of bi-excitons. The g⁽²⁾ function constructed out of these photons will be essentially void of all bi-exciton contributions, yielding the only the degree of clustering, with the familiar form of R:

We use the subscript TG to denote that time-gating has been applied. With the ability to remove all bi-exciton contributions, we are able to determine if a NQD is in fact well isolated without ambiguity, even if it has a large bi-exciton quantum yield. Furthermore, we can apply the degree of clustering, m, found by Eq. (2) into Eq. (1) to obtain the average Q_BX of the NQDs in the cluster as: $Q_{BX}=\left(R-R_{TG}\right)/\left(1-R_{TG}\right).$

Time-gating is a feature available on many Single Photon Avalanche Photodiodes (SPADs) where a digital input signal enables photon detection for a given time window (Fig. 1 ). Thus, a hardware based time-gating approach is well-suited for a measurement where the goal is to identify single emitters. While this hardware implementation is simple, it requires a separate photon correlation experiment to acquire g⁽²⁾ at different GDT values. Since such experiments are time consuming it becomes impossible to study R_TG as a function GDT, a study required to validate this approach. For this reason, here we utilize a software-based time correlated single photon counting (TCSPC) approach. Specifically, we record the photon arrival times originating from multiple detectors with respect to both the start of the experiment (macro-time) and to a common excitation sync pulse (micro-time) as depicted in Fig. 1(c). This allows for the simultaneous collection of both the PL decay dynamics and g⁽²⁾ function. Importantly, this approach allows for selective analysis of only photons that arrive after a certain time delay following the sync pulse, which enables the construction of a time gated g⁽²⁾ function for any arbitrary GDT value.

 

Fig. 1 Schematic of TCSPC implementations. (a) TCSPC setup allowing for hardware-implemented time-gating. (b) TCSPC setup allowing for software-implemented time-gating. (c) Details of TCSPC calculations. Photon arrival times are indicated by red circles while laser pulses are denoted by blue stars. PL lifetimes are determined by histogramming t values. Time-gating is implemented by using only photons arriving within the time-gated region, depicted by the yellow boxes. The gate-delay time (GDT) is also indicated. Anti-bunching plots are built up by histogramming the time differences, ΔT, between photons arriving on opposing channels.

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3. Results and discussion

We applied this software-based time-gating approach to CdSe/CdS core thick shell NQDs (a.k.a giant NQDs), which exhibit nonblinking PL with minimal residual intensity flickering [17,27,30,33]. In Fig. 2 , we display the PL decay dynamics, g⁽²⁾ before the time gating, decay of R_TG as the function of GDT, and g⁽²⁾ after application of GDTs for two different giant-NQDs (g-NQDs). While the observation that R is > 0.5 in both g-NQDs suggests that these g-NQDs have very high Q_BX, this measurement alone cannot rule out the possibility of NQD clusters. However, the decay of R_TG towards zero with the increase of GDT (Figs. 2(c) and 2(g)) and observation of near complete photon anti-bunching after time gating at 75 ns (Figs. 2(d) and 2(h)) provide clear evidence that these two NQDs are well isolated and R values in each case indeed provide a direct measure of the Q_BX.

 

Fig. 2 Single NQD data. Panels (a)-(d) and (e)-(h) correspond to different measurements/data sets. (a) PL lifetime from a single detector channel. Dash line: single exponential fit with time constant of 124 ns. (b) Standard g⁽²⁾ plot before time-gating. R = 0.50 (c) Plot of R_TG vs. GDT. Notice that the first 25 ns (before t = 0) are flat, this corresponds to the time before the sync pulse, due to the electronic delay in the system as is also seen in all lifetime plots. (d) g⁽²⁾ plot after time gating has been applied (GDT = 75 ns, R_TG(75) = 0.05). Note that most if this is due to cross-talk (sharp central spikes). (e) PL lifetime from a single detector channel. Dash line: double exponential fit with time constant of 23.7 and 110.5 ns. (f) Standard g⁽²⁾ plot before time-gating. R = 0.73 (g) Plot of R_TG as a function of GDT. (h) g⁽²⁾ plot after time gating has been applied (GDT = 75 ns, R_TG(7 5) = 0.13).

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However, R_TG values do not decay completely to zero as they should for single quantum emitters (cf. Figures 2(d) and 2(h)). Our analysis revealed that this residual value originates mainly from the sharp spike at zero time delay, which is composed of two peaks separated to one another by a time delay of ~30 ns. This doublet is the characteristic feature of cross-talk between the two SPADs of our HBT spectrometer. In general, several other factors such as the presence of dark-counts, background noise, and a pulse period that does not allow for complete NQD relaxation can also contribute to this residual value. The total contribution of these experimental artifacts together [10–13] can add error in the range of 2%-10% in determination of center peak area in our measurement.

Figure 2 further shows that R_TG for both g-NQDs decays exponentially with GDT (black dashed lines). While the R_TG of the first g-NQD (Fig. 2(c)), decays with a time constant of 17.5 ns, that of the second g-NQD, which possesses a higher Q_BX value (0.73 compared to 0.5) requires a longer decay time of 32.4 ns. This slow decay of R_TG demands longer GDT to reject all bi-exciton emission contributions in this measurement. The rate of decay of R_TG as a function of GDT can be related to the lifetime or quantum yield of the bi-exciton state (i.e., τ_BX or Q_BX). We calculate that the decay should follow an exponential dependence:

Additionally, our analysis of the decay of R_TG provides another powerful way to extract the recombination time of bi-excitons. By substituting the value of τ_X and α attained from Figs. 2(a) and 2(c) in Eq. (5), we determine τ_BX = 13.4 ns. While this τ_BX value is extracted without requiring any assumptions, it is close to a 14.0 ns lifetime that can be calculated from τ_X = 112 ns and Q_BX = 0.5, assuming the statistical scaling of Γ_Rad,BX. The τ_BX value extracted for the second g-NQDs (Figs. 2(e)-(h) also exhibits a similar agreement: τ_BX = 22.9 ns as calculated using Eq. (5), vs. τ_BX = 28.5 ns as calculated from a statistical scaling model (τ_X = 156 ns and Q_BX = 0.73). Our time-gating experiment is performed at low pump power (i.e., the average exciton population per pulse (μ) is below 0.1). The nearly single exponential decay shown in Fig. 2(a) indicates that bi-exciton emission is minimal, yet we can extract τ_BX via R_TG versus GDT data. In this way, our time-gated g⁽²⁾ approach presents a better and more accurate measurement of τ_BX in single-particle studies, as this extracted τ_BX value is free of contributions from higher order multi-excitons [34].

Thus far, we have considered isolated NQDs and demonstrated the power of our approach in rejecting multi-excitonic contributions to g⁽²⁾ measurements. However, the significance of our approach becomes even more evident when it is applied to a cluster of NQDs. In this case, our approach not only allows the extraction of the average Q_BX and τ_BX of NQDs in the cluster, it also provides some insight to the nature of clustering. Figure 3 provides a clear demonstration of this point.

 

Fig. 3 Single NQD data. (a) PL lifetime from a single detector channel. (b) Standard g⁽²⁾ plot, no time gating (R = 0.73). (c) Plot of R_TG vs. GDT. (d) g⁽²⁾ plot after time gating has been applied (GDT = 75 ns, R_TG(75) = 0.28).

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The conventional g⁽²⁾ measurement shown in Fig. 3(b) yields an R value of 0.73. While this value decays exponentially with GDT, the decay asymptotically approaches a value of 0.28. This provides clear evidence of clustering. A simple substitution of R_TG and R to Eq. (1) and Eq. (2) gives a value of m = 1.39 NQDs with an average Q_BX value of 0.63. Furthermore, by including the effects of clustering in the analysis for the decay of the R_TG, we extract the average τ_BX to be 6.2 ns.

Since a cluster of two identical dots will yield an R_TG value approaching 0.5, the value found here of R_TG = 0.28 (and thus m = 1.39) provides a clear indication that two (or more) non-identical dots are being examined. The cluster in this study could be composed of NQDs with unequal Q_X and/or unequal excitation probability. Specifically, for a cluster of 2 NQDs with equal excitation probability it can be shown that a difference in their Q_X value with a ratio of 1:0.2 can give rise to the R_TG of 0.28. On the other hand, for 2 NQDs with identical Q_X, a difference in excitation probability with the same ratio is necessary. The excitation probability difference can arise from a scenario in which 2 identical NQDs are separated by some distance and are excited by laser spot with Gaussian intensity distribution.

4. Summary

In summary, we have demonstrated a new method to successfully discriminate between clusters of similar particles and well isolated single emitters with a high bi-exciton quantum yield. Notably, this method has added utility by providing a means for estimating the size of a cluster, even if it contains particles having high Q_BX values, without the need for spectral rejection of multi-excitonic spectral lines. It can also be used as a powerful tool to determine the bi-exciton lifetime of a single particle with much less influence from higher order multi-excitons. Recently a post photon selection approach, in which first and second photon detection events in a g⁽²⁾ experiment are separated to extract the decay of BX is reported by Canneson et al. [33]. Although this approach can be considered as a more direct approach to extract BX decay, it is incapable of disentangling the contribution of multi-exciton emission and clustering in a g⁽²⁾ experiment. Furthermore, in the case of a NQD exhibiting PL intensity fluctuations, our approach can be extended to extract Q_BX, τ_BX and the degree of clustering for different emissive levels of g-NQDs that are often associated with different charge exciton states [27]. Additionally, while we have focused on colloidal quantum dot samples in this letter, this technique should be widely applicable to virtually any type of sample where single-particle analysis is relevant.

5. Methods

5.1 Data collection

Quantum dots with 5.5 nm CdSe cores having nominally 15 monolayers of CdS grown using a SILAR method [35] were drop cast onto glass coverslips. Pulsed laser excitation (405 nm, 70 ps pulse-width) was used to elicit PL (centered at 650 nm) from the NQDs, which was detected by a pair of SPADs (Perkin Elmer, SPCM-AQR14) in a custom setup with a HBT detection scheme as described above. Photon arrivals were recorded with a HydraHarp 400 TCSPC module (PicoQuant). A 532 nm long pass filter was used to reject laser scatter and a 775 nm short pass filter was placed in front of each detector to minimize cross-talk. The laser power was reduced to ~200 pW and focused to a near diffraction limited spot (NA 1.3, 100x). Values for μ are estimated to be μ ~0.1.

5.2 Data analysis

Time-gating was applied by removing all photons from the data stream having micro-time (t) values less than GDT as depicted in Fig. 1. Subsequently, g⁽²⁾ plots were created in the same fashion as is done without time gating: a histogram of time differences is created for photons arriving on different channels as depicted in Fig. 1(c).