Unraveling Size Dependent Bi‐ and Tri‐Exciton Characteristics in CdSe/CdS Core/Shell Quantum Dots via Ensemble Time Gated Heralded Spectroscopy
Einav Scharf, Rotem Liran, Adar Levi, Omer Alon, Nadav Chefetz, Dan Oron, Uri Banin
TL;DR
Researchers developed a new method to study multiexcitons in quantum dots, revealing how exciton interactions change with size and enabling clearer insights into their behavior.
Contribution
A novel ensemble time-gated heralded spectroscopy method is introduced to study biexciton and triexciton characteristics in quantum dot ensembles.
Findings
Biexciton binding energy measurements in CdSe/CdS QDs show a transition between attractive and repulsive exciton interactions.
Triexciton transitions and their spectral dynamics are resolved and isolated from higher-order multiexcitons.
The method enables robust, low-noise characterization of multiexcitons in quantum dot systems.
Abstract
Multiexcitons in quantum dots (QDs) manifest many‐body interactions under quantum confinement and are significant in numerous optoelectronic and quantum applications. Yet, the strong interactions between multiexcitons leading to rapid non‐radiative Auger decay introduce challenges for their characterization. While so far, the measurement techniques rely either on indirect methods or on single particle studies, herein a new method is introduced to study multiexcitons in QD ensembles utilizing spectrally resolved time‐gated heralded spectroscopy. With this approach, the biexciton binding energies is extracted in CdSe/CdS QD ensembles of several core/shell sizes, manifesting a transition between attractive to repulsive exciton‐exciton interactions. Additionally, for triexcitons, involving occupation of two excitons in the 1 s energy levels and one exciton in the 1 p energy levels, the open…
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TopicsSemiconductor Quantum Structures and Devices · Quantum Dots Synthesis And Properties · Strong Light-Matter Interactions
Introduction
1
Multiexciton (MX) states in colloidal quantum dots (QDs) serve as an ultimate model system for many‐body interactions of excitons under strong quantum confinement conditions. Unlike molecules, and more like bulk semiconductors, the QD states can occupy numerous excitons in close proximity and with strong overlap, without significantly altering the level structure.^[^ 1, 2 ^]^ However, unlike the bulk semiconductor limit, the quantum confinement condition induces strong Coulomb and exchange interactions with consequences of shifting the energy of MX states relative to the single exciton (X) state, and also invoking efficient Auger‐type non‐radiative interactions, the rate of which typically scales with the QD volume.^[^ 3, 4, 5 ^]^ In strongly confined QDs, non‐radiative Auger decay thus often effectively competes and overtakes the radiative decay of the MX states.^[^ 6 ^]^ This imposes limitations on the study and analysis of the energetic and dynamic characteristics of the MX states.
Beyond the fundamental many body physics, understanding the energetics and dynamics bears also important consequences for the utilization of MX states in diverse applications of QDs. First, in lasing, optical gain requires excitation of biexcitons (BXs).^[^ 7, 8, 9, 10 ^]^ The hastened Auger decay processes of the BX state though, counteract the radiative emission essential for lasing and hence numerous avenues, including rod architecture,^[^ 11, 12, 13, 14 ^]^ graded shell compositions,^[^ 15, 16, 17, 18, 19 ^]^ modifying the core/shell band alignment from type‐I (straddling) to type‐II (staggered),^[^ 20, 21 ^]^ giant shell QDs,^[^ 22, 23, 24 ^]^ quantum shell systems,^[^ 25, 26 ^]^ and most recently also coupled QD molecules,^[^ 27, 28 ^]^ have been pursued to manipulate and slow down the Auger decay rates. On the other hand, the increased Auger decay benefits the single photon emission characteristics of single colloidal QDs by quenching the BX emission and leading to strong photon antibunching without any optical filtering.^[^ 29 ^]^ MX states in QDs may also be relevant for light harvesting applications in photovoltaics and photocatalysis.^[^ 30, 31 ^]^
With these motivations in mind, we decipher herein open issues in the energetics and dynamics of MX states in QDs via introduction of a novel spectroscopic approach of ensemble time gated heralded spectroscopy. Considering first the simplest MX case of the BX state, the BX binding energy depends on the net Coulomb interaction between the charge carriers.^[^ 32 ^]^ An attractive (repulsive) X‐X interaction leads to a red (blue) shifted BX spectrum and a positive (negative) BX binding energy.^[^ 33 ^]^ In type‐I core/shell QDs, the electrons and holes are confined to the core, and the net X‐X interaction is usually attractive.^[^ 2, 33, 34 ^]^ This is also the case for core‐only QDs, in which the electrons and holes’ wave functions highly overlap.^[^ 1, 10, 35 ^]^ On the other hand, in a type‐II band alignment, the reduced attractive interaction between the separated electrons and holes is usually insufficient to overcome the strong repulsion between the same charge carriers (holes/electrons) that occupy the same region, resulting in a repulsive X‐X interaction.^[^ 21, 36, 37 ^]^ Herein we thus investigate CdSe/CdS core/shell QDs as an outstanding model system manifesting the transition between type‐I to (quasi) type‐II behavior based on core/shell dimensions.
Observing the X‐BX spectral shifts on the ensemble level is challenging as they are on the order of or smaller than the inhomogeneous spectral width, resulting in a significant overlap between the X and BX spectra.^[^ 38 ^]^ Prior studies rely on time‐resolved or quasi‐continuous‐wave (CW) power‐dependent photoluminescence and transient absorption measurements.^[^ 35, 39 ^]^ Examining the spectrum upon increasing excitation powers allows to follow the evolution of spectral peaks and features,^[^ 1, 40 ^]^ as in higher excitation power, the number of Xs increases leading to sequential state filling. On the ensemble level this X occupation will follow the Poisson distribution.^[^ 41 ^]^ Using transient absorption allows to study the spectrum and its dynamics at very short timescales, in which the MX signal is dominant.^[^ 42, 43 ^]^ Other works rely on low temperature measurements, which eliminate the thermal broadening, improving the spectral separation between the emitting states, although the inhomogeneous broadening still masks significant information, and further occurrence of charged exciton transitions also complicates the interpretation of the spectra.^[^ 44, 45 ^]^
The case of triexcitons (TXs) is even more intriguing, as a TX state involves occupation of an electron in the 1 *p_e_
- energy level and a hole in the 1 p 3/2 level, due to the twofold degeneracy of the 1 *s_e_
- energy level and driven by the repulsion between three holes in the 1s states (Figure 1e).^[^ 46 ^]^ The 1 p 3/2 − 1 *p_e_
- optical transition is higher in energy than the band gap X 1 s 3/2 − 1 *s_e_
- transition (marked by the blue and red arrows in Figure 1e, respectively), and is thus well separated from the X spectrum.^[^ 10 ^]^ Accordingly, emergence of a blue‐shifted peak indicates TX emission.^[^ 46 ^]^ Importantly, the TX emission can also arise from the recombination of an electron in the 1 *s_e_
- energy level (marked by the yellow arrow in Figure 1e), followed by relaxation of the 1 p electron and hole to the band edge.^[^ 47, 48 ^]^ The two TX transitions thus lead to emission in two colors, which can be utilized for two‐colored QD lasers.^[^ 49, 50 ^]^ Yet, when studying the TX emission, the 1 s 3/2 − 1 *s_e_
- transition is difficult to detect as it nearly overlaps the X and BX spectra.^[^ 48 ^]^
a–d) TEM images of the studied CdSe/CdS samples with core/shell radii of 2.2/1.5, 1.4/2.3, 1.8/5.2, and 2.2/5.5 nm (core/shell illustrations demonstrate the different sizes). e) Schematic of the configuration of the first energy levels in the QD, showcasing the optical transitions of an exciton (red arrow), biexciton (orange arrow), and triexciton (yellow and blue arrows) states. Notably, in the triexciton state, the 1 p energy level is occupied and there are two possible transitions: 1 s 3/2 − 1 se in lower energy (yellow arrow) and 1 p 3/2 − 1 pe in higher energy (blue arrow). f) Illustration of the experimental setup. An aggregate of QDs on a glass coverslip is illuminated by a pulsed laser, focused by an objective. The emitted light is collected by the objective and detected by a SPAD array at the output of the spectrograph. g) A fluorescence widefield image of the sample in the left panel, showcasing a typical 1.8/5.2 nm QD aggregate, marked by a circle. The right panel displays a bright‐field scanning transmission electron microscope (STEM) image of the same aggregate, containing 148 QDs. h) Power dependent spectra of 1.8/5.2 nm QD ensemble. The 〈N〉 values are calculated using the saturation power, extracted from Figure 2e. i) and j) 2D spectrum‐lifetime histograms of the ensemble in h extracted from the SPAD array measurements in low excitation power (i) and high excitation power (j). The black gaps are due to excluded noisy pixels. Brighter colors represent higher photon counts.
In order to more directly study MXs, so far usually single particle studies were deemed essential. When exciting single QDs with a pulsed laser, detection of two photons following a single excitation pulse clearly indicates BX emission, while detection of three photons indicates TX emission.^[^ 33 ^]^ Although the arrival time of multiple photons can be detected by splitting the emission signal to multiple single photon avalanche diodes (SPADs), it is challenging to precisely characterize the energies of the emitting states in the standard Hanbury‐Brown‐Twiss setup or its extensions.^[^ 47, 51 ^]^ An estimation of the emission energies can be done by filtering the signal in each SPAD or by diffracting the emission signal and scanning it with a SPAD.^[^ 48, 52, 53, 54 ^]^ This concept was used to measure the BX binding energy in single CdSe/CdS/ZnS QDs.^[^ 52 ^]^ Addressing the challenging TX state, in another work, the emission signal was split to four spectrally filtered SPADs, which allowed to distinguish between TXs that are emitted from the 1 s level and TXs that are emitted from the 1 p level.^[^ 48 ^]^ This study suggested that the 1 s 3/2 − 1 *s_e_
- transition was dominant in the TX emission. This improved the understanding of the energetics of multi‐excited states, yet it did not allow to construct the entire MX spectrum.
Lately, the promising method of heralded spectroscopy was introduced to study the electronic characteristics of single quantum dots, significantly improving the characterization of MXs.^[^ 33 ^]^ In this method, the spectrally resolved emitted light is detected by a SPAD array, providing simultaneous spectral and temporal information for each of the emitted photons along with photon statistics. This allows to post‐select events where multiple photons arrive in a single excitation pulse and therefore to study the spectrum and lifetime of each emitting state individually. Applying this approach on various single QDs revealed an attractive X‐X interaction in single CdSe/CdS/ZnS QDs,^[^ 33 ^]^ resolved the controversy regarding the BX binding energy of CsPbBr_3_ nanocrystals demonstrating that it is attractive,^[^ 39, 40, 43, 55 ^]^ and found two BX types in coupled QD molecules.^[^ 56 ^]^ Recently, this method was even extended to study TX emission in giant perovskite nanocrystals.^[^ 57 ^]^ These works therefore established the ability of heralded spectroscopy to resolve the ambiguity of previous methods in determining the emission energies of MXs.
These prior studies focused on single QDs to assign the X, BX, and TX in the cascaded photon emission. However, single particle studies require long integration times, ultra‐stable QDs, an elaborate statistical study, and the data can be noisy, especially when studying MXs that emit in low quantum yields.^[^ 48 ^]^ Furthermore, they may suffer from selection bias, as bright and stable QDs may not represent well the entire QD ensemble. Recently, a theoretical work laid the groundwork for measuring the TX lifetime and quantum yield in solution, demonstrating an opportunity to extract MX properties from an ensemble of QDs.^[^ 58 ^]^ Still, a complete spectral characterization of MXs in ensemble has yet to be addressed.
Herein, we introduce and utilize the ensemble time gated heralded spectroscopy method on small assemblies of QDs, using CdSe/CdS QDs as a model system. This approach is unintuitive as the detected multiple photon events can also originate from uncorrelated emission of Xs from multiple QDs.^[^ 58 ^]^ Yet, we combine the heralded spectroscopy approach with time gating to disentangle this complexity and obtain signals that are highly dominated by MX emission and selective to specific transitions. Advantageously, with this method, 10^4^‐10^6^ events of photon pairs and photon triplets may be collected within tens of seconds. These high signals improve significantly the photon statistics and signal to noise ratio and enable to extract the BX binding energy, to distinguish between the 1 s and 1 p TX emission, and even to observe MXs of higher orders. Additionally, by applying time gating with sub‐100 picoseconds (ps) resolution we resolve and find the lifetimes of the MX states, including the lifetimes of the TX decay pathways, a task which has hitherto eluded emission‐based studies. Applying this approach on small ensembles of CdSe/CdS QDs of different sizes, we show the crossover from a repulsive X‐X interaction in the quasi‐type‐II regime to an attractive interaction in the type‐I regime upon probing large cores with thin shells. Beyond the extraction of the MX characteristics for the model CdSe/CdS QD system, the approach opens up opportunities to study MX states in other QD systems which may be more challenging to address using alternative approaches or the single particle spectroscopy due to lower MX emission quantum yields. Especially considering the challenges of designing optimal hetero‐structured QD systems for demanding lasing applications, electroluminescent devices and additional applications, multiple parameters are varied in the structures which requires rapid analysis of their MX characteristics to derive reliable structure‐function correlations fed back to QD design.
Results and Discussions
2
Time Gated Heralded Multiexciton Spectroscopy Approach for QD Ensembles
2.1
A series of CdSe/CdS QDs of various core/shell dimensions (2.2/1.5, 1.4/2.3, 1.8/5.2, 2.2/5.5 nm core radius/shell thickness) were synthesized following a well‐established procedure.^[^ 59 ^]^ Figure 1a–d presents the transmission electron microscope (TEM) images of the various samples (the same images, size statistics, absorption and photoluminescence spectra are in Figure S1 (Supporting Information). To measure the optical properties of MXs in the QDs, a solution of CdSe/CdS QDs in 2.5% poly(methyl methacrylate) in toluene, was spin coated on a glass coverslip and the experiments were conducted on an inverted microscope with an oil‐immersion objective, as schematically presented in Figure 1f. A widefield photoluminescence image of the QDs, exemplified for the sample with core/shell radii of 1.8/5.2 nm, respectively, revealed bright spots of aggregates of QDs, along with dim spots of single QDs (Figure S2, Supporting Information). To utilize heralded spectroscopy on ensembles of QDs, we focused our study on the QD aggregates. The QD aggregates were placed at the focus of the objective, where they were excited by a pulsed 405 nm laser. The emitted photons were collected with the same objective.
As illustrated in Figure 1f, the emission light is then focused into a spectrograph with a grating that diffracts the emission signal onto a SPAD array detector, such that simultaneous recording of each single photon is recorded alongside its energy. To verify the characteristics of the studied aggregates, a full optical‐structural correlation was performed utilizing our previously reported on‐chip correlation method (Figure 1g).^[^ 60, 61 ^]^ Briefly, an optical characterization on a glass substrate with lithographically predefined markers is followed by extraction of the region of interest with a dual focused ion beam‐scanning electron microscope. The extracted area is then processed into a lamella for scanning transmission electron microscope (STEM) characterization. The left image in Figure 1g is an optical widefield image of a 1.8/5.2 nm QD aggregate on a marked substrate. The right image is the bright‐field STEM image of the very same aggregate, revealing that it consists of 148 QDs (the complete optical‐spectroscopic characterization of this aggregate is presented in Figure S3, Supporting Information).
To study MX states on such an ensemble, we measured the photoluminescence spectral‐temporal characteristics of the aggregates for a series of increased laser excitation powers. Figure 1h presents the power‐dependent spectra of another 1.8/5.2 nm aggregate. Increasing the excitation power influences the emission spectrum in two manners; 1) The main emission peak blue shifts from 1.95 eV to 1.97 eV; and 2) Additional spectral features evolve in the region above 2.1 eV. These observations indicate that additional emitting states contribute to the spectrum in higher fluences.
In order to resolve and assign the additional peaks that arise in the spectrum we utilize the two unique abilities of the SPAD array detector: time‐gated spectroscopy and heralded spectroscopy. Time‐gated spectroscopy correlates the emission energy with the arrival time of each photon to reveal the fluorescence decay dynamics of different spectral features. Figure 1i,j presents 2D spectrum‐lifetime plots of the aggregate in Figure 1h under low and high excitation powers (average number of generated Xs per QD per pulse of 〈N〉 = 0.13 ± 0.03, 7 ± 2, respectively). The black gaps are due to excluded noisy pixels, randomly distributed in the detector. This exclusion is reasonable considering the high resolution of the measurement (0.6 nm per pixel), relative to the broad spectral features (≈30 nm). At low fluence, a spectral peak near the band gap energy appears, with essentially a long lifetime (Figure 1i). At high fluence (Figure 1j), there are clearly two spectral features: The one near the band gap manifests now much faster decay, while the emergent shoulder peak in high energy exhibits even shorter lifetime. These changes are all due to the emergence of MX emission.
To gain the detailed spectral‐temporal information on the MX states, we now implement the powerful heralded spectroscopy approach. To this end, we post‐selected photon pair events to study BX emission, and photon triplet events to study TX emission. Figure 2a presents the spectra of the first and second photons in the post‐selected photon pairs (in blue and pink, respectively) in low excitation power, showcasing a blue‐shifted spectrum of the first photon, which is the BX photon. In previous works on single QDs, this shift was attributed to the BX binding energy, resulting from the repulsive X–X interaction.^[^ 52, 56 ^]^ However, in ensemble of QDs this assignment needs to be revisited and examined, as photon pair events may include BXs as well as two uncorrelated X emissions from different QDs. However, in this latter case, both the first and second photons in such pairs would reflect the ensemble inhomogeneous distribution (e.g., different emission due to different QD size), and should not manifest noticeable shifts.
Time gated heralded spectroscopy in QD ensembles. a) Spectral characterization of photon pair events under low excitation power. Blue and pink dots represent the 1st (“BX”) and 2nd (“X”) photons, fitted to multi‐gaussian models (in blue and red, respectively). b) First 20 ns of the total lifetime at low power (〈N〉 = 0.13 ± 0.03). The colored areas mark the temporal conditions applied on the arrival time of the photon pairs. Accordingly, the photon pair spectra in a) only feature pairs where the 1st photon arrived in the first 1 ns after the laser pulse (marked by a blue area), and the 2nd photon follows (pink area). c) Spectral characterization of photon triplet events under high excitation power. The 1st (TX, purple, 1 ns arrival gate), 2nd (BX, blue, 3 ns gate), and 3rd (X, pink, 200 ns gate) photons are fitted to multi‐gaussian models. The gray areas in a, c are normalized spectra of all photon events. The gaps are due to excluded noisy pixels. d) Total lifetime at high power (〈N〉 = 7 ± 2), showcasing the arrival times of the 1st and 2nd photons (purple and blue areas, respectively). e) The number of single photon events, photon pair events, and photon triplet events per second versus excitation power in the top, middle, and bottom panels, respectively (black dots). The colored lines are fits to the probability to generate at least one, two, or three Xs (pink, blue, and purple, respectively) according to the Poisson distribution (R 2 is 0.98, 0.98, and 0.95, respectively). The saturation power (150 ± 35 nW) is extracted from the fit of the top panel and used in the fits of the middle and bottom panels. 〈N〉 is the ratio between the used power and the saturation power.
Further insight is obtained by utilizing the time gating capability of the data from the SPAD array. Due to non‐radiative Auger processes, as well as the enhanced radiative rate in BX relative to X states, the BX ought to manifest a short lifetime.^[^ 62 ^]^ Thus, to enhance the relative contribution of BX emission in the signal, only photon pairs where the first photon arrived in the first 1 ns after the laser pulse were post‐selected. These temporal conditions are illustrated in Figure 2b, which presents the first 20 ns of the total lifetime measured for this sample (the repetition rate was 2 MHz). The blue area highlights the first 1 ns, as the temporal gate of the arrival time of the first photon, and the pink area presents the arrival time of the second photon, which is limited to 200 ns after the first photon to reduce contribution of dark counts (the full temporal conditions are detailed in the Experimental section). Notably, the BX energies, obtained from this analysis, are robust to small changes in these temporal gates. For example, increasing the chosen temporal gate of the first photons by 50%, changes the peak energy by less than 1 meV (Figure S4, Supporting Information).
A similar strategy allows us to extract spectroscopic information also for TX states, which are much more difficult to study on the single particle level due to the low number of such emissive events in typical QD systems. To this end, Figure 2c presents the spectra of post‐selected photon triplets in higher excitation power. Here the first photon, in purple, represents the TX emission (temporal gate of 1 ns; purple area in Figure 2d). The second and third photons represent the BX and X emission in blue and pink, respectively (temporal gates of 3, 200 ns, in the blue and pink areas in Figure 2d, respectively). The BX spectrum is still blue shifted from the X spectrum, as in the photon pairs spectra at low power, validating further this analysis. The X emission (in pink) has a more complex spectrum shape, as it consists of multiple emitting states, such as charged Xs and also BXs. Interestingly, the TX emission features two distinct peaks, both are blue shifted from the X and BX peaks. The high energy peak, which is associated to emission from the 1 p state, is visibly enhanced in comparison to the full spectrum in gray, clearly showcasing the significance of the post‐selection used in heralded spectroscopy for the observation of MXs.
Figure 2e presents the number of single photons, photon pairs, and photon triplets per second (in the top, middle, and bottom panels, respectively), as a function of the excitation power. In the three highest excitation powers the counts start to rise, assigned to transition to a quasi‐CW regime, where the multi‐excited QDs emit rapidly during the laser pulse (Figure S5, Supporting Information),^[^ 23 ^]^ hence they are neglected in this analysis. The power dependence plots are fitted to the probabilities to generate at least one, two, or three Xs, respectively, according to the Poisson distribution.^[^ 10 ^]^ The saturation power is extracted as a fitting parameter from the single photons’ fit in the top panel and is later used to fit the photon pairs and triplets in the middle and bottom panels. This shows that by applying a short time gating on the arrival times of the first photons, the detection ratio of MXs to multiple Xs is sufficiently high to produce a good fit (additional examples, demonstrating this, are in Figure S6, Supporting Information).
In order to further analyze and discuss the BX behavior in the QD ensemble, we first characterize fully the emission spectrum under low excitation fluence (〈N〉 = 0.13 ± 0.03), where the probability to generate TXs is significantly lower than the probability to generate BXs (by more than an order of magnitude). To extract the BX binding energy, we performed a time gated heralded analysis on the photon pair events, such that the first photons arrive in 2 ns bins and the second photons arrive later. Figure S7 (Supporting Information) presents the emission peaks of the first and second photons as a function of time, interestingly, both red shifting with time. Figure 3a,b demonstrates the spectra, achieved by this analysis, exemplified for the case where the first photons arrive in the first 2 ns (Figure 3b) and the second photons follow (Figure 3a). These spectra are well fitted to a multi‐gaussian model. To understand the contributing emitting states, we followed the emission of a single QD in low excitation power and found that its emission peak redshifts with time as well (Figures S8,S9, Supporting Information). By following the energies of the “on” and “off” states within its emission intensity trace (likely representing neutral and charged X emission),^[^ 63 ^]^ we were able to attribute slower (faster) and red (blue) shifted emission to the neutral (charged) X emission. Therefore, the apparent red shift with time in the single QD is a result of the transition between these two emitting states. Hence, the charged X state must be considered in the analysis of the emitting states in the QD ensemble as well.
Biexciton characterization in QD ensembles. a,b) Spectra of the 2nd and 1st photons (pink and blue dots, respectively) within the photon pair events at low excitation power (〈N〉 = 0.13 ± 0.03), with time gate of 2 ns for the 1st photons. The 2nd photons in a) are fitted to a sum of two gaussians (dashed black line). Its components represent neutral and charged X emission (black and red lines, respectively), and their peak energies are marked with the same color code under the spectra. Similarly, the 1st photons’ spectrum in b) is fitted to a sum of three gaussians (dashed black line). The black and red components are the same as in a), representing neutral and charged X emission, respectively. The blue line represents BX emission. c) and d) present the temporal evolution of these states (bins of 2 ns for the arrival time of the 1st photons in the pairs), as the relative integrated area under each fitted gaussian in the spectra of the 2nd and 1st photons (in c and d, respectively). In the 2nd photons in c the contribution of the X increases and that of the charged X decreases. In the 1st photons in d the contributions of the X and charged X increase and that of the BX decreases rapidly. Later the contribution of the charged X slightly drops. The lines are multi‐exponential fits.
Consequently, we fitted the spectrum of the second arriving photons in the pairs (corresponding to the singly excited state) to two gaussian components (Figure 3a), one in lower energy (1.942 eV, in black), to represent the neutral X, according to the energy of the X in longer times (Figure S10, Supporting Information), and another peak in higher energy (1.960 eV, in red) to represent the charged X emission. The first photons’ spectrum was fitted to three gaussian components (Figure 3b), one in high energy (1.979 eV, in blue) according to the energy of the first photons in short arrival times (Figure S4, Supporting Information), to represent the BX emission, and the two X and charged X components from Figure 3a, to account for emission from two singly excited QDs. This multi‐gaussian fit was applied for each of the time‐dependent spectra of the second and first photons, in order to follow the temporal evolution of the emitting states.
Figure 3c presents the relative integrated area under the fitted gaussian components of the neutral X (in black) and the charged X (in red) versus time for the second photons in the pairs (following the example in Figure 3a). It shows a simultaneous decrease in the relative contribution of the charged X and growth in the relative contribution of the neutral X, in accordance with the longer lifetime of the neutral X. To extract the rates of the changing contributions of the neutral and charged Xs, we applied a bi‐exponential fit, following a first‐order kinetic model. We used this analysis to estimate the lifetime of the charged X state. Therefore, the rate constant of the slow component was fixed to 1/105 ns^−1^, as 105 ns is the lifetime of the neutral X according to the total lifetime of all photons at 〈N〉 = 0.13 ± 0.03, presented in Figure S11 (Supporting Information) (the total lifetime in Figure S11, Supporting Information, exhibits three lifetime components of 2.4 ± 0.1, 13.8±0.2, 105±1 ns). The bi‐exponential fit of the time‐dependent relative area contributions, yielded a rate constant, analogous to a lifetime of 8±1 ns for the fast component. This rate is similar to the intermediate lifetime component in Figure S11 (Supporting Information), which was previously attributed to charged X emission.^[^ 28 ^]^
Figure 3d presents the relative integrated area under each of the gaussian components within the first photons’ emission (following the example in Figure 3b). It exhibits a fast decay of the BXs’ relative contribution (in blue), along with an increase in the relative contribution of the other two peaks. Then there is a slight decrease in the contribution of the charged X (in red). A mono‐exponential fit to the BX curve yielded a rate constant, corresponding to a lifetime of 4.7±0.1 ns, which is similar to the short lifetime component of all photons (Figure S11, Supporting Information), attributed mostly to BXs at the weak excitation regime.^[^ 28 ^]^ The X curve was fitted to a tri‐exponential fit with rate constants of 1/4.7, 1/8, 1/105 ns^−1^, reflecting the influence of all the emitting states on the change in the relative contribution of the X component. The dynamics of the relative contributions of the states in Figure 3d resemble the kinetics of consecutive first order reactions. Therefore, we treated the charged X state as the intermediate in such mechanism. Its curve was fitted to a difference between exponents, with formation component with a rate of 1/4.7 ns^−1^, according to the BX rate, and decaying components with rates of 1/8, 1/105 ns^−1^, according to the neutral and charged X rates. Interestingly, the same initial rate for the decay of the BX and the formation of the X components showcases that the BX state relaxes either into the neutral or the charged X states. Eventually, in longer times, the charged X component decays as well, as the neutral X component becomes the dominant emitting state.
Extracting Quantitative Binding Energies–the Case of Biexcitons
2.2
Following this understanding of the BX behavior in the aggregates, we extract the BX binding energy, the energy difference between the neutral X emission and the BX emission. We obtain a repulsive BX binding energy of ‐37 ± 4 meV for this ensemble.
We next utilize time‐gated heralded spectroscopy in QD ensembles to study and to accurately determine the BX binding energy, for a series of CdSe/CdS QDs of various sizes (TEM images in Figure 1 and Figure S1, Supporting Information). Given the freedom in tuning either core or shell sizes, it is clear that the rapid study enabled by the ensemble gated heralded spectroscopy is essential here. Figure 4a–d demonstrates the spectra of the first and second photons within the post‐selected photon pairs for the various samples. The sizes are noted in the top left corner of each panel, as the core/shell radii. Notably, the aggregates in Figure 4b–d showcase a blue shifted emission of the first photons’ (BX) spectrum, whereas Figure 4a showcases a red shifted emission of the first photons’ spectrum.
Size dependent behavior of multiexcitons in CdSe/CdS core/shell QDs. a–d) Photon pair spectra of samples from different sizes of CdSe/CdS QDs (core/shell illustrations demonstrate the different sizes). The spectrum of the 1st (BX) photons (blue) is blue shifted from the spectrum of the 2nd (X) photons (pink), for QD sizes 1.4/2.3, 2.2/5.5, and 1.8/5.2 nm (b, c, d, respectively). The 2.2/1.5 nm QDs exhibit a red shifted spectrum of the first photons, reflecting dominant attractive interaction. The gray area is a normalized spectrum of all photon events. The gaps are due to excluded noisy pixels. e) The BX binding energy of the various QD ensembles versus the X energy. f) The BX binding energy of the various QD ensembles versus the ratio between the core radius and the shell thickness. The BX binding energy becomes less negative in higher ratio. Inset: illustration of a BX in a quasi‐type‐II core/shell band alignment, where the holes in the valence band are confined to the core (blue) and the electrons in the conduction band are delocalized to the shell as well (red).
Figure 4e presents a summary of the BX binding energies of all the QD ensembles as a function of the X energy. For the larger 2.2/5.5, 1.8/5.2 nm QDs the BX binding energy was extracted following the analysis introduced above (Figure 3). For the smaller 2.2/1.5, 1.4/2.3 nm QDs the X energy shift with time was smaller, thus two components of neutral X and BX were sufficient to fit the data and extract the BX binding energies (Figure S12, Supporting Information, presents example of this analysis, Figure S13, Supporting Information, summarizes the charged X energies of the two larger QD samples). Additionally, a higher excitation power was required to observe sufficient BX signal in the smaller QDs. Thus 〈N〉 ≈ 1 was used (instead of 〈N〉 ≈ 0.1 for the larger QDs). The average BX binding energies were 2 ± 1, −13 ± 3, −29 ± 3, −37 ± 5 meV for the QDs of sizes 2.2/1.5, 1.4/2.3, 2.2/5.5, 1.8/5.2 nm, respectively. In prior work we studied 1.35/2.1 nm QDs on a single particle level and found a BX binding energy of ‐14 ± 6 meV, in line with the BX binding energy of the similar sized 1.4/2.3 nm QD ensemble study performed herein.^[^ 56 ^]^ Additionally, measurements of single 1.8/5.2 nm QDs revealed a BX binding energy of −31 ± 9 meV (Figure S14, Supporting Information). The similarity between the results of single QDs and ensembles of these sizes of QDs further supports the validity of utilizing heralded spectroscopy in small QD ensembles.
The size dependence of the BX binding energy is attributed to the quasi‐type‐II band alignment of the CdSe/CdS QDs (illustration of the charge carriers’ wave functions in a quasi‐type‐II system is in the inset of Figure 4f). On the one hand, when the core is bigger, the confinement of the holes decreases, reducing the X–X repulsion. On the other hand, when the shell is thicker, the much lighter electrons experiencing also a very small core/shell conduction band‐offset potential, delocalize further and the attractive interaction between the electrons and holes is reduced, enhancing the apparent X–X repulsion. Accordingly, in the transition from 1.4/2.3 nm QDs to 1.8/5.2 nm QDs the interaction becomes more repulsive, showcasing dominance of the effect of the thicker shell. To elucidate this dual size dependence, the BX binding energies can be compared to the ratio of the core radius to shell thickness, showcasing a consistent reduction in the repulsion with the increasing ratio (Figure 4f).
As mentioned, QDs of 2.2/1.5 nm exhibited an attractive BX binding energy. Comparing these QDs to the QDs of 2.2/5.5 nm, with the same core size, which exhibit strong X‐X repulsion, reveals yet again a significant dependence on the shell thickness. When the shell thickness is reduced from 5.5 to 1.5 nm, the electrons localize closer to the holes in the core, which enhances the attractive interaction between them. The overall attractive X–X energy is consistent with the typical BX binding energy in type‐I core/shell QDs.^[^ 33, 34 ^]^ Measuring an attractive BX binding energy is significant, as it showcases the ability of this method to observe the crossover from a quasi‐type‐II to a type‐I band alignment.^[^ 37 ^]^ Moreover, in an ensemble, detection of two photons where the first one is in higher energy might be confused with other effects, such as the size‐dependent quantum confinement and energy transfer.^[^ 64, 65 ^]^ Yet, observation of two photons where the first photon is in lower energy cannot be explained by these phenomena, further solidifying the validity of this technique.
Analysis of Triexciton Energetics and Branching Emission Dynamics
2.3
Following the quantitative understanding of the BX emission in the QD ensembles, we analyze the behavior of the 1.8/5.2 nm QD ensemble, presented in Figures 1, 2, 3, under higher excitation fluence (〈N〉 = 7 ± 2) in order to understand the TX emission. We applied a similar approach with a multi‐gaussian fit to the spectrum, combining heralded spectroscopy analysis with time gating. The spectrum of the first photons within the photon triplets, as appears in purple in Figure 2c, was fitted to 5 peaks (Figure 5a). The red and blue peaks represent the X and BX emission, respectively. As in the BX analysis, the first photons within the triplets may exhibit contribution of emission of Xs and BXs, due to multiple photons detection of emission from different QDs within a single excitation pulse. The single exciton peak energy extracted in this high excitation fluence is 1.960 eV (relative to 1.942 eV at lower fluence) as higher contribution of charging is expected. Indeed, the spectrum of the third photons in the triplets (pink spectrum in Figure 2c) is blue shifted relative to the X spectrum in lower power (pink spectrum in Figure 2a). However, to maintain a reasonable number of peaks we prefer to use in the fit procedure a slightly modified position for this effective “X” peak, at 1.960 eV.
Time‐gated heralded spectroscopy characterization of multiexcitons. a) Spectrum of the 1st photons within the photon triplets (from the analysis in Figure 2c) in purple dots, fitted to a sum of 5 gaussians (dashed black line: red – X and charged X; blue – BX; green, magenta, and cyan – MXs of higher order, TX and more). The peak energies are marked by lines below the spectrum. b) Illustration with the assignments of the emitting states of each of the 5 gaussians in a. The red and blue peaks are attributed to X and BX, respectively. The green and magenta peaks are TX emitted from the 1 s 3/2 − 1 se and 1 p 3/2 − 1 pe transitions, respectively. The cyan peak is a MX of higher order, for example a tetraexciton, as appears in the illustration. c,e,g) Time gated spectra of the 1st, 2nd, and 3rd photons within the photon triplets, over temporal ranges of 0–1, 0.4–3, and 1–3 ns, and time bins of 20, 50, and 100 ps, respectively. The two vertical lines in the range of 2–2.1 eV are detector artifacts due to excluded noisy pixels. Each time‐dependent spectrum is fitted to the 5 gaussians in a. d,f,h) Temporal evolution of the emitting states in the 1st, 2nd, and 3rd photons, respectively (time bins of 20 ps for the 1st and 2nd photons and 50 ps for the 3rd photons), fitted to mono‐exponential decays. The dashed gray line in d is the instrument response function (IRF). The color code of the decay curves is consistent with panels a and b.
Focusing on the first (TX) photons within the triplets, besides the X and BX components, the spectrum is well fitted to three additional peaks at higher energies (centered at 2.01 eV in green, 2.1 eV in magenta, and 2.175 eV in cyan). In order to assign these peaks, we performed time gated analysis for the photon triplets. Figure 5c presents the temporal evolution of the emission spectrum of the first photons in the triplets where each spectrum represents a 20 ps bin of the first 1 ns after the pulse, going from t = 0 in light purple to t = 1 ns in dark purple (the time‐gated build‐up of the spectrum in bins of 20 ps is in Figure S15, Supporting Information). The higher energy peak, associated with high‐order MX emission (TX and above) according to its high energy relative to the band edge peak,^[^ 10 ^]^ decays rapidly, likely due to the fast relaxation of these multi‐excited states. The lower energy peak red shifts, due to the relaxation of the blue shifted BX photons.
Similarly, Figure 5e(g) presents the time‐dependent spectra of the second (third) photons within the photon triplets in the range of t = 0.4‐3 ns (t = 1–3 ns), ranging from light to dark blue (pink), in time bins of 50 ps (100 ps; For further details see Section S2 in the Supporting Information). The two vertical lines in the spectra in the range of 2–2.1 eV are detector artifacts due to excluded noisy pixels. The spectra were fit to the 5 gaussians as in Figure 5a. Figure 5d,f,h presents the results of the temporal evolution of the intensity of each peak for the first, second, and third photons within the triplets, respectively (the color code is the same as in panels a and b; the time bins are 20 ps in panels d, f and 50 ps in panel h). The contribution of the low energy component in the first photons’ spectrum was negligible, and was thus disregarded in Figure 5d (the same was done for the high energy component in the second photons in Figure 5f and the two higher energy components in the third photons in Figure 5h).
Fitting these curves allows to extract the rate constants, reflecting the lifetimes of each of the spectral features in Figure 5a. Notably, a mono‐exponential decay is detected for each peak, further validating the analysis that yields the well‐distinguished emitting states (see Table S1, Supporting Information, for all the extracted lifetimes). The ability to isolate these states is significant when considering the large extent of their spectral overlap, as appears in Figure 5a.
Following these findings we assign the peaks: First, the red peak at 1.960 eV represents the X or the charged X, as illustrated by the red arrow in Figure 5b. Its lifetime of 6.3 ± 0.5 ns can be extracted from the time‐gated analysis of the third photons within the photon triplets (Figure 5h; or 6.1±0.2 ns according to the second photons in Figure 5f). Fitting the total lifetime of all the photons in this excitation power (〈N〉 = 7 ± 2) reveals a tri‐exponential decay with lifetimes of 0.70 ± 0.01, 8 ± 1, 100 ± 45 ns (Figure S11, Supporting Information), where the long component has a large error as it represents less than 1% of the decay behavior, as is characteristic of “delayed emission”.^[^ 66 ^]^ The lifetime of the X from the time gated analysis is similar to the lifetime of the intermediate component in the total lifetime, supporting its assignment as a charged X.^[^ 28 ^]^ This indicates that in high excitation powers, there should be a contribution of charged MX emission as well, yet it may be less emissive due to non‐radiative Auger decay.
The blue peak in Figure 5a corresponds to the BX state, as illustrated by the blue arrow in Figure 5b. Its contribution appears in the first, second, and third photons’ spectra (Figure 5d,f,h), and its average lifetime is 1.5 ± 0.1 ns (detailed results in Table S1, Supporting Information). This lifetime fits the fast component in the total lifetime of all photons at 〈N〉 = 0.13 ± 0.03 (2.4 ± 0.1 ns; Figure S11, Supporting Information). This component is dominated by BX emission, as in this power the probability to generate MXs of higher orders is low.
From the decays of the first photons within the triplets (Figure 5d), the lifetimes of the peaks centered at 2.01 and 2.1 eV were 323 ± 5 and 280 ± 30 ps, respectively. The lifetime of the peak centered at 2.175 eV was shorter, similar to the IRF (in a dashed gray line in Figure 5d). From the second photons’ spectral‐temporal analysis (Figure 5f), we extract similar lifetimes of 428 ± 3, 360 ± 10 ps for the 2.01 and 2.1 eV transitions, respectively, consistent with contribution of the high energy components also to this signal. The slightly longer values are expected considering the preselection of the time‐gated analysis, taking into account only photons with arrival time above 0.4 ns. Similarly, from the third photons’ spectral‐temporal analysis (Figure 5h), we extract a lifetime of 520 ± 20 ps for the peak centered at 2.01 eV, even longer due to consideration of later emitting photons. Interestingly, the BX and charged X states exhibited an analogous behavior to an intermediate in consecutive first order reactions. Thus, they were fitted to a difference of exponentials with formation lifetime of 520 ps. This showcases the relaxation cascade of MXs to form the BX and X states.
Finding such similar lifetimes for the 2.01 and 2.1 eV components is striking, especially without applying any constraints. It showcases that these components reflect indeed a decay from the same initial TX state. Therefore, we attribute the 2.01 eV component to emission of a TX through the 1 s 3/2 − 1 *s_e_
- transition and the 2.1 eV component to emission of a TX through the 1 p 3/2 − 1 *p_e_
- transition (green and magenta arrows in Figure 5b). These states should exhibit a similar observed lifetime as upon decay of the TX, either from the 1 s or the 1 p state, the remaining BX state only occupies the 1 s level, due to immediate (≈1 ps) relaxation of the 1 p exciton to the band edge.^[^ 67 ^]^ This is indeed well reflected in the close values extracted as the decay rates of both TX components.
The stark difference between the shorter lifetime of the 2.175 eV component and the similar longer lifetimes of the 2.01 and 2.1 eV TX components showcases that the high energy component (2.175 eV) reflects emission of higher‐order of MXs. The cyan arrow in Figure 5b demonstrates this assignment as a tetraexciton state, yet this state involves multiple high‐order MX states that cannot be temporally resolved by this method, as their fast lifetimes exceed the resolution of the detector. Nonetheless, we were able to distinguish between emission of TXs and MXs of higher order, which is very difficult in emission‐based measurements.
Examining the energy separation between the X and the two TX components provides further validation to their assignments (the energy spacings between all the emitting states in the 1.8/5.2 and 2.2/5.5 QDs are in Figure S16, Supporting Information). First, the separation of the 1 p peak (at 2.1 eV) from the 1 s X peak (at 1.942 eV) is in line with the literature.^[^ 48, 49, 68, 69 ^]^ Second, the 1 s TX emission is blue shifted by 68 meV from the neutral X peak at 1.942 eV, which is almost twice higher than the blue shift of the BX emission (37 meV). This shows that an additional X occupation leads to a similar effect of X–X interactions. Yet, the additional repulsion, in a TX state relative to the BX state, is slightly weaker (additional 31 meV) due to lower overlap between the wave functions of the charge carriers in the 1 p state and charge carriers in the 1 s state. This shift between the X and 1 s TX states is the TX binding energy. The TX binding energies are −61 ± 8 and −47 ± 3 meV, for the 1.8/5.2 and 2.2/5.5 nm samples, respectively. The stronger repulsion in the smaller QDs is consistent with the trend of the BX binding energies.
The power‐dependent study of the QD ensemble allowed to follow the evolution of the MX states with the excitation power. Interestingly, plotting the integrated area under each of the two TX gaussians as a function of the excitation power fits well to the probability to generate at least three Xs, according to the Poisson distribution (Figure S17a, Supporting Information). Additionally, plotting the area versus power on a logarithmic scale reveals linear trends with similar slopes (2.0 ± 0.2 and 1.9 ± 0.2 for the 2.01 and 2.1 eV peaks, respectively, see Figure S17b, Supporting Information), where the quadratic growth with excitation power, rather than cubic growth, is due to the onset of emission of MXs of higher order.^[^ 45 ^]^ In contrast, the onset of the peak at 2.175 eV is in higher excitation power, it does not fit well to the probability to generate at least three Xs, and the log‐log behavior of the area versus power showcases a larger slope (3.5 ± 0.7, Figure S17, Supporting Information). This yet again solidifies the assignment of the 2.01, 2.1 eV components as TX and the 2.175 eV component as a MX of higher order.
From this analysis, we find that the 1 s TX emission is stronger than the 1 p TX transition, with a contribution of 75%‐25%, respectively. This contribution is independent of the excitation power (Figure S18, Supporting Information), in line with branching decay kinetics with two competing emission pathways from the same electronic TX state. The dominance of the 1 s TX pathway stems from its higher rate due to the higher degeneracy in the 1 s state, with 2‐4 recombination pathways, in contrast to the single recombination pathway of the X in the 1 p level. Another factor that affects the ratio of the competing TX emitting pathways is the overlap between the electron and hole wave functions.^[^ 48 ^]^ Simulating the geometry of the 1.8/5.2 nm core/shell QD and solving the Schrödinger‐Poisson equations for single Xs using an effective mass approximation (following our previously reported procedure),^[^ 70 ^]^ yielded an electron‐hole overlap integral of ≈0.6 for an X in the 1 s state and ≈0.2 for an X in the 1 p state, neglecting multi‐carrier interactions. The higher overlap between the charge carriers in the 1 s state relative to the charge carriers in the 1 p state, contributes to the dominant 1 s 3/2 − 1 *s_e_ * TX transition. Considering the observed lifetime of the TX states (averaging the two extracted lifetimes yields ≈300 ps) and the branching 1 p to 1 s ratio, we can extract the lifetimes of the two TX transitions (details in Section S2, Supporting Information). The lifetimes for the 1 s 3/2 − 1 *s_e_
- and 1 p 3/2 − 1 *p_e_
- TX transitions are 0.4 and 1.2 ns, respectively.
Our results are somewhat different than the previous elegant work on single QDs, which reported that the 1 s TX emission dominates the emission entirely.^[^ 48 ^]^ In this work, the 1 s versus 1 p TX pathways were distinguished by spectrally filtering the MX signal. Accordingly, the photon statistics depend on the spectral position of the band‐pass filters and the purity of the emitting states. For example, positioning the filter for the 1 p 3/2 − 1 *p_e_
- transition in energy which is too high can bias the signal to detection of higher order of MXs, which emit in lower quantum yields. This can decrease the apparent contribution of the 1 p TX state. In contrast, our proposed method obtains the full spectrum of the MX states providing improved accuracy.
Finally, following the assignment of the high energy peak as a high‐order MX, the heralded spectroscopy method can be further extended to resolve events of 4 photons emission within a single laser pulse. Figure S19 (Supporting Information) presents such a case for the 1.8/5.2 nm QD sample. Interestingly, the first photons, exhibit a large contribution of the peak at 2.175 eV, along with a lower energy peak that can contain contribution of the 1 s transition of the tetraexciton. The second photons, mostly representing the TX emission, showcase a significant contribution of the peaks at 2.01 and 2.1 eV. Going to higher excitation power of 〈N〉 = 10 ± 2 even allows to distinguish 5 photon events, revealing two additional peaks centered at 2.25, 2.35 eV, representing much higher order MXs (Figure S19, Supporting Information). This example further demonstrates the strength of heralded spectroscopy in resolving MX states. Transitioning from single particles to small ensembles provides high photon rates, which enables to observe higher emitting states, and perform time gating in high temporal resolution. Combining this with the careful time‐gated analysis, provides reliable spectral‐temporal information for each QD sample.
Conclusion
3
To conclude, we demonstrated a new approach to utilize heralded spectroscopy for the study of MXs in QDs. By using this technique applied to small QD ensembles, we found the BX binding energies of CdSe/CdS QDs of several sizes, demonstrating a wide range of X‐X interaction strengths. Additionally, we characterized and assigned the two competing pathways of TX emission, and also observed MXs of higher order. Usually, these properties were so far tackled utilizing measurements of single particles. In this work, we used the time gating alongside the heralding technique as a “magnifying glass”, allowing to observe single particle properties on the ensemble level. Moreover, replacing single particle studies with an ensemble characterization produced much higher photon rates, which diminished the typical noise in single particle measurements. Other previously demonstrated ensemble approaches for the characterization of MXs have only provided partial information. For example, amplified spontaneous emission studies clearly presented amplification of the 1 s and 1 p transitions, indicating MX emission and allowing to extract their energies.^[^ 10, 69 ^]^ Yet, amplified spontaneous emission is a measure of optical gain, thus the extracted energies reflect the emitting states as well as the absorption of the excited states, which can affect the observed energies. Additionally, these studies have not tackled the MX decay dynamics, without complementary methods, such as transient absorption.^[^ 49 ^]^ In contrast to other ensemble techniques, our approach not only provides the energies and lifetimes of the emitting states, yet it can also decipher and distinguish between highly overlapping optical transitions. For instance, in amplified spontaneous emission studies the onset power of the 1 p transition clearly indicates TX emission, yet it does not distinguish between TXs and MXs of higher order. Moreover, our method allowed to extract the energies and assign four different states for the 1 s transition, the neutral and charged X, the BX, and the TX, providing crucial information regarding the many body interactions in the QDs. Thus, following the time‐gated heralded spectroscopy method can effectively yield MX properties of different QD systems with numerous design variables related to the heterostructure architecture, and can be applied even on systems in which the single particle characterization is more challenging. This will allow to obtain reliable data on the spectroscopy and dynamics of MX states in complex QD systems, with multiple variables such as core/shell dimensions etc. Such information can aid to achieve design principles for QDs tailored toward optoelectronic applications where MX characteristics are of relevance, including in lasing, light emitting diodes, single photon and quantum light sources, and even in photocatalysis.
Experimental Section
4
Optical Setup
The measurements were performed with an inverted microscope (Nikon ECLIPSE Ti) in the epi‐luminescence configuration. An oil‐immersion objective (×100 with a numerical aperture of 1.4, Nikon Achromat) was used to focus the excitation light and collect the emission signal. A 370 nm fiber‐coupled light‐emitting diode (Prizmatix) was used for widefield imaging for the selection of the aggregates. A 405 nm pulsed laser (EPL405, Edinburgh instruments) was used for point illumination (repetition rate of 2 MHz). Back‐scattered laser light was filtered by a 425 nm long pass dichroic mirror (DMLP425R, Thorlabs) and a long pass 460 nm filter (ET460lp, Chroma). Then the light was focused to the slit of a spectrograph (Acton SP‐2150i, Princeton Instruments) with a telescope (AC254‐300‐A, AC254‐100‐AB, Thorlabs). The light was diffracted by a grating in the spectrograph (300 grooves mm^−1^, blazed at 500 nm), then detected by a SPAD array (SPADλ, PI Imaging). The SPAD array was triggered by the laser, using a constant fraction discriminator (FLIM LABS) to turn the laser's NIM output to TTL. The IRF was 150 ps (gray dashed line in Figure 5d).
Time Gating
Time gating was used to enhance the contribution of MXs in the post‐selected multiple photon events, as well as to decrease the contribution of dark counts and avoid cross‐talk.^[^ 71 ^]^ Accordingly, photon pairs are considered only if the first photon arrived in the first 1 ns after the laser pulse and the second photon arrived between 0.8 and 200 ns after the first photon. The short‐time gate of the first photon is used to reduce the cases of detection of two Xs, that exhibit a longer lifetime (further details on the time gate of the first photon are in Figure S4, Supporting Information). The lower time gate of the second photon is in order to diminish the contribution of cross‐talk events and the higher time gate is to reduce the contribution of dark counts. For the smaller 2.2/1.5 and 1.4/2.3 nm QDs, which exhibited a shorter lifetime, a time gate of 60 ns was used for the second photons (instead of 200 ns). Applying the same reasoning, photon triplets are considered if the first photon arrived in the first 1 ns, the second photon followed it in times between 0.8–3 ns, and the third photon arrived in times of 0.8–200 after the second photon. In the photon triplets’ analysis, increasing the time gate of the second photon (BX) was done in order to increase the number of photon triplet events, and thus the signal‐to‐noise ratio in the time‐gated analysis in Figure 5. Similarly, in the time gated analysis of the BXs, presented in Figure 3, a time bin of 2 ns was used to increase the signal‐to‐noise ratio and obtain reliable fits of at least hundreds of photon pair events. Notably, the actual energies of the emitting states are determined by further fitting the spectra, a procedure which is independent to the specific chosen time bin.
Electron Microscopy
The image in Figure 1g is a bright‐field STEM image, acquired using a TITAN‐THEMIS transmission electron microscope operated at 300 kV. The images in Figure 1a–d are bright‐field TEM images, acquired using a Tecnai Spirit G2 operating at an acceleration voltage of 120 kV.
Statistical Analysis
The sample size was large, including 10^4^–10^6^ photon pair and triplet events, emitted from aggregates containing tens of QDs. Moreover, multiple aggregates were measured (12, 10, 17, and 10 aggregates of the 1.8/5.2, 2.2/5.5, 1.4/2.3, and 2.2/1.5 nm QDs). The collected data were post‐processed using MATLAB. The reported error values represent either standard deviations or 68% confidence intervals in the case of fitting results.
Conflict of Interest
The authors declare no conflict of interest.
Supporting information
Supporting Information
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