Apparent Versus True Carrier Multiplication Yields ... - ACS Publications

May 11, 2010 - John A. McGuire , Milan Sykora , István Robel , Lazaro A. Padilha , Jin Joo , Jeffrey M. Pietryga , and Victor I. Klimov. ACS Nano 201...
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Apparent Versus True Carrier Multiplication Yields in Semiconductor Nanocrystals John A. McGuire,†,‡ Milan Sykora,† Jin Joo, Jeffrey M. Pietryga, and Victor I. Klimov* Center for Advanced Solar Photophysics, Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ABSTRACT Generation of multiple electron-hole pairs (excitons) by single photons, known as carrier multiplication (CM), has the potential to appreciably improve the performance of solar photovoltaics. In semiconductor nanocrystals, this effect usually has been detected using a distinct dynamical signature of multiexcitons associated with their fast Auger recombination. Here, we show that uncontrolled photocharging of the nanocrystal core can lead to exaggeration of the Auger decay component and, as a result, significant deviations of the apparent CM efficiencies from their true values. Specifically, we observe that for the same sample, apparent multiexciton yields can differ by a factor of ∼3 depending on whether the nanocrystal solution is static or stirred. We show that this discrepancy is consistent with photoinduced charging of the nanocrystals in static solutions, the effect of which is minimized in the stirred case where the charged nanocrystals are swept from the excitation volume between sequential excitation pulses. Using sideby-side measurements of CM efficiencies and nanocrystal charging, we show that the CM results obtained under static conditions converge to the values measured for stirred solutions after we accurately account for the effects of photocharging. This study helps to clarify the recent controversy over CM in nanocrystals and highlights some of the issues that must be carefully considered in spectroscopic studies of this process. KEYWORDS Carrier multiplication, PbSe nanocrystals, charge separation, charged exciton, time-resolved photoluminescence, Auger recombination.

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least 2Eg; Eg is the energy gap) allows one to detect the effect by comparing population decays recorded for low-fluence excitation at low and high photon energies. The appearance of fast Auger decay at high spectral energies has been typically considered a signature of CM. The results of initial CM studies seemed to be mutually consistent indicating that this process occurs with high efficiency in NCs of a variety of compositions including PbSe,4,5,11,12,22 PbS,5 PbTe,7 CdSe,6 InAs,9,10 and Si.8 Data showing a dramatic enhancement in photoconductive gain in PbS NC photodiodes under illumination with high-energy photons were also interpreted in terms of CM.23 These results were challenged by other recent papers where the observed quantum efficiencies (QEs) of photon-to-exciton conversion were significantly lower than in earlier publications16,17 or CM was not detected at all.13,24 Some researchers have attributed these discrepancies to the influence of NC surface properties on CM. In ref 11, for example, this conclusion was drawn based on observed variations in apparent CM yields that showed a significant dependence on a specific chemical treatment applied to NC film samples. The effect of surface defects on the CM process was recently analyzed using tight binding calculations,25 which predicted a reduction of the CM energetic onset and an increase in the CM yields at near-threshold energies due to the contribution from impact ionization involving charges residing in intragap defect states. Experimental errors have also been invoked to rationalize the discrepancies in the apparent CM yields. Specific sources

arrier multiplication (CM) is a process in which absorption of a single photon produces multiple electron-hole pairs (excitons). This effect can potentially improve the performance of photovoltaic devices through increased photocurrent.1-3 Highly efficient CM in semiconductor nanocrystals (NCs) was first reported in 2004.4 This result was supported by follow-up studies of NCs of various compositions.5-12 However, eventually this process has become a subject of intense controversy because the CM yields measured in some of the recent studies have been significantly lower than those from earlier publications or the effect has not been observed at all.13-17 In the first report on CM in PbSe NCs, the effect was detected by analyzing carrier population decay following excitation with a subpicosecond laser pulse of variable photon energy pω.4 This method exploits the orders-ofmagnitude difference in single- and multiexciton lifetimes typically observed in colloidal NCs. Whereas intrinsic recombination of single excitons in these structures is due to radiative decay, multiexciton recombination is dominated by much faster nonradiative Auger decay.18 In PbSe NCs, for example, the respective time scales are 0.2-1 µs19,20 and 20-200 ps.4,21 The existence of a CM energetic onset (of at

* To whom correspondence should be addressed. E-mail: [email protected]. †

These authors contributed equally to this work.

‡

Present address: Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824. Received for review: 01/18/2010 Published on Web: 05/11/2010 © 2010 American Chemical Society

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of errors analyzed in the literature include the underestimation of absorption cross sections at high spectral energies14 and a transient shift of the band-edge absorption feature induced by exciton-exciton interactions.15 Another contribution to variations in reported CM efficiencies is the difference in the analyses of experimental data. One especially important factor in this case is the specific assumption made on the scaling between the number of excitons in a NC and the magnitude of the spectroscopic signature used to monitor the CM process. For example, in the case of photoluminescence (PL) measurements, the derived CM yields are directly dependent on the ratio of biexciton and single-exciton emission rates. However, for NCs this ratio is still not exactly known and the values used in recent CM studies vary from 2 to 5.13,16,17,26,27 A recent explanation for discrepancies in the CM literature proposed by us invokes photoinduced charging (photocharging) of NCs in which the NC core is left with an excess electron or hole that leads to extraneous non-CM related Auger decay signatures in measured dynamics.17 (This does not mean that the NC necessarily develops a net charge, as the core can become charged simply by transfer of an electron or a hole to the NC surface leaving the NC as a whole neutral.) We also suggested that the effects of photocharging can be suppressed if the NC solution is rapidly stirred, which prevents accumulation of long-lived charged NC species in the excited volume of the sample. Since earlier studies (except those in refs 13, 16, and 17) were conducted on unstirred (static) samples, we believe that uncontrolled photocharging is the most likely reason for large literature discrepancies and the overestimation of CM yields in earlier reports including those from our group. In this Letter, we conduct a detailed analysis of the effect of charging on CM measurements. We demonstrate that even though static solutions can show significant sampleto-sample variations in apparent CM yields due to variations in the likelihood of charging, the same samples measured under stirred conditions show comparable CM efficiencies. Further, central to analysis of time-resolved PL in the context of CM, we show that multiexciton emission rates scale as the square of the number of excitons per NC, N, or the product of the electron, Ne, and hole, Nh, occupancies. By generalizing this scaling to the case of charged NCs (i.e., the situation where Ne and Nh are unequal), we are able to relate the apparent CM yields observed in a partially charged NC ensemble to true CM efficiencies. Further, we can correct the static CM yields for photocharging (evaluated either from time-resolved or steady-state data), which allows us to reproduce the values measured for stirred samples. The analysis of true CM efficiencies indicates that in PbSe NCs the energetic onset of this effect is ∼2.5Eg, which is near the fundamental 2Eg limit and significantly lower (in terms of the band gap) than in bulk PbSe, where it is ∼6Eg.28 On the basis of the growth of QE with increasing photon energy, we estimate that the electron-hole pair creation energy (the © 2010 American Chemical Society

energy increment required to create an additional exciton) in NCs is from 2.3Eg to 3.25Eg, which is again significantly lower than in the bulk (∼5.3Eg according to ref 28). While for a given absolute photon energy, multiexciton yields in PbSe NCs do not exceed those measured for bulk PbSe films (at least for the NC sizes studied here), the fact that appreciable CM efficiencies are obtained for a significantly greater band gap makes CM in NCs of greater potential utility in photovoltaics and photocatalysis compared to CM in parental bulk solids. We study oleic acid-capped PbSe NCs prepared as described previously29,30 with mean radii (R) from ∼1.5 to ∼3.8 nm and band gaps from 1.085 to 0.617 eV. These sizes correspond to the regime of extremely strong quantum confinement, as even for the biggest NCs the energy gap is more than twice as large as that of bulk PbSe (0.28 eV). The PbSe NCs are dissolved to an optical density of 0.1 to 0.3 at 1.55 eV in either hexane or deuterated chloroform in a 1 mm thick cuvette with an airtight seal in an argon atmosphere. Time-resolved PL studies are performed by PL upconversion (uPL).31 The 1.54 eV pulses of a 250 kHz Ti: sapphire-based regenerative amplifier are stretched to 0.2-3 ps and used to gate emission in a β-barium borate nonlinear optical crystal. The 1.54 or 3.08 eV frequency-doubled pulses are used to excite the samples. The excitation spot diameter is ∼150 µm. The typical spectral resolution of the uPL system at the wavelength of the PbSe NC emission is ∼50 meV, which is comparable to the ensemble 1S emission peak width and much larger than multiexcitonic spectral shifts. The temporal resolution is defined by the cross correlation of the excitation and gate pulse widths and in all cases is shorter than 4 ps. The average number of photons, , absorbed per NC per pump pulse at the peak of the pulse spatial profile at the entrance face of the sample is ) jpσabs, where jp is the measured per-pulse pump fluence in units of photons per unit area and σabs is the absorption cross section. The absorption cross sections of PbSe NCs are taken from ref 32 and adjusted according to the refractive index of the solvent used to prepare a sample. In Figure 1, we display time-resolved PL data from samples with Eg of 0.617 eV (panel a) and 0.63 eV (panel b) that illustrate how dramatically sample stirring can affect measured dynamics. While the sample in panel a does not show any discernible static-stirred difference under 3.08 eV excitation, static and stirred dynamics are distinct for the sample in panel b; the static trace has a higher early time amplitude (A) and a lower long-time intensity (B) compared to the stirred trace. While the data in Figure 1b are taken at a relatively high fluence ( ) 1.4), the static-stirred difference for the 0.63 eV sample persists in the limit of low intensities (inset of Figure 1b) when the average number of photons absorbed per NC per pulse, , is much less than unity. Since the A/B ratio in the limit of f 0 is used to evaluate the CM efficiency,4 the observed increase in the A/B ratio in the static 2050

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FIGURE 1. Effect of sample stirring on PL dynamics. (a) PL dynamics for static (circles) and stirred (black solid line) conditions of PbSe NCs with Eg ) 0.617 eV excited at 3.08 eV ( ) 2.0) along with a trace (red dashed line) recorded using 1.54 eV excitation ( ) 0.19; stirred solution); this represents an example of a small (experimentally not discernible) static-stirred difference. Inset: the ratio of the early to late-time PL signals (A/B) as a function of for 3.08 eV (black crosses) and 1.54 eV (red circles) excitation. (b) Same as in panel a but for a sample with Eg ) 0.63 eV, which shows a significant static-stirred difference in PL dynamics ( ) 1.4). This difference persists in the limit of low pump intensities (inset). (c) Early PL decay (obtained by subtracting the long-time signal) for the sample in panel a for static (circles) and stirred (black solid line) conditions. Both traces can be fit to biexponential decays (blue dashed line) with time constants of 27(3) and 154(7) ps, consistent with tri- and biexciton Auger recombination measured under 1.54 eV excitation. (d) Same as in panel c but for the sample from panel b. The trace for the stirred sample shows biexponential decay (black dashed line) with time constants of 24(3) and 131(9) ps that are similar to those in panel c with the triexciton decay barely visible in the stirred case in panel d. On the other hand, the PL decay in a static sample shows two decay components with time constants of 18.7(1.4) ps and >500 ps (blue dashed-dotted line) in addition to the neutral biexciton decay. All data were acquired using femtosecond PL up-conversion31 with temporal resolution of e4 ps.

solution compared to that in the stirred case seems to indicate increased multiexciton production. However, such an explanation is inconsistent with a number of observations. For example, if the static sample indeed had a higher CM efficiency, this would only lead to an increase of the early time emission intensity, while the long-time PL signal, which reflects the total number of photoexcited NCs,4 would not change. Yet, the static sample in Figure 1b yields distinctly lower long-delay signal compared to the stirred one. Further, this sample measured under static conditions also exhibits additional time scales in the PL decay compared to the stirred case (Figure 1c,d) with faster relaxation at short delays and a slower decay at long delays. As explained below, all of these observations can be rationalized if we assume that in the static case the photoexcited NC ensemble contains charged excitations in addition to neutral ones. For a quantitative analysis of PL traces, one must understand how the emission rates of neutral and charged excitations scale with the number of electrons (Ne) and holes (Nh) © 2010 American Chemical Society

residing in a NC. In ref 17, we proposed a free-carrier model for describing the emission process in PbSe NCs. This model assumes that following intraband relaxation photoexcited carriers are distributed with equal probabilities over all eight lowest-energy degenerate states derived from four different L valleys.16,33 The likelihood that a given pair of conduction and valence-band states coupled by a dipole-allowed transition is simultaneously populated by an electron and a hole (i.e., the likelihood of forming a “bright” exciton) is then proportional to the product NeNh, implying that the emission rate, rem, scales as NeNh. In the case of neutral NCs (Ne )Nh ) N) this scaling reduces to rem ) rN ∝ N2. To experimentally verify the validity of the free-carrier model, we analyze the fluence-dependence of the early and long-time PL intensities measured for 1.54 eV excitation for which CM is not active. In Figure 2a, we plot the values of A and B as a function of for a sample with Eg ) 1.085 eV. Because of fast Auger decay, the long-time PL signal is entirely due to single excitons, and hence it can be calculated 2051

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modify this expression assuming that ri ) r3 for all i > 3, which yields A ) kr1(p1 + β2p2 + β3p3,∞), where β2 ) r2/r1, ∞ pi ) 1 -(p0 + p1 + p2). We further β3 ) r3/r1, and p3,∞ ) ∑i)3 use this formula with β2 and β3 as adjustable parameters to fit the early PL intensity as a function of (solid black line in Figure 2a). For the 1.085 eV sample shown in Figure 2a, this procedure produces β2 ) 4.12(0.15) and β3 ) 9.22(0.53) (all errors in this paper are the standard errors of fits to the data); both values match those expected for quadratic scaling of rN (4 and 9, respectively). We observe good quantitative agreement (particularly for β2) between our model and experimental measurements for a large number of samples with Eg from 0.617 to 1.085 eV (Figure 2b).35 These results validate the applicability of the freecarrier model to neutral multiexcitons and strongly suggest that it is also valid in the case of charged excitations (Ne * Nh) for which rem ) NeNh. To relate the multiexciton yield [η ) (QE/100%) - 1] to the low-fluence values of the measured A/B ratios, we consider the situation where QE e 200%. In the case of CM in an ensemble of all-neutral NCs, the early time PL is due to single excitons and biexcitons (r2 ) 4r1), and hence, for low fluences A ) k(1 - p0)[r1(1 - η) + r2η)] ) kr1(1 - p0)(1 + 3η). Following Auger decay, all of the initially excited NCs are populated with single excitons. Therefore, the long-time PL signal is B ) kr1(1 - p0). Using these expressions, we obtain η ) (A/B - 1)/3 and QE ) 100%(A/B + 2)/3. If we apply these formulas to the data in Figure 1b, we obtain η ) 0.41(0.01) and QE ) 141(1)% for the stirred sample. However, the same expressions produce an apparent yield of 1.12(0.04) and QE ) 212(2)% for the same sample measured under static conditions (for A/B > 4 we have to account for generation of triexcitons; in this case η ) (A/B + 1)/5 assuming that in the CM regime one has only two dominant exciton multiplicities36). This discrepancy can be accounted for if we assume that the static sample contains a certain fraction, f, of charged NCs generated via photocharging. On the basis of our measurements (see below), at the low excitation fluences used in CM studies ( < 0.05), for most of the samples f does not exceed 0.25, which allows us to consider only singly charged NCs.37 Additionally, we assume that the CM efficiency in these NCs is the same as in neutral ones38 and that charged NCs are not present in stirred samples, as they are swept from the excited volume between sequential laser pulses. To quantitatively evaluate the effects of charging on CM yields, we consider the situation where the CM QE is less than 200%. Under this condition, low-fluence excitation ( , 1) produces four species, neutral and charged excitons as well as neutral and charged biexcitons. The respective probabilities are p1 ) (1 - f)(1 - η)(1 - p0), p1* ) f(1 - η)(1 - p0), p2 ) (1 - f)η (1 - p0), and p2* ) fη(1 p0). Given the NeNh scaling of the radiative decay rates, the emission rates of charged species are r1* ) 2r1 and r2* ) 6r1 ) 1.5r2. Thus the early PL signal in a partially charged

FIGURE 2. Quadratic scaling of emission rates in PbSe NCs. (a) Measurements of short- (open squares, A) and long- (solid circles, B) time PL intensities versus and fits using the free-carrier model (lines) for a sample with Eg ) 1.085 eV using 1.54 eV excitation. The fits yield β2 ) 4.12(0.15) and β3 ) 9.22(0.53). (b) The ratio of the biexciton (open squares) and triexciton (solid circles) emission rates to that of a single exciton as a function of NC energy gap; horizontal lines mark the values for quadratic scaling (4 and 9). Across the range of sample sizes studied, we obtain r2/r1 ) 4.12(0.15) (the apparent increase in r3/r1 with decreasing Eg is discussed in a comment under ref 35).

as the product of the exciton emission rate (rem ) r1) and the number of initially excited NCs, B ) kr1(1 - p0), where p0 is the Poissonian probability that a NC is not excited34 and k is a proportionality constant. The probability p0 is given by p0 ) exp[-], where is the average NC occupancy related to by ) χ and χ is an adjustable parameter that accounts for attenuation of the pump pulse during propagation within a sample, spatial cross correlation between the PL and gate beams, the nonuniformity of the beam profile, as well as any uncertainty in the absorption cross sections. We determine χ by fitting the measurements of B versus (dashed blue line in Figure 2a; the other adjustable parameter in this fit is kr1). In the absence of CM, the early time PL amplitude can ∞ piri, where pi is the Poissonian be described by A ) k∑i)1 probability of having i excitons in a given NC immediately following photoexcitation. Since in our experiments we do not resolve multiexciton decays corresponding to i > 4, we © 2010 American Chemical Society

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decay by, for example, increased trapping at surface defects is unlikely, as the appearance of the additional fast time scale under static conditions always correlates with an increase in the amplitude of the early time PL signal. If we define an apparent multiexciton yield in the case of charging (ηapp*) in the same way as for an ensemble of solely neutral NCs [i.e., as ηapp* ) (A*/B* - 1)/3)], we obtain the following relationship between ηapp* and the true multiexciton yield, ηapp* ) [η + f (2 + η)/3](1 - f)-1 (we again point out that this expression is obtained assuming that true CM yields are identical in neutral and charged NCs, i.e., η* ) η). The above considerations indicate that the presence of even a small fraction of charged NCs can significantly increase the apparent CM efficiency compared to the real value. The influence of photocharging can be especially dramatic in the case of weak CM. For example, for η ) 10% even moderate charging with f ) 10% results in an apparent CM yield of 19%, which is nearly double the real value. For a known degree of photocharging, one can use the above model to derive true CM yields from the apparent efficiencies measured for static solutions. The value of f can be determined, for example, from the static-stirred difference in the long-time PL signals, f ) (B - B*)/B. Another approach is to measure the decrease in the time-integrated PL intensity in a static solution compared to that in a stirred one.40 Finally, the presence of long-lived charges in NCs leads to bleaching of the band-edge 1S absorption feature (inset of Figure 4a). The bleach magnitude (∆R) measured as the difference in NC steady-state absorption spectra under static and stirred conditions in the presence of 3.08 eV excitation normalized by the linear absorption coefficient of the stirred sample (R0) can also be used to evaluate the degree of photocharging. In deriving f from absorption data, one has to account for the 8-fold degeneracy of the bandedge electron and hole states as well as the mismatch of the effective propagation lengths at the excitation and the probe wavelengths. On the basis of these considerations, the |∆R|/ R0 value of 0.025 measured for the sample in Figure 4a ( ≈ 1) corresponds to f ) 0.19, which is in reasonable agreement with the value derived from PL quenching. To illustrate the method for extracting true CM efficiencies from static data, we consider the sample with Eg ) 1.085 eV in Figure 4a. The low-fluence limit of A*/B* for this sample (static solution) is 1.52(0.02), which corresponds to ηapp* ) 0.17(0.01) [QE ) 117(1)%]. The fraction of charged dots for this sample at pump intensities used in CM studies is 0.12(0.03) from PL time transients and 0.105(0.004) based on time-integrated PL data. Using the average value of 0.11, we calculate that the true CM yield is 8(1)%, which is close to the value seen in the stirred sample [10(1)%] and about a factor of 2 lower than the apparent CM yield measured for the static solution. In Figure 4b, we show multiexciton yields plotted as a function of photon energy normalized by energy gap for several samples measured under static (black solid triangles)

FIGURE 3. Sequences of recombination events in the case of CM in a neutral and a charged-core NC. (a) A biexciton created via CM in a neutral NC core decays via the Auger process (time constant τ2A) to a single exciton. The corresponding emission rate, which defines the PL intensity, changes from r2 ) 4r1 at early time (tA) to r1 at long time (tB). (b) The biexciton generated in a charged NC is “brighter” than that in a neutral NC and emits with a rate of 6r1. This charged biexciton first decays (time constant τ2*A) to a charged exciton (trion), which then decays (time constant τ1*A) to a single nonemitting charge. Thus, the presence of charged NCs leads to an increase of the early time PL intensity and a decrease of the longtime PL signal, which is accompanied by the appearance of two additional recombination time scales associated with charged biexcitons (τ2*A < τ2A) and charged single excitons (τ1*A > τ2A).

NC sample is A* ) kr1(p1 + 4p2 + 2p1* + 6p2*) ) kr1[1 + 3η + f(1 + η)](1 - p0). Because of Auger recombination, the long-time signal is entirely due to neutral NCs, and hence, B* ) kr1(1 - f)(1 - p0). These expressions indicate that the presence of excess charges increases the short-delay signal and simultaneously leads to reduction of the signal at long delays (compare Figure 3 panels a and b). These are exactly the trends observed experimentally if one compares the PL dynamics measured for stirred (f ) 0) and static (f > 0) samples (Figure 1b). The effect of photocharging can also explain the staticstirred difference in the measured early time dynamics shown in Figure 1c,d. In the case of neutral NCs, at the pump intensities used in these measurements ( ) 1.2-2), the initial PL relaxation should be due primarily to biexcitons and triexcitons. Indeed, the PL trace for the stirred sample in Figure 1d (and those measured at higher fluences; not shown) can be accurately fit by a biexponential decay with time constants of 131(9) ps and 24(3) ps that are consistent with Auger recombination of neutral biexcitons and triexcitons, respectively. The PL relaxation for the static solution can only be fit well with a triexponential that combines the neutral biexciton decay with two additional time scales (∼18.7(1.4) ps and >500 ps) that can be attributed to decay of charged excitations. Specifically, charged multiexcitons are likely responsible for the fast initial decay, while charged single excitons (trions) produce a final slower Auger decay component. An alternative explanation of the initial fast © 2010 American Chemical Society

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In Figure 4b, we also plot corrected CM yields (black open squares) that are calculated based on PL dynamics recorded under static conditions and the measurements of photocharging by time-integrated emission. We observe a remarkable correspondence between the corrected values of η and CM yields measured under stirred conditions, which provides further evidence that NC charging is the primary reason for static-stirred differences in apparent CM yields and the most likely reason for the overestimation of the CM QEs determined from earlier experiments that were conducted under static conditions. This further strongly suggests that measurements of stirred samples provide more accurate results for true CM yields compared to studies done on static solutions or film samples. In NC films, photocharging can be an especially serious problem because of the lack of simple means (such as stirring in the case of solution samples) for removing charged species from the excitation volume. However, we emphasize that, according to the results of our modeling, the existence of long-lived charges in NCs does not seem to suppress CM, but it does complicate accurate determination of actual CM efficiencies. The controversy over the reported values of CM efficiencies has led to discussion of whether CM is enhanced in NCs compared to bulk solids.16,28 A critical issue in this discussion has been how to account for the difference in the energy gap between NCs and the respective bulk solids. In the earlier reports on CM in NCs, QEs were analyzed using plots of QE versus (pω/Eg).4,5 Applying this representation to the present data (Figure 5a), one can see that both the CM threshold and the electron-hole pair creation energy appear reduced relative to bulk PbSe (data from ref 28). However, a more recent claim is that bulk and NC CM efficiencies should be compared side by side using plots of QE versus pω without taking into consideration the large differences in Eg.16,28 Since in this representation, CM efficiencies in NCs were lower than in the bulk, the authors of refs 16 and 28 concluded that CM in NCs is less efficient than in the parental bulk solid. In Figure 5b, we use the absolute photon energy representation to plot QEs from the present study along with those of bulk PbSe from ref 28. It appears that QEs for the NC sizes studied here do not reach the bulk values measured for the same photon energy. However, for the largest NCs (Eg ) 0.677 eV) studied with 4.64 eV photons, the multiexciton yield is ∼140%, which is only 30% lower than the value observed for the bulk sample (η ≈ 200%; ref 28) for a similar excitation energy. The fact that comparable CM efficiencies are observed despite the fact that the NC energy gap is ∼2.4 times greater than the bulk PbSe energy gap is important from the stand point of both fundamental physics and prospective applications in photovoltaics, where the value of practical importance is the output power. A convenient quantity for characterizing the utility of CM in practical applications is the product of the QE and the band gap, (QE/100%)Eg, (“energetic” figure of merit) since

FIGURE 4. Apparent versus true CM efficiencies in PbSe NCs. (a) The ratio of the early to late-time PL signals (A/B) as a function of for 3.08 eV (black crosses for the stirred sample and open blue circles for the static sample) and 1.54 eV (red solid circles) excitation of a sample with Eg ) 1.085 eV. Inset: Linear absorption (R0) of this sample as a function of wavelength (λ) in the region of the bandedge 1S feature (line) in comparison with the steady-state staticstirred difference (|∆R|) in R0 in the presence of 3.08 eV excitation with of ca. 1 (circles); the ratio of |∆R| to R0 at the absorption peak is 0.025, which corresponds to f ) 0.19. (b) Apparent multiexciton yields under stirred (red solid circles) and static (black solid triangles) conditions. The true multiexciton yields derived from the static measurements by accounting for measured degrees of NC core charging are shown by black open squares. Horizontal and vertical ovals mark samples with small and large static-stirred differences, respectively.

and stirred (red solid circles) conditions (3.08 eV excitation). We observe that if a sample shows a static-stirred difference in PL dynamics, the apparent CM yield for the static case is always greater than for the stirred case, as expected for CM in the presence of photocharging. Further, the static-stirred disparity does not seem to correlate with energy gap. Specifically, large static-stirred differences (ηapp* as large as ∼3η) can be observed for samples with either large or small band gap; compare data points in Figure 4b circled by vertical (large static-stirred difference) and horizontal (small static-stirred difference) ovals. Likewise, small static-stirred differences are also observed for samples of large or small band gap. We also observe that stirred samples with a similar energy gap exhibit similar CM yields with the general trend being the increase of η with increasing pω/Eg (i.e., decreasing energy gap). This is in contrast to the apparent CM yields, which show little correlation with energy gap on account of photocharging. © 2010 American Chemical Society

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FIGURE 5. Comparison of CM characteristics of PbSe NCs and bulk PbSe films. (a) QE of photon-to-exciton conversion for NCs (stirred samples) measured by time-resolved PL (open red circles) and transient absorption (TA; black crosses) in comparison to the measurements for bulk PbSe films (solid blue diamonds) from ref 28; the horizontal axis is photon energy normalized by Eg. (b) Same as in panel a but plotted as a function of absolute photon energy. (c,d) Same as in panel a but plotted respectively in terms of either the “energetic” figure of merit (defined as the product of QE/100% and Eg) or the CM figure of merit, QCM (see text for details); in both panels the horizontal axis is absolute photon energy.

this quantity determines the power-conversion efficiency of a photovoltaic device.3 Because of a larger band gap, CM of a given efficiency in NCs has greater utility than CM in the bulk parent compound. This fact is apparent from the plot in Figure 5c where we compare products (QE/100%)Eg, for PbSe NCs of different size and bulk PbSe for two different excitation energies. This plot indicates that for all NC sizes the photovoltaic figure of merit is greater in NCs than in the bulk and the observed enhancement is up to a factor of 2.2. We note, however, that while this comparison provides useful insights from the technological prospective, it is still not well suited for evaluating CM because it does not distinguish between the separate roles of the band gap and the CM QE. Specifically, even without CM (QE ) 100%), NCs show an enhancement in the “energetic” figure of merit simply because of a confinement-induced increase in the energy gap. From the physics standpoint, in comparing the CM efficiency between different materials one should explicitly account for the energy-gap because Eg represents an intrinsic energy scale in the CM process and also defines the width of a “time window” (∆tCM) during which CM can occur. Specifically, ∆tCM increases with increasing difference between Eg and the initial energy of a photoexcited carrier, E0 (defined by pω and optical selection rules). In models © 2010 American Chemical Society

where CM is due to competition between impact ionization (time constant τII) and carrier cooling (energy-loss rate rc), CM QE depends on the dimensionless quantity γ ) ∆tCM/τII, which links the multiexciton yield to both the photon energy and the band gap. Therefore, while the absolute carrier energy E0 (or pω) might provide a good basis for a comparative analysis of τII, in comparing CM QEs one should account for not only pω but also Eg. For example, for a material with equal electron and hole effective masses (for which the kinetic energy is partitioned equally between electrons and g)/2 r-1(E)dE, which clearly indicates holes), γ ) τII-1 ∫E(pω-E c g that the CM yield depends on both the photon energy and the band gap even in the case when τII is not dependent on E g. Furthermore, two important characteristics of CM, its spectral onset, pωCM, and the electron-hole pair creation energy, ε, directly relate to the band gap. From energy conservation alone, the minimum values of the threshold and ε are respectively 2Eg and Eg. However, momentumconservation considerations and phonon losses lead to two additional contributions to ε, EK (the “kinetic” term) and Eph, respectively.41,42 As a result, ε can be calculated as ε ) Eg + EK + Eph. In the free-particle approximation, EK ) 1.8Eg42 (i.e., it directly relates to Eg), while phonon losses are typically in the range of 0.5 to 1.0 eV.41,42 Given these 2055

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considerations, ε is at least ∼3Eg instead of just Eg as defined by energy conservation. At the same time, the CM threshold in materials with similar electron and hole effective masses (this closely describes the situation in PbSe) is at least 4Eg instead of the energy-conservation-defined value of 2Eg. It has been suggested that ε and pωCM can be reduced in NCs because of the relaxation of translational momentum conservation and suppression of phonon losses due to the discrete structure of electronic states (the “phonon bottleneck”).2 Although the phonon bottleneck has only been observed under exceptional circumstances,43 the relaxation of translational momentum conservation in NCs has been demonstrated in the universality of Auger recombination rates across NCs of direct- and indirect-gap semiconductors44 and almost band gap-independent Auger decay in pressure-dependent studies.45 The expected reductions of ε and pωCM in NCs if referenced to the band gap are indeed observed experimentally. In Figure 5a, we compare CM QEs for stirred NC samples (derived by both PL and transient absorption; the latter technique is explained, e.g., in ref 4) with those of PbSe bulk films from ref 28 plotted as a function of pω/Eg. From extrapolation of NC data near the CM onset, we find that pωCM in NCs is ∼2.5Eg, which is near the energy-conservation-defined limit of 2Eg and below the threshold in largegap bulk materials.41,42 It is also much lower than the CM threshold in bulk PbSe films (ca. 6 to 7Eg according to ref 28). We can further estimate ε in NCs based on the difference in QEs measured for the same sample at two different photon energies (3.08 and 4.65 eV). For NCs studied in this work, this estimate produces ε from 2.3Eg to 3.25Eg. On the basis of results from ref 28, ε in bulk films is ∼5 Eg, which is 1.5 to 2.2 times larger than the NC values. An important current challenge in CM research is to understand whether one can approach the ultimate limits in CM performance as defined by energy conservation. The multiexciton yield of an ideal (i.e., solely limited by energy conservation) CM material can be described by a sum of step functions with each step of 100% occurring at each extra Eg of energy. However, available experimental data for bulk materials indicate that the observed QE-vs-pω dependence is close to linear, which has been the reason for introducing the electron-hole pair creation energy as one of the parameters for characterizing CM. On the basis of this observation, we define the “ideal” CM performance as a linear growth of QE with inverse slope of Eg above the 2Eg threshold. The corresponding multiexciton yield can be expressed as η0 ) (pω/Eg - 2) for pω/Eg > 2. On the basis of this notion of ideal CM performance, one can introduce a CM figure of merit defined for a given excitation wavelength as the ratio of the real to the ideal multiexciton yields, QCM ) η/η0 ) η(pω/Eg 2)-1. This quantity provides a measure of how close to the ideal is the CM performance of a given material for a given photon energy pω > 2Eg. It is the latter that is of ultimate © 2010 American Chemical Society

importance in understanding the physics of CM and judging its enhancement in engineered structures. In bulk materials, the CM figure of merit is generally low because both ε and the CM threshold are significantly greater than the energy-conservation-defined values. As a result, in bulk PbSe films, QCM reaches a maximum value of only 0.13. The reduction of both ε and pωCM in NCs leads to an appreciable enhancement (by up to a factor of 2.2) in the CM figure of merit as evident from Figure 5d. To summarize, we have shown that a significant factor that has likely contributed to large discrepancies in reported CM yields and their overestimation in earlier studies is uncontrolled photocharging of NCs. This effect can be especially large in static samples resulting here in apparent multiexciton yields up to three times the true values. The differences in the fraction of charged NCs observed for samples with similar energy gaps suggests that photocharging is not solely a consequence of intrinsic (i.e., gap-dependent) properties of NCs and may be very sensitive to the identity of surface species and/or properties of NC environment. This may account for differences in apparent CM yields in nominally identical NC samples. The magnitude of the differences observed is similar to the magnitudes of discrepancies in earlier reports of CM in PbSe NCs. We have developed a model that provides a quantitative description of apparent CM yields in a mixed ensemble of neutral and charged NCs and used it to extract true CM efficiencies from measurements of static solutions with known degrees of photocharging. As a basis for comparison between different materials with regard to CM, we introduce a figure of merit, QCM, defined as the ratio of the measured multiexciton yields and those of the “ideal” CM material operating at the limits defined by energy conservation. The comparison of PbSe NCs to bulk films in terms of this figure of merit indicates an enhancement of CM in NCs compared to the bulk. Acknowledgment. This material is based upon work supported as part of the Center for Advanced Solar Photophysics, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES). J.M.P. and J.J. acknowledge support by the Chemical Sciences, Biosciences, and Geosciences Division of BES, U.S. DOE. J.A.M. and M.S. acknowledge support by Los Alamos National Laboratory Directed Research and Development Funds. REFERENCES AND NOTES (1) (2) (3) (4) (5) (6) (7)

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