Phonon-Mediated and Weakly Size-Dependent Electron and Hole Cooling in CsPbBr3 Nanocrystals Revealed by Atomistic Simulations and Ultrafast Spectroscopy

We combine state-of-the-art ultrafast photoluminescence and absorption spectroscopy and nonadiabatic molecular dynamics simulations to investigate charge-carrier cooling in CsPbBr3 nanocrystals over a very broad size regime, from 0.8 to 12 nm. Contrary to the prevailing notion that polaron formation slows down charge-carrier cooling in lead-halide perovskites, no suppression of carrier cooling is observed in CsPbBr3 nanocrystals except for a slow cooling (over ∼10 ps) of “warm” electrons in the vicinity (within ∼0.1 eV) of the conduction band edge. At higher excess energies, electrons and holes cool with similar rates, on the order of 1 eV ps–1 carrier–1, increasing weakly with size. Our ab initio simulations suggest that cooling proceeds via fast phonon-mediated intraband transitions driven by strong and size-dependent electron–phonon coupling. The presented experimental and computational methods yield the spectrum of involved phonons and may guide the development of devices utilizing hot charge carriers.

C sPbBr 3 perovskite nanocrystals (NCs) 1 have emerged as promising building blocks for optoelectronics, ranging from LEDs, lasers, or solar cells, to single-photon sources, owing to their high defect tolerance, high luminescence quantum yield, narrow luminescence line width, and ease of spectral tuning via (postsynthetic) composition control. 2 Recent reports pointing out surprisingly long lifetimes of hot charge carriers, 3 similar to those observed in thin-film leadhalide perovskites (LHPs), 4,5 furthermore reiterated prospects for a variety of high-efficiency applications relying on hot electrons, e.g., hot-electron catalysis, 6,7 hot-electron charge extraction, 8,9 or carrier multiplication. 10,11 At present, however, a lack of understanding of and control over charge-carrier cooling in LHPs limits the exploitation of such processes in devices.
In bulk and thin-film LHPs, several propositions have been made to explain the observed slow cooling, including arguments based on the phonon density of states (DOS), 12,13 Auger heating, 3,14,15 upconversion of acoustic to optical phonons, 16 surface passivation, 17 and formation of Froḧlich polarons, 18 i.e., quasiparticles comprised of a charge carrier and its self-induced lattice deformation. The latter process, enabled by strong electron−phonon coupling in the polar perovskite lattice, is thought to protect charge carriers from scattering with other charge carriers and phonons. 18 In addition to increased carrier diffusion lengths, 19 polaron formation may also dynamically compete with charge-carrier cooling on a (sub)picosecond time scales. 5,12,18,20 Along these lines, dissimilar charge-carrier cooling rates in CsPbX 3 , MAPbX 3 and FAPbX 3 (MA = methylammonium, FA = formamidinium, X = Cl, Br, I) have been associated with differences in polaron stabilization energies. 5,12,18,21 However, the notion that polaron formation slows down carrier cooling in perovskites 22 seems at odds with the finding of fast (<50 fs) cooling in anatase TiO 2 nanoparticles, 23 possessing a similar electron−phonon coupling strength as LHPs, with Froḧlich coupling constants α MAPbBr 3 (=1.5 − 1.9) < α TiO 2 (≈2) < α CsPbBr 3 (=2.6 − 2.8). 21,23 Moreover, reports on LHPs disagree on whether the organic cation accelerates 12,13 or slows down cooling, 16 and on the temperature dependence of cooling and polaron formation rates. 5,12,20,22 The picture becomes even more complicated in perovskite NCs: next to further clarifying the role of phonons 24 and composition, 25 the possibility of size-dependent cooling rates arising from quantum confinement (leading to a sparser electronic DOS) and dielectric confinement (in the solvent/ ligand environment) as well as ligand chemistry need to be addressed. While the description of confinement effects may borrow insights from the extensively studied II−VI, IV−VI, and III−V semiconductor NCs, 26−33 e.g., the discussion of "phonon bottlenecks" (postulated slow cooling via LO phonon emission due to a mismatch of energy-level spacing and phonon energy), it remains unclear to which extent such models can be applied to polar and ionic semiconductors with a possibly reduced probability of scattering between charges and phonons via polaron "protection". 5,21,22 Several important recent findings have both fueled the charge-carrier cooling debate and complicated the assessment whether cooling can be slowed down in perovskite NCs: (i) cooling rates strongly depend on excitation density: cooling in perovskites may be significantly slowed down at high excitation density, 13,16,25 due to overheated phonon modes (of similar importance as in many other polar semiconductors, e.g., GaAs), upconversion of acoustic to optical phonons, and a low thermal conductivity in perovskites; this requires "intrinsic" cooling rates of perovskites to be measured at low excitation densities; (ii) cooling rates are inconsistently defined: several ultrafast studies, including optical-pump THz-probe spectroscopy 12,34−36 and the transient-absorption (TA) spectroscopy reported here and elsewhere 25,37 clock the arrival time of cold carriers at the band edges; pump-push-probe spectroscopy is a variation on the theme, with altered IR oscillator strength of cold charges as the descriptor; 13,38 in contrast, other studies employing time-resolved photoluminescence (PL) or TA spectroscopy monitor the decay of the hot population and the carrier "temperature" via fitting the high-energy tail using a Fermi−Dirac or Boltzmann distribution, an approximation which becomes invalid in strongly confined systems; 37 finally, in femtosecond time-resolved two-photon photoemission (TR-2PPE) measurements 20 as well as computational studies, 39−41 the cooling rate was defined via the time evolution of the mean excess energy; in conclusion, comparisons of reported cooling rates are only meaningful if the same definition of cooling has been applied; (iii) charge-carrier-and state-resolved cooling has been elusive, as optical excitation and detection cannot easily discriminate electrons and holes, and the high and almost featureless DOS in perovskites essentially precludes a direct monitoring of the population of higher-energy states; (iv) while in CdSe NCs, control over the interaction of charges and phonons has been achieved via size confinement, 42,43 ligand chemistry, 41 or via tuning of the polar coupling through orbital symmetry and electron occupation, 42 similarly detailed studies in perovskite NCs are still lacking.
Finally, despite new insights from recent discussions of the role of polaron formation rates 20 and stabilization energies 21 in LHPs, their respective relevance to charge-carrier cooling remains poorly understood even in bulk: for example, if polaron formation would compete with cooling, then overall cooling rate constants should increase with decreasing initial excitation energy since charge carriers have less time to get dressed as a polaron and thus prolong the cooling process. However, currently, such a correlation is not unambiguously supported by experiments. 5,12,20 In this respect, it is important to note that previous models essentially rely on the harmonic and "time-averaged" polaron picture formulated by Froḧlich, 44 Feynman 45,46 and Osaka. 47 The importance of a dynamic picture and anharmonicity has been identified by Bonn et al. 48 who suggested that Froḧlich polaron transport additionally experiences "dielectric drag" induced by A-site cation rotations and rattling.
Here we take the first steps toward a truly time-dependent description via ab initio molecular dynamics (AIMD) supported by state-of-the-art ultrafast PL and TA spectroscopy. Time-domain simulations and experiments reveal that excitedstate dynamics are governed by lattice vibrations, i.e., phonons, mediating strong time-dependent fluctuations of the electronic band structure. This dynamic picture provides an alternative explanation for the slow electron cooling close to the conduction band: without the need to invoke polaron formation, a slow final stage of the electron cooling process arises from suppression of intraband transition rates close to the conduction band edge. Our time-domain description allows us to address open questions regarding charge-carrier cooling in perovskite NCs, particularly the role of phonons and the dependence on the NC size. To this end, we extend the typically studied size regime, from the weak-to-intermediate confinement regime (4.1−12.3 nm, via ultrafast PL and TA spectroscopy), down to the intermediate-to-strong confinement regime (0.8−4.0 nm, via atomistic density-functional theory (DFT) calculations). Phonon-mediated cooling successfully explains three key experimental and computational findings: (i) a weak size dependence of cooling rates, due to a similar phonon DOS for NCs larger than 1.7 nm; (ii) comparable cooling rates of electrons and holes, in line with both conduction and valence band states predominantly formed by orbitals derived from the lead-halide sublattice; (iii) intraband gaps near the band edges modulated at frequencies of ∼15, ∼80, and ∼140 cm −1 , corresponding predominantly to bending and stretching modes of the PbBr 6 4− cage. 21 Figure 1a depicts our structural model of a 2.9 nm CsPbBr 3 NC, with Cs and Br atoms terminating the surface to emulate a surface capping of cations (e.g., oleylammonium, didodecyldimethylammonium) and anions (e.g., bromide, oleate, phosphonates). As shown recently, 49,50 such a model (Cs 200 Pb 125 Br 450 in case of a 2.9 nm NC) successfully describes the most common experimental observations of luminescent CsPbBr 3 NCs, at a minimum computational cost. From the projected electronic density of states (PDOS) displayed in Figure 1b, we infer that the band-edge orbitals are mainly formed via Pb and Br atomic contributions, with Pb dominating the conduction band and Br the valence band, respectively. Cs contributes significant orbital character only to higher energy states, i.e., at about >0.5 eV excess energy. We furthermore verified that the electronic structure is devoid of localized midgap (trap) states in the displayed energy range.
After having determined the ground-state electronic structure, see Figure 1b, we follow its adiabatic time-dependent evolution in Figure 1c via AIMD simulations performed within a canonical ensemble at 300 K, employing the computational framework previously developed for CdSe, 51 PbSe, 51 and PbS NCs 41 (see ref 41 and Supporting Information for details). Briefly, for each time step of the nuclear trajectory of the AIMD, we compute the adiabatic electronic structure at the DFT level of theory. In other words, we assume that the ground-state trajectory at room temperature also samples the excited-state potential energy surface. Owing to the soft nature of the perovskite lattice, the constituent atoms (especially the lighter Br and Cs) undergo large displacements from their Nano Letters pubs.acs.org/NanoLett Letter equilibrium positions, translating into significant fluctuations of the electronic structure. Note that the energies of molecular orbitals fluctuate by several times k B T, i.e., ≫0.026 eV, and energy levels frequently cross in time (see also Figure S3 for NCs of different sizes). Such drastic dynamics of the electronic structure, a clear manifestation of strong electron−phonon coupling, raises important questions regarding charge-carrier cooling, such as what is the time scale and mechanism of charge-carrier cooling? Is the cooling rate different for electrons and holes? Also, related to discussions of phonon bottlenecks in confined systems, 13,18,52,53 is there a size dependence for perovskite NCs?
To address these questions, we will now study charge-carrier cooling in colloidally dispersed CsPbBr 3 NCs experimentally, via femtosecond PL and TA spectroscopy. Monodisperse CsPbBr 3 NCs were synthesized according to a procedure laid out by Maes et al., 54 see Supporting Information for details. Figure 2a shows the linear absorption spectrum of the different sizes used. Given an estimated Bohr radius of 7 nm, 1 they range from strongly confined 4.1 nm NCs to weakly confined 12.3 nm NCs, the former matching with the largest size used in the theoretical calculations. After photoexcitation, we study charge cooling in these colloidally dispersed NCs via ultrafast PL 55,56 (see Figure 2b) and TA spectroscopy (see Figure 2c), respectively. While the former monitors the photoinduced rise of the PL intensity and the latter the change in absorbance ΔA, respectively, both techniques feature a time resolution of ∼150 fs, broadband detection, and tunable excitation with welldefined excess energy to the band gap. The measured ΔA signal originates from Pauli blocking (bleach) and spectral shifts, sometimes denoted as band gap renormalization in bulk materials. Importantly, the signal is composed of contributions from either electrons or holes. In contrast, PL measures the radiative recombination of electron−hole pairs implying that a  signal can only be observed if both electron and hole occupy the relevant energy levels involved in the transition. As such, combining both PL and TA might elucidate differences in the cooling of electrons and holes.
We note that photoexcitation in NCs always creates an electron−hole pair, a combination that can fuse to form a Coulombically bound exciton, which might have different properties than isolated charges. It was shown, however, that excitons dissociate rapidly in LHPs due to the limited binding energy compared to room temperature. 57 The average number of absorbed photons per NC, often denoted as ⟨N⟩, is kept sufficiently low (⟨N⟩ ≪ 1) in both TA and PL experiments to avoid nonlinear effects such as optical gain, Auger recombination, and high-density phonon bottlenecks, all of which were observed earlier on similar NCs and in bulk perovskite materials. 58−60 This regime also corresponds to the calculations presented further, where only single charges are present in the NCs.
Since spectral shifts around the band gap cloud the interpretation of the ΔA and PL signal as a direct metric for the population at that edge, we spectrally integrated the ΔA and PL signals, see Figure S2. For 12.3 nm NCs, Figure 2d shows the resulting kinetic traces of the ingrowing bleach, i.e., the band-edge population normalized at 3 ps, for different excess energies (E excess ) relative to the band gap. Note that the E excess defined here contains both electron and hole excess energy. Upon increasing E excess , we observe a slower ingrowth corresponding to the increased time carriers need to dissipate the larger excess energy. For E excess = 0.65 eV, Figure 2e compares the ingrowth of the absorption bleach to the ingrowth of the integrated PL signal under similar excitation conditions. The almost perfect match indicates that electrons and holes cool at equal rates in these CsPbBr 3 NCs. Figure 2f shows the resulting cooling times for all sizes, allowing three conclusions: (i) cooling times appear size-independent in the range of ca. 4−12 nm; (ii) the cooling time depends about linearly on the excess energy, indicating a fixed electron−hole pair energy dissipation rate of 2.02 eV/ps; (iii) the similarity of TA and time-resolved PL kinetics indicate that both electrons and holes cool on equivalent time scales, and we derive a dissipation rate of about 1.0 eV/ps per carrier.
We note that the observed excess-energy-independent dissipation rate is incompatible with a picture in which polaron formation efficiently competes with cooling. 20 In the latter case, the dissipation rate should decrease for high initial excess energy, since cooling would be slowed down once the polaron has formed. 5 The absence of such a trend in our data suggests that polaron formation is not (the only mechanism) responsible for slow cooling in CsPbBr 3 NC. To understand this potentially surprising result, we may recall that polarons in CsPbBr 3 are large, i.e., energetically weak, with reported formation energies on the order of 0.01−0.15 eV, changes to the Pb−Br bond length by ∼1%, bond angles altered by ∼10°, and polaron sizes of ∼3 nm. 21 In light of these moderate (timeaveraged) quantities, the phonon-mediated time-dependent fluctuations of the electronic structure (see Figure 1c and the discussion below) by up to 0.5 eV are large and likely influencing the cooling rates as well. Consequently, cooling may not only be determined by the relative rates of chargecarrier cooling and polaron formation but instead/additionally by the strength of electron−phonon coupling.
To describe cooling within a dynamic (i.e., time-domain) picture, and to obtain atomistic insights into the apparent indifference of electron and hole cooling rates as well as the observed size independence, we performed nonadiabatic molecular dynamics (NAMD) simulations on CsPbBr 3 NCs from 0.8 to 4.0 nm (see Supporting Information for details). As in several other works on condensed-phase systems, we utilize the fewest-switches surface-hopping approximation with neglection of back reactions 61 to simulate carrier cooling. Taking the dynamic electronic landscape in a 4.0 nm large Cs 490 Pb 343 Br 1176 NC as a starting point (see Figure 3a), we initially excite either an electron or a hole to orbitals 0.5 eV above the respective band edges and subsequently follow the relaxation toward the band edges, indicated schematically by the white arrows in Figure 3a. Driven by large nuclear displacements at 300 K, the heavily time-dependent conduction band electronic structure leads to complicated relaxation patterns of the absolute energy of the electron, see Figure S4 in the Supporting Information. However, referencing the electron's energy or hole's energy to the conduction band minimum (CBM) or valence band maximum (VBM), respectively, at each point in time, allows us to follow the more insightful relaxation of the "excess energy" per charge carrier, see Figure 3b and Figures S4 and S6.
After an initially rapid cooling stage both for electrons and holes, in line with the similarity of PL and TA data, the subsequent cooling stage from about 1 ps proceeds more slowly: while the majority of the holes has quickly reached the band edge, this does not hold for the electrons, see Figure 3c. By fitting Maxwell−Boltzmann distributions to each of the charge populations, we find that holes have cooled within 2 ps to within 50 meV of the VBM, while even at the end of our time window at 4 ps, electrons remain excited in orbitals about 150 meV above the CBM, i.e., about 6 times k B T, see Figure S5 and Table S1. Such "persistent energetic electrons" have been reported experimentally before in MAPbI 3 , where electrons with 250 meV excess energy decayed with a time constant as large as 100 ps. 5 Our results for the CsPbBr 3 NC suggest that slow electron cooling in perovskites close to the CBM may be more general and at play even in the low carrier-density regime (one charge per NC) studied here. 58 However, the amount of retained excess energy may depend on the specific halide, 13 Asite cation, 13,25,37 or (nano)size of the crystal.
To aid the comparison to the variety of reported ultrafast studies, we plot our data in two different representations: (i) energy-resolved populations at selected snapshots in time (see Figure 3c), which may be compared to, e.g., femtosecond timeresolved two-photon photoemission (TR-2PPE) measurements, 20 and (ii) time-resolved kinetics at selected excess energies (see Figure 3d), which may be compared to, e.g., TA and time-resolved PL, as presented above, or PPP studies. The latter techniques monitor the arrival time of cold charges at the band edges, corresponding to the traces at 0−0.1 eV excess energy in Figure 3d. Band edge arrival times on the order of 1 ps for both the electron and hole are in qualitative agreement with our own TA results (∼0.5 ps for 0.5 eV excess energy per carrier, see Figure 2d) and those of other groups. 14,25 An even more quantitative agreement is found between the higher excess-energy transients in Figure 3d and those deduced from a TR-2PPE study by Evans et al. 20 Alternatively, we may monitor the average excess energy (white lines in Figure 3b), observable, e.g., in TR-2PPE measurements, 20 which can be fitted with the decay model proposed by Prezhdo et al., 62 extended by a biexponential component (see Supporting  Information). A similar biexponential fit has been employed Nano Letters pubs.acs.org/NanoLett Letter before 16 to describe the conceptually related cooling model accounting for scattering with LO and acoustic phonons. 24,59 For the initial cooling stage at (within the first ∼1 ps and at >∼0.1 eV excess energy), we find cooling time constants t cool,e = 0.69 ps for the electron and t cool,h = 0.75 ps for the hole, corresponding to rate constants of 0.72 eV/ps and 0.67 eV/ps, respectively. This is in excellent agreement with the 0.64 eV/ps electron cooling rate found in a TR-2PPE study 20 on CsPbBr 3 single crystals and the 0.8 eV/ps charge-carrier cooling rate found in THz experiments on CsPbBr 3 NCs, 34 but a factor 1.5 and 2 lower than the cooling rates in CsPbBr 3 NCs determined via TA in this and a previous 25 study, respectively. We suggest that the small discrepancy with the latter studies may be accounted for when also including spin−orbit coupling in the calculations 63 and the fact that cooling rates in the TA studies were defined via band-edge arrival times and fitted with singleexponential kinetic models. Overall, we observe rapid initial electron and hole cooling, slow electron cooling within 0.1 eV of the conduction band edge, confirm the ballpark of previously obtained cooling time constants, but reiterate (see above) the importance of referring to the same metrics when comparing the various cooling time regimes reported in literature.
To start rationalizing the data, we now drastically simplify the complex time-dependent electronic structure in Figure 3a and fit the kinetic traces in Figure 3d with a cascade model comprised of six static levels between 0 and 0.5 eV excess energy, arbitrarily spaced by 0.1 eV, see Figure 3e. As we will show now, this simplification (see Supporting Information for a discussion of applicability) captures the observed fast initial and slow subsequent relaxation. Specifically, we can now fit the electron and hole cooling kinetics in Figure 3d when allowing both relaxation (k down,i ) and re-excitation (k up,i ) transitions between adjacent levels, representing "cooling" and "heating", respectively. Figure 3f summarizes the fit result: at all excess energies, the downward rates are similar for the electron (on average 0.47 eV/ps) and hole (on average 0.60 eV/ps). In contrast, the contribution of the re-excitation rate to the total rate (k up,i /(k up,i + k down,i )) significantly increases toward the band edges: for the hole from less than 1% at 0.5 eV up to 14% at 0.1 eV excess energy, and for the electron from 21% at 0.5 eV to 33% at 0.1 eV excess energy. Nano Letters pubs.acs.org/NanoLett Letter Re-excitation rates approaching relaxation rates straightforwardly explain the large population of higher-energy states observed after ≥1 ps, i.e., after completion of the initial cooling stage, especially in the case of electrons. We note here that we can exclude Auger heating as a possible origin, 3,15 as we study the single charge-carrier picture, differing from the exciton and other many-particle excitations studied upon optical excitation. Instead, persistent energetic electrons from re-excitations appear due to the phonon-mediated fluctuations of the band structure, culminating in frequent state crossings close to the CBM (see Figure 3a). Upon a state crossing, the carrier acquires excess energy if it does not immediately relax to the new lower-energy state. This thus corresponds to a "heating" event in the static cascade model. Its likelihood increases with the occurrence of state crossings. Phonon-mediated energy fluctuations naturally also explain the observed variations in overall cooling rates: whereas higher-energy states (∼0.5 eV above the band edges) are more coherent and facilitate the initial rapid cooling stage, the final stage (within ∼0.1 eV of the band edges) proceeds slower due to incoherently modulated state energies, enabling re-excitation to higher-energy states via state crossings; the latter is more pronounced for the conduction band-edge states than for the valence band-edge states, yielding slower electron cooling close to the CBM.
Accepting the qualitative suggestions by the cascade model, we now present a comparison of experiment and theory, attempt to further understand possible size effects (affecting both nuclear and electronic degrees of freedom), and resolve the spectrum of involved phonon modes. Remarkably, contrasting a large number of reports on size-dependent excited-state dynamics in II−VI and IV−VI semiconductor NCs, 28 only a few pioneering studies reported on sizedependent charge-phonon couplings 64−66 and their involvement in charge-carrier cooling. 67 Elucidation of phonon-driven excited-state dynamics is even more needed in the soft perovskite NCs. Computationally, we compare seven NCs from 0.8 to 4.0 nm edge length (see structural models in Figure 5a), and experimentally from 4.1 to 12.3 nm (see Figure  S1). To the best of our knowledge, this constitutes the first report with matching computational and experimental sizes, and the most thorough (combined) size dependence so far.
Figures 4a−c demonstrate the close match of experiment and theory: first, and following previous reports, 16,58,59 we identify the TA bleach to the blue of the band gap as a proxy for warm charge carriers (see Figure 4a). In Figure 4b, we then monitor the cooling kinetics of warm carriers in 4.1 nm NCs by plotting the decay of the bleach fwhm. While such analysis has been applied previously, we acknowledge its limitations (see Supporting Information for a detailed discussion) and regard the transient "excess" fwhm (with respect to the longterm limit, e.g., at 1 ns) as a lower bound for the real electron and hole excess energy. As the rise of the band gap bleach above, the transient fwhm can be fitted with a biexponential decay, representative of a two-stage cooling process. The slow stage involves 10 meV excess fwhm decaying over 10 ps back to thermal equilibrium (∼60 meV fwhm at 1 ns). Figure 4c shows that the computational cooling data for electrons in essentially equally sized 4.0 nm NCs (see also Figure 2) can likewise be fitted with biexponential decay (see above and Supporting Information for details), yielding an amplitude of ∼100 meV with the experimentally found long time constant of 10 ps. In other words, warm electrons within 100 meV of the CBM are persistent in time, decaying only at a rate of 0.01 eV/ps back to thermal equilibrium. With the TA and computational results as lower and upper bounds, respectively (see Supporting Information for details on the CBM energy referencing), we conclude that the slow cooling stage involves electrons that are between 10 and 100 meV warmer than thermal equilibrium, i.e., described by a carrier temperature between ∼400 and ∼1500 K, and persisting for ∼10 ps. Figure 4d,e show the computed size-dependent cooling dynamics for the electron and hole, respectively, in CsPbBr 3 NCs from 0.8 to 4.0 nm. Fits yield weakly size-dependent  Figure 2f. Shaded regions depict the error margins, estimated to be ±15% and ±30% in experiment and theory, respectively. Black and gray dashed lines (depicting linear fits to electron and hole data between 1.7 and 4.0 nm) serve as a guide to the eye. Nano Letters pubs.acs.org/NanoLett Letter initial cooling time constants t cool (see Figure 4f), consistent with a weak phonon bottleneck effect due to quantum confinement. Slightly slower initial cooling for smaller NCs confirms and extends previous results on strongly confined CsPbBr 3 39 and weakly confined MAPbBr 3 NCs. 24 Only the smallest 0.8 nm NC departs from the weak size dependence, as here both electron and hole cooling proceed significantly slower, in 1.4 and 1.8 ps, respectively, yielding an electron− hole pair cooling rate of 0.63 eV/ps. However, we note that this particular NC is barely larger than one CsPbBr 3 unit cell; in fact, due to its ultrasmall size, the Cs:Pb:Br stoichiometry strongly deviates from 1:1:3 and essentially resembles the Cs 4 PbBr 6 motif. When excluding the strongly off-stoichiometric 0.8 and 1.2 nm NCs, the average initial cooling time constants t cool for CsPbBr 3 NCs between 1.7 and 4.0 nm are 0.90 and 0.93 ps for the electron and hole, respectively, corresponding to a size-averaged initial electron−hole pair cooling rate of 1.10 eV/ps.
In the second and slower cooling stage, electrons reach thermal (quasi-) equilibrium at the CBM via dissipating the last 20−100 meV excess energy over a time scale of about 10 ps, for all sizes (see fits in Figure 4d). The size-averaged persistent electron excess energy amounts to 42 meV (or 800 K). The lack of a size dependence suggests that the slow cooling stage is not a result of quantum confinement and a phonon bottleneck due to a sparse DOS, as discussed for CsPbBr 3 and MAPbBr 3 . 39 Instead, as introduced above, we interpret the persistent warm electron population as a fingerprint of efficient phonon-mediated re-excitation in the vicinity of the CBM. Potentially, such an assignment may also be a candidate for (at least partially) explaining the persistent hot electrons (at up to ∼100 ps) found in bulk LHPs, 5,22 i.e., materials lacking quantum confinement. In contrast to electrons, holes in our CsPbBr 3 NCs do not feature such a clearly discernible slower cooling stage: the remaining excess energy after the initial cooling amounts to only 27 meV on average, i.e., about k B T, as expected for the "conventional" picture of a static, unperturbed band structure.
Singling out predominant phonon modes that mediate excited-state dynamics requires an inspection of the phonons DOS and the interaction of charges and phonons. For the perovskite NCs from 0.8 to 4.0 nm (see Figure 5a), the bottom panels in Figure 5b show that the phonon DOS is almost sizeindependent. Three LO phonon peaks at ∼80, ∼100, and ∼140 cm −1 can be discerned, related to Pb−Br−Pb bending and Pb−Br stretching, 21,68,69 and a broad low-energy peak consistent with local polar fluctuations, e.g., Cs + -induced deformations of the PbBr 6 4− cage, as previously observed via Raman spectroscopy. 68 Only NCs equal or smaller than 1.2 nm exhibit significantly altered phonon spectra; this is similar to the case of CdSe, where a transition from bulk-like phonons to molecular-type vibrations has been observed at 2 nm, a size at which crystallites contain only a single complete zincblende unit cell. 70 Altered phonon spectra for the smallest NCs are furthermore expected given their increasing off-stoichiometry which approaches the Cs 4 PbBr 6 motif at 0.8 nm. Overall, the weakly size-dependent phonon DOS for NCs > 1.2 nm matches well the weakly size-dependent initial cooling rates of electrons and holes. Nano Letters pubs.acs.org/NanoLett Letter The phonon-mediated coherence loss between neighboring intraband states, expressed here as a "phonon influence spectrum" for electrons and holes, is depicted in the upper and middle panels of Figure 5b, respectively (see Supporting Information for calculation details). Briefly, the phonon i n fl u e n c e s p e c t r u m i s c a l c u l a t e d a s , where C(t) = ⟨ΔE(t)-ΔE(0)/⟨ΔE 2 (0)) is the normalized autocorrelation function for an intraband transition of energy E ij between neighboring states j = i + 1 and i, with ΔE = E ij − ⟨E ij ⟩ denoting the fluctuation with respect to the statistically averaged transition energy ⟨E ij ⟩. At high excess energy (0.2−0.5 eV above the respective band edge), the phonon influence spectra are of low amplitude and almost featureless. This indicates that, for both charge carriers, the initial cooling proceeds via transitions between states whose energy difference fluctuates only a little, at least on the time scale of the cooling process. Such slight incoherence is consistent with thermally activated cooling rates 12 due to momentarily lower energy separations (and thus increased adiabatic coupling) between intraband states. Along these lines, the significantly increased phonon influence for smaller NCs (see Figure 5b) should also yield enhanced nonadiabatic couplings and faster cooling. However, the opposite trend is observed, with a weak deceleration of cooling for very small NCs (see Figure 4c). We reconcile both results by acknowledging that phonon bottlenecks (due to a sparser DOS) 39 likely annul the size dependence of the nonadiabatic couplings and even slow down cooling for very small NCs.
In the final cooling stage, i.e., within 0.2 eV of the band edge, the phonon influence increases markedly, and the decoherence between states occurs at frequencies covering and exceeding the entire spectral range of the phonon DOS. Predominantly involved are the Pb−Br−Pb bending and Pb−Br stretching vibrations, at around 80, 110, and 140 cm −1 , next to low-and high-frequency contributions. Whereas the involvement of LO phonons is intuitive, higher frequencies may originate from combination bands and overtones, and lower frequencies may be a signature of difference frequencies, surface modes, or Cs +induced local polar fluctuations deforming the PbBr 6 4− cage. 68 Smaller NCs exhibit sharper features in the phonon influence spectrum, in line with the sharper "molecular-type" vibrations observed in the phonon DOS (see bottom row in Figure 5b). Interestingly, the relative increase of the phonon influence at low energy (within 0.2 eV of the band edge) as compared to high energy (0.2−0.5 eV above the edge) is more pronounced for larger NCs. Together with a larger number of state crossings stemming from a tighter energy level spacing, it is conceivable that larger NCs can sustain persistent energetic carriers at rather high excess energies. While further work will be required to investigate the matter, our phonon influence spectra suggest that not only thermal broadening 41 (involving the CBM-VBM interband transition) but also charge-carrier cooling (involving intraband transitions) is governed by strong charge-phonon interactions in CsPbBr 3 NCs, consistent with the thermal activation of both processes. 12,71−73 A tempting future extension of our work would be to make use of the obtained atomistic insights in CsPbBr 3 NCs and study chargephonon coupling in related material systems. For example, we anticipate that it could be a worthwhile endeavor to attempt controlling the observed phonon influence in LHP NCs via appropriate tuning of the composition, stoichiometry, or surface termination.
In summary, we address open questions regarding the rate and origin of slow charge-carrier cooling in LHPs by studying CsPbBr 3 NCs via both experimental (ultrafast laser spectroscopy) and computational chemistry methods (NAMD). Studying a very broad combined size regime, from 0.8 to 12 nm, i.e., covering the strong to weak confinement regime, we establish that cooling proceeds initially at about 1 eV ps −1 carrier −1 , decreasing only weakly with decreasing size, and roughly independent of excess energy and carrier type (i.e., electron or hole). Within about 100 meV of the CBM, however, electron cooling is significantly slowed down, with rates decreasing by more than 1 order of magnitude. The agreement of our NAMD simulations with the recently reported persistent energetic electrons in LHP thin films is surprising: while the latter results have been suggested to arise from a competition between polaron formation and cooling, our NAMD simulations on perovskite NCs suggest that slow cooling close to the CBM may not require polarons and is, at least in part, a result of phonon-mediated fluctuations of the electronic structure. Spectral analysis shows that a wide spectrum of thermally populated phonons induces decoherence of the electronic structure, especially within about 0.2 eV of the CBM, and predominantly mediated by deformation, bending, and stretching modes of the PbBr 6 4− cage at about 15, 80, 110, and 140 cm −1 . Our study reiterates the importance of strong charge-phonon coupling in LHPs, yet emphasizes the dynamic nature of the interaction. Formulated in the timedomain, excited-state dynamics are intuitively tracked, and results can be directly compared to a variety of ultrafast spectroscopies. As our combined experimental−computational approach may be straightforwardly extended to other materials systems, we hope that it may aid the fundamental understanding of excited-state dynamics in polar semiconductors in general and accelerate the development of devices utilizing hot charge carriers.

* sı Supporting Information
The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.9b05051. Materials; nanocrystal synthesis; sizing analysis via TEM; time-resolved and spectrally resolved TA and PL data; TA bleach at high excess energy and a short time scale; computational chemistry methods; timedependent electronic landscape; energy-and stateresolved electron cooling kinetics; Maxwell−Boltzmann fit to electron and hole populations from NAMD simulations; fit of the decay of the average excess energy in NAMD simulations (PDF)