Low threshold and efficient multiple exciton generation in halide perovskite nanocrystals

Multiple exciton generation (MEG) or carrier multiplication, a process that spawns two or more electron–hole pairs from an absorbed high-energy photon (larger than two times bandgap energy Eg), is a promising way to augment the photocurrent and overcome the Shockley–Queisser limit. Conventional semiconductor nanocrystals, the forerunners, face severe challenges from fast hot-carrier cooling. Perovskite nanocrystals possess an intrinsic phonon bottleneck that prolongs slow hot-carrier cooling, transcending these limitations. Herein, we demonstrate enhanced MEG with 2.25Eg threshold and 75% slope efficiency in intermediate-confined colloidal formamidinium lead iodide nanocrystals, surpassing those in strongly confined lead sulfide or lead selenide incumbents. Efficient MEG occurs via inverse Auger process within 90 fs, afforded by the slow cooling of energetic hot carriers. These nanocrystals circumvent the conundrum over enhanced Coulombic coupling and reduced density of states in strongly confined nanocrystals. These insights may lead to the realization of next generation of solar cells and efficient optoelectronic devices.

generated per additional gap of photon energy above the CM threshold). These numbers are substantially better than in the most widely studied systems, PbS and PbSe, which had once been thought to offer the greatest potential for CM. Importantly, the efficiencies reported here are most favorable in the smallest dots, which suggests room for further improvement. This is again in contrast to the case of traditional inorganic NCs in which carrier cooling rates increase with increasing confinement, thereby negating the gains in Coulomb interactions and relaxation of momentum conservation that come with increased confinement and that would otherwise be expected to lead to increased CM efficiencies.
It should be noted that Klimov's group has shown a CM threshold below 2.2 eV in PbSe/CdSe core/shell NCs (C.M. Cirloganu et al., Nat. Comm. 5, 4148 (2014)), which partly undercuts one of the main claims here, namely that the CM threshold is lower than in traditional inorganic NCs. However, the slope efficiencies were still only 25%. At least as important as the numbers is the fact that such core/shell NCs are fundamentally more complicated than the perovskite NCs studied here.
The study of CM has struggled somewhat since the initial reports of high efficiencies in inorganic NCs were found to be erroneous. CM has been overtaken as a topic of study by singlet fission for a variety of reasons including perceptions of potential efficiencies and the fact that there have been clearer paths to design of molecules for singlet fission. There will be interest in the present work in boosting prospects for CM in a relatively simple material system. As such, this is a valuable contribution to the field. It should also be noted that the data in the present manuscript is very thorough, and great care appears to have been taken to obtain clean data.
The one general shortcoming I see in the paper is the absence of a quantitative explanation for the observations. That the CM efficiencies in FAPbI3 NCs are greater than in the bulk was to be expected given the authors' previous paper on the phonon bottleneck in lead halide perovskite nanocrystals. In lines 28-30, the authors write that "these NCs solve the conundrum over the contrasting requirements of enhancing Coulombic coupling and the density of states in strongly confined NCs." The results don't really solve the conundrum; they just avoid it by taking advantage of slowed cooling. Like many papers on CM, the paper offers more in the way of materials characterization than new scientific insights. A particularly glaring deficiency is the absence of discussion of the effective masses of electrons and holes. In the conventional picture, for a system characterized by single conduction and valence bands and in the case that the effective masses of electrons and holes are equal, the threshold for CM is 3Eg. In other lead halide perovskites, the effective masses of electrons and holes are similar to each other (see, for example, Ref. 27 of the present manuscript and L.M. Herz, ACS Energy Letters 2, 1539 (2017)). As noted in lines 152-154 of the present manuscript, the authors of Ref. 27 did not observe MEG in CsPbI3 under excitation at a photon energy of 2.65 Eg. In fact, it was in terms of the similar electron and hole masses that the failure to observe MEG in Ref. 27 was understood. From this perspective, the low CM threshold reported here is unexpected. It may be that the conventional picture is not relevant here, but such a statement would need to be justified.
The rest of my comments are about specific details of the paper. Panels c and d in Supplementary Figs. 5 -7 are somewhat confusing. In particular, what is the physical meaning of using a linear fit for the A data, i.e., for ? With the exception of the 7.5 nm NCs excited below the MEG threshold, the fits are clearly statistically inconsistent with the data insofar as few of the error bars cross the fit line. Moreover, the inconsistencies are in different directions for different data sets. In Figs. 5c and 6d the A data seem roughly linear; in Fig. 5b the data appears to have a sub-linear dependence on fluence; and in Figs. 6c, 7c, and especially 7d, the A data appear to have a super-linear dependence on fluence.
The authors should state that the FA in FAPbI3 refers to formamidinium the first time FAPbI3 is mentioned in the manuscript, not just in the methods section.
I presume that the solid red and blue fits to RPOP are fits of Eq. 1. This is implied but not stated explicitly.
Reviewer #3 (Remarks to the Author): The manuscript "Low threshold and Efficent Multiple Exciton Generation in Halide Perovskite Nanocrystals" describes multiple exciton generation in colloidal FAPbI3 nanocrystals. The claimed multiple exciton generation (MEG) threshold is very low (2,25*Eg), being potentially relevant for photovoltaic applications. The positive effect on power conversion efficiency of solar cells is demonstrated by calculations for different MEG thresholds, showing that only the threshold of 2*Eg is relevant. The efficiency of the MEG is relatively high due to the phonon bottleneck and slow cooling of hot carriers. The latter has been experimentally studied by the femtosecond transient absoption (TA) measurements, where the band edge bleaching shows a risetime of hundreds of femtoseconds. The dynamics of multiple excitons is studied by TA spectroscopy as well, where the band edge bleaching at pumping photon energies above 2*Eg exhibits a fast component at the time scale of 100 picoseconds. The conclusions are supported by experiments in a convincing way. The experiment description is detailed enough for reproduction.
• All reviewer comments are displayed in dark orange italics.
• All our responses are displayed in black.
• Sentences indicating changes to the manuscript are blue. Response: We are delighted that reviewer shares our enthusiasm on the discoveries and greatly appreciate the reviewer's recommendation for publication of our work. The reviewer's critical comments are very helpful for improving our manuscript. We have carefully considered the reviewer's comments and would like to reply as follows: Comment 1: The authors always refer their results to multiples of the bandgap energy Eg. In the case of quantum confined nanocrystals, it would be more appropriate to refer to the exciton energy instead. If for simplicity reasons you want to keep the Eg nomenclature, you should at least comment on the exciton binding energy of these perovskite nanocrystals and, if this is nonnegligible, relate these results to multiples of the exciton energy.
Response 1: We thank the reviewer for raising this point. In the pioneering reports on MEG studies, the Eg of NCs is defined by the 1s exciton absorption gap. We therefore used the Eg nomenclature to be consistent with these studies and for easy comparison by the readers to these works.
The value of Eg in our work is determined by the bleaching peak position of the exciton transition from transition absorption spectra, therefore, it corresponds to the excitonic absorption energy (i.e., exciton energy) and already includes the exciton binding energy (i.e., right third term of Eq.1 in SI). Furthermore, the calculated exciton binding energy is ~14-25 meV for our large to small FAPbI 3 NCs, is negligible as compared to exciton energies and therefore has a negligible effect (<1.5%) on the determination of MEG threshold in terms of Eg. Thus the exciton energy can be used as the bandgap energy here. For clear understanding, we have therefore added "The small binding energies (~14-25 meV for our large to small FAPbI 3 NCs) is negligible compared to the exciton energy. Here, we simply used the exciton energy (determined by TA spectra shown in Supplementary Fig. 3) as the bandgap energy." in Note 1 of SI.

Comment 2:
Calculating the efficiency of the MEG process, the authors should clarify whether they refer to 2-exciton generation only or if they consider also possible higher order MEG (generating a number of excitons >= 3). These high-order processes should be taken into account, as they use as excitation energies also values that are higher than 3 times the bandgap energy.
Response 2: We thank the reviewer for these constructive comments. We are not referring 2exciton generation, higher order MEG is also considered. We have revised our previous unclear statement to the following sentence in the Supplementary Note 3 in SI: "For an ideal MEG process, β follows a staircase function, i.e., when the photon energy reaches the N × Eg, maximum MEG yield is N (solid line in Supplementary Fig. 10a). For the non-ideal case (dashed lines in Supplementary Fig. 10a), as in our experiment, β increases linearly with /Eg, the slope represents the MEG slope efficiency ( )." And recalculated Supplementary Fig. 10 based on the above method as shown below.
Supplementary Fig. 10 (a), multiple exciton generation yield as a function of hv/Eg for an ideal case (max, solid line) and two non-ideal cases as examples (dashed lines). Calculated PCE under AM1.5 solar illumination as a function of E g for MEG threshold of (a) 2E g and (b) 3E g at different MEG efficiencies.

Comment 3:
In the balance calculation reported in Fig. 3c, do you exclude the generation of 3 or more excitons (when allowed by the energy conservation)?
Response 3: We have included the generation of 3 or more excitons, please refer to the response to comment 2.
Comment 4: The authors should clarify in the text the method they have used to calculate the MEG QY.
Response 4: Following the reviewer's suggestion, we have revised accordingly as shown below: " is determined by the PB intensity ratio at the beginning of pump excitation to that at a delay time of 4 ns." in Fig. 2 caption is changed to: " is determined by the PB intensity ratio at the beginning of pump excitation at a delay time of ~2 ps to that at a delay time of 4 ns and fitted with Eq. 1 (solid lines)." and "The R POP ratios at the y-intercepts at above and below MEG threshold photon energy excitation thus facilitate the determination of MEG QY (Fig. 2d-f)." on page 7 of the main manuscript (paragraph 1, line 7) is changed to: "MEG QY is determined by the ratio of the y-intercepts of fitting curves of R POP (using Eq. 1) with <N 0 > << 0.1 at above and below MEG threshold ( Fig. 2d-f)."

Reviewer #2:
General Comments: In the present manuscript, the authors explore the implications of their recent work (M.J. Li et al., Nat. Comm. 8, 14350 (2017)) showing a phonon bottleneck in the cooling of hot carriers in lead halide perovskite nanocrystals (NCs). The slowing of hot carrier cooling observed in their earlier work suggests that the efficiency of carrier multiplication (CM) should increase in lead halide perovskite nanocrystals compared to the bulk material. The observation of slower cooling with increasing confinement in their earlier report opens the possibility that the efficiency of CM could be higher in lead halide perovskites than in traditional inorganic NCs like PbS and PbSe. The present manuscript illustrates that these expectations are born out. Specifically, the threshold for the onset of CM in FAPbI3 NCs is found to be as low as 2.25 ± 0.1 times the gap in 7.5 nm diameter NCs, while the slope efficiencies are observed to be as high as 75% (i.e., an average of 0.75 extra electron-hole pairs are generated per additional gap of photon energy above the CM threshold). These numbers are substantially better than in the most widely studied systems, PbS and PbSe, which had once been thought to offer the greatest potential for CM. Importantly, the efficiencies reported here are most favorable in the smallest dots, which suggests room for further improvement. This is again in contrast to the case of traditional inorganic NCs in which carrier cooling rates increase with increasing confinement, thereby negating the gains in Coulomb interactions and relaxation of momentum conservation that come with increased confinement and that would otherwise be expected to lead to increased CM efficiencies.
The study of CM has struggled somewhat since the initial reports of high efficiencies in inorganic NCs were found to be erroneous. CM has been overtaken as a topic of study by singlet fission for a variety of reasons including perceptions of potential efficiencies and the fact that there have been clearer paths to design of molecules for singlet fission. There will be interest in the present work in boosting prospects for CM in a relatively simple material system. As such, this is a valuable contribution to the field. It should also be noted that the data in the present manuscript is very thorough, and great care appears to have been taken to obtain clean data.

Response:
We are delighted that reviewer shares our enthusiasm on the discoveries. The reviewer's critical comments are very helpful for improving our manuscript. We have carefully considered the reviewer's comments and would like to reply as follows: Comments 1: It should be noted that Klimov's group has shown a CM threshold below 2.2 eV in PbSe/CdSe core/shell NCs (C.M. Cirloganu et al., Nat. Comm. 5, 4148 (2014)), which partly undercuts one of the main claims here, namely that the CM threshold is lower than in traditional inorganic NCs. However, the slope efficiencies were still only 25%. At least as important as the numbers is the fact that such core/shell NCs are fundamentally more complicated than the perovskite NCs studied here.
Response 1: We thank the reviewer for pointing out this reference that we were not aware of. There are many reports on determining MEG thresholds of semiconductor nanostructures from many different groups using different methods. Here we only compare with the classic semiconductor nanocrystals (i.e., PbS and PbSe NCs), which have been thoroughly investigated using the transient absorption method. The MEG thresholds were found to be at ~ 3Eg, and are consistent with our control experiments. We therefore refer them as "benchmark materials in the MEG NCs discussion". Apart from the core-shell nature, we note that the method to determine the MEG threshold in this paper (Nat. Comm. 5, 4148 (2014) was based on the TRPL approach. Nonetheless, we have added this reference in Table S2 of SI as an example of core/shell NCs.
We would also like to point out that our claim is on "surpassing the highest MEG efficiencies in strongly confined incumbents", and not the MEG threshold. We are referring to the slope efficiency.
We have therefore cautioned our claim "surpassing the highest MEG efficiencies in strongly confined incumbents" and revised it to "surpassing the highest MEG slope efficiencies in strongly confined incumbents" in manuscript.

Comment 2:
The one general shortcoming I see in the paper is the absence of a quantitative explanation for the observations. That the CM efficiencies in FAPbI3 NCs are greater than in the bulk was to be expected given the authors' previous paper on the phonon bottleneck in lead halide perovskite nanocrystals. In lines 28-30, the authors write that "these NCs solve the conundrum over the contrasting requirements of enhancing Coulombic coupling and the density of states in strongly confined NCs." The results don't really solve the conundrum; they just avoid it by taking advantage of slowed cooling.

Response 2:
We thank the reviewer's valuable comment. To make our statement clearer to readers, we have changed "these NCs solve the conundrum over the contrasting requirements of enhancing Coulombic coupling and the density of states in strongly confined NCs." into "These perovskite NCs circumvent the conundrum over the increased carrier cooling rates and reduced density of states which reversed the gains of enhanced Coulomb coupling and relaxation of momentum conservation in strongly confined conventional NCs." on page 3-4 of the revised manuscript. The comment on "quantitative explanation for the observations" is addressed in the next comment. In the conventional picture, MEG is believed to be more favourable for systems where the ratio of the electron/hole (or hole/electron) effective mass is large. This is based on the simple assumption that the lighter particle can take most of the excess excitation energy, when m e >>m h or m h >>m e , the MEG threshold would be close to 2Eg.
However, there are also several detailed theoretical calculations (based on atomistic semiempirical pseudopotential method) predicting that systems with small and similar electron/hole effective masses are more favourable for slow hot-carrier cooling and therefore more efficient MEG (e.g., J. Phys. Chem. Lett. 2013, 4, 317;Chem. Phys. Lett. 2010, 496, 227).
As highlighted in Prof Eran Rabani's computational and theoretical work (J. Phys. Chem. Lett. 2013, 4, 317):"It turns out that the intuitive assumption that the lighter particle takes most of the excitation energy is, in fact, incorrect. Indeed, transitions where the lighter particle takes the excess energy are much stronger than other transitions. However, the density of singly excited states, where the heavy particle takes the excess energy, is much larger. Thus, the effective oscillator strength of such transitions is larger, often by 2 orders of magnitude. Because the average Coulomb coupling of the heavier particle is significantly lower, the overall efficiency decreases when the two masses differ, consistent with the results shown in Figure 1." Another recent computational work (J. Phys. Chem. Lett. 2017, 8, 3032) also predicted the lowthreshold (close to 2Eg) MEG in MAPbI 3 NCs and the stronger Coulomb coupling between the initial single-exciton and final-biexciton states and longer hot-carrier cooling of highly excited states is more favourable for MEG. The calculated MEG process is on the time scale of tens fs, which agrees well with our measured <90 fs. Supplementary Fig. 13 (a), Optimized cubic structure of FAPbI 3 . (b), DFT calculated band structure of FAPbI 3 .
We have now included DFT calculations ( Supplementary Fig. 13). The electron and hole effective mass were calculated to be 0.14m 0 and 0.20 m 0 , respectively. The small and similar effective mass of the electron and hole in the FAPbI3 are thus favourable for slower hot-carrier cooling and more efficient MEG, which is in agreement with the theoretical predictions.
Based on the reviewer's suggestion, we have added the following discussion on the mechanism of efficient and low-threshold MEG in our perovskite NCs: "From our density function theory (DFT) calculations ( Supplementary Fig. 13), the electron and hole effective masses are found to be 0.14m 0 and 0.20m 0 , respectively. The small and similar effective mass of the electron and hole are predicted to be favourable for slow hot-carrier cooling and more efficient MEG. 36, 37 Furthermore, another recent computational work 38 also demonstrated low-threshold MEG (close to 2Eg) in methylammonium lead iodide (MAPbI 3 ) NCs. The authors attributed that the stronger Coulomb coupling between the initial single-exciton and final-biexciton states, and the longer hot-carrier cooling of highly excited states were more conducive for MEG. Their calculated MEG process was on the time scale of tens of fs, which agrees well with our measured value of <90 fs. Although, PbSe or PbS also have similar small carrier effective masses, the high MEG threshold can be explained by their faster hot-carrier cooling (Fig. 4c) and/or reduced density of final states in strongly confined NCs. We therefore ascribe the origins of the low threshold and efficient MEG in our intermediate-confined perovskite NCs to arise from the small and similar carrier effective masses, slow hot-carrier cooling and appropriate quantum confinement effect." on page 11 of manuscript and Supplementary Fig. 13 in the revised SI. Response 4: We thank the reviewer for his/her valuable comment.
The linear fit of A is only guide for eyes showing a possible trend that may exist in the figures. We acknowledge that the initial PB signal (i.e., A) only grows almost-linearly at low pump fluence. At high pump fluence, it will saturate. We agree with the reviewer that our linear fit for A maybe misleading to the readers, we have therefore deleted these fits.
Our MEG yield is determined at extremely low pump fluence (i.e, the y intercepts when <N 0 > << 0.1), the inconsistent trends of A at high pump fluence would not affect the determination of MEG yield by A/B. Actually, the A/B data have already been provided in Panels e of Fig. 5-7. Therefore, all the different trends of A at high pump fluence (<N 0 > > 1 to 2) will not affect the determination of MEG QY.
The following are the possible explanations for these different trends of A for different sized NCs.
The state-filling-induced transient absorption change can be expressed as: Where , (ℏ ) is the contribution of transition ℏ to the ground-state absorption at the probe wavelength, ( ) are occupation numbers of the electron (hole) states involved in the transition. In the case of the twofold degeneracy in NCs, ideally, the sum of average populations of the lowest electron and hole states as a function of initial average e population can be presented as: Where, <N> is the average number of generated excitons per NC, which equals to <N 0 >MEG QY , <N 0 > is the average absorbed photons per NC. Therefore, at larger <N> at higher pump fluence or MEG QY >1, PB is easier to saturate (see Figure below), which is consistent with our observations of <N 0 > dependence of A (Fig. 5d) for smallest NCs with highest MEG QY . Figure R1. Initial TA amplitude as a function of pump fluence <N 0 >.
It should also be noted that the above red and blue curves are calculated based on the constant MEG QY and neglect the multiple exciton interactions at higher carrier densities. However, at higher pump fluence, the MEG QY may be changed (may become larger due to stronger carrier interactions at higher carrier density), therefore the experimental may derivate from the pure state-filling model.
In addition, the fast surface trapping (reported on the time scale of tens to hundreds of fs) usually exists in NCs following photoexcitation. Therefore, there is also a competing process between hot-carrier cooling, MEG and surface trapping. As discussed in our previous paper (Nat. Comms., 8, 14350, 2017), the slower hot-carrier cooling or the presence of the phonon bottleneck could be due to fewer surface traps. Nonetheless, details of competing process between hot-carrier cooling and surface trapping in perovskite NCs are still not clear, which will be investigated in the future. The superlinear increase of PB signal at higher pump fluence  for larger NCs with weak MEG effect could be due to the saturation of surface traps.
Therefore, these analysis suggest that the carrier density-dependent competing channels bestows a sublinear or superlinear behaviour of the initial peak PB signal with pump intensity, resulting in different trends for different sized NCs under different photon energies. The detailed investigation of these deviations at high pump fluence is beyond the scope of this work.

Comment 5:
The authors should state that the FA in FAPbI3 refers to formamidinium the first time FAPbI3 is mentioned in the manuscript, not just in the methods section.
Comment 6: I presume that the solid red and blue fits to RPOP are fits of Eq. 1. This is implied but not stated explicitly.
Response 6: This is correct. To make our statement clearer to readers, we have revised accordingly as shown below: " is determined by the PB intensity ratio at the beginning of pump excitation to that at a delay time of 4 ns." in Fig. 2 caption is changed to: " is determined by the PB intensity ratio at the beginning of pump excitation at a delay time of ~2 ps to that at a delay time of 4 ns and fitted with Eq.1 (solid lines)." and "The R POP ratios at the y-intercepts at above and below MEG threshold photon energy excitation thus facilitates the determination of MEG QY (Fig. 2d-f)." in page 7 of manuscript is changed to: "MEG QY is determined by the ratio of the y-intercepts of fitting curves of R POP (using Eq.1) at <N 0 > << 0.1 at above and below MEG threshold ( Fig. 2d-