Multiexciton Absorption Cross Sections of CdSe Quantum Dots

Letter pubs.acs.org/JPCL

Multiexciton Absorption Cross Sections of CdSe Quantum Dots Determined by Ultrafast Spectroscopy Nils Lenngren,*,†,‡ Tommy Garting,†,‡ Kaibo Zheng,† Mohamed Abdellah,†,¶ Noel̈ le Lascoux,†,§ Fei Ma,†,∥ Arkady Yartsev,† Karel Ž ídek,† and Tõnu Pullerits*,† †

Department of Chemical Physics, Lund University, Box 124, 221 00 Lund, Sweden Department of Chemistry, Qena Faculty of Science, South Valley University, Qena 83523, Egypt

¶

S Supporting Information *

ABSTRACT: Multiexciton absorption cross sections are important for analysis of a number of experiments, including multiple exciton generation and stimulated emisson. We present a rigorous method to determine these cross sections using transient absorption (TA) measurements. We apply the method to CdSe quantum dots (QDs) and core−shell (CdSe)ZnS QDs. The method involves measuring TA dynamics for various excitation intensities over a broad time range and analyzing the experiments in terms of a kinetic multiexciton model taking into account all contributions to the signal. In this way, we were able to quantify exciton and multiexciton absorption cross sections at different spectral positions. The absorption cross sections decrease with increasing number of excitations, qualitatively in agreement with the state-filling effective mass model but showing a slower decrease. The cross sections for single-exciton to biexciton absorption range between 57 and 99% of the ground to single-exciton cross section. SECTION: Spectroscopy, Photochemistry, and Excited States

S

charging can occur on the same time scales, complicating the analysis.26 One way to avoid this problem is the core−shell QD structure, especially a gradient core−shell structure, where the core material changes gradually into a shell material with higher band gap.27−29 For quantitative analyses, one also needs to have a good knowledge of the oscillator strengths of the transitions that contribute to the signal. For photoemission, the emittive transitions were carefully analyzed,30 leading to relatively low MEG yields based on the photoemission dynamics studies.18,19 However, such an analysis is missing for transient absorption (TA). Here, all possible signal components have to be considered. The intuitive assumption that the relative amplitude of the fast component can be directly related to the yield of the MEG has to be rigorously tested. In this work, we present a detailed analysis of the absorption cross section of multiple excitons. All components of the pump−probe signal (the bleach, stimulated emission, and excited-state absorption) are taken into account and quantified. The study shows that the simple state-filling effective mass model can qualitatively describe the observed trends but should not be used for quantitative calculations. The studied QDs had a core diameter of either 3 (S samples) or 5 nm (L samples) and either no shell (sample S0 and L0), a

emiconductor nanocrystals are attracting broad scientific attention from both fundamental and applied research.1−3 Of special interest are nanocrystals with quantum confinement,4 called quantum dots (QDs), with intriguing optoelectronic properties such as quantized energy levels and a sizetunable band gap. Quantum confinement occurs if the size of the nanocrystal is similar to or smaller than the exciton Bohr radius.5 Potential applications of QDs are related to both emission and absorption of the nanocrystals. Klimov and co-workers have demonstrated optical gain in QDs at size-tunable emission wavelengths demonstrating the feasibility of QD-based lasers,6 and the Kambhampati group has accomplished spectral control of QD optical gain7 and applied it to optoelectronics.8,9 Tunable absorption spectra are of interest for solar cell applications.10,11 In addition, in the context of solar energy, the potential of QDs in breaking the Shockley−Queisser12 thermodynamic limit via so-called multiple exciton generation (MEG) is particularly attractive.13−16 Via this effect, one can “harvest” the excess energy of the absorbed photons, which otherwise would be converted to heat and inevitably lost.17 While the practical potential of the effect in drastically improving the efficiency of the solar cells is apparent, the reported yields of the process are contradictory,18−22 and understanding of the basic principles is still obscure.23−25 MEG yields are usually determined from the relative amplitude of the characteristic fast multiexciton dynamics of photoemission or photobleaching. Surface trapping and QD © 2013 American Chemical Society

Received: July 18, 2013 Accepted: September 16, 2013 Published: September 16, 2013 3330

dx.doi.org/10.1021/jz401522h | J. Phys. Chem. Lett. 2013, 4, 3330−3336

The Journal of Physical Chemistry Letters

Letter

Figure 1. (Main plot) Absorption and fluorescence spectra of sample L0 in toluene (dashed lines), its TA at longer delay times (solid lines), the spectral position of the pump (circles), and the wavelengths where the kinetics were analyzed (arrows). (Inset) Absorption spectra of samples S0, S5, and S300 (offset vertically) with the pump wavelength at 470 nm (gray line) and the probe wavelengths of 526, 540, and 575 nm (arrows). The arrow colors show the relative probe positions: at the band edge (red), at the first absorption peak (green), or at the blue edge of the first absorption peak (blue).

Four different excitation intensities were used for each sample (see Table S1 in the Supporting Information) giving rise to different TA kinetics ΔA(I,t) (see Figure 2 and Figure S1, Supporting Information). The bleach amplitude increases for higher excitation intensities, displaying intensity-dependent

thin shell (sample S5), or a thick shell (sample S300). Though not an exhaustive set, samples were chosen to give a variety of core and shell sizes. Absorption spectra of these samples are given in Figure 1. Experimental details of the TA setup and sample preparation are given in the Experimental Methods section. Multiple exciton absorption cross sections were determined from the TA characteristics of the samples. Formation of excitons in CdSe QDs causes dominantly a bleach in the TA signal at the lowest exciton level, which decays on the time scale of nanoseconds. Although a single exciton in a QD typically recombines on a nanosecond time scale, multiple excitons feature a significantly shorter lifetime due to Auger recombination. Therefore, a signal originating from multiply excited QDs can be clearly resolved from TA kinetics. The treatment of TA kinetics can be simplified if the shape of the TA spectra does not change significantly with time, meaning that the kinetics at a single wavelength are representative of the entire spectrum. Probing sample L0 with a white light continuum confirmed that this holds after about 1 ps, while at subpicosecond times, the TA spectra go through rapid changes, reflecting relaxation of the exciton to the lowest state.31 In the current study, we do not analyze these processes. We perform single-wavelength analyses of sample L0 at two representative wavelengths and use narrow-band probes (bandwidth typically around 15 nm) for the other samples. The wavelengths are at the band edge, where stimulated emission should play a role, at the first absorption peak, or on the blue side of the first absorption peak. Like the sample choices, the probe wavelengths are intended to provide some representative examples in lieu of a complete study.

Figure 2. TA kinetics for sample L0 probed at 592 nm for the four pump intensities, normalized to the I0 trace at times longer than 6 ns (open symbols), and fits (lines). The normalization factors are given next to each trace. Faster kinetics from multiexcitons can be clearly identified for earlier times. All curves overlap at times over 1 ns, which corresponds to the one-exciton signal. At shorter times, the presence of different amounts of biexciton and even other multiexciton decays causes a spread in the curves. 3331



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Following eq 3, we plot ΔA0, calculated from all points where t > 6 ns, as a function of corresponding relative excitation intensities (see Figure 3 and Figure S2, Supporting

kinetics that is attributed to the creation of multiple excitons via a sequence of absorptions. MEG via impact ionization can be disregarded because the photon energy of the pump (≤1.5Eg for all samples) is well below the threshold of the process, 2.5Eg in CdSe.24 The number of excitons per QD is quantized, with states having discrete lifetimes that can be identified from different regions in the TA kinetics.32 For sample L0, the signals from the two lowest pump intensities are very similar when the signals are normalized at the long-time region. This means that the amount of multiple excitons in these two cases is negligible, and we can take the I0 curve as reference kinetics corresponding to one exciton in each excited QD. The pump intensities for the other samples were likewise chosen such that I0 would correspond to negligible amounts of multiple excitations. At long times (t > 6 ns), we only have one exciton per excited QD, independent of excitation intensity. At higher excitation intensities, two and more excitons per QD are generated. We can clearly recognize the faster biexciton decay in the higher excitation intensities at subnanosecond times and even triple and higher multiple excitons at even shorter time scales. Because electronic dephasing in QDs is expected to be fast,25 we ignore possible coherent effects33,34 in exciting with short laser pulses. In the absence of coherent effects, the multiple exciton populations initially follow a Poissonian distribution32,35 PN =

e−⟨N ⟩·⟨N ⟩N N!

Figure 3. (Left) Kinetic model containing six levels describing the multiexciton dynamics and corresponding TA signal. PN is the population of the N-exciton level, A and E represent absorptive and emittive transition strengths between neighboring levels, τN are the multiexciton Auger recombination times, and τ1 is the one-exciton decay time. (Right) Maximum, average, and minimum TA signal (crosses) ΔA0 rescaled from t > 6 ns (see Figure 2) as a function of excitation intensity (lower axis) and average number of excitons per QDs (upper axis) for sample L0 probed at 592 nm, fits according to eq 3, and maximum signals from the one-exciton K1 (from the three higher-intensity measurements). See Figure S2 (Supporting Information) for other samples and probe wavelengths. See the text for details.

(1)

Information). At each intensity, we have, depending on the sample, six or eight time points where ΔA was measured. Experimental noise causes a spread of the data points. We perform exponential fits to eq 3 based on either the minimum, the maximum, or the average value, giving three different values of ⟨N⟩0 and enabling us to calculate the initial populations from eq 1 (see Supporting Information, section 2). From the excitation intensity and ⟨N⟩0, we can also obtain σabs. (For fitted ⟨N⟩0 and σabs, see Tables S4 and S5, Supporting Information.) We point out that in this cross section determination procedure, we do not need to know the concentration of the QDs.40,41 Even for the highest intensities, we cannot discern more than five components to the signal decay. There might be several reasons for this. For example, the TA signal induced by higher than four-exciton excitation might be shifted away from the probe wavelength. Furthermore, the signals from higher excitons might be decaying so quickly that they cannot be resolved by our equipment. The Auger recombination lifetime of higher excitons might also become comparable with a relaxation rate of carriers in QDs. In this case, we cannot resolve the carriers as we measure the TA bleach of the lowest excited state. Due to the limited number of components, we model the population of the tetraexciton state as the sum of all higher populations, leading to a six-level model for quantifying the TA kinetics; see Figure 3. In this model, we explicitly follow the kinetics of populations of up to tetraexcitons and include all possible signal contribution including bleaching, stimulated emission, and absorption due to all populations PN, where N is the level index. The population dynamics are calculated using simple Pauli master equations, where we only allow downward population transfer with rate constants kN = 1/τN.42 Lifetimes are determined from experiment (see below). The initial populations for each excitation intensity are calculated from the

where ⟨N⟩ is the average number of excitons per QD, N is the number of excitons, and PN is the fraction of QDs with N excitons. Because the optical density of the sample is low (≤0.25 at the pump wavelengths), we can further assume that the excitation intensity does not vary in the probe volume, and we can use ⟨N⟩ = σabs·I for the average number of excitons per QD, where σabs is the absorption cross section at the excitation wavelength and I is excitation intensity in units of the number of photons per pulse per excitation area. From eq 1, we can represent the fraction of excited QDs, Pexc, as36,37 ∞

Pexc =

∑ PN = 1 − P0 = 1 − e−⟨N ⟩ = 1 − e−σ

abs·I

N=1

(2)

Clearly, if we know Pexc, we can obtain ⟨N⟩ and thereby determine σabs. In the long-time region, where only one exciton per QD remains, the signal intensity ΔA(I,t > 6 ns) is proportional to Pexc and decays biexponentially38,39 with time. From fitting, we find the lifetimes τ1,α and τ1,β and the amplitudes α and β of these two exponential decay components, where the amplitudes are normalized so that their sum is 1. We can now rescale the signals at long time to the corresponding signal at t = 0, which we call ΔA0(I), and represent all signals and ⟨N⟩’s via the reference intensity I0 and the corresponding ⟨N⟩0, obtaining ΔA 0(I ) =

ΔA(I , t > 6 ns) α e−t / τ1,α + β e−t / τ1,β

= ΔA 0,max ·(1 − e−(I / I0)·⟨N ⟩0)

(3)

where ΔA0,max is the largest possible single-exciton signal rescaled to t = 0, corresponding to the case if all QDs contain one exciton, that is, the sum in eq 2 is equal to 1. 3332



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Figure 4. Relative absorption cross sections for the N → N + 1-exciton transitions. The error bars for I4 were calculated by repeating the fitting procedure with the time constants fixed at their maximum (minimum) values as estimated from a semilogarithmic plot. The dotted lines show the behavior predicted by the state-filling model and the constant model. (a) Sample L0 probed at the band edge, calculated from three different pump intensities. For corresponding plots of the other samples and probe wavelengths, see Figure S3 (Supporting Information). (b) The S samples calculated from the highest pump intensity. (c) The L0 sample probed at the band edge, compared with the L0 sample probed away from the band edge and the S5 sample at the band edge. All data were calculated from the highest pump intensity. (d) The energy levels of CdSe QDs at the band gap with the experimental pump and probe transitions displayed.

concluded that E10 ≈ 0.02A01. The relative emission strengths for biexcitons and triexcitons were obtained based on literature rad18 rad 30 and krad For the tetraexciton, data as krad 21 ≈ 4k10 32 ≈ 1.2k21 . we extrapolated the lower exciton stimulated emission rates in rad rad terms of krad 10 , giving k43 ≈ 1.1k32 . In this way, it will be possible to extract the absorption strengths from the KN values. In sample L0, there is a minor initial fast component observed in the TA decay even when the pump intensity is kept well below the limit for creating multiple excitons. This is attributed to possible trapping of excitons45 and/or exciton relaxation through the band. The component was subtracted from all decays before any further analysis was performed. The relaxation time constants τN for the N-exciton levels and associated prefactors were fitted one at a time from the slowest to the fastest by global exponential fits to all measurements of each sample in the appropriate time region. (See Supporting Information section 2.) By comparing these fits to eq 4 and the time evolution given by the kinetic model, a system of linear equations is obtained that can be solved to retrieve the KN coefficients. (See Supporting Information section 3 for details.) This was done independently for each intensity, each giving its own set of KN coefficients. Our method allows us to check the consistency of the calculated maximum possible initial signals from the oneexciton level K1 (if all QDs contain exactly one exciton) with the experimental data. The exponential fits to eq 3 should approach the maximum initial signal value for I → ∞. This was done for the three fitted sets of ⟨N⟩ (from the average, the upper boundary, and the lower boundary of ΔA0) in Figure 3. The populations corresponding to each fitted ⟨N⟩ give different values of K1 for each intensity. The results are presented in

Poisson distribution according to eq 1. Such a kinetic model has an analytic solution where the population dynamics of the levels are given as a sum of up to four exponentials with time constants τN. Because the lower levels receive an influx from the higher levels, the prefactors of these exponentials can be either positive or negative. Finally, the TA signal due to the N-exciton population PN(t) is obtained as SN (t ) = (AN , N + 1 − EN , N − 1 − A 01) ·PN (t ) = KN ·PN (t ) (4)

where AN,N+1 is absorption from the level N to the level N + 1, EN,N−1 is the stimulated emission from level N to N − 1, and A01 is the ground-state bleach.31 Coefficients KN defined by the second equation can be seen as the signal from the level N if all population is on that level, PN = 1. This is the maximum possible signal from that level. The total signal from an ensemble of QDs is a sum of the signals from the individual max exciton levels Stot(t) = ∑NN=1 SN(t). Each KN in eq 4 is a sum of three components. We know A01; it is the absorbance at the analysis wavelength. The Einstein coefficient for stimulated emission is in general the same as that for absorption after one has taken the Stokes shift into account. When probing away from the band edge, stimulated emission can thus be assumed to be negligible. At the band edge, however, the fine structure of the band edge transition complicates the situation because the lowest level is a dark state.43,44 The Einstein coefficient for stimulated emission was calculated from the radiative lifetime obtained from the long component of the kinetics together with the quantum yield of the QDs (see Supporting Information section 4 for details). We 3333



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Figure 3 and Figure S2 (Supporting Information). The K1 coefficients from the populations arising from the average ⟨N⟩ give a maximum signal that is, for all intensities, in very good agreement with the initial fit. Having shown that only the averaged ⟨N⟩’s are consistent with the K1 calculated from them, we can justify only considering these values in the following analyses. To further estimate the robustness of the calculations, prefactors were also fitted with τN held fixed at the maximum and minimum time values that could be reasonably estimated from the data. KN values were calculated for both extreme cases using the population distribution from the average ΔA0. The spread in their value serves as an estimate of KN error and is shown as error bars in Figure 4. After having calculated KN values and their errors, we turn to absorption cross sections. Absorption strengths for each N are calculated from KN according to eq 4 and then recalculated to absorption cross sections (the absorption cross section of the ground state at the excitation wavelength was calculated from ⟨N⟩, and the absorption spectrum is known) and presented in Figure 4 and Figure S3 (Supporting Information), where the data points represent the relative absorption cross section for creating the (N + 1)th exciton. We present the data as ratios of the multiexciton to single-exciton cross sections in order to focus on the behavior characteristic to multiexcitons; absolute values can be calculated in the manner of Cho et al.46 At the higher exciton levels, we consider the highest-intensity case to provide the most reliable value (hollow circles in Figure 4a and stars in Figure S3, Supporting Information). We discuss the results starting from the effective mass description of the electronic structure of CdSe QDs.47,48 We show the relevant electron and hole energy levels together with the pump and the probe transitions in Figure 4d. The pump pulse excites mainly the 1P3/2(h) to 1Pe transition. The pumpgenerated state rapidly relaxes to what corresponds to the band edge 1S3/2(h) to 1Se transition, thereby filling the transition in the pump wavelength region. Consequently, we can assume that the pump transition does not depend on the number of excitons that are present in the system, which means that the Poisson condition of independent excitation events holds. Pumping close to the band edge would lead to deviations from Poissonian statistics already at ⟨N⟩ = 1,33 but when pumping away from the band edge, the population was reported to be Poissonian after relaxation to the band edge at least until ⟨N⟩ = 10.49 The Poisson model is further justified by 1P3/2(h)− 1Pe states having high degeneracy. Nevertheless, it is possible that small deviations occur at our highest intensities, which will be a source of error. All changes due to the excitations are expected to occur in the probe region. Two distinct models have previously been used to describe absorption cross sections of multiexcitons. In one case, it is assumed that the cross section is constant, which corresponds to an infinite number of states that can be excited and contribute to the signal.13,14 We will call this the “constant model”. The second possible model, the so-called state-filling effective mass model, is based on calculations of degeneracy of the 1Se and 1S3/2(h) states of CdSe QDs. In the effective mass description of CdSe, the 1Se electron state has multiplicity 2, and the absorption should drop to half of the original value after the first excitation. Filling both 1Se electron states (two excitations) should reduce the absorption to zero.50 The predictions of the two models are shown as dotted lines in Figure 4b and c.

For small QDs (Figure 4b), the absorption cross section drops rapidly when the number of excitations increases. The behavior qualitatively follows the state-filling model. However, the band edge absorption cross section drops to negligible levels only after three excitations. Shell thickness has no noticeable effect on the relative absorption cross sections, but the spectral position of the probe has; when going away from the band edge toward higher energies, clearly more states are available for absorption than the simple effective mass model predicts, which becomes especially prominent for triply and quadruply excited QDs. The absence of a shell thickness effect indicates that charging of the QDs is not significant because charging would be expected to decrease with increasing shell thickness. The large QDs (Figure 4c, sample S5 probed at the band edge also included for comparison) deviate more from the behavior predicted by the state-filling model. Like for small QDs, more states are available for absorption at higher energies; in fact, the constant model is a satisfactory approach when the probe wavelength is tuned away from the band edge. A more detailed description of CdSe QD electronic structure than the state-filling model indicates greater complexity. For example, exciton−exciton interaction causes a shift in the absorption spectrum of the biexciton; the shift is close to 13 meV and independent of QD size.50 A biexciton shift of the absorption spectrum at the band edge would at least partly explain the σ12 shift from 0.5σ01 predicted by the state-filling model. It has been pointed out that even in the atomistic description of QDs, the general properties of the absorption spectrum in terms of S states, P states, and so forth (see Figure 4d) are preserved. However, because thousands of valence electrons are involved in the transitions, each level corresponds to many more transitions.51 The effect is clearly more obvious for the large QDs with a larger number of valence electrons and for higher transition (probe) energies where the density of states is larger. Under these conditions, the QDs have “bulk-like” behavior in multiple exciton absorption.52 In this Letter, we have developed and used a method to obtain multiexciton absorption cross sections in semiconductor QDs with different sizes and shell thicknesses and probed at different spectral positions. The ratio of multiexciton to singleexciton cross sections changes depending on the properties of the QDs; as a common pattern, it decreases with increasing number of excitations, qualitatively in agreement with the statefilling effective mass model but showing a slower decrease. In general, both the constant and the state-filling model can be applied in some cases as a reasonable estimate. However, for establishing actual values of absorption cross sections, calculations such as those presented here are needed. Our method has an internal consistency check enabling significant narrowing-down of the possible errors. Such a method is of big interest to MEG yield determination or any other studies where knowledge of the number of multiexcitons is essential.

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EXPERIMENTAL METHODS Lumidot CdSe QDs of 5 nm diameter (sample L0) dispersed in toluene were bought from Sigma-Aldrich and diluted. Core− shell (CdSe)ZnS QDs with a 3 nm effective core diameter and two different shell thicknesses were synthesized by us using a previously described method17,29,53,54 by heating CdO and ZnO with oleic acid as the capping agent in 1-octadecene solution first to 180 °C until the solution cleared and then to 3334



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300 °C, injecting Se and S with trioctylphosphine as the capping agent, and letting QDs grow for either 5 s (sample S5) or 5 min (sample S300) before rapid cooling, purification, and transfer to toluene solution. The growth time determines the shell thickness. CdSe QDs of 3 nm diameter (sample S0) were synthesized by us using a similar method,17,55 where ZnO and S were not included; the growth time was 2 min, and the injection temperature was 260 °C. The samples were characterized using an Agilent 8453 UV−visible spectrophotometer and a Spex 1681 0.22m spectrometer. The resulting absorption and fluorescence spectra are shown in Figure 1. Two TA setups were used, one for sample L0 and one for the other samples. The setup for sample L0 has been described in detail elsewhere56,57 as has the other setup.11 In short, sample L0 was excited using a 150 fs pump pulse at 505 nm, shown as circles in Figure 1. The other samples were excited with an 80 fs pulse at 470 nm. In all experiments, the pump beam had a significantly larger cross section than the probe beam (typical fwhm diameters being 1 mm and 150 μm). Cuvettes were 0.5 (sample L0) or 1.0 mm (other samples) thick. The band edge transition optical density was kept below 0.3. The pulse intensity was varied by using neutral density filters and denoted in the text as factors of experiment-specific reference intensities I0(experiment). For sample L0, after the pump pulse, a white light continuum probe interacted with the sample, providing TA spectra at delay times from subpicoseconds up to 22 ns (see Figure 1). Other samples were probed to 10 or 12 ns. For sample S5, narrow-band-width probe pulses at 540 and 575 nm were used in separate experiments. For the other samples, only 540 (S300) or 526 nm (S0) probe light was used.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Multiexponential fits, calculations of the initial population distribution, extraction of the KN coefficients, cross section for stimulated emission, and absorption cross sections from different pump intensities. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (N. Lenngren). *E-mail: [email protected] (T. Pullerits). Present Addresses §

N. Lascoux: ILM, Université Claude Bernard Lyon 1, France. F. Ma: Biophysics of Photosynthesis Division, University of Amsterdam, The Netherlands. ∥

Author Contributions ‡

N. Lenngren and T. Garting contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financed by the Swedish Research Council, the Swedish Energy Agency, the Crafoord Foundation, and the Knut and Alice Wallenberg Foundation. N. Lascoux acknowledges a guest researcher’s fellowship from the Wenner-Gren Foundations. We acknowledge technical assistance by Pavel Chábera, fruitful discussions with Khadga Karki and collaboration within nmC@LU. 3335



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Multiexciton Absorption Cross Sections of CdSe Quantum Dots

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