NANO LETTERS
Modified Power Law Behavior in Quantum Dot Blinking: A Novel Role for Biexcitons and Auger Ionization
2009 Vol. 9, No. 1 338-345
Jeffrey J. Peterson and David J. Nesbitt* JILA, National Institute of Standards and Technology and UniVersity of Colorado, Department of Chemistry and Biochemistry, UniVersity of Colorado, Boulder, Colorado 80309 Received October 14, 2008; Revised Manuscript Received November 29, 2008
ABSTRACT Single photon detection methods are used to acquire fluorescence trajectories from single CdSe/ZnS colloidal quantum dots (QDs) and analyze their blinking behavior. Although the “off-time” distributions follow ideal power law behavior at all wavelengths and intensities, significant deviations from power law behavior are observed for the “on-times”. Specifically, with improved time resolution, trajectory durations, and photon statistics, we report a near-exponential falloff of on-time probability distributions at long times. Investigation of this falloff behavior as a function of laser wavelength and power demonstrate that these deviations originate from multiexciton dynamics, whose formation probabilities can be very low on a “per laser pulse” basis, but become nearly unity on the time scales of the longest on-times. The near quadratic, power-dependent results indicate the predominant role of biexcitons in the long time on-to-off blinking dynamics, which can be interpreted in terms of an Auger ionization event. In conjunction with Poisson modeling of the photon statistics, the data is consistent with QD ionization efficiencies of order ≈10-5 and highlight a novel role for biexcitons and Auger ionization in QD blinking.
In the 25 years since their discovery, it is now widely appreciated that colloidal semiconductor quantum dots (QDs) have significant potential as superior fluorophores due to their continuous absorption spectra, large absorption cross sections, and superior photostability.1 In fact, these unique optical properties enable one to routinely detect emission from a single QD over extremely long time periods (∼1000 s), enabling unprecedented dynamic range in the study of the emission properties of single fluorophores and opening new possibilities for QDs in applications where only one emitter is required. For example, single QDs have been used as real time fluorescent labels for tracking of biological processes2 and as single photon sources suitable for quantum cryptography.3 Despite many advantages, one of QDs’ significant limitations is the presence of fluorescence intermittency, or blinking. Blinking describes a phenomenon, first observed in QDs over 10 years ago,4 in which the QD does not emit a steady stream of photons while under continuous laser excitation, but rather randomly switches between a fluorescent bright state (on) and a nonfluorescent dark state (off). Despite its importance for both fundamental scientific inquiry as well as technological applications, the origin of QD blinking remains an intensely debated subject.5 In QDs, it is commonly believed to be the result of a charging process, * To whom correspondence should be addressed. 10.1021/nl803108p CCC: $40.75 Published on Web 12/15/2008
2009 American Chemical Society
whereby an electron (or hole) in an excited electronic state leaves the core material and resides in a trap state on or near the QD surface. Once the core QD has become ionized, very efficient nonradiative pathways are opened that are several orders of magnitude greater than radiative relaxation rates and render the QD nonfluorescent; upon reneutralization, the QD returns to normal fluorescence cycling. Within this picture, the duration of on and off events reflects the timescales on which the QD charges and reneutralizes; by compiling histograms from the durations of on and off events, one can determine the rate at which these processes occur. Surprisingly, it is experimentally observed that the durations of on and off events are not exponentially distributed as expected for charging and reneutralization occurring via single rate constants, but instead are highly nonexponential and indicate that the rates of QD blinking vary by several orders of magnitude.6,7 Specifically, the probability density of both on and off time distributions decay following a power law P(τon/off)∝ τon/off-m, where a power law exponent m ≈ 1.6 ( 0.2 is typically observed for both on and off times over as many as 6 orders of magnitude in time. Thus, one outstanding challenge to understand the microscopic origin of blinking behavior has been to develop physically motivated models that predict such broadly distributed rates of QD blinking. One early model attributed blinking to the creation of a QD biexciton,8 a rare event formed by the absorption of
multiple photons that results in the presence of multiple electron-hole pairs within the QD. Because of the presence of excess carriers, very efficient nonradiative Auger relaxation channels are opened that can rapidly ionize the QD by ejecting an electron to a surface trap state. The ejected electron is presumed to reside in the surface trap until it returns to the core QD, restoring normal laser-induced fluorescence cycling. The biexciton model provided a compelling physical picture for QD ionization and, by positing an exponential distribution of trap state energies and a return rate that is exponentially dependent on trap state energy, was shown to predict power law behavior for the off time distributions.9 However, it was also quickly noted that a model based solely on Auger ionization is qualitatively inconsistent with the power law behavior of the on-time distribution.9 The fundamental reason is that even though electrons ejected to different trap states may well occur with significantly different rate constants, the total unimolecular rate constant for electron ejection would be given by the sum of individual rate constants. Thus, one expects that the overall decay of the on-state would still be described by a single rate constant, which would incorrectly predict exponential distributions for the on-times. This failure to predict power law distributions for on-times is common to any model invoking a static distribution of trap states and has motivated other models that consider the role of dynamic processes, such as fluctuating tunneling barriers or a diffusing trap state energy.7,10,11 Further insight into the origin of QD blinking behavior has been gained by recent reports in the literature that describe deviations from universal power law behavior and thus requiring an additional refinement to the general picture of power law dynamics.12-14 For example, Knappenberger et al. observed an exponential cut off of the on-time power law behavior at ∼5 s when using laser excitation ∼100 meV above the QD’s absorption edge, highlighting the potential role of excitation energy in QD blinking processes.13 Pelton et al. observed short time cut offs of on-time power law behavior at ∼1 ms that provide support for a diffusion controlled mechanism of blinking processes.14 Here, we report deviations from the power law behavior of on-time probability decays that highlight the role of biexcitons and Auger ionization in the long time QD blinking processes. Specifically, utilizing a high detection efficiency confocal apparatus and long ∼1000 s fluorescence trajectories in order to obtain large statistical samples times with high 1 ms temporal resolution, we observe that the on-time power law behavior of single CdSe/ZnS QDs becomes truncated and exhibits an exponential cut off at long times. The exponential falloff is quadratically dependent on laser power, which is consistent with the formation of biexciton species. Although the probability of multiexciton events can be extremely low on a “per laser pulse” basis, we use simple Poisson statistics to demonstrate that the formation of biexcitons (or higher excitonic) species is a highly probable event on the much longer time scales of blinking, indeed, at even lowest feasible laser excitation intensities on the nW level (i.e., 1 nW ≈ 2.2 W/cm2). In the context of an Auger model for exciton Nano Lett., Vol. 9, No. 1, 2009
promoted ionization of the QD, this data can be used to extract experimental estimates for the efficiency of the charge ejection event. Experimental Section. The experimental apparatus consists of a confocal microscope, coupled with multiple laser excitation sources and single photon detection. The circularly polarized excitation light is expanded by a telescope before entering an inverted microscope (Olympus IX-70), reflected to the objective by a dichroic mirror, and focused at the sample surface by a 100×, 1.4 NA oil immersion objective (Olympus PlanApo).15 QD fluorescence is collected through the same objective, passed through the dichroic mirror to separate the sample fluorescence and excitation light, and focused through a 50 µm pinhole. The emission is further split into two paths by a dichroic mirror (Chroma 580DCSP) that has ∼50% transmission at the QD ensemble emission peak, passed through a pair of band-pass filters (Chroma 580-70M) in order to suppress light outside QD emission range, and detected by a pair of avalanche photodiodes (Perkin-Elmer SPCM-AQR14). A series of flip mirrors allow selection between one of multiple different laser excitation sources. Pulsed excitation at 434 nm is provided by a GaN diode laser (Picoquant, LDH 440) with a pulse width of ∼70 ps and repetition rate of 5 MHz. Continuous wave (cw) excitation at 488 nm (Novalux Protera-488-15) and 532 nm (QED GLS-5) are also utilized. The laser power is attenuated by a series of neutral density filters to achieve laser powers of ∼20 nW-2 µW after the microscope objective, corresponding to intensities of ∼40 W/cm2-4 kW/cm2. CdSe/ZnS QDs are purchased from NN-Laboratories (average core diameter of ∼3.3 nm) and exhibit an absorption peak at 561 nm, emission peak at 576 nm, and measured fluorescence quantum yield of 36 (5) %. After ozone cleaning glass coverslips for 45 min, samples are prepared by spin casting dilute concentrations of QDs in a 0.5% (g/mL) poly(methylmethacrylate)/toluene solution, resulting in QDs embedded in a thin ∼10 nm polymer film. Images are created by raster scanning a piezoelectric stage (PI P-517.3CL) in 39 nm steps through a 10 × 10 µm area, revealing typical surface densities of 0.3 QDs/µm2 and near diffraction limited spot sizes (full width at half-maximum, fwhm ≈ 240 nm). After locating individual QDs in the image, the stage is repositioned such that a single QD is centered in the laser focus and fluorescence trajectories are obtained for 500-1000 s, corresponding to ∼108 absorbed photons. Single photon detection is performed with a time-correlated counting card (Becker-Hickl TCSPC-134), allowing time-correlated, timetagged fluorescence measurements with ∼50 ns macrotime resolution and enabling the data to binned post acquisition in any desired time interval. All trajectories are binned and analyzed at 1 ms in the current report and all measurements are performed at room temperature. Results. Fluorescence trajectories obtained from single QDs exhibit digital blinking behavior and histograms compiled from the fluorescence intensities reveal two distinct populations corresponding to the on and off states, as shown in Figure 1. The off-state intensity includes primary contributions from the detector dark counts and autofluorescence 339
Figure 1. A 5 s sample subset of a much longer 1000 s fluorescence trajectory obtained from a single QD, exhibiting digital blinking behavior (λexc ) 434 nm, I ) 230 W/cm2). On the right-hand side, a histogram of count rates compiled from the entire fluorescence trajectory is displayed.
from the polymer film and the distribution of intensity levels is well described by a Poisson distribution. The on-intensity level averaged over several QDs is in good agreement with expectations based on laser intensity, QD absorption cross section, fluorescence quantum yield, and the measured collection efficiency of the confocal microscope.16 A bin time of 1 ms is chosen such that the smallest on-state intensity levels corresponding to the lowest laser powers (20 nW) occur at least 6σ above the average background level, ensuring the on- and off-states remain well separated. A small fraction (∼10%) of fluorescence trajectories are excluded from further analysis due to anomalously large background levels or and/or intensity histograms exhibiting multiple rather than 2-state behavior in the on-state distributions. These trajectories most likely originate from (i) low probability occurrences of more than one emitters (either QDs or other impurities) within the laser focus or (ii) multiplestate intensity behavior in isolated QDs. The second possibility is particularly intriguing, and proves relevant in ongoing electric field studies in our laboratory of single QD emission frequencies, fluorescence lifetimes, and intensities.17 However, for simplicity in our model analysis of the long time falloff behavior, we focus on the large majority of “binary” 2-state QD emitters. Fluorescence trajectories are analyzed by a threshold analysis. Briefly, a threshold intensity is selected; intensities above the threshold are considered on, intensities below the threshold are considered off. For the cleanest separation of events, the threshold is selected to be approximately midway between the lowest on-intensity and the highest off-intensity. However, deviations from this specific choice of threshold have no impact on the results reported from our analysis. A trajectory is analyzed to determine the set of all on and off events and a histogram describing the number of occurrences Ni of each event duration τi is compiled. The shortest possible event is limited by the bin time used to generate the trajectory (1 ms) with the longest possible event limited by the duration of the acquisition (≈1000 s). We calculate probability distributions P(τon/off) (formally a probability density, with units of s-1) for on- and off-times by dividing the number 340
Figure 2. Probability distributions of on-times (left panel) and offtimes (right-panel) determined from a single QD fluorescence trajectory (λexc ) 434 nm, I ) 120 W/cm2). The solid red lines are fits to the data, as described in the text; the dashed red line indicates the power law component from the fit of the on-time distribution.
of occurrences of a given event by the average time duration between closest preceding and following events P(τ1) )
2Ni [(τi+1 - τi) + (τi - τi-1)]
(1)
By analyzing probability densities rather than (unitless) probabilities, one reduces the limitations from poor sampling statistics of long time events due to finite trajectory duration. This in turn can extend the dynamic range of events sampled by an additional 2 orders of magnitude in time and 3 orders of magnitude in probability.9,18 A typical probability distribution for the on- and off-times determined from a single trajectory is shown in Figure 2. As expected, the off-time distribution follows a straight line on a log-log scale and can be described by a power law of the form P(τoff) ) A0τ-m off . In contrast, the on-time distribution appears qualitatively different; at short times the on-time distribution follows a straight line on a log-log scale (i.e., power law behavior), but at long times it begins to exhibit clear downward curvature. The on-time distribution can be well described by a modified power law of the form P(τon) ) A0τ-m on exp(-τon/τfall-off), where the exponential component appears as a falloff of the power law behavior (on a log-log scale) with the 1/e time constant τfalloff corresponding roughly to the “knee” in the on-time distribution. It is worth noting that the statistical appearance of this additional structure in the long on-time distributions was greatly enhanced by experimental modification of the data acquisition to permit detection of up to ≈107 photon events and longer time trajectories up to 1000 s. This observation of a nonpower law falloff in on-time probability distributions was initially quite surprising and therefore its robust presence was verified in over 100 individual QDs, in QDs spun-cast in different polymer hosts and on bare glass substrates, and in QD samples provided by other commercial suppliers. In order to investigate whether the falloff is a light-induced effect, we measured the on- and off-time probability distributions at various laser excitation intensities (Figure 3). The off-time probability Nano Lett., Vol. 9, No. 1, 2009
Figure 3. On-time (left panel) and off-time (right panel) probability distributions measured under pulsed laser conditions at λexc ) 434 nm and laser intensities of 230 (red), 120 (blue), and 66 W/cm2 (green). The solid lines are fits to the data as described in the text.
Figure 4. Average falloff rate at λexc ) 434 nm determined from ∼40 QDs at each power level, error bars are determined from standard deviation of the mean. The inset shows the same data plotted on a log-log scale, illustrating quadratic dependence of the average falloff rate on laser power. Solid lines are fits to the data as described in the text.
distributions follow an identical power law that is independent of all investigated laser intensities with an average power law exponent moff ) 1.6 ( 0.1 determined from 166 QD trajectories. By way of contrast, however, the on-time probability distributions are strongly laser intensity-dependent. All samples follow a power law at short times, but exhibit a clearly decreasing falloff time τfalloff with increasing laser intensity. We also note a very slight decrease in magnitude of the on-time power law exponent with increasing laser power. This could arise from systematic missing of blinking events faster than the 1 ms bin time or simply parameter correlation between power law and falloff exponents. Further simulations are currently underway to ascertain the origin of this effect.19 The intensity dependence of the on falloff times is summarized in Figure 4, which plots the average falloff rate (1/τfalloff) versus laser intensity determined from ∼40 QDs at each intensity level. The falloff rate exhibits a quadratic dependence on laser intensity with an exponent m ) 1.9 (1) determined from least-squares fit on a log-log plot. Nano Lett., Vol. 9, No. 1, 2009
The quadratic dependence of this falloff time on laser intensity is strongly suggestive of a multiphoton process, therefore motivating reexamination of the role that biexcitons may play in QD blinking on the long-time scale. In order to create a biexciton, a QD must absorb two photons before exciton relaxation occurs, rendering the process strongly dependent on the nature of photon absorption and necessitating that the two cases for (i) pulsed laser and (ii) cw laser excitation be considered separately. First, we consider the case of pulsed excitation. In a pulsed excitation experiment, the laser pulse width τpulse (currently ≈70 ps) is typically much less than the fluorescence lifetime τfluor (approximately 30 ns), whereas the time duration between pulses ∆Trep (in the present study, ≈200 ns) is typically much greater than τfluor. Therefore, the relevant physical quantity one must consider is the average number of excitons per pulse 〈Nexciton〉. This can be calculated in a straightforward manner using the on-axis peak intensity of the laser from 〈Nexciton〉 ) (∆Trepσabs/Ephoton)I0 where I(r) ) I0 exp(-2r2/ω02) and ω0 is the 1/e electric field radius at the focused waist, σabs is the cross-sectional area for photon absorption, and Ephoton is the energy of a single photon. In terms of the more experimentally accessible quantities such as the total number of photons per pulse (Nphoton) and the intensity based on the 2 fwhm laser beam area (A ) πDfwhm /4), the average exciton number for Gaussian beam excitation of a QD is easily shown to be 〈Nexciton〉 ) ln 2Nphoton(σabs/A). Assuming these to be independent events, the probability P of creating n excitons per pulse (with n ) 0, 1, 2...) is given by a Poisson distribution Pµ(n) ) e-µ
µn n!
(2)
where µ is the average number of excitons per pulse, that is, µ ) 〈Nexciton〉. Since P is a probability per pulse, 1/P reflects the average number of pulses per n exciton formation event. When multiplied by the laser pulse repetition rate, this represents the characteristic 1/e time for creation of an n exciton state. At the typical powers used in single molecule confocal microscopy, this probability for multiple exciton formation can be quite low on a per laser pulse basis. For a 5 MHz pulsed laser with an average of 50 W/cm2 illumination intensity (see Figure 5a), for example, 〈Nexciton〉 ≈ 0.075, and the probability of creating a single exciton in a single pulse is only P ≈ 7.0%. These conditions translate into much lower probabilities for bi- and triexciton formation of ≈ 0.26% and 0.0066% per pulse, respectively, which are sufficiently rare to ensure single exciton dynamics in the vast majority of absorption events. However, converted into the equivalent 1/e time for an n exciton creation event, this becomes not so improbable on much longer time scales for blinking. At these same excitation conditions, for example, single excitons are created on ∼1 µs time scale, biexcitons are created on a ∼10 µs time scale and triexcitons are created on an ∼1 ms time scale. This is most convincingly illustrated in terms of the n exciton formation probability as a function of time, as plotted in Figure 5a. For a 1 MHz average excitation rate over a period of g100 µs, a QD has a g 99% probability of 341
Figure 5. (a) Probability of creating one or more single (black), bi- (red), and triexciton (blue) as a function of time delay after any chose start time in a pulsed laser experiment (λexc ) 434 nm, I ) 50 W/cm2, ∆Trep ) 200 ns, σabs ) 4.8 × 10-15 cm2). (b) Probability of creating one or more multiexcitons at various laser on-axis peak intensities, as a function of time in a pulsed laser experiment (λexc ) 434 nm, ∆Trep ) 200 ns, σabs ) 4.8 × 10-15 cm2).
having experienced at least one biexcitonic event. Considering that the duration of on-time events observed from QD blinking trajectories are orders of magnitude greater (i.e., from the bin time of 1 ms up to 10-100 s), one can conclude with near certainty that biexcitons are created on the time scale of each QD blinking event observed. This Poisson analysis can be readily generalized to include the role of multiexciton species (e.g., biexciton, triexciton, etc.) The characteristic 1/e time of forming any multiexciton species τmultiexciton is given by τpulsed multiexiton )
∆Trep 1 - e-Nphoton
σabs A
(
1 + Nphoton
(
σabs A
2∆Trep σabs Nphoton A
)
2
)
≈
for Nphoton
σabs , 1 (3) A
where the approximation is valid at low average exciton probability. Equation 3 predicts that at low laser power (i.e., low Nphoton), the rate of multiexciton formation (1/τmultiexciton) scales quadratically with laser power, as one expects for a multiphoton process. In Figure 5b, the intensity dependence of forming one or more multiexciton species in typical experimental conditions is shown, again illustrating that at even the lowest laser intensities there is a near unity probability of forming multiexciton species on the time scale of typical on-to-off blinking events. We now turn to the case of cw laser excitation. In order to form a biexciton, the QD must absorb two photons before an emission event occurs and the relevant physical quantity to consider is the average number of excitons per fluorescence lifetime. If at the simplest level one assumes a constant delay between excitation and fluorescence emission events (approximated by the mean value τfluor), the derivation for the 1/e time for forming a multiexciton species would be identical to that of the pulsed laser analysis. The result would be eq 3 where ∆Trep is replaced by τfluor and Nphoton now representing the number of incident photons per fluorescence ˙ ˙ lifetime NCW photon ) Nphotonτfluor where Nphoton is the rate of laser excitation (equal to the laser power divided by the energy 342
of a single photon). A more realistic model allows an exponential distribution of photon emission times. Assuming an exponential distribution of photon emission times and a Poisson distribution of photon absorption times, it can be shown that the characteristic 1/e time for forming a multiexciton species is given by
τCW multiexiton )
(
(
)
σabs +1 A ≈ 2 σabs N˙photonτfluor A 2τfluor σabs , 1 (4) for Nphotonτfluor 2 σ A ˙Nphotonτfluor abs A
2τfluor N˙photonτfluor
(
)
)
Again the approximate expression corresponds to the limit of low exciton probability per fluorescence lifetime and predicts a near quadratic dependence of the inverse lifetime on the laser excitation rate. In addition to the pulsed excitation studies at 434 nm, we have also explored blinking statistics for two cw laser excitation scenarios with the results at λexc ) 488 and 532 nm and at a series of laser powers summarized in Figure 6. The general trends observed at λexc ) 488 and 532 nm are essentially identical to those observed for λexc ) 434 nm for comparable average laser intensities. Specifically, the ontime probability distributions follow a power law at short times and exhibit a falloff at long times that is clearly laser intensity-dependent. Similarly, the off-time probability distributions follow a straight power law with an average slope of moff ) 1.6 ( 0.1, identical to the 434 nm pulsed excitation scenario and independent of laser intensity. The intensity dependence at each λexc is summarized in Figure 7, which plots the average falloff rate (1/τfalloff) versus absorption rate in order to account for differences in photon energy and QD absorption cross section at the different excitation wavelengths. The data at each excitation wavelength exhibit a near-quadratic dependence on laser power, consistent with predictions of eqs 3 and 4 and providing evidence for a multiexciton mediated rolloff in blinking events at long times. Nano Lett., Vol. 9, No. 1, 2009
number of excitons (per pulse or fluorescence lifetime), one predicts in the case of pulsed excitation ∆Trep pulsed τfalloff
) Pionize[1 - e-〈Nexciton〉(1 + 〈Nexciton 〉)] ≈ Pionize〈Nexciton 〉2 for 〈Nexciton 〉 , 1 (5) 2
with similar results τfluor cw τfalloff
)
Pionize〈Nexciton 〉2 Pionize〈Nexciton 〉2 ≈ for 〈Nexciton 〉 , 1 2(〈Nexciton 〉 + 1) 2 (6)
Figure 6. On-time probability distributions measured for cw excitation at λexc ) 488 nm (left) and λexc ) 532 nm (right) at various laser intensities, as indicated in the figure. Solid lines are fits to the data as described in the text.
Figure 7. Left panel: average falloff rate at λexc ) 434, 488, and 532 nm determined from ∼40 QDs at each absorption rate, error bars are determined from standard deviation of the mean. Right panel: same data plotted on a log-log scale, illustrating quadratic dependence of the average fall rate on laser power. Solid lines are fits to the data as described in the text.
Conversely, for a fixed absorption rate there is a clear increase in the falloff rate with increasing photon excitation energy, reminiscent of the wavelength dependence observed by Knappenberger et al.13 Discussion. The observation of an exponential falloff of on-time probability distributions that is quadratically dependent on laser power at all investigated excitation wavelengths suggests a common origin of these effects, consistent with a biexciton formation and Auger ionization model. The Poisson analysis demonstrates that such biexcitonic states are being formed with essentially unity probability on the time scale of blinking events. What remains to be quantified is the magnitude of their impact on QD blinking. In the context of a model for QD blinking in which the off-state is due to a charged QD species, one can estimate the probability that a biexciton leads to charge ejection, that is, the efficiency for QD ionization. It is worth noting that since the falloff times are much longer than the time scale for biexciton creation, it is immediately evident that such events rarely lead to ionization. If we implement the ionization efficiency Pionize in eq 3 by simply scaling the probability of biexciton formation and rewrite the expressions in terms of the average Nano Lett., Vol. 9, No. 1, 2009
for cw excitation. In either scenario, one predicts a quadratic scaling of the number of ionizations per pulse (∆Trep/τfalloff) or per fluorescence lifetime (τfluor/τfalloff) versus the average number of biexcitons per pulse (or per τfluor). Plotted in Figure 8 is the number of ionization events per pulse (left panel) or fluorescence lifetime (right panel) versus 0.5〈Nexciton〉2 observed over a series of laser powers, demonstrating excellent agreement with predictions at each of the three wavelengths. The solid curved lines represent the best fit predictions to the data from eqs 5-6 with the dashed lines representing the predicted limiting case for 〈Nexciton〉 , 1. Of more quantitative interest, the ionization efficiency per excitation event, Pionize, is given by the slope of both dashed and solid lines as the average number of biexcitons approaches zero, as predicted from eqs 5-6. Nonlinear least-squares fit to the data yield values of Pionize ) 2.8 (2) × 10-5 at 434 nm, 5.2 (6) × 10-5 at 488 nm, and 2.4 (2) × 10-5 at 532 nm. Though the data suggest differences in Pionize as a function of excitation wavelength, we have not performed sufficient tests for limited sampling statistics at the different power levels. However, the data are all consistent with ionization efficiencies on the order of 4(2) × 10-5. Auger relaxation rates have been measured in semiconductors QDs and are found to occur on a ∼10 ps time scale.20 This lifetime is several orders of magnitude smaller than a typical QD fluorescence lifetime (τfluor ≈ 30 ns), which could
Figure 8. Determination of biexciton ionization efficiency for pulsed (left panel) and cw (right panel) excitation. The vertical axis represents the probability of ionizations per pulse (or per τfluor), the horizontal axis represents the average number of biexcitons per pulse (or per τfluor) and the slope of the line is approximately equal to the biexciton ionization efficiency. Error bars reflect standard deviation of the mean values. Solid lines are fits to the data, dashed lines indicate the biexciton ionization efficiency in the limit of low carrier density. 343
therefore in principle contribute to a large biexciton ionization efficiency. However, there is no a priori reason to expect an equivalence between the Auger relaxation and Auger ionization rates (krelax and kionize, respectively) because, although both processes involve the same electronically excited initial state, the final states are quite differentsin the case of Auger relaxation the final state is the ground electronic state of the “core” QD, in the case of Auger ionization the final state is a trap state at the surface or outside of the QD. By analogy with fluorescence quantum yields, the ionization efficiency is given by Pionize ) kionize ⁄ (krelax + kionize)
(7)
which illustrates the fact that ionization and (nonionizing) relaxation processes compete for removal of the biexcitonic state. If relaxation is indeed extremely rapid, this argues for a relatively inefficient ionization process, which is in agreement with the measured presented here. Finally, we compare our results with existing reports in the literature. Several techniques have been used to evaluate multiexciton ionization efficiency in QDs.21-23 As one strategy, sensitive electrostatic force measurements of species excited in a relatively low intensity laser regime are used to measure light promoted charge ejection from individual QDs. These methods yield ionization efficiencies of 10-6-10-5, which are in reasonable agreement with the values measured from the current study.21 This regime corresponds to very low probabilities for triexcitonic and higher (n > 2) events, which would be most comparable to our experimental conditions in the limiting case of 〈Nexciton〉 , 1. As another approach, higher intensity light sources have been employed to achieve near unity probability for multiexcitonic excitation in each QD with the resulting fraction of species undergoing charge ejection representing the ionization efficiency. In this regime, much higher ionization efficiencies of 10-2-10-1 have been reported,22,23 that is, significantly larger than currently observed. However, it is quite likely that higher levels of exciton formation might also be expected to lead to enhanced ionization efficiency, which have not been considered in our model. Since these latter results are obtained in the limit closer to 〈Nexciton〉 ≈ 1, this may well account for the much larger efficiencies observed. It would be interesting to extend these studies over a much wider range of excitation intensities, specifically providing data with which to explore the dependence of ionization efficiency on the exciton number for a single QD. Long time falloffs in the on-time probability distributions have been previously reported with different, and sometimes contradictory, sensitivities to a variety of experimental variables, including excitation intensity, temperature, and excitation wavelength.7,12,13,24 For example, Stefani et al. reported falloffs of on-time probability distributions that were linearly dependent on laser intensity,12 whereas Knappenberger et al. found on-time falloffs to be independent of laser power.13 Furthermore Knappenber et al. observed a frequency dependent threshold of photon excitation energies ∼100 meV above the QD’s absorption edge at which exponential falloffs of the on-time probability distribution occur. This threshold was interpreted as arising from the opening of a 1-photon 344
channel associated with electron ejection to the QD surface. With regard to the frequency threshold, for the range of QD sizes and excitation wavelengths explored in the present work the minimum photon energy investigated already corresponds to excitation ∼121 meV above the QD absorption edge. Therefore, the appearance of exponential falloff times in our data at all wavelengths is entirely consistent with these previous studies, though clearly it would be interesting to explore this threshold hypothesis further with longer wavelength excitation. With regards to the intensity-dependence of on-time falloffs, we note that Stefani et al. also reported an important role of the QD’s surrounding environment: on insulating glass, the falloff was linearly dependent on laser intensity; on a conducting substrate, the falloff was exponentially dependent.12 The current work was done with QDs embedded in a thin polymer matrix and one may reasonably expect a different response in the different environment. Although our proposed biexciton model accounts for the clear nonlinear behavior reported herein, any environmental effects were not included and this intriguing sensitivity gives direction to future refinements and theoretical efforts ongoing in our laboratory.19 With regards to the power independent fall-off times previously reported13 at first glance they would seem inconsistent with the quadratic laser power dependence of the falloff rates clearly demonstrated in the present work. However, based on a closer inspection of the model, we can at least suggest a relatively simple reason for their power independence in falloff times. Specifically, eq 4 predicts that the falloff rate increases quadratically with the average number of excitons at low laser powers, but begins to roll over when the probability of creating a biexciton per pulse (or per fluorescence lifetime) becomes sufficiently large. In the limit where the probability of creating a biexcition (or multiexciton) per pulse approaches unity, the predicted falloff rate will saturate and become independent of laser power. In fact, even though we have been intentionally working at nW to sub µW average laser powers to minimize biexcitonic effects, the data plots in Figure 8 already show evidence for such a deviation away from pure quadratic behavior at the highest values explored. The experimental conditions described by Knappenberger et al. occur in laser power regimes considerably above the saturation limit predicted by eq 4, which may begin to help explain the lack of power dependence in their observed falloff times. Indeed, if we assume that biexciton saturation was present in these previous studies, the observed range of falloff times from Knappenberger et al. would translate into an estimated ionization efficiency of ∼10-6, which is in reasonable agreement with the results reported herein. Clearly more work would need to be performed to make such comparisons suitably quantitative. However, it does highlight the serious and perhaps underappreciated experimental challenges of maintaining sufficiently low biexciton formation rates in QDs (and thus probability of biexciton ionization events) over time scales stretching out to seconds and longer. In conclusion, we have measured an exponential falloff component of the on-time probability distributions generated Nano Lett., Vol. 9, No. 1, 2009
from QD blinking trajectories obtained under low power excitation conditions in the nW to µW range. The data at these low powers reveal a clear quadratic dependence on incident laser intensity, which we have tentatively attributed to low probability biexciton formation followed by inefficient Auger ionization. We do not observe these biexcitons directly, though their time averaged concentrations at these power levels would be exceptionally challenging to detect. However, we analyze the data based on simple Poisson statistics, which allows us to calculate the number of biexciton (or higher exciton) species that are formed during the time interval corresponding to the observed falloff time, from which one can estimate a QD biexciton ionization efficiency of ∼10-5. These results complement previous reports multiexciton ionization in QDs and highlight the potential importance of biexcitons in QD blinking processes under typical single molecule laser illumination conditions. References (1) Klimov, V. I. Semiconductor and Metal Nanocrystals: Synthesis and Electronic and Optical Properties; CRC Press: Boca Raton, FL, 2003. (2) Dahan, M.; Levi, S.; Luccardini, C.; Rostaing, P.; Riveau, B.; Triller, A. Science 2003, 302, 442. (3) Michler, P.; Imamoglu, A.; Mason, M. D.; Carson, P. J.; Strouse, G. F.; Buratto, S. K. Nature 2000, 406, 968. (4) Nirmal, M.; Dabbousi, B. O.; Bawendi, M. G.; Macklin, J. J.; Trautman, J. K.; Harris, T. D.; Brus, L. E. Nature 1996, 383, 802. (5) Cichos, F.; Von Borczyskowski, C.; Orrit, M. Curr. Opin. Colloid Interface Sci. 2007, 12, 272. (6) Kuno, M.; Fromm, D. P.; Hamann, H. F.; Gallagher, A.; Nesbitt, D. J. J. Chem. Phys. 2000, 112, 3117. (7) Shimizu, K. T.; Neuhauser, R. G.; Leatherdale, C. A.; Empedocles,
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S. A.; Woo, W. K.; Bawendi, M. G. Phys. ReV. B 2001, 63, 205316. (8) Efros, A. L.; Rosen, M. Phys. ReV. Lett. 1997, 78, 1110. (9) Kuno, M.; Fromm, D. P.; Hamann, H. F.; Gallagher, A.; Nesbitt, D. J. J. Chem. Phys. 2001, 115, 1028. (10) Kuno, M.; Fromm, D. P.; Johnson, S. T.; Gallagher, A.; Nesbitt, D. J. Phys. ReV. B 2003, 67, 125304. (11) Frantsuzov, P. A.; Marcus, R. A. Phys. ReV. B 2005, 72, 155321. (12) Stefani, F. D.; Knoll, W.; Kreiter, M.; Zhong, X.; Han, M. Y. Phys. ReV. B 2005, 72, 125304. (13) Knappenberger, K. L.; Wong, D. B.; Romanyuk, Y. E.; Leone, S. R. Nano Lett. 2007, 7, 3869. (14) Pelton, M.; Smith, G.; Scherer, N. F.; Marcus, R. A. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 14249. (15) Mention of commercial products is for information only; it does not imply National Institute of Standards and Technology recommendation or endorsement, nor does it imply that products mentioned are necessarily the best available for the purpose. (16) Fomenko, V.; Nesbitt, D. J. Nano Lett. 2008, 8, 287. (17) Peterson, J. J.; Nesbitt, D. J. University of Colorodo, Boulder, CO. Unpublished work. (18) Hodak, J. H.; Fiore, J. L.; Nesbitt, D. J.; Downey, C. D.; Pardi, A. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 13351. (19) Baker, T. A.; Peterson, J. J.; Nesbitt, D. J. University of Colorodo, Boulder, CO. Unpublished work. (20) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011. (21) Krauss, T. D.; O’Brien, S.; Brus, L. E. J. Phys. Chem. B 2001, 105, 1725. (22) Son, D. H.; Wittenberg, J. S.; Alivisatos, A. P. Phys. ReV. Lett. 2004, 92, 127406. (23) Kraus, R. M.; Lagoudakis, P. G.; Muller, J.; Rogach, A. L.; Lupton, J. M.; Feldman, J.; Talapin, D. V.; Weller, H. J. Phys. Chem. B 2005, 109, 18214. (24) Wang, S.; Querner, C.; Emmons, T.; Drndic, M.; Crouch, C. H. J. Phys. Chem. B 2006, 110, 23221.
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