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Auger Ionization and Tunneling Neutralization of Single CdSe/ZnS Nanocrystals Revealed by Excitation Intensity Variation I.Yu. Eremchev,*,† I.S. Osad’ko,† and A.V. Naumov†,‡ †

Institute for Spectroscopy RAS, Troitsk, Moscow 108840, Russia Moscow State Pedagogical University, Moscow 119991, Russia

‡

ABSTRACT: The dependence of the average times in on- and off-states of the fluorescence of a single semiconductor colloidal core/shell nanocrystal (so called quantum dot) on the intensity of the continuous laser excitation was studied. The mathematical expressions for the intensity dependence for on- and off-average times are obtained in the framework of a modified charging model, taking into account both the Auger ionization and tunnel neutralization mechanisms for transitions between neutral (on-state) and the ionized (off-state). The experiment with a single nanocrystal CdSe/ZnS shows that average on-time is proportional to 1/I2, whereas the average offtime is nearly independent of the intensity, I. These results agree with processes of Auger ionization and tunneling neutralization of the core in core/shell nanocrystals.

1. INTRODUCTION Blinking of fluorescence intensity under CW-laser excitation is the main feature of light emission from single core/shell semiconductor nanocrystals (NCs). Reasons for this blinking have been discussed intensively in the literature. Mainly two different mechanisms for blinking fluorescence are discussed extensively. The first mechanism was proposed by Efros and Rosen 20 years ago.1 In accordance with their model, blinking in fluorescence of core/shell NCs emerges because of processes of ionization and neutralization in the NC excited by CW laser light. If the core of the NC is neutral, the NC emits bright fluorescence. Intervals with bright fluorescence are on-intervals. If the core of the NC is ionized, the NC emits fluorescence with weak intensity. Intervals with weak intensity are off-intervals. Later, this model was named the “charging model”. The track of fluorescence intensity in the charging model looks like a random telegraph signal (RTS). Such blinking fluorescence was observed by many groups2−7 as well as in our measurements.8,9 Appearance of the second approach to the problem of blinking fluorescence was stimulated by observation of fluorescence tracks which were unlike RTS: fluorescence intensity fluctuates in both on- and off-intervals the way we cannot distinguish unambiguously on- interval from offinterval.10 To explain such fluctuations, Frantsuzov et al.11 proposed that fluctuations of the intensity in fluorescence tracks result from stochastic fluctuations of nonradiative transitions in NCs resulting from the existence of multiple recombination centers (MRC) in the NC. The MRC model ignores ionization−neutralization processes. One stated that the MRC mechanism is universal for NCs.12 To date, a huge number of fluorescence tracks of various types have been measured in many works carried out by various © 2016 American Chemical Society

groups. It seems that tracks of RTS-type were found in the majority of the observations. Such tracks can be described by the charging model. Therefore, mechanisms of ionization− neutralization in single NCs irradiated by CW laser light deserve comprehensive discussion and careful experimental study. Efros and Rosen1 proposed an Auger process for ionization of NCs. The process involves two excitons. The first exciton disappears, and its energy is transferred to the electron−hole pair of the second exciton. Such a process of Auger ionization results in a single time, tA, of ionization and exponential distribution for on-times. However, experiments carried out later by several groups2−7 revealed power-law distribution t−α with 1 < α < 2 for on-intervals. Such a distribution could not be explained by Efros−Rosen theory. An explanation of the power-law distribution for on-intervals was proposed by Osad’ko.13,14 He assumed that electron−hole pairs, j, localized in the core on the core/shell interface can exist in NCs. These pairs, j, take an active role in the processes of Auger ionization. Charge from jth electron−hole pair can be transferred to the shell if the pair borrows the energy from the exciton existing in the core. In such a model of Auger ionization, we have, in fact, several doorways, j, for the Auger ionization and therefore several ionization times, tAj . Because the average duration of the on-interval created via jth doorway depends on the ionization time tjA, several exponential on on on distributions Won j = exp(−t/tj )/tj , where tj is the average duration of on-times can be found. In accordance with refs 13 and 14, summation of the exponential distributions results in Received: June 30, 2016 Revised: August 17, 2016 Published: August 24, 2016 22004

DOI: 10.1021/acs.jpcc.6b06578 J. Phys. Chem. C 2016, 120, 22004−22011

Article

The Journal of Physical Chemistry C power-law distribution t−α found in experiments.2−7 Moreover, the value of the exponent α depends on radial distribution of the jth localized pairs near the core−shell interface.14,15 Rate of Auger ionization does not depend on temperature. However, the Bawendi group4 has shown that increasing the temperature from 10 K changes the view of on-time distribution. This fact proves that not only Auger processes can take part in the ionization of the core. Tunneling ionization involving energy of phonons will depend on temperature and can be used14 for the explanation of changes in the on-time distributions found in ref 4. It should be noted that tunneling ionization and Auger ionization will exhibit different dependence of the average on-time, τon, on the intensity of laser excitation. As to the mechanism of neutralization, we have to make choice between two mechanisms: (i) Auger neutralization16,17 involving excitons created in the charged core (trions) and (ii) neutralization of the core via direct tunneling of the charge from the shell. These mechanisms of the neutralization will exhibit different dependence on the excitation intensity. Therefore, they can be discriminated in experiments with changes of the excitation intensity as well. The influence of excitation intensity in distributions of on-/ off-intervals in blinking fluorescence of single CdSe/ZnS and Si nanocrystals was carefully studied in ref 18. Distribution of on-/ off-intervals demonstrated truncated power-law and power-law dependences, respectively. It was shown that on-interval durations were shortening with excitation power increasing, and the possible reasons for this effect were discussed. In spite of the performed investigations, the physical mechanisms of on-/off-transitions is still open. The main goal of the present paper is to make a new attempt to clarify mechanisms resulting in on-/off-jumps in CdSe/ZnS NC fluorescence intermittency. For this purpose, power dependences of the average on-/off-times were studied experimentally and theoretically in the framework of a modified charging model, taking into account both the Auger processes and tunnel transitions.

Figure 1. Absorption and fluorescence spectra for bulk sample with CdSe/ZnS NCs. The vertical arrow shows excitation wavelength (red edge of absorption), and two vertical dashed lines indicate spectral region where fluorescence was registered.

CF Nikon 100×, 0.95 NA) were captured using a highly sensitive cooled CCD camera with inner electronic multiplication (Andor Luca). Electronic multiplication (about 70 in our case) considerably reduces the readout noise (effectively to zero). The dark noise is 50% and dark count rate ∼22 Hz), 50:50 interference beam splitter (Thorlabs BSW10R) in a Hanbury Brown and Twiss intensity interferometer scheme and a device for correlated multichannel photon counting in picosecond diapason (ID Quantique id800; with time resolution ∼81 ps). Analysis of photon antibunching data was performed by using custom-built software; it allows us to obtain fluorescence tracks and photon pair time difference distribution (start−stop correlator) using values of measured photons’ absolute arrival times.

2. EXPERIMENT The measurements were performed via the technique of fluorescence microscopy of single objects using a custom-built wide-field epi-fluorescence microscope.19 This allowed us to register a large number of single-NCs in the field of view of the microscope in each experiment at the same time. NC fluorescent trajectories were obtained by recognition and analysis of single NC fluorescent images in the sequence of fluorescence microscope images (CCD camera frames), using custom-built software. Each fluorescent trajectory consisted of 20 000 sequential measurements (20 000 consecutive frames in CCD camera) with an exposure time of 20 ms. Excitation of single NCs was carried out using tunable (in the range of 565 to 635 nm) continuous dye-laser Coherent CR 599 pumped by a solid-state laser Coherent Verdi V6 (532 nm). Variable neutral-density filters were used to alter the input power. A Thorlabs power meter was used to control excitation laser intensity. NCs were excited at a wavelength of 603 nm, which corresponds to the red edge of the absorption band of NCs under study (see Figure 1). A dichroic beam splitter (Thorlabs DMLP0605) and band-pass interference filter (Semrock 628/32) were used to filter the scattered laser light. Fluorescent images of single NCs in the field of view (diameter ∼27 mkm) of microscope objective (nonimmersion

3. RESULTS First, we need to determine which of the bright blinking spots belong to fluorescence of genuine single NCs. To find answer to this question, start−stop photon correlator should be measured for the bright spot of interest. The telling feature of fluorescence from a single NC is observation of photon antibunching. Figure 2 shows such a correlator. 22005

DOI: 10.1021/acs.jpcc.6b06578 J. Phys. Chem. C 2016, 120, 22004−22011

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The Journal of Physical Chemistry C

insufficient statistics. Therefore, we took for statistical analysis fluorescence track parts without these long off-intervals. These analyzed track parts are marked by red lines in Figure 3 and are shown in Figure 4 on an enlarged time scale. 2. Two very long dark intervals were observed in tracks with excitation intensity of 65 W/cm2 (Figure 3a) and 520 W/cm2 (Figure 3d). The dark interval on Figure 3a (starting from ∼100 s) is not due to a photobleaching process because this NC was emitting again in subsequent measurements. It is caused probably by the existence of a deep trap (with slow transition rate). The long “dark” state at Figure 3d is probably due to the NC photobleaching, because the NC was not emitting in subsequent measurements (for time interval greater than 350 s). 3. Fluorescence intensity in on-states grows nonlinearly with excitation intensity increasing (∼4.5 × 104 at 65 W/cm2; ∼9 × 104 at 130 W/cm2; ∼1.5 × 105 at 260 W/cm2 and ∼2 × 105 at 520 W/cm2) It is probably result of nonradiating Auger recombination contribution growing with excitation intensity increasing. 4. The fluorescence track shown in Figure 3b demonstrates two types of blinking. Blinking presented in time interval of 0− 50 s differs considerably from random telegraph signal predicted by “charging model”.1 This blinking looks like the MRC model predictions.11 However, blinking on time scale of 50−200 s looks like RTS predicted by the charging model. RTS is a more typical signal for fluorescence from NCs we have studied. Tracks shown in Figure 3a,c,d can be described by a modified charging model. Manifestation of two types of models for blinking in a fluorescence track of the same NC shows that this NC can change its physical behavior in the course of one experiment. Such behavior of NCs relates to the combined model introduced in ref 17 for the explanation of blinking fluorescence from NCs of A- and B-type.21 Figure 4a−d shows fluorescence tracks for four intensities of excitation of NC (65, 130, 260, and 520 W/cm2) and photon distribution functions w(N,n) calculated for these tracks. Tracks shown in upper panels of panels a−d relate to time intervals marked by red lines in Figure 3 (60 s for a, b, and c and 50 s for d). Tracks presented in the lower panels show fluctuations of the intensity on shorter time intervals of 6 s. First consider the power dependence of average fluorescence intensity (with 20 ms acquisition time), which can be calculated (with the help of the eq 1) by averaging of photon distribution functions w(N,n) presented in Figure 4, where N is photon number.

Figure 2. Illustration to the analysis of single and a few nanocrystals contributions into fluorescence track. Measured track (a) exhibit the pronounced blinking with off-states close to background noise signal, that usually considered as a proof of luminescence of a single quantum emitter. Start−stop correlator measured during the time interval, when the nano-object under study is in the bright state with luminescence intensity below (b) and above (c) the dashed line on the fluorescence track.

The correlator in Figure 2b shows that simultaneous emission of two photons is very nearly impossible. This fact is a telling feature of the emission from a single emitter. The correlator in Figure 2c shows that there is a marked probability to emit simultaneously two photons. Hence, intensity above the dashed line includes photons solely from two uncorrelated emitters (see e.g. theoretical description in ref 20). All fluorescence tracks analyzed in this work were obtained for the same single NC CdSe/ZnS. Figure 3 shows fluorescence tracks obtained at four intensities of laser excitation: In = 2n65 W/cm2 (n = 0, 1, 2, 3). These tracks exhibit the following peculiarities: 1. At least two types of off-intervals (short and long) are in fluorescence tracks shown in Figure 3b,c. We could not carry out statistical treatment of these long off- intervals because of

∞

⟨N ⟩n =

∑ Nw(N , n) N =0

(1)

Dependence of the average fluorescence intensity on the intensity, In of excitation is shown in Figure 5. The two left points in Figure 5 show that average fluorescence intensity increases if we increase excitation intensity from I0 = 65 W/cm2 to I1 = 130 W/cm2. However, if we continue to increase excitation intensity, average fluorescence intensity becomes weaker. The two right points in Figure 5 show this fact. Such nonmonotonic dependence of fluorescence intensity on the intensity In of excitation can be understood if we look at the photon distribution functions presented in Figure 4. Shapes of the distributions shown in Figure 4a,b are almost similar, but the photon distribution function shown in Figure 4b is shifted to large values of N

Figure 3. Fluorescence tracks measured at various intensity of excitation: 65 W/cm2 (a), 130 W/cm2 (b), 260 W/cm2 (c), and 520 W/cm2 (d). 22006

DOI: 10.1021/acs.jpcc.6b06578 J. Phys. Chem. C 2016, 120, 22004−22011

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The Journal of Physical Chemistry C

Figure 4. Fluorescence tracks and corresponding photon distribution functions measured at various intensity excitations: 65 (a), 130 (b), 260 (c), and 520 W/cm2 (d). Tracks shown in upper panels of panels a−d relate to time intervals marked by red lines in Figure 3. The lower panels of panels b−d show fluorescence tracks on enlarged time scale of 6 s. The line in photon distribution in panel a shows Poisson distribution.

excitation intensity, the shape of the photon distribution function changes considerably: the peak situated at large N diminishes and then disappears. The center of gravity of the distribution function of the new shape without the right peak, i.e. ⟨N⟩, diminishes. The two right points in Figure 5 show this fact. This result can be also explained as follows: The average NC fluorescence intensity depends on (i) the ratio of time spent in on-state to the total experimental time and on (ii) NC fluorescence intensity in on-state. These two competing contributions depend on power in different ways. The first term decreases with power,18 but it is compensated by a linear increase of the second term at low excitation power. However, at high excitation intensities, the power dependence of fluorescence intensity in on-state strongly deviates from the linear curve. These two competing contributions lead to the observed single NC average fluorescence intensity power dependence behavior.

Figure 5. Dependence of the average fluorescence intensity on the intensity of excitation, i.e., dependence of center of gravity of the distribution functions shown in Figure 3 on laser intensity.

relative to the distribution function shown in Figure 4a. This shift is linearly proportional to the increase of the excitation intensity. Therefore, the average fluorescence intensity increases with excitation intensity, In. If we continue to increase 22007

DOI: 10.1021/acs.jpcc.6b06578 J. Phys. Chem. C 2016, 120, 22004−22011

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The Journal of Physical Chemistry C Consider now fluorescence tracks. The whole time scale in the lower panels represented in Figure 4 is 6 s. Attribution of the interval to on- or off-type is not the problem in these panels. Comparing tracks represented in panels b−d of Figure 4 with each other we clearly see shortening of on-intervals at the increase of the excitation intensity, as was previously found in ref 18, whereas durations of off-times are practically independent of the excitation intensity. By measuring durations τon,off of on/off times in the track of 60 s represented in Figure i 4a and in tracks of 6 s depicted in Figure 4b−d, we find a set of values τon,off (n) where n is the number of fluorescence track. By i using the standard equation τnon,off =

1 Nn

Figure 7. Process of ionization in the Efros−Rosen charging model.

Figure 7a shows annihilation of an exciton with rate Γ0 with emission of one photon. If two excitons occur in the core, as Figure 7b shows, one of them can recombine with transfer of its energy to the hole of the second exciton. The hole acquiring additional energy leaves the core for kth trap in the shell. If laser light creates again an exciton in the charged core, this exciton (trion) can recombine with transfer of its energy to the electron existing in the charged core (Figure 7c). This is an additional nonradiating channel for annihilation of the exciton in the charged core. Therefore, recombination of trions is realized with rate Γk > Γ0,22,23 and the quantum yield of fluorescence in the off-state diminishes as compared with quantum yield in the on-state, creating on/off blinking. However, such a charging model predicts a single ionization time, tA. If we take into account Auger ionization involving two excitons, we cannot explain the power-law distribution of on-times observed in experiments. Therefore, Osad’ko13,14 proposed the following modification of the Efros−Rosen theory. He assumed that some core atoms on the core/shell interface can have after its excitation wave functions j localized on the core/shell interface, as Figure 8 shows.

Nn

∑ τion,off (n) i=1

(2)

for the average time, we find the following values for the average on times in four tracks related to various excitation intensities: τon n = 1.5 s (a), 0.29 s (b), 0.136 s (c), and 0.07 s (d). The duration of 6 s for tracks with excitation intensities 130, 260, and 520 W/cm2 was enough to estimate the average values (because 6 s period is much longer than the corresponding average times). Consequently, the duration of 6 s was not enough for tracking excitation intensity 65 W/cm2 for estimation of average value (∼1.5 s), and we choose 60 s interval. Nevertheless, we checked the influence of analyzed fluorescence track interval duration on obtained average time. We found that the influence is negligible. The average off-times in four tracks related to various excitation intensities are τoff n = 0.143, 0.144, 0.087, and 0.098 s. Dependences of the average on-times (circles) and off-times (triangles) on the excitation intensity In where In = 2nI0 (n = 0, 1, 2, 3) and I0 = 65 W/cm2 are shown in Figure 6. Solid and dashed lines show functions τon,off = const/In2 and τon,off = const/In, respectively.

Figure 8. Auger ionization in accordance to modified charging model13,14 with hole escape from surface state j.

If exciton energy is transferred to hole j localized on core/ shell interface, hole j will be ejected to shell trap k and the core will be ionized. The value of ionization time tAj will depend on which hole takes part in the Auger ionization. This modified charging model was able to explain power-law distribution of on-times.13−15 Consider now how an energy diagram will look in the modified charging model. The theoretical diagram for core/ shell NCs that could be able to describe blinking fluorescence with on/off-times which depend on the excitation intensity in the way we have found in our experiment is depicted in Figure 9. Here 0, ex, j, and ex + j are the ground state of the NC, oneexciton state, state with jth e/h pair localized in the core on the core/shell interface, and the state with one exciton and one jth e/h pair, respectively. Lex and Lj describe the intensity of exciton level and jth level pumping by CW-laser light via

Figure 6. Dependence of the average on-times (circles) and off-times (triangles) on the excitation intensity, In. Solid and dashed lines show functions τon,off = const/In2 and τon,off = const/In, respectively.

Obviously, the average on-times measured in our experiment depend on the square of the intensity In. The average off-times are nearly independent of the intensity In.

4. THEORY AND DISCUSSION In accordance with the Efros−Rosen charging model,1 only excitons take part in the process of Auger ionization. This process is shown in Figure 7. 22008

DOI: 10.1021/acs.jpcc.6b06578 J. Phys. Chem. C 2016, 120, 22004−22011

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The Journal of Physical Chemistry C

power law in the distribution of off-times is realized because of the few ways of neutralization of the NC.3,5 This allows us to change the power-law dependence on a sum of a few exponential distributions with average values τon,off which cover 3−4 orders of magnitude.3,13−15 Consequently, it allows us to compare the average on-/off-times power dependencies obtained in experiment with the corresponding mathematical expressions, eqs 3 and 4. The power dependence for onintervals in eq 3 fits the corresponding experimental points in Figure 6 well. This means that ionization in our NCs is realized via an Auger mechanism. If the second term dominates in eq 4, τoff k depends inversly on the excitation intensity. Such dependence is shown by a dashed line in Figure 6. If the first term dominates in eq 4, the average duration of off-times does not depend on the intensity. Triangles in Figure 6 show such independence on the excitation intensity. Hence, neutralization in our NCs is described by a tunneling process shown by the white arrow in Figure 9. Note that there is a trend of a small decrease in average off-times with excitation power. Perhaps increase of laser intensity leads to NC heating. This can be result of increase in time spent in offstate and as consequence to an increase in nonradiating transitions in the NC and hence to small increase in phononassisted tunneling transitions responsible for off-times. However, it should be investigated in detail in future work. Our result concerning the absence of Auger neutralization in our NCs correlates with the result obtained in ref 24. There it was shown that the distribution of off-times in NC CdSe/CdS/ ZnS does not depend on the size of the NC. It should be noted that the theory predicts and experiment shows25 that the rate of Auger process is size-dependent: the rate increases if the size of the NC decreases.

Figure 9. Energy diagram for the core of NC irradiated by CW-laser light. The four left states describe neutral core with excitations L and relaxations Γ. The two right states describe the ionized core. Thick arrows show processes of ionization and neutralization.

creation of free electron/hole pairs and after their relaxation either to exciton state or to the jth state. GAjk describes the rate of the escape of the charge from the jth pair in the core to the kth trap of the shell. Levels k and ex + k relate to the core state with a charge in kth trap of the shell and the exciton state with the charge in kth trap. Existence of the charge in kth trap manifests itself in the difference of values of nonradiative transitions in the neutral and the charged core. It was shown in ref 22 that a trion in the charged core decays two times faster than an exciton in the neutral core. Γj describes the rate of the annihilation of the jth e/h pair. Neutralization of NC can be realized, at least, in two ways: (i) by tunneling of the charge with rate gk from the shell; this tunneling is shown by a white arrow in Figure 9, and (ii) with the help of Auger process with rate gAk ; annihilation of a trion in the charged core with returning charge from trap k to the core owing to the trion energy is shown by red arrow. Six rate equations for the probabilities p0, pex, p1, pex+j, pk, and pex+k to find the core of NC in one of six possible states shown in Figure 9 describe quantum dynamics of NCs irradiated by CW-laser light. These six equations enable one to calculate transition from the on-state to off-state and back. The following equations for the average on- and off-times have been found with the help of these six equations:

5. CONCLUSION We measured blinking fluorescence of the same single colloidal core/shell nanocrystal CdSe/ZnS at four CW-laser excitation intensities In = 2n65 W/cm2 (n = 0, 1, 2, 3) at ambient conditions. The inverse square intensity dependence was observed for average on-times, whereas average off-interval duration was practically independent of the excitation intensity. To explain the observed phenomena the modified charging model was applied, which includes • Transitions from on-state to off-state caused by Auger ionization of the core in core/shell nanocrystal • Transitions from off- to on-states caused by tunneling and/or Auger neutralization of the core. Power dependence for average on-times τon ∝ I−2 n observed in our experiment proves that transitions from on- to off-states are realized with the help of an Auger ionization mechanism. Intensity-independent average off-time, τoff, proves that neutralization of the core is realized by direct tunneling of the charge from the shell. We must note that there is a trend of a small decrease in average off-times with excitation power. This is probably due to nonradiating transition probability growth with laser intensity incising and consequent NC heating. It should be noted that the possibility to change and control the NC blinking statistics by variation of laser excitation intensity has great potential for colloidal quantum dot (QD) applications (e.g., for super-resolution microscopy with colloidal QD labels).

(Lj /Γj)(Lex /Γ0) 1 ≈ Gj(Lj /Γj)(Lex /Γ0) on = Gj (1 + Lj /Γj)(1 + Lex /Γ0) τj (3)

1 τkoff

= gk

Γk +

gkA

Lex + Γk +

≅ gk + gkA

gkA

+ gkA

Lex Lex + Γk + gkA

Lex Γk + gkA

(4)

Derivation of these equations is presented in the Appendix. Efficiency rate, Lj, of the jth pair creation and efficiency rate, Lex, of the exciton creation are linear functions of the intensity, In, of the CW-laser. Therefore, eq 3 demonstrates const/In2 dependence on the laser intensity. The truncated power-law in the distribution of on-intervals is realized because of the existence of several doorways for ionization,13−15 and the 22009

DOI: 10.1021/acs.jpcc.6b06578 J. Phys. Chem. C 2016, 120, 22004−22011

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The Journal of Physical Chemistry C

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Off-States

APPENDIX: DERIVATION OF EXPRESSIONS FOR AVERAGE VALUES OF ON- AND OFF-TIMES

It is obvious that Poff k = Pk + Pex+k describes the probability of the finding core of a NC in an ionized state, i.e., in off-state. Evolution in time of these probabilities is described by the following equations:

On-States

It is obvious that pon j = p0 + pex + pj + pex+j describes the probability of finding the core of NC in neutral on-state. Evolution in time of these probabilities is described by the following four equations:

Peẋ + k = −(Γk + gkA )Pex + k + Lex Pk Pk̇ = ΓkPex + k − (Lex + gk )Pk

Peẋ + j = −(Gj + Γj + Γ0)Pex + j + Lex Pj + LjPex

By summing both equations, we arrive at the following equation:

Pj̇ = Γ0Pex + j − (Γj + Lex )Pj + LjP0

Pk̇off = −(gk Pk + gkA Pex + k)

Peẋ = ΓjPex + j − (Lj + Γ0)Pex + Lex P0 P0̇ = ΓjPj + Γ0Pex − (Lex + Lj)P0

Pj̇on = −GjPex + j

Γk + gk

Pk + Pex+k, we can express the probabilities Pk and Pex+k via Poff k :

(2A)

Rate constants Γ0, Γj, and Gj are responsible for fast dynamics in NCs. Rate constants Lex, Lj ≪ Γ0, and Γj determine slow dynamics of on/off jumps in NC. Because we are interested in slow on/off dynamics, we may set Ṗ1j = Ṗ j = Ṗ 1 = 0. Then the first, second, and third lines in eq 1A appear as follows:

Pk =

Pk̇off = −

1 τkoff

Γj

Pex =

Lex P0 , Γ0

Pex + j =

Lj Lex P0 Γj Γ0

■

(5A)

1 on Pj τjon

Pkoff

τkoff

Pkoff

(12A)

= gk

Γk + gkA Lex + Γk + gkA

+ gkA

Lex Lex + Γk + gkA

Lex Γk + gkA

(13A)

AUTHOR INFORMATION

*E-mail: [email protected]. Phone: +7 495 851 02 36. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by Russian Science Foundation (Project 14-12-01415 “Statistical methods for single nanoobjects luminescence study: Fundamental aspects and applications of spectromicroscopy of single molecules and quantum dots in material science and nanotechnology”). Authors acknowledge Prof. Alexey Kalachev and The Kazan E. K. Zavoisky Physical-Technical Institute of the Kazan Scientific Center of the Russian Academy of Sciences for the opportunity to use the device for correlated multichannel photon counting ID Quantique id800 (purchased under support of RSF Project 14-12-00806 “Developing basic units for long-distance quantum communication”) which was used for the measurements of second-order photon correlation functions of luminescing quantum dots as described in section 2 of the present paper.

(6A)

Inserting this expression for P0 into eq 5A for the probability Pex+j and the later to eq 3A, we arrive at the following equation for finding the distribution of on-times: Pj̇on = −

Lex + Γk + gkA

Corresponding Author

P jon (1 + Lj /Γj)(1 + Lex /Γ0)

Lex

is the average duration of off-intervals. Here we used the inequality Lex ≪ Γk + gAk .

Inserting these equations into the expression for the probability Pon j , we find the following relation between P0 and Pon j : P0 =

1

≅ gk + gkA

(4A)

we arrive at the following simple expressions for the probabilities: P0 ,

Pex + k =

for finding distribution of off-times. Here

(3A)

By solving these equations we can express pex, pj, and pj+ex via P0. Taking into account that

Lj

Lex + Γk +

Pkoff , gkA

By inserting these expressions into eq 10A, we arrive at the following equation:

−Γ0Pex + j + (Γj + Lex )Pj = LjP0

Gj , Lex , Lj <