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Comparison of Carrier Multiplication Yields in PbS and PbSe Nanocrystals: The Role of Competing Energy-Loss Processes John T. Stewart,*,† Lazaro A. Padilha,† M. Mumtaz Qazilbash,† Jeffrey M. Pietryga,† Aaron G. Midgett,‡,§ Joseph M. Luther,‡ Matthew C. Beard,‡ Arthur J. Nozik,‡,§ and Victor I. Klimov*,† †

Center for Advanced Solar Photophysics, C-PCS, Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States ‡ Chemical and Material Sciences Center, National Renewable Energy Laboratory, Golden, Colorado 80401, United States § Department of Chemistry and Biochemistry, University of Colorado at Boulder, Boulder, Colorado 80309, United States ABSTRACT: Infrared band gap semiconductor nanocrystals are promising materials for exploring generation III photovoltaic concepts that rely on carrier multiplication or multiple exciton generation, the process in which a single high-energy photon generates more than one electron−hole pair. In this work, we present measurements of carrier multiplication yields and biexciton lifetimes for a large selection of PbS nanocrystals and compare these results to the well-studied PbSe nanocrystals. The similar bulk properties of PbS and PbSe make this an important comparison for discerning the pertinent properties that determine efficient carrier multiplication. We observe that PbS and PbSe have very similar biexciton lifetimes as a function of confinement energy. Together with the similar bulk properties, this suggests that the rates of multiexciton generation, which is the inverse of Auger recombination, are also similar. The carrier multiplication yields in PbS nanocrystals, however, are strikingly lower than those observed for PbSe nanocrystals. We suggest that this implies the rate of competing processes, such as phonon emission, is higher in PbS nanocrystals than in PbSe nanocrystals. Indeed, our estimations for phonon emission mediated by the polar Fröhlich-type interaction indicate that the corresponding energy-loss rate is approximately twice as large in PbS than in PbSe. KEYWORDS: Carrier multiplication, multiple exciton generation, Auger recombination, hot exciton cooling

S

promoted to the conduction band following a collision with a high-energy carrier.14 Because of fast energy losses due to phonon emission and constraints imposed by combined requirements of energy and momentum conservation, the efficiency of impact ionization in bulk materials is relatively low, especially in the spectral region relevant to solar energy conversion.15,16 It has been expected that in NCs, CM is enhanced as a result of relaxation of momentum conservation17 and reduced phonon emission rates due to the “phonon bottleneck”.18−20 Consequently, NCs have been proposed as a material’s platform for the realization of concepts of CMenhanced solar cells.21 Following the first report on the observation of spectroscopic signatures of CM in PbSe NCs,5 these types of structures have become a model system for both experimental studies of MEG5,9−11 as well as testing various theoretical models of this process.22−27 On the basis of the most recent measurements of CM in PbSe NCs,10,28,29 accounting for extrinsic effects such as photocharging, the efficiency of this process is enhanced compared to bulk PbSe30 as inferred from the reduction of both

emiconductor nanocrystals (NCs) have emerged as a novel material’s platform for testing the concepts of generation III solar cells.1−4 These nanomaterials have been speculated to be more efficient solar energy converters as compared to their bulk counterparts due in part to enhanced carrier multiplication (CM) or multiple exciton generation (MEG), the process whereby absorption of a single high-energy photon results in more than one electron−hole pair. The small dimensions of these NCs (on order of a few nanometers) confine the carrier wave functions to distances that can be much smaller than the bulk exciton radii, the phenomenon referred to as quantum confinement. The spatial confinement results in increased Coulomb interactions between carriers, and processes mediated by Coulomb interactions such as nonradiative Auger recombination and CM are enhanced in quantum confined nanostructures compared to bulk semiconductors.5,6 Materials designed to take advantage of efficient CM should have an energy gap that is smaller than that found for the traditional Shockley−Queisser limit and could potentially enhance the single-junction device power conversion efficiency to above 40%.7−9 Hence, there has been considerable interest in exploring CM in nanocrystals made from infrared energy gap materials such as Si, PbSe, PbTe, and PbS.5,10−13 In bulk semiconductors, CM has been explained by impact ionization, a process in which a valence-band electron is © 2011 American Chemical Society

Received: September 27, 2011 Revised: December 6, 2011 Published: December 12, 2011 622

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becomes progressively larger with increasing NC size, that is, decreasing confinement energy. We argue that this discrepancy is likely due to the differences in the rates of competing energy relaxation channels. Specifically, we estimate the energy-loss rate for phonon emission mediated by the polar Fröhlich interaction. We find that the cooling rate is approximately a factor of 2 larger in PbS compared to PbSe due to a larger longitudinal optical (LO) phonon energy and stronger electron−phonon polar coupling. Thus, although multiexciton generation rates in these materials might be similar, the faster carrier cooling in PbS NCs narrows down the time window when CM can occur and decreases the CM efficiency. The samples studied here are high quality, well-passivated PbS NCs grown via air-free solution-based synthesis.36 In a typical procedure, Pb oleate is made by mixing PbO (2 mmol, 0.446 g), 1-octadecene (ODE) (10 g), and oleic acid (amount determines NC size). Pb oleate is heated to 110 °C under vacuum and then to 120 °C under N2. We then inject bis(trimethylsilyl)sulfide (1 mmol, 210 μL in 4 mL of ODE) and let it cool for 30 min to one hour. The Schlenk line is removed and the NCs are precipitated in an argon glovebox. In Figure 1, we display absorption and photoluminescence (PL)

the CM spectral onset and the electron−hole pair creation energy if both are expressed in terms of the band gap energy (Eg). While the use of Eg-normalized photon energies in evaluating the CM efficiencies is still a subject of debate,30,31 the fact that PbSe NCs are superior to bulk PbSe with regard to CM-related enhancement of the “energetic output” (evaluated, for example, from the product of Eg and the QE24,28 or calculated power conversion efficiencies7,8; QE is the quantum efficiency of photon to exciton conversion) is almost universally accepted in the community. Despite improvement relative to bulk PbSe, the CM yields in PbSe NCs are still lower than those expected based solely on energy conservation. On the other hand, only by approaching the energy-conservation defined limit one can obtain an appreciable improvement in the power conversion efficiency of practical devices.7−9 Therefore, a search for new types of NC materials with enhanced CM performance is an important direction of current research in this area. This search could be simplified through the development of a better understanding of which properties of bulk semiconductors are relevant for determining their CM efficiencies in nanocrystals. This is particularly important as the theory of CM and competing relaxation pathways in NCs is still under development. An initial approach to elucidating the factors that define CM efficiencies could be to examine two semiconductors that are nearly identical in their bulk form except for a few key characteristics and then attempt to draw correlations between those characteristics and the measured CM performance for the nanocrystal form of the same materials. While empirical in nature, even a relative description of the relevant bulk properties relevant for NCs would be very important for both experimental searches for new efficient CM materials as well as providing direction to future theoretical efforts to describe CM and the competing processes. To this end, in this Letter we present experimentally determined CM quantum efficiencies and biexciton lifetimes for a large collection of PbS NCs and compare these results to the more well-studied PbSe NCs.9,10,28,29,32 PbS and PbSe semiconductors are nearly ideal for this sort of comparison because they share many similarities in their bulk form. They are both IV−VI direct gap semiconductors; both have rock-salt cubic lattice structure; furthermore, due to small effective masses of electrons and holes and large dielectric constants,34 they both exhibit strong confinement even for relatively large NC sizes of more than 10 nm. Just as importantly, PbS and PbSe have a few key differences in material properties such as the phonon energies and the strengths of the electron−phonon polar coupling; the latter parameters are important in CM as they define the rate of important competing processes: intraband cooling via phonon emission. It is apparent that some of the similarities between PbSe and PbS are relevant when considering multicarrier interactions because we observe in the collection of PbS sizes studied (those with an energy gap in the range of Eg = 0.6 − 1.2 eV) that PbS and PbSe NCs exhibit very similar Auger recombination lifetimes of biexciton states. Since multiexciton generation is the inverse of Auger recombination, this observation together with the similar band structure35 suggest that PbS and PbSe NCs should have similar CM time scales, and thus, similar CM efficiencies. However, our results show a surprising disparity in multiexciton yields in PbS and PbSe NCs. Specifically, we observe that CM yields are systematically lower in PbS NCs compared to PbSe NCs and the deviation

Figure 1. Absorption (solid lines) and photoluminescence (dashed lines) spectra for two PbS samples with energy gaps Eg = 0.84 eV and Eg = 0.71 eV, lower and upper, respectively. The inset is a TEM image of the Eg = 0.84 eV sample, which shows that the NCs are nearly spherical, highly monodisperse particles.

spectra for two PbS NC samples with energy gaps Eg = 0.84 ± 0.01 eV and Eg = 0.71 ± 0.01 eV. Both samples show a wellresolved 1S transition, which couples the band-edge electron (1Se) and hole (1Sh) states. A fairly small line width of this transition (∼100 meV) indicates a high monodispersity of our samples, which is also evident from the transmission electron microscopy (TEM) studies (see an example of a TEM image in the inset of Figure 1 of the sample with Eg = 0.84 eV). To measure biexciton lifetimes and the CM efficiency, we use two ultrafast time-resolved spectroscopic techniques: transient absorption (TA)5 and PL up-conversion (uPL).10,37 In TA measurements, the samples are excited with 100 fs (FWHM), 3.1 eV second-harmonic pulses from a 1 kHz Ti−sapphire amplifier. The pump-induced absorption changes (Δα1S) are 623

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probed with variably delayed pulses from an optical parametric amplifier tuned to the 1S absorption maximum. The Δα1S signal is dominated by state-filling-induced absorption bleaching and its temporal evolution tracks the decay of the 1S state occupation factor.5 Because of a high, 8-fold degeneracy of the 1S levels in PbS, for the range of pump fluences studied here the 1S bleach is directly proportional to the average occupancy of the NCs, ⟨N⟩ (the average number of excitons per NC). In the uPL experiment, emission from the sample is frequency mixed in a nonlinear optical crystal (beta barium borate) with variably delayed, 2 ps (FWHM), 1.55 eV gating pulses from a 250 kHz Ti−sapphire amplifier. The upconverted signal is spectrally filtered with a monochromator and detected using a photomultiplier.37 The instantaneous emission intensity measured in this way is proportional to the product of occupancies of electron- and hole-emitting levels and scales as ⟨N⟩2 in the case of neutral excitations.10 Multiexcitons can be distinguished from single excitons based on a significant difference in recombination time scales.5 In PbS and PbSe NCs, intrinsic single-exciton lifetimes are on the order of hundreds of nanoseconds to microseconds. Further, we use only well passivated samples that do not show any fast dynamics due to surface trapping. Therefore, in our measurements limited to 1−1.5 ns, the single-exciton signal is essentially “flat”. Multiexcitons, however, have lifetimes that are less than a nanosecond for the range of samples studied and are observed as a fast initial component decaying to the singleexciton background (see Figure 2). In all time-resolved spectroscopic studies, we actively stir NC samples to make sure that the results of CM measurements are not affected by photocharging.10,28,29,38,39 Multiexcitons can be generated via two processes: CM following absorption of a single photon or the absorption of multiple photons from the same laser pulse. In the case of low photon energy excitation (1.55 eV), CM is not possible in the samples studied here because the photon energy is below the energetic onset of this process (∼2.8Eg; see below). Therefore, multiexcitons can only be generated via multiple-photon absorption at sufficiently high fluences when the initial NC occupancy ⟨N0⟩ is approaching unity or exceeds it. ⟨N0⟩ can be determined from the product of the NC absorption crosssection (σ) and the pump fluence (j): ⟨N0⟩ = σj, where j is measured in units of photons per cm2. We use absorption cross sections from ref 40 after introducing a solvent dependent correction in the dielectric confinement factor.41 The distribution of initial NC occupancies in the photoexcited NC ensemble immediately following photoexcitation (t = 0) can be described by a Poisson distribution,41,42 pn = ⟨N0⟩ne−⟨N0⟩/n!. The average exciton multiplicity (the average number of excitons per NC in a subensemble of photoexcited NCs) can be related to ⟨N0⟩ by ⟨Nx⟩ = ⟨N0⟩/(1 − p0).43 In the case of CM, when multiexcitons are generated by single photons, ⟨Nx⟩ provides a direct measure of the QE of photonto-exciton conversion: QE = lim→0 . In the case of TA measurements, ⟨Nx⟩ can be found from the ratio of the signals before (t = 0) and following Auger recombination (τ ≫ τA): ⟨Nx⟩ = a/b (Figure 2a,b). Since the PL signal scales quadratically with NC occupancy,10 the ratio of the early- to late-time signals measured by uPL (we denote it A/B) is more sensitive to changes in the average exciton multiplicity compared to TA. Specifically, when the photoexcited NC ensemble contains only single excitons and biexcitons A/B = 3⟨Nx⟩ − 2, and hence, ⟨Nx⟩ = (A/B + 2)/3.10,28

Figure 2. TA and PL traces for a PbS sample with Eg = 0.71 eV. Panels (a,b) show TA dynamics while panels (c,d) show uPL. The panels on the left show data for 1.55 eV excitation. Differently colored traces represent different excitation fluences. The dynamics recorded for low fluences (⟨N0⟩ ≪ 1) are essentially flat indicating a high quality of surface passivation in our samples. The insets show the measured a/b (A/B) ratio of the early- to late-time signal (symbols) that is compared to the ratio calculated assuming Poisson distribution of NC occupancies (thin dashed line). The panels on the right correspond to excitation at 3.1 eV. These data show a greater than unity a/b (A/ B) ratio even in the limit of zero fluence, which is a signature of carrier multiplication. We fit the a/b (A/B) ratios (insets) to a line and use the zero-fluence intercept as the measure of the quantum efficiency. In TA, the quantum efficiency is simply a/b, while in uPL the quantum efficiency is (A/B + 2)/3. For both TA and uPL we find the quantum efficiency to be 1.18(0.03) for this sample.

In Figure 2, we show an example of NC population dynamics measured using TA (top panels) and uPL (bottom panels) with 1.55 and 3.1 eV excitation for a sample with Eg= 0.71 ± 0.01 eV. The traces recorded using 1.55 eV pump photons for low fluences (⟨N0⟩ ≪ 1) (Figure 2a,c) are nearly “flat” on the time scale of our measurements (1−1.5 ns), indicating that surface trapping is insignificant in these well passivated NCs. Also in the case of the 1.55 eV excitation we find excellent agreement of the measured a/b and A/B ratios with those calculated for Poisson statistics (dashed thin lines in the insets to Figure 2a,c). To study the regime of CM, we use 3.1 eV excitation and measure carrier dynamics by applying both TA (Figure 2b) and uPL (Figure 2d). As mentioned above, the efficiency of CM can be evaluated from the ratio of the early to late-time signals measured in the limit of progressively decreasing pump fluences. We extrapolate the zero fluence limit of the early to late ratios by using a linear fit (solid thin lines in insets to Figure 2b,d). For the sample shown, we find QE = 1.18 ± 0.03 for both TA and uPL. The agreement between these two measurements confirms the conclusion of ref 10 that the results of a quantitative assessment of CM yields do not depend on whether carrier dynamics are monitored using TA or PL if one assumes quadratic scaling of PL intensity with ⟨N⟩. 624

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similar to that in PbS NCs with the same confinement energy. This leads us to conclude that the rates of the inverse process, multiple exciton generation, in these two materials if evaluated as a function of Ec should also be similar, as both Auger recombination and CM are described by the same matrix element.14 This consideration might further suggest a similarity in the CM efficiencies in NCs of these two compositions. Surprisingly, however, both the measured CM yields and the overall size dependent trend observed in PbS NCs are different from those in PbSe NCs. The results of QE measurements for PbS NCs are summarized in Figure 4 and compared to PbSe

We extract the biexciton Auger decay lifetimes (τA) from moderate-fluence scans (⟨N0⟩ ∼ 0.5) recorded using 1.55 eV excitation by fitting the early time decay to a single exponential with a constant single-exciton background. The same lifetimes are recovered in the low fluence scans measured with 3.1 eV pump photons, confirming that the initial fast component in this case is indeed due to biexcitons produced via CM. The measured biexciton lifetimes for PbS NCs of different sizes are displayed in Figure 3. The closed circles represent TA

Figure 3. Biexciton Auger-decay lifetimes as a function of nanocrystal volume (a) and confinement energy (b). In (a), we observe that biexciton lifetimes of PbS as measured by TA (solid circles) and uPL (open circles) agree very well with PbSe (gray shaded area encompasses the ± single standard deviation). The biexciton lifetimes of PbS follow the same volume scaling observed in NCs of other compositions. In (b), biexciton lifetimes of PbS and PbSe are shown to agree very well over a large range of confinement energies. This implies PbSe and PbS have similar Auger rates and suggests that the rate of the inverse process, multiexciton generation, should be similar.

Figure 4. Quantum efficiency of photon-to-exciton conversion as a function of confinement energy and normalized photon energy; for all data the photon energy is 3.1 eV. In (a), carrier multiplication yields of PbS NCs measured by TA (close circles) and uPL (open circles) are contrasted against previous measurements of PbSe nanocrystals (gray shaded region). Despite their similarities in the bulk, PbS and PbSe show strikingly different quantum efficiencies for large NCs (low confinement). In (b), the CM yields are plotted as a function of hv/Eg. For small NC sizes (hv/Eg is small), PbSe and PbS show similar QEs. However, as the size gets larger, the QE of PbS flattens out while the QE of PbSe increases. The inset in panel (b) shows a low fluence uPL time scan at Eg = 1.12 eV, clearly showing a greater than unity A/B ratio for hv < 3Eg.

measurements while open circles are uPL measurements and each data point in the plot represents an individually synthesized sample. For easy comparison, the gray shaded region encompasses the ± single standard deviation of over forty measurements of PbSe NCs.10,28,29 In Figure 3a, we plot the lifetimes as a function of NC volume. For both PbS and PbSe NCs, the Auger lifetimes scale linearly with NC volume as was first observed for CdSe NCs44 and then for NCs of a variety of other compositions.6 While following the same size dependence, the Auger lifetimes in PbS NCs are systematically longer than in PbSe NCs of the same radius. Interestingly, when we replot the same data as a function of confinement energy (the difference between the NC- and the bulk-material band gap, Ec = Eg(NC) − Eg(bulk)), we observe a very close correspondence between the two data sets (Figure 3b). These results suggest that the strength of the carrier−carrier interaction responsible for Auger decay in PbSe NCs is very

NCs. In Figure 4a, we plot the QEs as a function of confinement energy, while in Figure 4b we plot them as a function of normalized photon energy (hv/Eg). The open circles are uPL measurements on PbS NCs while close circles are TA measurements. In the same plots, we also show PbSe measurements as gray shaded regions. All data are collected using the 3.1 eV excitation and different data points represent separately synthesized NCs of varying size. We note that for smaller NC sizes, the QEs measured for PbS and PbSe NCs are approximately similar. In fact, it is particularly noteworthy that we observe greater-than-unity QEs for photon energies below three energy gaps (hv < 3Eg). Specifically, we measure QE = 1.06 ± 0.03 for PbS NCs with Eg = 1.12 ± 0.01 eV, which 625

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kloss, and further assuming that CM is weak (η ≪ 1), we can present the CM yield as η ≈ T/τCM,28 where T is the time window during which CM is possible. The latter is determined by the time of carrier relaxation from the initial “hot” state, Ehot, to the CM threshold, ECM (defined by energy conservation and hot [(dE)/ selection rules), and it can be calculated from T = (∫ EECM (kloss(E))]. If kloss is not strongly energy dependent, the integral reduces to T = ΔE/τCM (here, ΔE = Ehot − ECM), which further leads to η ≈ ΔE/(klossτCM). This relationship allows us to present the electron−hole-pair creation energy, εeh (the energy required to generate a new electron−hole pair after the CM threshold is reached) in terms of τCM and kloss. Specifically, using the definition εeh = δ(ΔE)/δη, we obtain εeh = klossτCM. The electron−hole-pair creation energy is a convenient characteristic of the CM process as it captures the competition between carrier−carrier interactions leading to CM and “parasitic” energy losses due, for example, to phonon emission. The smaller values of εeh correspond to stronger CM performance and they can be achieved by reducing kloss and/ or τCM. The energy-conservation dictated limit of εeh is Eg. We can use Fermi’s golden rule to gain some insight on how τCM compares between PbS and PbSe. We observe a close correspondence between biexciton Auger lifetimes in PbSe and PbS NCs, which according to the Fermi golden rule requires that the product of the Coulomb coupling matrix element squared and the density of final single-exciton states (gx) must be similar in these materials. With the impact ionization picture of CM, the rate of this process (i.e., the inverse of τCM) is proportional to the square of the same matrix element but the density of single-exciton states is replaced with that of biexcitons (gxx). According to theoretical calculations, gxx scales as a power of gx: gxx ∝ (gx)n, where n > 2.25,27 These considerations suggest that the characteristic time constants of CM and Auger recombination are connected by the following relationship: 1/τCM ∝ (1/τA)(gx)n−1. On the basis of the experimental observation of the similarity of Auger lifetimes in PbS and PbSe NCs, we might expect that the CM time constants for these two materials are related by τCM(PbS)/ τCM(PbSe) = [gx(PbSe)/gx(PbS)]n−1. As both electrons and holes are slightly heavier in PbS than in PbSe, the density of single-exciton states in PbS is larger than in PbSe, implying that gx(PbSe)/gx(PbS) < 1. This further suggests that τCM(PbS)/ τCM(PbSe) < 1, that is, the rate of CM is faster in PbS NCs than in PbSe NCs. On the other hand, our experiments indicate that CM yields are lower in PbS than in PbSe NCs. This is an indication of a difference in the competing energy relaxation channels (i.e., kloss) which favors CM in PbSe NCs. One channel competing with CM is hot-carrier cooling via phonon emission. In ionic bulk crystals, intraband relaxation is dominated by the polar Fröhlich-type coupling of charge carriers to LO phonons. The strength of this coupling can be characterized by the constant of polar interactions, αF, which is determined by the difference between material-specific static and high-frequency dielectric constants, κ0 and κ∞, respectively: αF = (e2/ℏ)[m/(2ℏωLO)]1/2(1/κ∞ − 1/κ0), where ℏωLO is the LO phonon energy, e is the electron charge, and m is the carrier effective mass.48,49 In the case of the conduction band electron, for PbS, αF = 0.33 which is greater than the polar-interaction constant for PbSe, αF = 0.22.34 An additional factor that defines the energy loss rate due to phonon emission is the LO phonon energy itself, as it determines the number of phonon emission events required to dissipate a given amount of energy. The LO phonon energy for

corresponds to hv/Eg = 2.77. Previously, nonzero multiexciton yields were also observed for sub-3Eg phonon energies in PbSe and PbS NCs.10,45 These are important observations because optical selection rules applied to materials with similar electron and hole effective masses, such as PbS and PbSe, predict that the CM spectral onset is at least 3Eg,46,47 which is close to the energy of the symmetric 2Ph-2Pe transition in PbSe and PbS NCs.23 The observations of measurable CM for sub-3Eg photon energies might be indicative of the important role of asymmetric, nominally forbidden transitions in defining the CM threshold in these NCs. As the NC size becomes larger and hv/Eg increases, the QE in PbS NCs becomes progressively smaller compared to values measured for PbSe (shaded area). For samples that are sufficiently large such that hv/Eg > 4, we find that the multiexciton yield (η = QE − 1) in PbS NCs is at least a factor of 2 smaller than that in PbSe NCs. To highlight this difference, in Figure 5, we compare directly the TA dynamics and a/b

Figure 5. Comparing time dynamics and quantum efficiencies for PbS and PbSe nanocrystals (NCs) with a very similar band gap as measured by TA. The band gap of the PbSe NCs are Eg = 0.75(1) eV and the PbS NCs are Eg = 0.73(1) eV. The curves show low fluence (⟨N0⟩ ∼ 0.14) time dynamics for both PbSe (black line) and PbS (red line) excited at 3.1 eV, clearly demonstrating the disparity in quantum efficiencies observed for larger NCs. The inset shows the a/b ratio as a function of fluence as well as the linear fit used to extract the quantum efficiency for each sample.

ratios measured side-by-side for PbSe and PbS nanocrystals of similar energy gaps, Eg = 0.75 ± 0.01 eV and Eg = 0.73 ± 0.01 eV, respectively. This data clearly demonstrates the disparity in CM yields in PbSe and PbS NCs at small energy gaps (i.e., large NC sizes). For the largest sizes, we even observe a change in the size-dependent trend as the QE starts to flatten out and even decreases with increasing the hv/Eg ratio. This is the opposite of the behavior seen in PbSe NCs. In general, CM yields are determined by the competition between carrier−carrier interactions leading to generation of new electron−hole pairs and alternative energy-loss channels such as phonon emission. Introducing a characteristic CM time, τCM, and the energy-loss rate due to non-CM related processes, 626

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Letter

PbS is 28 meV while for PbSe it is 17 meV.34 Together, these parameters imply that energy loss rate due to phonon emission, kLO ∼ ℏωLOαF, is more than twice as large in PbS than in PbSe. On the basis of a simple expression derived earlier for the case of weak CM, this difference would immediately translate into an approximately two-fold reduction in the CM yield in PbS compared to PbSe even if the characteristic times of the CM process in these two materials were similar. These bulk-semiconductor arguments are directly applicable to hot-carrier relaxation in NCs only in the case of fairly large NC sizes when the interlevel spacing is smaller than ℏωLO and carrier cooling occurs via bulk-like single-phonon processes.50 However, a similar relationship between the cooling rates in PbSe and PbS NCs is also expected for more complex energy relaxation scenarios such as multiphonon emission invoked in ref 51. The rate of this process is also directly linked to the electron−phonon coupling constant and the LO phonon energy and is expected to be higher in PbS NCs than in PbSe NCs. Indeed, a comparison of experimental measurements of 1P-to-1S relaxation in these two materials indicates that the 1S filling time is shorter in PbS NCs45 than in PbSe NCs51 when evaluated as a function of the 1S−1P energy difference, a quantity that is directly proportional to the confinement energy. Thus, both simple estimations of phonon emission rates based on bulk-material parameters as well as direct measurements of cooling times suggest that the energyloss rate in PbS NCs is enhanced compared to that in PbSe NCs. This is likely at least one of the factors which explains lower CM yields in PbS NCs as compared to PbSe NCs. As discussed above, in addition to the overall reduction in the CM efficiencies, PbS NCs exhibit a peculiar dependence of QE on hv/Eg, in which the initial increase in QE with hv/Eg (i.e., with increasing NC size) is followed by a flattening of the QE and even an apparent drop for biggest sizes (R > 8 nm). This is in contrast to a monotonous increase of the QE with hv/Eg observed for PbSe NCs. Since in both cases we refer to the measurements conducted for the same excitation energy (3.1 eV), the observed difference in dependences on hv/Eg suggests that PbS and PbSe NCs have distinct dependences of CM yields on particle size. The fact that in PbSe NCs all data points are aligned along the same line in the QE versus hv/Eg representation is indicative that εeh is almost size independent. This might imply that 1/τCM and kloss are both either size independent or characterized by similar size dependences. On the other hand, a progressive deviation of PbS QEs from PbSe results with increasing NC size suggests that εeh increases in larger particles, that is, CM becomes less efficient. Since over the whole range of NC sizes studied here we have observed similar size dependences of Auger lifetimes in PbS and PbSe NCs, it is expected that the values of τCM in these materials should also exhibit similar size dependences. This further suggests that the distinct size-dependent trends in CM in PbSe and PbS NCs are due to the difference in the behaviors of kloss. As before, we can link this difference to distinct LO-phonon energies and polar constants. Specifically, as ℏωLO is greater in PbS, the inter levelspacing is expected to approach the phonon energy at smaller sizes for PbS NCs than for PbSe NCs. This would result in a quicker increase of the contribution from fast single-phonon processes to kloss in PbS NCs with increasing NC size and a corresponding increase in εeh leading to progressive deviation of CM QEs from those in PbSe NCs.

An additional contribution to the size-dependent trend observed in PbS NCs might be associated with the sizedependent change in the NC band-structure resulting from breaking of the spherical symmetry. According to two-photon measurements52 in PbS NCs, for example, the asymmetry increases with decreasing NC size. The resulting enhancement of asymmetric, nominally parity-forbidden optical transitions can reduce the CM threshold and thus increase its efficiency in small PbS particles. The effect of band-structure asymmetry is weaker in PbSe NCs52 and likely does not appreciably influence the size dependence of CM yields. Thus, both the decreasing band-structure asymmetry with increasing NC size (leading to increased τCM) and the increasing contribution from singlephonon emission (leading to increased kloss) can both contribute to the observed increase of εeh in PbS NCs of larger sizes . To summarize, we evaluate Auger recombination times and CM yields in a selection of differently sized PbS NCs using two different femtosecond techniques, TA and uPL, and compare them with measurements on PbSe NCs. We observe a close correspondence between Auger decay times in these two types of the NCs that suggests that both of these materials are characterized by similar strengths of carrier−carrier interactions. Since Auger-type carrier−carrier coupling is also responsible for multiexciton generation, this might further imply the similarity in characteristic times of the CM process (τCM) and also CM performance. However, our measurements indicate that CM yields in PbS NCs are systematically lower than in PbSe NCs and this deviation increases with increasing NC size. We argue that this difference in behaviors is indicative of the different rates of energy-loss mechanisms competing with CM (kloss). Specifically, simple estimations for the polar, Fröhlich-type electron−phonon coupling indicate that the energy-loss rate due to phonon emission is more than a factor of 2 higher in PbS than in PbSe. Further, as NC size is increased, the contribution from fast single-phonon processes is expected to “turn-on” earlier in PbS than in PbSe NCs, which might explain a progressive deviations between CM yields observed for these two materials. Our studies highlight the importance of considering a material’s dependence of various energy relaxation channels (phonon and nonphonon related) when addressing the problem of CM. This notion might be especially important in future theoretical studies of this process since so far most theoretical works have focused on analyzing the multiexciton generation mechanisms while often treating the intraband relaxation rate as an adjustable parameter. Energy relaxation rate arguments can also be important when searching for new candidates that might exhibit highly efficient CM. Specifically, PbTe is a promising material with regard to CM performance as being structurally similar to PbS and PbSe it exhibits a weaker electron−phonon polar coupling and a smaller LO phonon energy compared to both of these materials. So far, there has been only one reported study of CM in PbTe NCs;13 however, these earlier experiments did not account for photocharging. Therefore, re-evaluation of CM performance of PbTe NCs using high-quality, well-passivated samples and taking into account extrinsic effects such as photocharging can help to verify whether the effects of carrier− phonon polar coupling considered in the present study can indeed influence CM efficiencies in NCs. 627

dx.doi.org/10.1021/nl203367m | Nano Lett. 2012, 12, 622−628

Nano Letters

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Letter

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (J.T.S.) [email protected]; (V.I.K.) [email protected]

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ACKNOWLEDGMENTS J.T.S., J.M.P., J.M.L., A.J.N., and V.I.K. acknowledge support of the Center for Advanced Solar Photophysics (CASP), an Energy Frontier Research Center (EFRC) funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES). L.A.P. is supported by the Los Alamos National Laboratory LDRD program. J.T.S and M.M.Q. are supported by LANL Director’s Postdoctoral Fellowships. M.C.B. and A.G.M. were supported by the Solar Photochemistry program within the Division of Chemical Sciences, Geosciences, and Biosciences, BES, DOE. We would like to thank CASP EFRC members for useful discussions and Wan Ki Bae for TEM characterization.

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dx.doi.org/10.1021/nl203367m | Nano Lett. 2012, 12, 622−628