Long-Range Exciton Diffusion in Two- Dimensional Assemblies of Cesium Lead Bromide Perovskite Nanocrystals Authors: Erika Penzo, Anna Loiudice, Edward S. Barnard, Nicholas J. Borys, Matthew J. Jurow, Monica Lorenzon, Igor Rajzbaum, Edward K. Wong,1 Yi Liu, Adam M. Schwartzberg, Stefano Cabrini, Stephen Whitelam, Raffaella Buonsanti, & Alexander Weber- Bargioni This document is the unedited Author’s version of a Submitted Work that was subsequently accepted for publication in ACS Nano, copyright © American Chemical Society after peer review. To access the final edited and published work see https://doi.org/10.1021/acsnano.0c01536. Penzo, Erika, Anna Loiudice, Edward S. Barnard, Nicholas J. Borys, Matthew J. Jurow, Monica Lorenzon, Igor Rajzbaum, et al. “Long-Range Exciton Diffusion in Two-Dimensional Assemblies of Cesium Lead Bromide Perovskite Nanocrystals.” ACS Nano 14, no. 6 (May 27, 2020): 6999– 7007. doi:10.1021/acsnano.0c01536. Made available through Montana State University’s ScholarWorks scholarworks.montana.edu Long-Range Exciton Diffusion in Two- Dimensional Assemblies of Cesium Lead Bromide Perovskite Nanocrystals Erika Penzo,1* Anna Loiudice,2 Edward S. Barnard,1 Nicholas J. Borys,1† Matthew J. Jurow,1,3 Monica Lorenzon,1 Igor Rajzbaum,1 Edward K. Wong,1 Yi Liu, 1,3 Adam M. Schwartzberg,1 Stefano Cabrini,1 Stephen Whitelam,1 Raffaella Buonsanti,2 Alexander Weber-Bargioni1* 1 The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 2 Institute of Chemical Sciences and Engineering of the École Polytechnique Fédérale de Lausanne, CH 1015, Switzerland 3 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Corresponding Authors * Erika Penzo: erikapenzo@gmail.com *Alexander Weber-Bargioni: awb@lbl.gov 1 ABSTRACT Förster Resonant Energy Transfer (FRET)-mediated exciton diffusion through artificial nanoscale building block assemblies could be used as an optoelectronic design element to transport energy. However, so far nanocrystal (NC) systems supported only diffusion lengths of 30 nm, which are too small to be useful in devices. Here, we demonstrate a FRET-mediated exciton diffusion length of 200 nm with 0.5 cm2/s diffusivity through an ordered, two-dimensional assembly of cesium lead bromide perovskite nanocrystals (CsPbBr3 PNCs). Exciton diffusion was directly measured via steady-state and time-resolved photoluminescence (PL) microscopy, with physical modeling providing deeper insight into the transport process. This exceptionally efficient exciton transport is facilitated by PNCs’ high PL quantum yield, large absorption cross-section, and high polarizability, together with minimal energetic and geometric disorder of the assembly. This FRET-mediated exciton diffusion length matches perovskites’ optical absorption depth, thus enabling the design of device architectures with improved performances, and providing insight into the high conversion efficiencies of PNC-based optoelectronic devices. KEYWORDS: perovskite nanocrystals, exciton diffusion, optoelectronics, Förster resonant energy transfer, excitonic transport 2 Energy transport at the nanoscale plays a critical role in a plethora of natural systems: for instance, the photosynthetic process relies on Förster Resonant Energy Transfer (FRET) to transport energy along a few μm of diffusion length,1 as well as being the primary mechanism of energy transport between proteins or different moieties of the same protein.2 A detailed understanding of the FRET mechanism is pivotal to the realization of artificial systems with efficient, long-range energy propagation. Quantum dots (QDs) solids have recently gained a lot of attention, thanks to the advantages of engineering 3D arrays with nanoscale building blocks, the QDs, which are well known for their exceptional optical properties, as well as for the possibility to easily tune such properties by changing their size, composition and surface chemistry.3-5 In addition, their self-assembly in close- packed systems facilitates the communication between neighbouring QDs, by enabling FRET of excitons that are able to hop onto adjacent, non-excited QDs, which results in the transport of the excitonic energy for multiple steps before the exciton recombines. Understanding FRET-mediated exciton diffusion is critical for enhancing the performances of optoelectronic devices, such as flexible organic light-emitting diodes (OLEDs), solar cells, and light modulators,6 by engineering the diffusion appropriately for the desired application. For instance, solar cells largely benefit from the migration of excitons towards the charge-separation interfaces.5, 7 Conversely, in light-emitting devices a large diffusion of exciton is detrimental to the efficiency, since it prevents the exciton from radiatively emitting from the QD where it formed, thus risking his non-radiative trapping onto adjacent layers.5,7 Artificial nanoscale building block assemblies are characterized by relatively short FRET-mediated exciton diffusion lengths, typically on the order of 10 nm in organic semiconductors,8 30 nm in inorganic nanocrystal (NC) solids,9 and up to 50-70 nm in 3 perovskite nanocrystals (PNCs) assemblies,10 which is a major limitation to the exploration of exciton-based optoelectronic phenomena and to the development of optoelectronic devices. Here we show that, within close-packed, two-dimensional assemblies of isoenergetic PNCs, FRET- mediated exciton diffusion lengths reach 200 nm, close to the light absorption depth of 200-400 nm for this class of materials,11 and the so far longest reported FRET-mediated exciton diffusion length in a NC system. The Förster equation describes the structural and optoelectronic requirements between a donor- acceptor system to maximize the FRET rate:12 1 𝜏 2π𝜇 𝜇 κℏ r 𝑛 1 𝜏 is the hopping rate between Donor (D) particle and Acceptor (A) particle with 𝜏 the average time for an exciton to hop from donor to acceptor. Maximizing this hopping rate requires (1) minimizing inter-particle distances r and the refractive index n, while (2) maximizing the spectral overlap between donors and acceptors 𝜇 𝜇 , and (3) aligning the dipole moments to maximize the orientation factor κ . Additional parameters critical to experimentally study the transport are maximizing the polarizability of individual emitters to enhance (4) photon absorption and hence exciton creation, (5) the fluorescent quantum yield for an optimized signal-to-noise ratio, and (6) a flat energy landscape between the particles (small inhomogeneous line broadening). Importantly, these 6 parameters affect the exciton diffusion multiplicatively, meaning that if one parameter is poorly optimized, exciton diffusion can be completely suppressed. Hence, most studies of exciton transport through nanoscale building block assemblies find exciton diffusion length of 6-30 nm (diffusion coefficient of 0.2-1210-3 cm2/s) for chalcogen-based quantum dot 4 (QD) assemblies,7, 9, 13 and of 3-50 nm for organic systems where the exciton transport occurs via singlet emitters and therefore FRET.14-17 The short diffusion lengths are attributed to one or several of these parameters to be limiting, for instance the spatial disorder in QD assemblies. PNCs are a fascinating NC class with the potential to excel in all of these parameters. As bulk semiconductors, lead halide perovskites have emerged as a promising class of materials for low- cost, solution-processable optoelectronics,18-21 demonstrating thin-film solar cell power conversion efficiencies exceeding 25%22,23 and LEDs with 20% external quantum efficiency.24-27 In the form of NCs, the all inorganic cesium lead halide (CsPbX3, X = I, Br, or Cl) PNCs provide high optical tunability as a function of composition,28-31 size,30,32-35 and shape,32,33,36 with impressive exciton generation efficiency,37 and scalable solution-phase processes.38-41 Although the quantitative use of the FRET equation is beyond the scope of this work, we note that in terms of FRET-mediated transport they qualitatively optimize all 6 parameters: (1) The inter-particle distance r is determined by the ligand (oleic acid and oleylamine) and amounts to 2 nm. (2) The spectral overlap is high due to narrow emission linewidths and small Stokes shifts.28 Furthermore, their defect-tolerant optical emission (responsible for record efficiencies in QD solar cells)29 maintains the symmetric and narrow emission bands compared to chalcogen based NCs,21 and hence the high spectral overlap. The transition dipoles (3) between adjacent PNCs have been reported to be well-aligned as well,42 due to their cubic shape directing the overall orientation in a self-assembled layer. PNCs have also demonstrated a combination of high photon absorption crosssection (4) and near unity quantum yield (5).43 A flat energy landscape (6) among NCs can be obtained by controlling the NC size in a weakly quantum confined regime where the energy bandgap is primarily determined by the chemical composition, so as to reduce the thermalization 5 of excitons towards NCs with smaller bandgap, which would limit the back transfer onto larger bandgap NCs and therefore decrease the sites availability for the exciton random hopping.28 In this work we show that indeed, FRET-mediated exciton transport through self-assembled close-packed PNC mono layers reaches over 200 nm with a diffusivity of ~ 0.1 cm2/s, which is in terms of diffusion length one order of magnitude higher and in terms of diffusivity two orders of magnitude larger than previously reported for chalcogen based QDs.9 We demonstrated these values by studying the exciton diffusion in a close-packed monolayer of CsPbBr3 PNCs. Importantly, in order to create a model system for the study of such diffusion, we took particular care in the sample preparation and developed a strategy to deposit only one monolayer of PNCs per sample. The ordered self-assembly of PNCs in a monolayer causes the exciton propagation to be restricted to two dimensions only, allowing us to directly image the exciton motion on a 2D plane without any loss of data in the third dimension. Moreover, this 2D PNCs assembly can be replicated theoretically with a 2D random walk, thus simplifying the interpretation of experimental data within a statistic framework. The diffusion length and the diffusivity were determined via stead state PL microscopy mapping and the diffusivity using time resolved PL mapping, respectively. The diffusion was modeled using continuum and discrete representations of exciton hopping to provide a physical interpretation of our experiments, showing that albeit demonstrating record FRET mediated exciton diffusion length, our system is still in a sub-diffusive regime. 6 DISCUSSION To simplify the direct visualization of exciton transport we confined the PNCs in a self- assembled monolayer on a Si wafer with about 1 nm of native oxide and coated with ~10 nm of a hydrocarbon polymer, deposited via plasma-polymerization of methane, to prevent both exciton quenching into the Si and wave-guiding along the dielectric layer. We made two kinds of samples, a close-packed sample to study the exciton diffusion, and a control sample with a sparse layer of small groups of PNCs separated by at least 20 nm to prevent FRET based diffusion. The NCs examined in this study were cubic CsPbBr3 PNCs with an average side length of 10 nm, synthesized according to an established procedure.28 Spin-coating from a toluene solution on a surface functionalized with a –CH terminated polymer reproducibly yielded close-packed monolayers with an inter-particle spacing of ~2 nm, which is determined by the ligand length (see Fig. 1, Methods, and the SI for a detailed description of the sample preparation methods). In order to stabilize the PNCs during optical characterization (conducted at room temperature and under ambient atmosphere), we adapted a recently reported passivation process:44 after spin-coating the solution of PNCs, we deposited ~3 nm of aluminum oxide by plasma-assisted atomic layer deposition (ALD). The passivated samples emitted stable PL upon illumination with continuous wave (CW) laser powers as high as 2000 W/cm2 or with pulsed laser fluences as high as 50 μJ/cm2. 7 b) 200 nm Figure 1. Deposition of controlled PNC 2D architectures. (a) CsPbBr3 PNCs in toluene were spin-coated onto a Si substrate functionalized with a –CH terminated polymer and coated with 3 nm of aluminum oxide deposited by ALD to prevent degradation during measurements. (b) SEM micrograph of a close-packed monolayer of PNCs. The blue circle shows the size (FWHM ~200nm) of the excitation laser spot used in subsequent optical experiments. Exciton diffusion in NC solids and organic semiconductors has predominantly been studied indirectly by spectroscopic techniques, which can only provide a coarse estimate of the diffusion length.7, 45 Recent microscopy-based approaches allow for the direct measurement of diffusion dynamics including the two main quantities that characterize exciton diffusion: the diffusion length and the diffusion coefficient (or diffusivity). 9 Figure 2 illustrates the basic principles of the visualization of exciton diffusion. In order to measure the diffusion length and coefficient, we start by exposing the PNCs monolayer to a perpendicular laser beam, whose size is kept as close as possible to the diffraction limit. The laser beam generates a local population of excited states in the PNCs monolayer, with an initial spatial distribution that matches the intensity profile of the excitation spot. The PL maps were recorded 8 with a CCD camera, after sample excitation with a 450nm laser, whose spot was kept as close as possible to the diffraction limit, with a FWHM of 240nm (Supplementary Fig. S18). If the motion of an exciton is allowed, for instance by FRET-induced hopping onto a nearby PNC, then it is possible that such exciton will travel before radiatively recombining. Our sample preparation greatly enhances the probability of such hopping by promoting the self-assembly of the PNCs in a close-packed monolayer. Excitons can therefore propagate within the 2D plane of the monolayer, which ultimately results in a radial expansion from the excited state distribution (Figure 2a) and in a broader spot of the collected PL with respect to the excitation spot. Conversely, if the distance increases with respect to the close-packed case, the FRET-hopping is not allowed, and the emission can only occur from the same PNC that was initially excited (Figure2b). In order to better understand this effect, we report in Figure 2b-c two images of PNCs, one in a close-packed lattice and the other one when the PNCs are spincoated onto a non-engineered substrate, thus resulting in a random distribution where each PNCs is far enough from others (at least 20nm) to hinder FRET transport. The collected PL from such samples are reported in Figure 2e,f. Here, we can appreciate the significantly different results of the excitation with the same spot, shape and energy: the spatial extent of the PL map collected from the close-packed sample is much broader than the excitation spot, whereas in the sample in which excitons cannot hop the spatial extent of the PL map resembles the excitation spot, only slightly enlarged due to the convolution of the point-spread function of the microscope with the spatial profile of the excitation spot PNC (i.e., a convolution of a diffraction-limited point source with the diffraction-limited spot of a focused laser beam; see Supplementary Fig. S18). A schematic of the set-up is shown in Supplementary Fig. S19. In order to provide a more quantitative comparison between the two images, in Figure 2g we report a line 9 scan through both PL maps as extracted from Figures 2e-f. Both spots are radially symmetric, so we can take any direction in each spot and compare their relative cross-sections. In first order approximation it is possible to describe the diffusion profile within a Gaussian framework. By subtracting the variances of the Gaussian fits with and without diffusion (close-packed vs sparse monolayer), we obtain a first estimate of the diffusion length, as explained in detail in the Supporting Information (see section S2). In order to provide a theoretical support for our data, we perform a detailed set of calculations, which are reported in the Supporting Information (see Sections S3 to S6). Specifically, we model the steady state diffusion using two complementary approaches, namely: i) continuum equations (S4) and ii) microscopic simulations (S5). First, we approximate the processes of creation, hopping, and recombination of optically excited excitons in nanoparticles by considering the statistic of classical bosonic particles on a lattice, in steady- state conditions. This description results in the derivation of a continuity equation, where the concentration of excitons is the physical quantity that undergoes a spatial variation with its own diffusion constant. The equation is solved both numerically and semi-analytically, with agreement between the two solutions in the linear regime, i.e., the condition of low excitation power in which the optical measurements were conducted. The semi-analytic solution is particularly useful to simulate the trends in non-linear regimes, that is, when the excitation power is large enough to prompt a power-induced change of the exciton diffusion profile. The modelling via continuity equation ultimately indicates that our PL profiles are consistent with a mean excitonic diffusion constant of the order of 0.5 cm2/s, or (224 nm)2/ns, which results in a characteristic exciton diffusion length of about 200 nm (consistent with the simple estimate in which we approximate profiles as Gaussian distributions). 10 In addition, we perform a set of discrete-time and continuous-time Monte Carlo simulations, where we model excitons as classical particles, able to undergo various processes, sitting with fixed positions within a 2D square array, similarly to the experimental sample. Using a model that assumes a completely uniform film of NCs, subtle but significant variations occur between the tails of measured and predicted profiles. However, as seen in the SEM images, vacancies occur in the experimental film (Fig. 1b). These vacancies can inhibit further exciton propagation. Hence, in order to provide a more realistic description of our system, we carry out Monte Carlo simulations in presence of both spatial and energetic disorder. Spatial disorder is taken into account by arranging the available site in a non-close-packed fashion, so that the hopping of an exciton onto a neighbour PNC may be inhibited due to excessive distance from the closest neighbour, whereas energetic disorder takes into account the deviation from perfectly isoenergetic particles and hence the presence of larger or smaller PNCs, resulting in (slightly) smaller or larger bandgap energies. When such disordered conditions are included in the model, the predicted PL profiles reproduces the experimental PL profile more closely (Fig. 2g). Importantly, the excitation power in all measurements was kept sufficiently low so as to remain in the linear excitation regime (see Methods and Supplementary Fig. S23). The presented interpretation of deconvolving the Gaussian distributions and extracting the diffusion lengths is only correct in the regime where PL intensity scales linearly with the excitation power. At higher pump fluences, exciton-exciton annihilation starts taking place, thus introducing an additional term in the diffusion equations that may lead to significantly broader PL profiles that could be interpreted as enhanced exciton diffusion, while actually it is related to a steep exciton 11 concentration gradient and the effective diffusion length for each exciton is still the same. A detailed discussion is reported in the Supporting Information. 12 Figure 2. Direct measurement of steady-state exciton diffusion. (a) When PNCs are assembled in a close-packed monolayer, the distance between NCs is minimized, allowing for efficient FRET- mediated exciton diffusion. (b) When PNCs are spatially separated, FRET-mediated exciton diffusion is inhibited. (c) SEM micrograph of a close-packed monolayer of PNCs. (d) SEM micrograph of a sparse monolayer of PNCs. (e) Normalized PL intensity profile emitted by the close-packed PNCs monolayer when excited with a diffraction-limited laser spot with wavelength 450 nm. (f) Normalized PL intensity profile emitted by the sparse PNCs monolayer when excited with a diffraction-limited laser spot with wavelength 450 nm. (g) PL profile cross-sections of panel (e) (black dashed line) and (f) (blue dashed line) together with simulated PL profile cross-sections for a square lattice of nanoparticles with a vacancy fraction of 20% (green line), and for a sparse sample of nanoparticles on which hopping cannot occur (cyan line). The dashed red line corresponds to the excitation laser profile cross-section. The inset shows the main figure on a logarithmic vertical scale. CsPbBr3 PNCs are not strongly quantum confined in the size range used in this study (10 nm average cube side);28 as a result, our close-packed monolayers mostly constitute a flat energy landscape for exciton diffusion, which is a requirement to maximize FRET in a NC solid. In the presence of energetic heterogeneity, excitons travel downhill in energy and thermalize onto NCs with smaller bandgaps. The process of back transfer onto larger bandgap energies has a lower rate and is less likely to take place, thus limiting the availability of viable neighbours for an exciton to hop on. Because of this more efficient funneling of excitons from high-energy to low-energy states than vice-versa, signatures of this inhomogeneity-driven process can be discerned in the temporal evolution of the PL spectrum after pulsed excitation. The additional energy transfer relaxation 13 channels of the high-energy sites accelerate their relaxation rates, yielding energy-dependent excited-state lifetimes, where higher-energy NCs have shorter lifetimes than those with lower- energies. Thus, following pulsed excitation, the PL spectrum typically shifts to lower energies as the higher-energy NCs relax more rapidly than the low-energy NCs. Experimentally, this behavior can be resolved as time-dependent shifts in the PL spectrum to lower energy or emission energy- dependent relaxation kinetics. In colloidal semiconducting NCs, this energetic heterogeneity largely arises from the polydispersity of the sample where the larger quantum dots are less quantum confined, have lower-energy excited states, and thus serve as sites where an exciton can easily transfer to, but from where it cannot easily leave.46,47 In Figure 3, we used time-resolved PL (TRPL) spectroscopy to assess the disorder in the energetic landscape in our films. As shown in Fig. 3a, during the assembly process, energy minimization pushes the PNCs of different size to the edges of the ordered, close-packed regions. The result is that the central regions of the film are significantly less polydisperse than the edges. Examples of a uniform area (top dashed rectangle) and of a non-uniform area (bottom dashed rectangle) are highlighted in Fig. 3a. As seen in Figs. 1b and 2c, the exciton diffusion studies were performed in the regions with monodisperse PNCs; in these portions of the film time-resolved PL spectroscopy measurements displayed identical decay kinetics for the high and low energy components of the PL spectrum (Fig. 3b) and, accordingly, the PL spectrum does not change with time (Supplementary Fig. S20). In contrast, in the edge regions with maximal polydispersity, signatures of excitons getting funneled in lower-energy bandgap NCs were subtle but detectable: the lower energy emission (2 nm band; centered at 529.2 nm) decays slower than the higher-energy emission (2 nm band; centered at 502.6 nm) and the PL spectrum exhibits a clear evolution to 14 lower energies following pulsed excitation (Supplementary Fig. S20). These results indicate that within the central regions of the film, the site-to-site variations in energy are minimized thus strongly reducing, or possibly eliminating, exciton funneling towards low-energy sites. Therefore, only at the edges, where the polydispersity is maximal, we see evidence of an energetic landscape where the energy transfer towards smaller bandgap NCs is more efficient. Rather, in the central regions, the weak quantum confinement combined with the assembly process play a pivotal role in achieving the long exciton diffusion distance. Future work will be dedicated to a detailed investigation of the redshift observed in defectual areas. Figure 3. Probing the energy landscape by time-resolved PL spectroscopy. (a) SEM micrograph of a close-packed monolayer of PNCs showing ordered areas made of uniformly sized 15 PNCs (blue dashed rectangle) and disordered areas made of PNCs of different sizes at a crack in the film (orange dashed rectangle). (b) PL intensity as a function of time at two emission wavelengths, 529.22 nm (blue solid line) and 500.56 nm (orange solid line), measured on an ordered area made of uniformly sized PNCs. The overlap of the curves indicates equivalent PL lifetimes at both emission wavelengths. (c) PL intensity as a function of time at two emission wavelengths, 529.22 nm (blue solid line) and 500.56 nm (orange solid line), measured on a disordered area made of PNCs of different sizes. The orange curve displays the slower decay of the low energy portion of the PL spectrum, due to exciton migration to smaller bandgap PNCs. To better understand and measure the dynamics of the long-distance exciton diffusion, time- resolved PL microscopy was employed to track the temporal evolution of the PL spatial profile as it expanded from its initial state. Areas of the close-packed monolayer of monodisperse PNCs were excited with short laser pulses and the magnified PL emission (100) was collected by a single mode fiber (with a diameter of 5 μm). To gain spatial resolution, the collection fiber was mounted on a translation stage that systematically scanned the fiber aperture in the focal plane of the microscope. At each fiber position, a full time-resolved PL transient was recorded, which allowed us to track how the excited state expanded with time. The temporal evolution of the normalized PL profile is shown in Figure 4a, where the 0 nm distance corresponds to the position of the fiber, which is aligned with the center of the PL spot. Distances greater than zero are reported as the distance between the center of the PL spot and the center of the image of the fiber aperture on the sample plane. The spatially and time-resolved PL trace in Figure 4a clearly displays the expected diffusive broadening, which can be quantified by calculating the PL profile width at each time slice (Fig. 4b). The rate at which the variance of the spatial PL profile (Gaussian fit) rises with 16 time is sub-linear, indicating a sub-diffusive process, which, as discussed above, is very likely due to PNC voids in the film. In these dynamic measurements, as reported in previous work,9 the average exciton diffusion length is calculated as the increase in the PL profile variance from its initial state (right after excitation) to the average lifetime of the system. The average lifetime of these PNCs in a close-packed film is 1.14 ns (as calculated as the time to reach a decay of 37% or 1/e of the PL intensity, see Supplementary Fig. S21), which, using the dynamics in Figure 4b, yields an average exciton diffusion length of 194 nm that is in very good agreement with that derived from our steady-state measurements. The PL lifetime was also calculated at each point (a PL lifetime map is shown in Supplementary Fig. S22) and was observed to decrease as the reciprocal of the PL intensity (Fig. 4c). This spatially dependent change in dynamics is an expected effect of the diffusion process, which is driven by the exciton density gradient, and results in a larger net outward exciton flux in areas with a large exciton population. The minimum lifetime was found to be 1 ns, and grew to 1.3 ns 500 nm away from the excitation intensity maximum. The majority of the signal (>90%) lies between -500nm and 500nm from the excitation maximum; as a result, the lifetimes calculated for values <-750nm and >750nm result very scattered, since the diffusion outside this region is almost completely exhausted and the signal is weak and indistinguishable from noise (especially considering that we are collecting with a single mode fiber with a diameter of 5 μm). Importantly, we highlight that the measurements were performed with excitation intensities that are deep in the linear regime of the power-dependence of the PL (see Supplementary Fig. S23), therefore we exclude the possibility that the central decrease in lifetime is due to higher order non-radiative recombination processes (e.g. Auger recombination). 17 The diffusion coefficient (or diffusivity) was calculated as the first derivative of the PL profile variance increase in time and is shown in Figure 4d. The effective diffusivity was found to decrease with time, from 0.5 cm2/s right after excitation to ~ 0.1 cm2/s at the end of the diffusion process. These values of FRET-mediated exciton diffusion length and diffusivity are the largest reported so far for NC solids. Chalcogen-based QDs demonstrated diffusion lengths in the range of 20-30 nm and diffusivities between 0.2E-3 cm2/s and 1.5E-3 cm2/s.9 The system here reported demonstrated one order of magnitude higher diffusion length and about two orders of magnitude higher diffusivity. Significantly, the exciton diffusion dynamics in our PNC films is comparable with that of unbound charge carriers in other bulk perovskite materials. Individual crystals of CsPbI2Br perovskite measured by pump-probe microscopy revealed a diffusivity of 0.27 cm2/s,48 and thin films of polycrystalline hybrid perovskite measured by transient absorption microscopy showed a diffusivity in the range of 0.05 cm2/s to 0.08 cm2/s, with diffusion lengths of 220 nm in the 2 ns experiment time.49 18 Figure 4. Probing exciton diffusion dynamics by time-resolved optical microscopy. (a) Time evolution of cross-sectional PL intensity profile. (b) PL profile variance increase as a function of time (blue dots). The green solid line shows the fit to the power law Aꞏtα (A = 0.3527 cm2/s , α = 0.53). (c) Space-resolved PL lifetime calculated from the signal in (a) as the time for 37% or 1/e decay of PL intensity. (d) Diffusivity (or diffusion coefficient) as a function of time calculated as the first derivative of the PL profile variance variation in (c). CONCLUSIONS In conclusion, ordered monolayers of isoenergetic PNCs were fabricated via controlled self- assembly. This system demonstrated extremely efficient FRET-mediated exciton diffusion, which 19 was directly characterized by steady-state and time-resolved PL microscopy together with an analytical and statistical model that granted a deeper understanding of the exciton diffusion dynamics. Our measurements directly capture a diffusion length of 200 nm and a diffusivity of 0.5 cm2/s. This diffusion length is ten times longer and the diffusivity is two orders of magnitude larger than previously reported values for films of chalcogen-based QDs,9,50 and comparable to that of charge-carrier diffusion in thin films of polycrystalline hybrid perovskite. Demonstrating such long-range diffusion, our PNC system is ideally suited to study FRET processes and FRET- mediated energy transfer on length scales that are easily accessible and therefore easy to optimize. Moreover, long exciton diffusion establishes an additional design element for next generation PNC-based optoelectronic devices.21, 51 Further progress towards even longer-range exciton diffusion may be achieved by improving PNCs energetic uniformity as well as by optimizing the protective ALD-based process that prevents perovskite degradation. Additionally, fabricating PNC assemblies with increased complexity, for example deliberately varying inter-particle distance in certain assembly portions or by positioning PNCs with decreasing bandgap next to each other forming an oriented energy funnel, could demonstrate ways to move excitons to predetermined positions. Overall, we showed that ordered assemblies of isoenergetic PNCs support FRET-mediated exciton diffusion with exceptional lengths, which can be used to better the performances of PNCs-based optoelectronic devices. 20 METHODS PNCs synthesis and characterization CsPbBr3 nanocubes were synthesized by a procedure adapted from the original report.28 All chemicals were purchased from Sigma-Aldrich and used as received without further purification. Cs2CO3 (1.2 mmol) was added to 10 mL 1-octadecene and stirred at 120 °C under vacuum for 1 hour. Oleic Acid (2 mmol) was injected under nitrogen atmosphere and resulting mixture was stirred at 120 °C for two hours until fully dissolved. In a separate container, PbBr2 (0.19 mmol) was added to 5 mL 1-octadecene and stirred at 120 °C under vacuum for 1 hour. Oleic acid (1.6 mmol) and oleylamine (1.5 mmol) were injected under nitrogen atmosphere. The resulting mixture was stirred at 120 °C for two hours until fully dissolved, then was heated to 165 °C. To this preheated lead solution was added 0.4 mL of hot Cs2CO3 solution under nitrogen atmosphere with vigorous stirring. Reaction was stirred for 5 seconds and cooled rapidly in ice bath until reaction mixture solidified. After freezing, reaction mixture was warmed to room temperature and transferred into centrifuge tubes. The mixture was centrifuged at 8,500 rpm, for 10 minutes. The supernatant was discarded, and the pellet was redispersed in anhydrous hexane (6 mL). An equal volume of tert-butanol was added to precipitate the NCs, and the mixture was centrifuged at 12,000 rpm for 15 minutes. The supernatant was discarded, and the pellet was redispersed in toluene. These solutions were then centrifuged for 5 minutes at 700 rpm, and the pellet was discarded to remove large aggregates. The supernatant was transferred to a glove box for film deposition. PNCs monolayer fabrication 21 PNCs monolayers were prepared by spin coating (1,500 rpm, 45 s) from a colloidal suspension of nanocubes in toluene onto Si wafers coated with 10 nm of a –CH terminated polymer, which is sufficiently thin to prevent lateral wave-guiding but thick enough to prevent exciton quenching by the silicon. The concentration was 3 g/l for the close-packed monolayer and 60 mg/l for the sparse monolayer. The hydrocarbon polymer was deposited by polymerizing methane in a plasma chamber (40 mTorr, RF power 100 W, 10 °C, Oxford Instruments). Aluminum oxide (3 nm) was deposited by plasma assisted atomic layer deposition at 40 °C (Oxford Instruments). Films were characterized by SEM (Zeiss). Steady-state PL microscopy The setup for steady-state PL microscopy is shown in Supplementary Figure S3a. The 450 nm CW diode laser source was collimated and then focused to a diffraction-limited spot by a 100X 0.95 NA objective lens. The back aperture of the objective was overfilled to assure diffraction-limited performance. Emission from the sample was collected by the same objective and additionally magnified 5.3X for a total magnification of 530X and imaged on a CCD camera (QSI SI 660 6.1mp Cooled CCD Camera) with pixel size 4.54 μm, which provided an effective imaging pixel size of 8.63 nm. A 490 nm long-pass dichroic filter (Semrock Di03-R488-t1) and two 496 nm long-pass edge filters (Semrock) were used to remove the excitation laser beam from the PL signal. The laser beam was imaged through the 490 nm long-pass dichroic filter (Semrock) and a 498 nm short-pass edge filter (Semrock) to remove the PL signal. Measurements were performed at 45 W/cm2, which corresponds to the probability of one absorbed photon per 1,705 nanocubes during the 1.15 ns average lifetime. 22 Time-resolved PL spectroscopy The setup for time-resolved PL microscopy is shown in Supplementary Figure S3b. The pulsed laser source (center wavelength 465 nm with a 2.5 nm bandwidth; 5 ps pulse duration; 40 MHz repetition rate) was collimated and focused by a 100X 0.95 NA objective lens. The back aperture of the objective was overfilled to assure diffraction-limited performance. Emission from the sample was collected by the same objective. A 490 nm long-pass dichroic filter (Semrock) and a 496 nm long-pass edge filter (Semrock) were used to remove the excitation laser beam from the PL signal. The PL spectral components were separated with a monochromator (Princeton Instruments Acton 2300i) and detected by a single-photon counting avalanche photodiode (MPD PDM series) connected to a time-correlated single-photon counting unit (Picoharp 300). The temporal resolution was approximately 50 ps as determined by the FWHM of the instrument response function. Measurements were performed at laser fluence 2.5 μJ/ cm2, which corresponds to the average probability of one absorbed photon per 32 nanocubes per pulse. Time-resolved PL microscopy The setup for time-resolved PL microscopy is shown in Supplementary Figure S3c. The pulsed laser source (center wavelength 465 nm with a 2.5 nm bandwidth; 5 ps pulse duration; 40 MHz repetition rate) was collimated and focused by a 100X 0.95 NA objective lens. The back aperture of the objective was overfilled to assure diffraction-limited performance. Emission from the sample was collected by the same objective and imaged on a single-mode fiber (P1-405P-FC-2, Thorlabs) attached to a translation stage (Attocube ECS series) that scanned the emission focal 23 plane. The fiber mode field diameter was 2.5 – 3.4 μm at 480 nm; the stage was moved in 5 μm steps corresponding to 50 nm at the sample. A 490 nm long-pass dichroic filter (Semrock) and two 496 nm long-pass edge filters (Semrock) were used to remove the excitation laser beam from the PL signal. The laser beam was imaged through the 490 nm long-pass dichroic filter (Semrock) and a 498 nm short-pass edge filter (Semrock) to remove the PL signal. The signal was detected by a single-photon counting avalanche photodiode (MPD PDM-series) connected to a time-correlated single-photon counting unit (Picoharp 300). The temporal resolution was approximately 50 ps, as determined by the FWHM of the instrument response function. Measurements were performed at laser fluence 5 μJ/ cm2, which corresponds to the average probability of one absorbed photon per 16 nanocubes per pulse. The sample was mounted above the objective lens on a piezoelectric scanning stage. Samples were scanned during the course of the measurements (~30 min) over an area of 5×5 μm to avoid photobleaching or photodamage. Physical modeling We simulated the processes of exciton creation, recombination, and hopping at continuum- and microscopic levels of resolution. We convolved the resulting excitonic profiles with the optical point-spread function in order to calculate observed PL profiles. Full details of these calculations are given in SI sections S3 to S6. 24 For Table of Contents Only 25 ACKNOWLEDGMENT This work was performed at the Molecular Foundry supported by the Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02- 05CH11231. M.J.J. and Y.L. were also supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DE-AC02-05CH11231 within the Inorganic/Organic Nanocomposites Program (KC3104). A.W.-B. was supported by the U.S. Department of Energy Early Career Award. We thank Prof. Alexander Holleitner (Technical University Munich) and Prof. Ian Sharp (Technical University Munich) for insightful discussions. ASSOCIATED CONTENT Supporting Information. Supporting Information Available: the file contains detailed description of the sample preparation and of the estimation of the diffusion length with a Gaussian approximation, a comprehensive section of the theoretical modeling via both continuity equations and Monte Carlo simulations, and Supporting Figures. (PDF). This material is available free of charge via the Internet at http://pubs.acs.org. 26 A preprint version of this manuscript is deposited in arXiv: Erika Penzo; Anna Loiudice; Edward S. Barnard; Nicholas J. Borys; Matthew J. Jurow; Monica Lorenzon; Igor Rajzbaum; Edward K. Wong; Yi Liu; Adam M. Schwartzberg; Stefano Cabrini; Stephen Whitelam; Raffaella Buonsanti; Alexander Weber-Bargioni; Long-Range FRET- Mediated Exciton Diffusion in 2D Assemblies of CsPbBr3 Perovskite Nanocrystals. 2020, 2003.04353. arXiv. https://arxiv.org/abs/2003.04353. AUTHOR INFORMATION Present Addresses † Department of Physics, Montana State University, PO Box 173840, Bozeman, MT, 59717 Author Contributions The manuscript was written through contributions of all authors. R.B. and A.W.-B. conceived the initial work. A.L. and I.R. performed initial experiments. E.P. prepared the samples, conducted the diffusion experiments and analysis, and wrote the manuscript. S.W. developed theoretical modeling and analysis. A.L. and M.J.J. synthesized the nanocrystals with supervision of R.B. and Y.L. E.S.B. and N.J.B. supported the experimental measurements and data analysis. E.P., E.K.W., and E.S.B. built the microscope. A.M.S. and S.C. supported the sample preparation and provided input into the data interpretation. E.P., E.S.B., N.J.B., M.J.J., A.M.S., S.W., M. L. and A.W.-B. helped with writing the manuscript. All authors have given approval to the final version of the manuscript. 27 REFERENCES 1. Scholes, G. D.; Fleming, G. R.; Olaya-Castro, A.; van Grondelle, R., Lessons from Nature about Solar Light Harvesting. Nat. Chem. 2011, 3, 763-774. 2. Piazza, F.; Sanejouand, Y.-H., Long-Range Energy Transfer in Proteins. Phys. Biol. 2009, 6, 046014. 3. Pietryga, J. M.; Park, Y.-S.; Lim, J.; Fidler, A. F.; Bae, W. K.; Brovelli, S.; Klimov, V. I., Spectroscopic and Device Aspects of Nanocrystal Quantum Dots. Chem. Rev. 2016, 116, 10513- 10622. 4. Boles, M. A.; Ling, D.; Hyeon, T.; Talapin, D. 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Schwartzberg, Stefano Cabrini, Stephen Whitelam, Raffaella Buonsanti, and Alexander Weber-Bargioni S1. PEROVSKITE NANOCRYSTAL (PNC) minated polymer (Supplemental Fig. S1) and/or with so- SAMPLE PREPARATION FOR EXCITON lutions in solvents other than toluene. Hexane and octane DIFFUSION MEASUREMENTS BY PL solutions deposited only patches of multilayers separated MICROSCOPY by wide empty regions. Development of a reproducible and controllable deposition technique was necessary for Exciton diffusion can be controlled by modulating the establishing consistent PNC monolayers. Simple drop NC assembly. Minimizing NC-NC distance (R) is es- casting, due to the lack of control over the drying process, sential to maximizing the rate of FRET (FRET rate is 6 yielded unacceptable sample-to-sample variability in theinversely proportional to R ). In a 3D NC solid, excitons final film morphology. Although very effective in forming can move in any direction within the solid, as long as highly ordered 2D assemblies of conventional semicon- the NCs are physically close enough and with sufficient ducting QDs, methods depending on the interface be- spectral overlap. Confining the NCs to 2D reduces the tween immiscible solvents (e.g. Langmuir-Blodgett tech- available paths for exciton hopping to a space directly niques) were not applicable to PNCs because of their accessible to imaging by optical microscopy. instability (solubility) in polar solvents[1]. Spin-coating The solution concentration and the spinning param- demonstrated reproducible results and allowed control of eters were optimized for close-packed monolayer depo- the density of PNCs in the film by modulating either sition; all samples were deposited at 1,500 rpm for 45 spin-coating speed or solution concentration. The mor- seconds and the film morphology was adjusted by vary- phology of the film was mostly determined by function- ing the concentration of PNCs solution. A closed-packed alization of the substrate surface and by the solvent in monolayer without portions of an extra layer and with which the PNCs were dispersed. We achieved the best minimum empty regions was consistently obtained when control of film morphology when the substrate surface spin-coating from a ca. 3 g/l solution in toluene of was functionalized with a -CH terminated polymer and CsPbBr3 PNCs at 1,500 rpm for 45 seconds. A sparse the PNCs were dispersed in toluene. Under these con- monolayer was obtained when spin-coating from a ca. 60 ditions, spin-coating from a dilute solution would yield mg/l solution of CsPbBr3 PNCs at 1,500 rpm for 45 sec- separated patches of PNCs, with patch sizes controllable onds. A closed-packed monolayer of NCs was not achiev- all the way to individual PNCs. The distance between able without the surface functionalization with -CH ter- individual patches increased as the PNC concentration was reduced. Conversely, deposition from a concentrated solution yielded a continuous monolayer of PNCs, spo- radically covered by patches of a second layer. The size of the patches increased, and the distance between them decreased, with increasing solution concentration. For very concentrated PNC solutions, two complete monolay- ers were formed. The number of layers steadily increases with increasing PNCs concentration in solution, following the same process of patch expansion. The spin-coating speed had a similar effect on film morphology. High spin- coating speeds would reduce the film thickness or increase the spacing between NCs patches and reduce the patches’ size, while low spin-coating speed would have the oppo- site effect. Adjusting the spin-coating speed had a more limited impact on the film morphology. Adjusting the 200 nm NP concentration was the most effective way to control the PNCs film density. All samples used in this study were screened by SEM to ensure consistent morphologies FIG. S1: SEM micrograph of PNCs deposited from a toluene and to compensate for batch to bath variability common solution (concentration ca. 3 g/l) by spin-coating (1,500 rpm to PNCs. for 45 s) on a Si wafer. Alumina coating by ALD was necessary to ensure sam- 2 ple stability under illumination and to prevent long-term degradation due to air moisture. Unprotected PNCs samples were found to be unstable under illumination with a focused laser beam even at low excitation power (below 100 W/cm2). The main signatures of this in- stability were a rapid decay of photoluminescence (PL) intensity together with changes in the PNCs film mor- phology, observed by SEM after exposure to the laser beam. S2. ESTIMATION OF EXCITON DIFFUSION LENGTH BASED ON GAUSSIAN APPROXIMATION The average exciton diffusion length of our system can be calculated according to: √ FIG. S2: Histogram of the PL profile sigma measured on a L = σ2 − σ2 , (S1) sparse monolayer of PNCs (light blue) and on a close-packeddiff diff no−diff monolayer of PNCs (coral). The average values of the two distributions are (167±18) nm and (260±22) nm respectively. where σ2diff is the exciton distribution variance in the presence of diffusion (measured on the closed-pack films) and σ2no−diff is the exciton distribution variance without diffusion (measured on the sparse films). The PL in- approximated exciton transport within nanoparticle ar- tensity profile measured in a far-field microscopy system rays as classical stochastic processes at mean-field (Sec- is given by the convolution of the single-emitter point tion S4) and microscopic (Section S5) levels of detail, re- spread function (PSF) with the excited-state population spectively. In Section S6 we compare simulation results density (i.e. the exciton distribution). The PL inten- with experiments. sity profiles and their underlying excited state population profiles are well approximated by Gaussian functions, so Readers interested solely in a comparison between the that the additive rule of variances upon convolution ap- experimental profiles described in the main text and sim- plies. The exciton diffusion length as defined above can ulated profiles should focus on Section S6. In this section be determined from the difference in the measured widths the parameters used in simulations are chosen to match of the PL profiles since the effect of the PSF convolu- those of our experiments: we consider a square nanopar- tion cancels out. The steady-state intensity PL profile ticle grid of lattice constant 10 nm; a laser source that is was measured on the close-packed and sparse samples an Airy profile of full width half-maximum (FWHM) 240 repeating the measurement in multiple locations on the nm; a laser source intensity low enough that no exciton- same sample and across several samples made from the exciton interactions occur; a point-spread function for same solution of PNCs. For each measurement the PL received light that is an Airy profile of FWHM 270 nm; intensity profile was fitted with a Gaussian function and excitons of lifetime ≈ 1 ns; and an exciton hopping rate the profile width was extracted as the variance of the such that their diffusion constant, on a pristine lattice, is 2 Gaussian fit. The two width distributions are shown in ≈ 0.5 cm /s. By contrast, in Section S4 and Section S5, Supplemental Fig. S2 for the sparse and close-packed in which we describe in detail the simulation methods films; their mean values 〈σ2 2 used, we use a variety of parameters, chosen for conve-close−packed〉 and 〈σsparse〉 were used to calculate the average exciton diffusion length for nience or to make contact with results described in the our system according to literature. For instance, we sometimes approximate the √ laser source to be a Gaussian function, to illustrate dif- 〈 〉 〈 〉 − 〈 〉 ferences with the Airy function, or we vary the excitonLdiff = σ2 2close−packed σsparse , (S2) hopping rate or laser beam intensity in order to illustrate important trends that inform our understanding of the and it was found to be 200 nm. processes under study. We have done this for complete- ness and to provide the detail required to replicate the results described here. S3. MODELING EXCITON PROCESSES To summarize, Sections S4 and S5 detail the meth- ods used and make contact with results given in the lit- This supplement describes in detail the methods we erature; Section S6 contains the results specific to the used to model exciton hopping in nanoparticles. We present experiments. 4 4 (a) (a)(b)100 100 100 100source source (b) (c) source source 3 (c) 0 c(r), analytic cc(/rc)0, analytic (10) c(r), analytic cc(/rc)0, analytic (1)10 c(r), numeric cc(/rc)0, numeric 10(1) c(r), numeric cc(/rc)0, numeric (1) 100 c/c01(0a0) (1) (1) c/c 1.4 (1) 1.40 J = Y, h = Z 0 (0b) (12).4 (1) J = Y, h = Z (2) 2 1.4 2 J0= Y, h = Z (2) (1c)(r) 2 J0= Y, h = Z (2) (1c)(r)10 1 10 2 0 sim sim 10  s < 102 10 (3)10 10w 16 ln(2) 10  s < 102 (3) 2 2 w 161 c 11 Gw 10  s < 10 c (31) Gww 16 ln(2) 10  s < 10 (3) w 16 ln(2)ln(2)10 10c 10 Aw̃ (2) c J h c 10 (4) 10−1 Aw̃ (2)c (2) J h c (4) (2)c J h (4) J h (4) 0 0 (2)bb rb rr (15).12.2 bb rb rr (15).12.2 (2) 2 sim c0 1f0( c 10 10 2 r)4 sim 104 c0 rr f10(0r) 4 40.5 bbr2 rb 2 1(50) −2 10 bcbr02 rb 2 rr (5) ċ = D c c µc + (x) (6) ċ = D c c µc + (x) (6)1/2 1/20  102 2 20 ċ = Dr c c2 µc + (x) (62) ċ = Dr2c c2 µc + (x) (6)10 s < 10 (3) 10 s < 11002 − (3)2 u = 2 (7) 3 u = 2 (7) 3 10  s = 2 2 (7) 2<310 10 (3)s <1u010 s < 11003 (3) 10 (3)s <u =1202 (7) (3)10 10 t/t0 = 10 (8) t/t 30 = 10 (8)10 6 3 310 6 t/t0 = 10 10 6(8) 1−4 10 6 t/t0 = 10 (8) 1 102 100 102 104 1031002 0 2 4 3 1 1J h 8 10 (41)0 10 kB− −4 0 4 8 J T 10 k T 8 h10 (9) 2 0 2 B 44 0 −8104 −4281(04)0 0 104 8 21804 4(09) 2 2 1 0 1 k T 2 2 1 0 1 k T2 10 10 0 102 104 3 1.5 .5 B 3 (9) 3 101.5 10 1.51B 0 3 10 (9) 41028 100 102 104 0 J h 0rr ((µµrmm))J(4) h J rh(µm) (4) rr ((µrµm m))0J(4) h r (µm) (4) 0 0 0 bb rFbIG. S3:rrThe Gaussian function G(w5(r)) andbtbhe Airyfurnbction A ⇣w̃(rrr) plotted on lin⌘ear-linear(5(a)) and linear-log (b) scales ⇣ ⌘ FIG. 2: Numerical solutions to Eq. F(BIG1(w). =2fo0:.r2 µamN)r.uaBmny gseeerttiicnogafw̃l s≈soo0ul.7ur2ct8eiwonwinestceatnnoasriErtainqe(g.exsf)(orB=t0h1e)(tw0ifnoeoxfrudpniacmtiorneasnrnt2sog/iho(ea2nvewoleef2qsu)saol uwinrdctihtessa)itnh(st1ahel)fontwhseiitrtimeh(axsax)itm=ut0mhve(a 2 20ilnuseecx,dapilmeenrs/io(2nwles)s units) (s1h)ow that the scale r2 and2asseen in panel (a) the two functions indeed look similar on that scale. However, panel (b) shows how different the tails ofand shape of profiles changeċa=s Dwe mcoavnedtfhrcesohtmwaopftueµnhcecotiof+rlipnreoa 2 2bb rb nrs are(.fixrTlhe)tesosectthahilasenbmngba(toe6tċen)ra=ltsiont(Dweh5eare)berxmpreecroigmviembb entfsrercd.oesmWcribeteµhdcecino+rntlhrisnisiedpaa(epxrer)t,aobeGc(tah5uas)euebstnhbs(eoi6apn)onlinitns-s(oep5aure)bradcrefeungwimithe.wWidethconsider a G(5a)ussian source with widthrb ctionrrof our parameter w = 0.4 µm. (a) The maxipmauromapmticesxitsecairntoAwniry=cfuon0nct.ci4oenn.µTtmrhais.tfui(onacnt)iocnThishabenotghmetshaexdsioemuprcueemnofdraednxiactieiotnou‘nfpeltoc’ nboyntcheensu0 utbrpstaroatntieo-bmnouocndvhinananngogpefarsrtoicdmlees,ptaehndnedisletihneceearupon 0 upon moving from the linear function convolved with the resulting ex2citon profile,2via Eq. (S1x8),(tµomfor)m the observed profile. W2h(e2n)computi2ng the tailsto the nonlinear regime. (b) The prrofi2lteowthidethn2sontleind x of(µm) (2) ċ = D c steadyc-state prµoefiaclterso+iwrteiasgriimċdm(pxs=oer)tt.ahDn(tebnr)sotċqTtuo=chaapre(peD6rpor)xroriomcofiat2tleecothfweµtpihdocietnt+ch-ss2sporuteaec(dnexfµdu)pncctrtio+ownfilaesrċad((Gtxs=(hau6)tesh)sDioeanur:stqseeuercSadurp(oep6tler)tmoeceodnttallioFnfiegµt.)hSc7ien.+stohuec(ex)profile (t(h6e)outer dotted line) in the large-0 limit. (c) Profiles’ full width altarhugael=-fm20alximimitu.m(cv)aPlureofi(flewsh’ (mf7u))llcwhaidntgheaatcchR ⌘ 2 u oarl=df imn2galxyi.mEuqmuavtailounep(afwrahm(7e))tecrhsa:ntgheealcecnogrtdhisncgallye. Equation parameters: the lengthscale parameter combination of Eq. (B1) is paramµewter/SD4c.om=MbE8iAn,Nain-tFidIoEincLaDotfiEnQEgUqAt.hT(aIBOt1Nt)Shies sRou⌘rceµw2aInid/d/IttDihme=isscalg8esr,e[i3an].tdeiTrchatethitanerngmt(ihn3e)aµtedxtehscciertibosenos utdhreic↵eseulwfsI-ii/odInth is greater than th0 0 (3e) exciton di↵usion length (and so the low-power profiles laernegtohnl(yansldigshotlythberlooawd-eprotwhearnptrhoefilse=osu2arrce)o.nTldyehstserluicpgtihoowtnlyeofrbepxrcaoitraoand(sm7e. )reTttheheratiensrmtPhine=ρs=do6eus42crrcibe0e),s.pwaThirhicehpower parameter is P = 640, whichu must be smaller than unity to be in thme ulisntebaA=re. rs32Eemgffeaimcltlievre. dtehsacrniptuionnitoyf extcoitobne(b7ie)nhatvhioer linea=annihilation of (bosonic) particles. A suimilar term (plus (7) t/t0 =u 10 (8) t/t0 = u p1r 3h0igrh2eegr-iomrdeer. nonlinearities) w(8ou)ld a(ls7o)be present in apn The creation, hopping, and recombination ofwlase⌘r- w eff2ec⇡tive0d.4es5crµipmtion if particles a(r4e)instead fewrmo⌘nicw(i.e.2 ⇡ 0.45 µm (4) induced excitons in nanoparticles results from the quan- if only one exciton per nanoparticle is permitted). We3 tum mechanics of light-matter interactions. In this sec- address this case in Section S5. 3 Eq. (B12), except near the edge ofEtqt/kh. TttiBeo(nBs=wi1em2a1p)u0p,lr3aoextixmiocaneteptbtheosnexeparorcestsaehrey tea/drttg0ief=aocft1s0(8) tt./k Ts by c2onsiderin2g thhteBe sim (8) t/t = 10 (8) T1hu0el3taertmioΦn( ⇣x)bdoes2xcribes t2hae⌘rinytena2s(rit8ty0i)offathcet2lsa.ser beam ⇣ ⌘ 0 (9) 0 (9) 2 2 where artifacts associated with itswphesertraitoeisdaicrtibfaocutnsdasrsieosciat Dr c c µc =tics of classical bosonic parteicdles.NwCoiottnhseideitrthstahetpdeirfr-eisocdaaitlcipnobsgit0oioseunpxnxpad=cae(rxi,areyn)s/do(n2wtthhee)tDwcoNr-odoinmtcceenesntiohtnrcaaaltstuibroseµntsrcact=e. exp fiaelilndg0spacearn/d(2wthe) concentration field fusion, creation, self-destruction, and pair annihilation of We shall consider profiles with radial symmetry in the are apparent. The dotted squareasrhecolaawsspiscpalatbhroseonntisc.cpaarltTeiclheosenondaolattttieincde (tsesheqeeuk.gam. rRTaeefn. sn[2h]e)o.rwdspelastncher,eiabnsdecdwaeliewnillowSnreitcetΦi(oxin) =BtΦh01fe(krc/mowT)na.nfiHnremreersΦ0dtieshsactribed in Section B 1 confirms thatB B Writing down a master equation for this set of processes, proportional to the las(e9r )power output; f(r/w) is a func-which we compare our simulationws hwicikhthBTweexpceormimpeanrtealour Esiqm.u(lBat1i2o)nsdewpeiktnh (9)taking the continuum steady-state limit,(9an)dRigno⌘ring B dtTisoenxoc√nponletyariinuminpgeotnhntealtalhserebseaiEnmgqw(l.5ied)(thBpap1ar2raa)mmedteeertpewer;nacndodsmo-nly upon the sing(l5e) parameter com- data, which is well away from the simdautfllaua,cttuiwoathninigcbho 2 2 2 2 noixissebtweormeulsn,lwdae-wgeaty fbrionmattiohne simulµatwio rn/≡Db:oxx+byouisntdhe- radialbcoinoraditn(ia9ote)n. WRe ⌘shalµl cwon-/D: p side⇣r cases in which f i⌘s an Airy function orpits Gaussian ⇣ ⌘ D∇2c(x)− µc(x)− ρc(x)2 = −Φ(cx)=. (S3/) exappproximra2t/io(n2.(wThe)2A)iry functio(n6)is0 c = 0/ exp r2/(2(w)2) (6) Here c(x) is the concentration (number per unit area) ( )22J1(r/w) of particles (excitons) at spatial location x = (x, y) on Aw(r) ≡ , (S4) r/w a two-dimensiZonal substrate. We shall regard (S3) as an effective steady1-state description of the optic⇣al sipgnal ⌘ w( ) (7) w( ) (7)0 0 w0here J Z1 is t1he first-order Bess0el function o⇣f thepfirst k⌘ind.0 0 prod(urced)2b/y2 laser-induced nanopartic2lec (r ) = e d⇠0 ⇠0e(⇠ ) /2excitons; to do 2 2Kc0(r0⇠)0=Re(rTIh)e p/a20ram0 eter wso we make the additional assumption that p0hotons are 0 (r ⇠ ) . d⇠q0ua⇠n0teifies(t⇠he)w/id2tKh of the⇠l0aser beam. 0 0The Gaussian function is 0 R(BI102()r ⇠ ) . (B12) emitted by isotr0opic one-body exciton decay, i.e. that 0 ( ) the optical signal at position x is proportional to c(x). r2 This description is approximate in several respects, as Gw(r) ≡ exp − . (S5)2w2 we shall describe, but several features of its solution, and in particular the approximate shape of the exciton pro- The functions Gw(r) and Aw̃(r) have equal widths at half The asymptotic decay of c(r) far frTofimlhe eprtoahdsueycemsdo, puprtroocvitedieccinadsnieghctaiyntootfhec4w(.ror)kiInfgnasrtoeffrropourmolattthhieoeirnmsoabxuiemrtucwmeevcaealunne wlihnenew̃ar≈ 04a.7.n2d8wI,nnbouttnedrliiffpneoreslauabrt-ion between linear and nonlinear experiments. stantially in their tails; see Supplemental Fig. S3. In this be obtained by evaluating the sourbce-forTbeheteatrienarmdediinal(blSy3) tsehyvatmaclou-uaplteisntogDtdheescrsiboesutrhceed-iff-reesurpapldemiaenlltryweegsoiyftmeneu-se the Gaussian profile for the pur- regimes metric version of Eq. (B1), whichmienftursiitocnwoovfeerdxsciiitmoonnes.nosTifhoisnEissqa.n (approximation: exciton poses of illustration. When comparing with experimental 00 0 hopping is00generally0 sub-diffusive Bon1)sm, alwl lhenigcthhscailnes twdaota,dhoimweveenr,swieounsse the Airy function. reads r2c + rc (µ/D)r2c = 0. rUeapdosnra2cre+scarlcingo(fµ/D)r2Wc h=en0.thUe pnoonlianeraersctaelrinmg iosfpresentWinheEnqt.h(eBn1o)ntlhineenar term is present in Eq. (B1) then r this becomes the modified Besserl ethqiusatbioecnom(oefs otrhdeermodtifiheedsBhaepsseeloefqtuhaetioenxc(itoofnorpdreorfile cthaengshesapwe itohf stohuerceexciton profile changes with source zero); its solution, boupnded at infinizteypr,oi)s;pitrsopsolrutitoionna,l btoupndepdowaetrinfi0n.itypC,oinsspidroepr oartGioanuaslstiaon sopuorcweerofw0.idtChonws,idie.er. a Gaussian source of width w, i.e. K0(r) / exp(r)/ r, with ⌘K0(µ/rD) ./Vexiepw(edro)n/ r,w(rit)h=⌘0Gw(µr/)D(s.eeV(ieBw3e)d). oIn thel(inre)a=r reg0Gimwe(,rw) h(seene (B3)). In the linear regime, when a linear-log plot the tails of the proafilieneaarer-ltohgerpelfotrethael-tailsthoef tphaerapmroefiteler caorme bthinearetifonrePal⌘- ⇢t0hwe4/pDar2amisestmeracllo,mthbeination P ⌘ ⇢ w40 /D2 is small, the most straight, with gradient (seemFosigt.s1tr)a, ifgrhotm, wihthichgradiemnatximu(mseexFcigto. n1)c,ofnrcoemntwrahtiicohn, whmicahxoimccuumrseaxtctithoenocroi-ncentration, which occurs at the ori- in principle the parameter combinatiinonprµin/cDipcleanthbeepraeraadmetegricno,misbci(n0a)ti/onµ0/(DseecaEnqb. e(Bre1a2d)). Wghinen, is 0c(i0s)la/rge,0 t(hsene Eq. (B12)). When 0 is large, then o↵. (In general, the linear-log ploot↵s.er(vIens gtoenaercacle,ntuh-e lince2ar-locg apnldotEsqe.rv(Bes1)toreadcucceenstuto- c2 c and Eq. (B1) reduces to ate features in the tails of profiles;atseefeFaitgu.re4s).inInthoeutrails of profiles; see Fig. 4). In our 2 experiments we found however thaetxspuecrhimdein↵tussiwve ftoauilnsd however that such di↵⇢ucsiv=eta0iGlsw(r), (B13)⇢c 2 = 0Gw(r), (B13) were obscured by the fact that the wAeirey ofubnscutiroend sboyurtchee fact that the Airy 1f/u2nction source 1/2 has considerable width beyond its firhsatsmcoinismiduemra,balendwbidyth beaynodndsoitcs(0fi)rs/t m0inim. uTmhu, sa,nads bshyownaindFsigo.c2((0a))/, plo0ttin. gThus, as shown in Fig. 2(a), plotting the fact that observed profiles werethceonfavcotlvtehdatwoitbhsetrhveed prco(fi0)le/sw0 eforer acosnevrioelsveodf nwuimthertihcael expce(r0i)m/en0tsfocraarrsiedrieosuotf numerical experiments carried out point-spread function of our optics;psoeientS-escptrieoandBfu5n.ction of oautrdoi↵petircesn;tseseouSreccetiionnteBns5i.ties 0 inadticdait↵eesretnhte soonusrecte oifntensities 0 indicates the onset of the nonlinear regime via the changethoef gnroandlienneta.r regime via the change of gradient. As we move from the linear to theAnsonwleinmeaorverefgriomme,the linear to the nonlinear regime, We will use the term ‘di↵usive broadWeneiwngil’l tuosedetshceritbeerm ‘dtih↵euswiviedtbhrooafdtehneinegx’ ctiotodnespcroibfiele chtahnegews.idtIhn othf ethlieneaxrciton profile changes. In the linear the type of broadening of the excittohneptryopfieleofreblarotiavdeentoing orfetghime extchiteonwipdrtohfileofretlhaetiveexctioton prreogfiimlee isthdeetweirdmthineodf the exciton profile is determined the source seen in Fig. 1. the source seen in Fig. 1.by the source width w and the ebxycittohne dsoecuaryce lewnigdtthh w and the exciton decay length c(0)/0 c(0)/0 ffwwhhmm ((µµmm)) ffwwhhmm ((µµmm)) 4 0 100 10 I(ntroducin)g a rescaled concentration field via c ≡ source source Φ w2/D ĉ brings (S7) to the form c(r), analytic cc(/rc), analytic (1) 0 0 c(r), numeric cc(/rc)0, numeric (1) c/c0 (1) J = Y, h = Z (2) − − 2 2 22 2 J = Y, h = Z (2) ∇̂ ĉ− P ĉ −R ĉ = −f(r̂). (S8)10 10 10  s < 102 (3) c 10  s < 10 2 (3) (2) J h (4) Here we have introduced the parameter combinations c J h (4)0 − − ✏bb ✏rb ✏4 4 rr (5)10 10 ✏ ✏ ✏ (5) 4rrċ = Dbbr2crb⇢c2 µc + (x) (6) ρΦ0w  2 ċ = Dr2c ⇢c2 µc + (x) (6) P ≡ (S9)10 s < 10 (3) D2 ✏u = 2 (7) ✏u = 2 (7) − − t/t 3 0 = 10 (8) and 10 6 10 6 t/t0 = 10 3 (8) J h −8 −4 (4) k− B T 0 4 8 kB✏T 8−4 (09)(9) 4 8 µw2 ✏ R2 ≡ . (S10) r (µm) r (µm) D ✏bb ✏rb ⇣ ⌘ 4 2FIG. S4: ✏rNr umerical (dashed (r5e)d) [Eq. (S11)] and semi- The combination P ≡ ρΦ0w /D is the nonlinearity or (x) = exp r2/(2w2 analytic (blue) [Eq. (S15)] solutions to the lineari0zed ver- powe)r paramet(e1r). When P is large the beam is powerful ċ = Dr2csi 2on⇢ocf Eq. µ(Sc3+) agr(exe,)providin(g6a) check on our numerical in the sense that the term nonlinear in c is important. procedure. The parameter combinations are P = 0 and When P is small we are in the linear regime, where the R2 ≡ µw2/D = 0.32; see Section S4 B. Exciton conxce(nµtrma-) term in ρ in E(q2.)(S3) may be ignored. The experiments tion curves have been normalized by their value at the origin, c(0✏ = 2 (7) reported in this paper are performed in the linear regime. )u≡ c0. The (normalized) model source is shown grey; for √ simplicity it is Gaussian with width parameter w = 0.I4/Im. The paramet(e3r)R ≡ µw2/D is a ratio of lengthscales.µ 0 The dotted box shows the region relevant to typical experi- `beam ≡ w is the√lengthscale associated with the beam mental mea3surements, in which the measured intensitypspans profile. `hop ≡ D/µ is the lengthscale on which ant/t0 = 10 (8)two or three orders of magnitude. w0 ⌘ w 2 ⇡ 0.45eµxmciton that li(v4e)s for characteristic time µ−1 will hop before it dies. Thus when R = `beam/`hop is large, the k T b⇣eam diamet⌘er is much greater than the distance overB B. Scaling(a9Dn)arly2scis ⇢c2 µc = expwhicrh2/a(2twyp2)ical exciton will diffuse. When R is small, ✏ 0 the beam diameter is much less than the exciton hopping A scaling analysis shows that (S3) is governed by two distance. For the experiments reported in the main text parameter combinations. Eq. (S3) can be written the lengths `be(a5m) and `hop are comparable. ( ∇2 − ρ ) ( 2 − µ ) Φ ( r )0 ⇣ ⌘ c c c = − f p . (S6) D D Dc = w 0/⇢ exp r2/(2(w0)2) (6) C. Numerical solution of Eq. (S3) If we choose to measure lengths in units of the beam width w, and introduce coordinates (x̂, ŷ) ≡ w−1(wx,(y)), In general, E(q7.) (S3) must be solved numerically. To do then (S6) becomes so we simulated numerically the time-dependent version ( ) ( ) ρw2 µw2 Φ w2 of the equation on a 2D periodic grid, using a forward-∇̂2c− c2 − c = − 0 f (r̂) . (S7) different method and a five-point Laplacian stencil: D D D [ ] cx,y(t+ ∆t) = cx,y(t) + ∆t −ρcx,y(t)2 − µc −2x,y(t) + Φ0f(r∆x /w) + D∆ ∆2t x [cx+1,y(t) + cx−1,y(t) + cx,y+1(t) + cx,y−1(t)− 4cx,y(t)] . (S11) Here cx,y(t) is the exciton concentration at grid point ulated until the steady state was reached. We confirmed (x, y) at time t. We measured spatial distances in mi- the accuracy of our numerics by comparing the steady- crons, and took ∆x = 50 (i.e. we have 50 lattice spac- state solution of Eq. (S11) to a semi-analytic solution of ings to the micron). We set the timestep ∆t = 10−5, Eq. (S3) in a certain limit (Section S4 D): see Supplemen- which was small enough to maintain numerical stability. tal Fig. S4. We usually began simulations with a spatial profile c(r) equal to that of the source term Φ0f(r∆ −2 x /w), and sim- 5 (a) (b)100 100source source (c) 0 c(r), analytic cc(/rc)0, analytic (1)10 c(r), numeric cc(/rc)0, numeric (1) 100 c/c0 (1) (1) 1.40 J = Y, h = Z ( 12).4− −2 J0= Y, h = Z (2) (1c)(r)10 2 10 sim √10  s < 102 (3) √ − − 10  s < 102 (3) w 1w6 ln1(62)ln(2)c 110 1 c 10 (2) J h (4) c0 (2) c J h (4) c − − ✏4 4 bcb ✏rb ✏rr (15).12.2 (2) −2 sim 100 10 ✏ċ = Dbbr02 ✏c rb 2 ✏rr (5)10 − ⇢c µc + (x) (6)1/2Φ  −20 ċ = Dr2c ⇢c2 µc + (x) (6)10 s < 11002 (3) ✏u = 2 (7) 10  s < 102 10 (3)s <✏u =1202 (7)3 (3) 10−3 − t/t = 10 (8)− 10 6 − 0 6 2 0 2 4 t/t 3 0 = 10 (8) 1 10 10 10 10 J h10 −3 10 1 −−8 −−4 (4) kBT− −2 0 2 40 4 8 (9)kB✏T 8−4 0 104 10 10 103 1.5 0 1.5 ✏ 3 (9)Φ 10 −28 100 102 104 0 J h rr ((µµmm))J(4) h r (µm) (4) Φ0 Φ0 ✏bb ✏rb ✏rr (5) ⇣ ⌘ FIG. S5: Numerical solutions to Eq. (S3) for a range of source intensitiesΦ(x0) s=how0 etxhpatthr2e/s(c2awl2e)and shap(1e)of profiles change as we move from the linear tċo=thDe✏rno2ncline✏⇢arc2regiµmc✏e+.rrFo(rxt)he pu✏rbpb(o6s)es o(✏f5ri)bllustra✏triron we consider a G(5a)ussian source with widthbb rb parameter w = 0.4 µm. (a) The maximum exciton concentration changes dependence upon Φ0 upon moving from the linear to the nonlinear re3gim.5e. (b)ċT=heDprrofi2lec wid⇢tch2s tenµdcto+warċd 2 2 x (µm) (2) (xs=)thDersqucar(e6r)⇢ocotof µthce+souc(ex)profile (t(h6e)outer dotted line) in the large-Φ0 limit. (c) The full widths at ha✏luf-m=a2ximum value (FWH(M7)) of the profiles change accordingly. Equation parameters: the lenrgth/scawle parameter combination (S10) is R ≡ µw2/D = 8, indicating that theI/sIource width is great(3e)r than the exciton0 diffusion0length (and s3o low-power profiles are only slightly broader tha✏n t=he2source). The power p(a7ra)meter (S9) is P = 64Φ0,u which must be much smaller than unitty/tto✏=ube=10in32 the linear regime(8. (7)2.5 0 ) 0 pw ⌘ w 2 ⇡ 0.45 µm (4) 2 k T t/t = 10 3 0 ⇣ (8) ⌘ − t/t B 0 = 10 3 (t9iv)er2t(o8)solv2ethe equation in t2he ab2sence of the term in2 D c ⇢c µc = exp r /(2w )✏ 0 ρ, i.e. in the linear lim0 it. The linear limit is appropriate10 10 102 4wh1e0n tkheTsource intensity Φ0 and the resulting maximumB 3.5 kBT exciton concentration is small. (9) (5)Φ ✏ ✏ 0 In (29D) the solution of Equation (S3) can be obtainedrF by the mepthod of G⇣r ⌘een2’s fun0c2tions, and isr0/w 3 (1) c = 0/⇢ exp∫ r /(2∞ ∫(w ) )Φ ∞ ( (6) √ )w R = 0.56 0c(x, y) = dy′ dx′Φ (x′)2 + (y′)2 R = 1.13 2πDw 2.5 R (= 2.81 √ (−)∞ −∞ (7) ) c × K λ (x− x′)20 + (y − y′)2 . (S12) (2R) = 22.5 √ c0 2 Here λ ≡ µ/D is the reciprocal of the characteristic−2 0 2 4 lengthscale for exciton diffusion, and K0 is the zeroth or-10 10 10 10 der modified Bessel function of the second kind. Writing  ′ ′10 s < 102 Φ (3) u ≡ x−x and v ≡ y− y , and passing to plane polar co-0 ordinates via the transformations (u, v) = ξ(cos θ′, sin θ′) and (x, y) = r(cos θ, sin θ) gives FIG. S6: The full width at half maximum rF of a series ∫ ∞ of normalized exciton profiles, plotted relative to the source Φ0c(r, θ) = ξdξK (λξ) (S13) J 0width w h(= 0.15 µm), for a range of s(o4ur)ce intensities Φ0. D 0 Distinct curves correspond to distinct choices of exciton dif- ∫ 2π dθ′ (√ ) fusion constant D; the resulting dimensionless parameters × Φ r2 + ξ2 + 2rξ cos(θ − θ′) . R ≡ w2µ/D are shown. The dot√ted lines left and right indi- 0 2π cate the width of the source and 2 times that value, respec- ✏ ✏ The source Φ in our experiments is an Airy function, butbb tivrebly. The ✏smrraller is R the broader is (th5e)exciton profile in for the purposes of checking our numerics we replace it the linear (small-Φ0) regime. At large Φ0, in the√strongly non- by a Gaussian function. In this case (S13) can be reduced r2 linear r2egime, all profile widths tend to a value 2 times that 2 2 2ċ = D c of t⇢hce sourceµ. cW+hether(xno)nlinear broa(d6en)ing or narrowing to a single integral. Setting Φ(r) = Φ e−(x +y )/(2w )0 we occurs depends therefore on the value of R. have ∫ Φ ∞0 2 c(r, θ) = e−r /(2w 2) −ξ2 2dξ ξe /(2w )K0 (λξ) D 0 ✏D. =Se2mi-analytic solution of the(li7n)earized version ∫u 2π× dθ ′ −2 ′ of Eq. (S3) erξw cos(θ−θ ). (S14) 0 2π As a benchmark for our numerics (Section S4 C) and to ∫The inner integral can be carried out√using the formula2π t/gtain=insi1g0ht3into the properties of Eq(.8(S)3), it is instruc- dθ exp (α cos θ + β sin θ) = 2πI 2 0( α + β2), where 0 0 kBT (9) ✏ c(0)/Φ0 ffwwhhmm ((µµmm)) 6 I0 is the zeroth-order modified Bessel function of the first source intensity increases, the profiles broaden. From kind. This result allows us to write (S14) in the mani- Eq. (S17) we see that in the limit of large intensity, the festly θ-independent form spatial profile has the shape c(r) ∝ G (r)1/2√w = G √ w 2(r), Φ which is a Gaussian with a width 2 times that of the0 −r2c(r) = e /(2w 2) (S15) source. As shown in Supplemental Fig. S5(b), the outer ∫D∞ ( ) blue profiles indeed tend to this shape (shown by the × 2dξ ξe−ξ /(2w2) rξK (λξ) I , outer dotted gray line). Plotting profiles’ full width at0 0 w20 half maximum value in Supplemental Fig. S5(c), we see that they tend to the expected width in the limit of large which we can evaluate numerically. In Supplemental Φ . Fig. S4 we show that, in the relevant parameter regime, 0 Nonli√near narrowing can also be seen, when the decaythe steady-state limit of the numerical procedure (S11) length D/µ is large compared with the source width agrees with the semi-analytic solution (S15), except near w. In this case the dimensionless parameter R < 1, and the edge of the simulation box (where artifacts associ- the profile width in the linear regime can be broader than ated with periodic boundaries are apparent). The dotted the profile width in the strongly nonlinear regime, which square shows the scale on which we typically compare our √ tends to a value 2 times that of the source; see Supple- simulations with experimental data, which is well away mental Fig. S6. Note that these width comparisons refer from such artifacts. to shapes of normalized profiles, those scaled by their val- Note that rescaling space and the concentration field ues c(0) at the origin: profiles generated at large values in the manner described in Section S4 B confirms that of Φ are generally broader than those generated at small (S15) dep√ends only upon the single parameter combina- 0 R ≡ Φ0, in the sense that greater exciton density is generatedtion µw2/D: away from the origin. ∫ ∞ ( ) ( ) −r̂2ĉ(r̂) = e /2 2 dξ̂ ξ̂e−ξ̂ /2K0 ξ̂R I0 r̂ξ̂ . (S16) 0 F. Observed profiles are a convolution of the exciton profiles and the optics’ point-spread function E. Interpolation between linear and nonlinear regimes Intensity profiles I(r) observed in experiment are not the exciton profiles c(r) themselves, but are instead the convolution of the exciton profile and the point-spread When the nonlinear term is present in Eq. (S3), the function S(r) of the optics [3]: shape of the exciton profile changes with source power Φ0. Consider a Gaussian source of width w, Φ(r) = I(r) = (c ? S) (r) Φ0Gw(r) [see Eq. (S5)]. When Φ0 is large, such that the ∫ ′ ′ parameter combination P ≡ ρΦ w4/D2  1, Eq. (S3) = dx dy c(x′, y′)S(x− x′, y − y′). (S18)0 can be approximated near its core as The optics plays a dual role in our experiments: it gives ρc2 = Φ0Gw(r), (S17) rise to an Airy-function source profile Φ(r) of FWHM 240 nm on the substrate, and it gives rise to the point- 1/2 from which we get c(0) ∝ Φ0 . Thus, as shown in Sup- spread function S(r) for received light, an Airy func- plemental Fig. S5(a), plotting c(0)/Φ0 for a series of nu- tion of FWHM 270 nm, that appears in (S18). Airy merical experiments carried out at different source inten- functions are sometimes approximated as Gaussian func- sities Φ0 indicates the onset of the nonlinear regime. The tions, because the cores of the two profile types have value of the gradient of the function in the high-power similar shapes [Supplemental Fig. S3(a)]. However, the regime depends on the types of nonlinearities present tails of the two functions differ markedly [Supplemental (e.g. it differs for bosonic and fermionic excitations), Fig. S3(b)]. This difference is significant when computing but the qualitative change can be used to determine the steady-state profiles, as shown in Supplemental Fig. S7, extent of the linear regime. particularly as regards inflation of the tail of the profile. As we move from the linear to the nonlinear regime, the We carried out the convolution (S18) numerically. width of the exciton profile changes. In the linear regime The trends described previously, such as the broaden- the width of the exciton profile is determined (for the ing of profiles at large source power, can be seen in I(r) model diffusio√n equation) by the source width w and the much as in c(r), with quantitative differences: see e.g. decay length D/µ. In Supplemental Fig. S5(b) we show Supplemental Fig. S8. a series of normalized exciton profiles (blue) that result from (S3), for the range of choices of source power Φ0 shown in panel (a). In (b), the inner profile corresponds S5. MICROSCOPIC SIMULATIONS to the case of lowest power, and is slightly larger than the source (the inner dotted gray line) by virtue of the To complement the approach of Section S4 we simu- diffusive broadening seen in Supplemental Fig. S4. As lated space-dependent exciton dynamics using discrete- 7 (a) Gaussia(na ()sao)uGraGcueasussiasina ns osouurcrece (b) A(b(ibr)y) AfuAirnyirc fytu infoucnnti costnoio usnor ucsorecuerce 100 100100 100 1000 source sosouurrccee sosuourcrceseource c(r) cc(r(r)) c(cr()r) c(r) (c(c⋆⋆GGσ))((rr)) (c ⋆ A(cσ̃)⋆(r(c ⋆ Gσ)(r) σ (c ⋆ Aσ̃)(rA ) σ̃)(r) (c ⋆ A )(r) ) (c ⋆ A )(r) (c ⋆ A σ̃ σ̃)(r)σ̃ −1 −1 10−1 10−101 110−1 100−1 −10−2 −2− 10 2 10 2 −2 10−210 10 −10 −3 10−33 −3 − 10 −3 −1.5 0 1.5 3 103 − −3 −1.5 0 1.5 310 −3 −1.5 0 1.5 103 3 −3 −1.5 0 1.5 3 − − r (µm)3 1.5 0 r (1µ.m5 ) 3 − − rr((µµmm)) 3 1.5 0 ⇣ ⌘ rr((µµmm)) 1.5 3 ⇣ ⌘ FIG. S7: Gaussian arn(dµAmiry)functions used fo(rxt)he=sourceexppro⇣filerΦ2/(r()2wan2d) t⌘he convorrl(u(1t(µ)iµomnm()S)18). (a)(xN)um=ericael xsoplutio⇣nr2t/o(2w2) ⌘ (1) Eq. (S3) for a Gaussian source Φ(r) ∝ Gσ(r) of width σ =0 0.4 µm (gr2ay dott2ed) gives an exciton profile c(r) (blue 0). Subsequent 2 2 convolution with a Gaussian (red dotted) ora(nxA) i=ry⇣fu0ncetxiopn (redr s/o⌘l(i2d)wpr)oduce distin(c1t)curves. (b)N(uxm)er=ical s0oeluxt⇣piontor /(2w )⌘ (1) Eq. (S3) for an Airy function source Φ(r) ∝ Aσ̃(r), withσ̃ =20.7σ (g2ray dotted), gives an exciton profile c(r) (blue) whose tails2 2are markedly different to the tailsof(cx(r)) =with a0Gexaupssian sxrou(/µrc(me2.)wSub)sequent convo(l1u)tio(2n)with an Airy(xfu)nc=tion g0ixveex(sµptmhe)solrid /(2w ) (2) (1) red line. x (µm) (2) x (µm) (2) (a) (b) x (µm) I/I0 (2) (3) I/I0 (3)A. Fermionic statistics x (µm) (2) 100 1.4 I/I0 (3) I/I0 (3) c(r) (c ⋆ A)(r) p p −1 1.2908510 w 0 ⌘ w 2 ⇡ 0.4W5eµcmonsider a two(4-d)imensional subwst0r⌘atewof 2na⇡no0p.a4r5-µm (4) 1.2 I0/I p ticles whose pos(i3ti)ons are fixed. In Secti0on S6Ip/wIe take (3)−2 Airy source 010 −1/2 w ⌘ w 2 ⇡ 0t0h.e45na⇣µnmoparticles t⌘o s(i4t )in a square arrawy, s⌘imwilar 2to⇡th0e.⇣45 µmΦ ⌘ (4)0 Dr2c ⇢c2 µc = exepxeprimern2ts/(r2ewpo2r)ted in thDerm2acinte⇢xct2; in µthcis=sectionewxpe0 r2/(2w2) 10−3 1 p 0also c⇣onsider trangu⌘lar arrays and disorderepd arrange- ⇣ ⌘ 10−2 100 102 104 10−0 2⌘100 2 4Drw2 w 10 10 0c ⇢c2 2⇡µc0=.45µmeexnpts. Wre2m/(o2dw(e4l2)e)xcitons Dasrcl2acssicwa⇢lcp⌘2artwicµlecs2,=a⇡ble0t.o4e5xpµmr2/(2w2) (4)Φ (c) 0 Φ 0 0 (d) 0 undergo various processes. Each nanoparticle ca (5) n be oc- (5) 100 100 ⇣cupied by an ⌘exciton A or be vacant ∅, i.e. we assume ⇣ ⌘ r2 2 ⇣ ferm2ionic ex2c⌘iton statistircs.2 Inthis case the normalized− D c ⇢− c µcp= 0exp r /(2w 2 ⇣ ⌘2 2 (5)10 1 0 10 1 0 ex2citon p0rofil )es broad(eD5n) at hcigh ⇢pocpwer eµvecn =inthe 0abe-xp r /(2w ) c⇣= 0⌘/⇢ exp sren/c(e2( 2 2 0 2 ⇣ o wf ⇣e)x)citon ⌘hop(6p)ing. Thec b=roaden0i/n⇢geixspdifferren/t(2(w ) ) (6) (x) = 0 exp r2p/(2w2) (1) (x) = exp r20 /(2w2) (1)10−2 10−2 in detail to t⌘hat of the bosonic statisptics considere⇣d in ⌘ c = 0/⇢ exp r 2 Sec/ti(o2n(w0 2 2 0 2S4). C) onsid I(r)/I(0) w(2)() I(r)/I(0) (5) er (e6x)citon creatcio=n with0r/a⇢teeΦxp(r),r /(2(w ) ) (6) (7)(2)− − w( ) (7) (5) 10 3 10 3 −3 −1.5 0 1.5 3 −3 −1.5 0 1.5 3 Φ(r) rr((µm) p I/I ∅0 r (µm)) ⇣ (3) ⌘I/I0 (3)−−−→ A, ⇣(S19) ⌘ ⇣ w( ) (7) p w() (7) c = 2 ⌘/⇢ exp r2/(2⇣(w0 2 2 0 22p 2 ) )2p⌘ (6) c = 0/⇢ exp r /(2(w ) ) (6) FIG. S8: Observed pro(fixl)e=s I0(erx)p vawrry0/⌘(i2nwwa)02s⇡im0.i4l5a(µr1m)fash(ixo)n=t(o40)exp wwrh0/e⌘(r2eww Φ) 2(⇡r)0.=45(µ1Φm) 0f(r) is(4t)he laser source as in Section S4, exciton profiles c(r), with quantitative differences. (a) Sim- and exciton self-destruction with rate µ,⇣ ⌘ ⇣ ⌘ ilar to Supplemental Fig. SD5r(ax2)(c,µmb)⇢uc2twµcit=han eA(2xpir)y-rf2u/(n2wct2i)oDn0 rx2 (µcm) (2)⇢c2 µc = 0 exp r2/(2w2) µ source of width 0.24 µm. (b) The full width at halfwm(ax)imum (7) A −→ ∅. w(()S20) (7) (FWHM) of exciton- and observIe/Id0 profiles behav(e3s) similarly, I/I (5) 0 (3) (5) but are numerically different. Profiles c(r) and I(r) are shown p The stochastic process defined by (S19) and (S20) is ain (c) and (d), respectivewl0y⌘, wove2r⇡lap0id.45oµnm G⇣aussia(4n) 0ref⌘ere0nce p p 2 2 w ⌘ w 2 ⇡t0w.4o5-µsmtate ⇣ dyn(a4)mic⌘s with steady-state solution curves. c = 0/⇢ exp r /(2(w ) ) (6) c = 0/⇢ exp r 2/(2(w0)2) (6) ⇣ ⌘ ⇣ ⌘ Dr2c ⇢c2 µc = 2 2 20 exp r /(2w ) Dr c ⇢c 2 µc = 0 exp r2/(2w 2) Φ(r) w( ) (7) w( ) c(r)(7=) , (S21) Φ(r) + µ (5) (5) p ⇣ ⌘ p ⇣where c(⌘r) is the density of excitons (A-particles) at po- c = 0/⇢ exp r2/(2(w0)2) (6) c = 0/⇢ exp sr2it/io(2 0n(wr)2.) We a(6s)sume that the process of destruction pro- time and continuous-time Monte Carlo algorithms. duces a photon, and so the time-averaged exciton density w() (7) w() (7) r (µm) ⇣ ⌘ (x) = exp r2/(2w20 ) (1) c(r)/c(0) (2) c(0)/Φ0 I/I0 (3) p w0 ⌘ w 2 ⇡ 0.45 µm (4) ⇣ ⌘ r (µDmr2) c ⇢c2 µc = exp r20 /(2w2) ⇣ ⌘ (x) = 0 exp r2/(2w2) ((51)) p ⇣ ⌘ c = 0/⇢ 2 Ie(xrp)/I(0r) /(2(w0)2) ((62)) fwhmr((µµmm)) ⇣ ⌘ (x) = exp r2/(2w2w( ) (7) 0 ) (1)I/I0 (3) p x (µm) (2) w0 ⌘ w 2 ⇡ 0.45 µm (4) ⇣ ⌘ I/I0 (3) Dr2c ⇢c2 µc = 20 exp r /(2w2) 0 pw ⌘ w 2 ⇡ 0.45 µm (4) (5) ⇣ ⌘ Dr2c ⇢c2 µc = 0 exp r2/(2w2) p ⇣ ⌘ c = 2 0 20/⇢ exp r /(2(w ) ) (6) (5) w( ⇣ ⌘ ) (7) pc = 0/⇢ exp r2/(2(w0)2) (6) w() (7) 0/µ = 10 2 (1) 0/µ = 10 2 0/µ = 1 (1) 8 (2) 20/µ = 10 0/µ = 1 (1) 0/µ = 10 2 (2) (3) 100 100 100 0/µ = 1 /µ = 102 (2) sign(E) (3) (4)0 10−1 10−1 10−1 /µ = 1020 (3) E(arb.) (5)sign(E) (4) 10−2 sign(E) 10 −2 10−2(4) E(arb.) ↵ = 0.01 (5) (6) 10−3 E(arb.) 10−3 (5) 10−3 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 r(µm) ↵r=(µ0m.)01 ⇢(a) ⇠r(eµmK)1I1(a) (6) (7) ↵ = 0.01 (6) FIG. S9: Numerically computed steady-state radial exciton profiles c(r) (cyan) and the exact solution (S21) (red dashed) for the 1 processes (S19) and (S20), together with the model Gaussi⇢a(na)so⇠ureceKp1rIo1fi(ale) (blue dotted). ShowKn1in⌘seKt/ahre the(t7w)o-dimensional (8) K I (a) images from which the ra⇢d(iaa)l⇠preofil1es1are computed. (7) f (9) K ⌘ K/h1 (8) K1 ⌘ K/h1 1 (8) 2400 101 for Gaussian Φ(r). We havehI@tfo/t@≈0i2πw Φ0/µ for small (10) Φ0/µ. Equations (S22) and (S23) are plotted in Supple- 350 f mfental (F9i)g. S10. We take the rate of self(-9d)estruction to 100 be µ = 0.5 ns−1 and the beah@mf/p@wowi er parameter to be (11) 300 h@f/@0i h@fΦ/@0=0i ((P10/)24.4) ns−1, where P is measure(d10i)n microwatts 250 (µW), which we estimate to be characteristic of our ex- − 10 −1 N ⌘ 3 −2 −1 0 1 2 −3 −2 −1 periments. These behaviors arer +usbeful diagnostics of the (12)10 10 10 10 10 10h@f/@wi10 10 10 100 101 102 h@fo/n@sweti o(f11n)onlinear behavimor,⌘an(bdarl)Φ l/oNw u(1s1)to verify that (13)0(µW) Φ0(µW) experiments reported in the main text are done in the linear regime. N ⌘ r + b (12) FIG. S10: FWHM (left) a⌘nd normalized intensity (righ 3 (14) Nt) ⌘ r + b (12) m (b r)/N (13) of exciton profiles c(r) produced by the stochastic processmes ⌘ (b r)/N (13) (S19) and (S20), for varying beam power Φ0. We take µ = −1 B. Exciton hopping is subdiffusive in the presence0.5 ns . Note that the FWHM continues to broaden with 20 k T (15)of energetic diBsorder source power, unlike the case of 3bosonic exciton statistics. (14) 3 (14) In the experiments reported40inkBtThe main text we believe (16) is proportional to the steady2-0sktBaTte photoluminescence in- that su(b15d)iffusive motion of excitons arises from vacancies tensity. 20 iknBTthe nanoparticle array. In this section(15w)e recall some We simulated these processes using continuous-time features of exciton subdiffusion brought about by another Monte Carlo [4] and a squ40akrBeT-lattice nanoparticle ar- mechan(1is6m) , energetic disorder, that has been quantified ray. Comparison with (S21) provides a simple bench- 40 ikn Tother studies [3]. Hopping on a rough e(n16e)rgy landscapeB mark against which to check the calculation of radially- leads in general to subdiffusive behavior at short times averaged profiles. As shown in Supplemental Fig. S9, the and diffusive behavior at long times [5, 6]. time- and radially-averaged profile c(r) is proportional To make contact with these results we carried out to the source profile Φ(r) at low power, and broadens as Monte Carlo simulations of an exciton moving between beam power is increased. For the purposes of illustration nanoparticles on a two-dimensional substrate; see Sup- we take the source Φ(r) = Φ0Gw(r) to be Gaussian with plemental Fig. S11. Simulation boxes had periodic full width at h√alf maximum intensity (FWHM) 240 nm, boundaries in both dimensions. We considered spatially i.e. w = (120/ 2 ln 2) nm. ordered substrates, in which nanoparticles were arranged For fermionic statistics the width of the profile grows as a close-packed lattice with inter-particle separation logarithmically with power at high power: we can solve a = 8 nm – see Supplemental Fig. S11(a) – and spa- the equation c(r ) = c(0)/2 to yield the FWHM, F ≡ tially disordered substrates, such as that shown in Sup-0 2r : plemental Fig. S11(b). These we generated by perform-0 √ ( ) ing short constant-volume Monte Carlo simulations of the Φ nanoparticles themselves, assuming they were hard discs0 F (Φ0) = 2w 2 ln + 2 . (S22) µ with radii drawn from a truncated Gaussian distribution peaked about 8 nm. We found (shown below) that at We can also calculate the total integrated intensity constant particle density the averaged exciton transport ∫ ( ) properties were not strongly affected by the presence of 2 µ+ Φ0 spatial disorder.Itot = dθ rdr c(r) = 2πw ln , (S23) µ Once the substrate was generated, we performed ex- FWHM(nm) I/I0 c(0)/Φ0 I/I0 I/I0 time 0, var = 0 time 0, var = 0 9 (a) (b) (c) 11 PP((xx)) 0044 66 88 1100 xx((nnmm)) FIG. S11: Examples of spatially ordered (a) and disordered (b) substrates used for illustrative exciton-hopping simulations. The black traces show trajectories taken by two simulated excitons. Nanoparticle colors indicate their energies; red shades and blue shades are high and low in energy, respectively. Panel (c) shows the distribution of inter-nanoparticle distances seen in panel (b). citon hopping simulations using a discrete-time Monte (corrected for periodic boundaries); averages 〈·〉 are taken Carlo algorithm. We selected at random a nanoparticle, over initial conditions, waiting times and (where appro- and created an exciton on that nanoparticle. We then se- priate) realizations of energetic and spatial disorder; and lected at random any neighbor (up to a cutoff distance) Nsteps is the number of Monte Carlo steps taken. Sim- of that nanoparticle, and proposed to move the exciton ple considerations indicate roughly the exciton diffusion to that nanoparticle. Following Ref. [7] we accepted this constant expected. Take the nanoparticle radius to be proposal with a probability designed to ensure that the a ∼ 10 nm. Assume the characteristic rate for an exciton exciton jump from i to j happens with rate to hop from nanoparticle to nanoparticle is τ−10 (R 6 0/a) , ( ) ( ) where τ0 ∼ 10 ns, and assume that the Förster radius R6 01 R0 R(i→ j) = min 1, e−β(Ej−Ei) . (S24) is of order 10 nm [3]. Then the (long)-time exciton diffu-τ 60 Rij sion constant is roughly D = 1 R0 a2 ≈ 10−4 cm22τ a /s.0 Here i and j are the nanoparticle identities; τ is the mean This scale of this result is consistent with the exciton dif-0 −4 2 exciton lifetime; R is the Förster radius; R is the dis- fusion constant of 3 × 10 cm /s reported in Ref. [3];0 ij tance between nanoparticles i and j; E and E are the the precise numerical value of this result is sensitive toi j −1 the ratio R /a to the sixth power, and upon insertion of bandgaps of nanoparticles i and j; and β ≡ (kBT ) . For 0different values (e.g. R = 12.5 nm and a = 8 nm) we spatially disordered substrates the combination R 00/Rij obtain the numbers shown in Supplemental Fig. S12(a). can be greater than or less than unity, and so it is con- venient to write (S24) as In that figure we show D from Eq. (S27), as a function ( ) of time, for four different values of energetic disorder (on6 1 R ( )→ min −β(E −E ) a spatially uniform lattice). A constant value indicatesR(i j) = min 1, e j i , (S25) τ R diffusive motion, which is reached at times that increaseij as the roughness / (kBT ) of the energy landscape in- 6 with Rmin ≡ minij Rij and τ ≡ τ0 (Rmin/R0) . With creases. For e.g. nanoparticles for which  ≈ kBT , we time measured in units of τ we accepted the move from estimate the diffusive approximation made in Eq. (S3) i to j with probability to be valid only on timescales exceeding about 100 ns ( ) ( ) [in general the D appearing in (S3) could be thought of6 → Rminp (i j) = min 1, e−β(Ej−Ei) , (S26) as a spatial and temporal average over the microscopicacc Rij behavior shown in Supplemental Fig. S12(a)]. which is ≤ 1. Otherwise, the proposed exciton move In Supplemental Fig. S12(b) we show the long-time was rejected. We considered a Gaussian distribution of diffusion constant obtained for particular values of sub- nanoparticle energy levels E with variance 2, P (E ) ∝ strate energetic disorder, normalized by the value for no( ) i i exp −E2/(22) . energetic disorder. For a spatially ordered lattice (greeni The exciton diffusion parameter is line), the fall-off of D with / (kBT ) is less rapid than for ( ) discrete Gaussian disorder on a 1D lattice (blue dotted〈 2〉 〈 2〉 2 6 line), D()/D(0) = exp(−β2 2 −1[∆x(t)] + [∆y(t)] a Rmin  )(1+erf(β/2)) [6]. This D(t) = · , (S27) 4N τ R makes physical sense, because energetic ‘traps’ caused bysteps 0 0 the proximity of nanoparticles with unusually high and where ∆x(t) and ∆y(t) are the time-dependent distances low energies are geometrically harder to avoid in 1D than (in units of a) traveled in each dimension by excitons in 2D. The simulation result also shows a more rapid 1 P (x) 0 4 6 8 10 x(nm) 10 (a) 1022 (b) 10 0 10 0 ϵϵ = 0 10 = 0 ϵϵ==0.02.k2kBTBT ϵϵ==kBkTBT 0.5 ϵϵ==2k2BkTBT 2255 1 10−−1 11001 10 1 0.05 ddiissoordrdereerded(t(=t =11××110033 nnss)) oorrddeererded(t(t= 4=33× 104 nnss)) 11DD(discrete) 22DD(cts.) 100 1100−−22 100100 102 104 00 00..55 11 11.5.5 2 2 100 102 104t (ns) ϵϵ((kkBTBT )) t (ns) FIG. S12: (a) The diffusion parameter (S27) as a function of time, for four different values of energetic disorder (on a triangular lattice). A constant value indicates diffusive motion, which is reached at times that increase as the roughness / (kBT ) of the energy landscape increases. (b) Long-time diffusion constant obtained for particular values of substrate energetic disorder, normalized by the value for no energetic disorder. For a triangular lattice (green line) the fall-off of D with / (kBT ) is less rapid than for discrete Gaussian disorder on a 1D lattice (dotted blue line), but more rapid than for a continuous Gaussian surface in 2D (solid blue line) [6]. The presence (red line) or absence (green line) of nanoparticle spatial disorder (at constant area) has little effect on the fall-off of D with energetic roughness. (a) (b) 1 100 I(r) I(r) Φ(r) Φ(r) expt. expt. 10−1 0.5 10−2 10−3 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 r(µm) r(µm) FIG. S13: The observed photoluminescence profile (black dashed line) on a sparse nanoparticle substrate is a convolution of our laser source Φ(r) (blue line), which is an Airy function with FWHM 240 nm, with the point-spread function of the optics for received light, whdicohuibs alen Awir_ypfaunrcatimoneotfeFrW=0H.M5*2070.2n4m0./Tshqertg(r2ee.n0*lilnoegis(2th.0e )n)u*m5e0r;ic /a/l icno nuvonluittsio noIf (mr) uof nthmese two Airy functions, Edq.o(uS1b8l)e. Tphsifs_cwon_vopluatrioanmmeattcehre=s 0th.e5*o0bs.e2rv7e0d/profile, providing a baseline from which we can assess theeffect of exciton hopping [see Supplemental Fig. S14]. Panels (a) ansdq(br)t(s2ho.0w*llionega(r2-li.n0e)a)r*5an0d; l/in/einar -ulong iptlso tos,fr emspuec tnivmely. fall-off with / (kBT ) does D for a continuous Gaussian described in Section S6. surface (solid blue line), D()/D(0) = exp(−β22/2) [6]. This hierarchy also makes physical sense: a continuous surface is less likely to give rise to particularly abrupt energy changes (traps) than are discrete energy levels S6. COMPARISON WITH EXPERIMENTAL DATA drawn from a Gaussian distribution. We found that the presence or absence of spatial disor- In this section the parameters used in simulations are der of nanoparticles (at constant nanoparticle areal cov- chosen to match those of our experiments. We consider erage) has little effect upon D() [compare red and green a square nanoparticle grid of lattice constant 10 nm (or a lines in Supplemental Fig. S12(b)]; the same is not true continuum approximation thereof); a laser source that is of spatial disorder at varying nanoparticle coverage, as an Airy profile of full width half-maximum (FWHM) 240 I/I0 DD(n(nm 2 m2//nnss)) D(ϵ) I/I D0 D((ϵ0)) D(0) 11 (a) (b) 0 (c)1 10 100 I(r) I(r) c(r) Φ(r) Φ(r) Φ(r) expt. expt. expt. 10−1 10−1 0.5 10−2 10−2 10−3 10−3 −2 −1 0 1 2 −2 −1 0 1 2 −2 −1 0 1 2 r(µm) r(µm) r(µm) FIG. S14: The observed photoluminescence profile (black dashed line) on a dense nanoparticle substrate is broader than that shown in Supplemental Fig. S13, on account of exciton hopping. The source Φ(r) here is as in Supplemental Fig. S13. The green line I(r) results from the solution c(r) of Eq. (S3), for D ≈ 1 (cm)2/s, convolved with the point-spread function S(r), per (S18). The scale of tdhe=b0ro.a0de9ning is consistent with that seen in experiment, supporting our estimate of the basic rateof exciton hopping. However, profile5shapes are not identical, indicating that exciton hopping in experiment is not perfectly described by the diffusion equation. Panels (a) and (b) show linear-linear and linear-log plots, respectively. Panel (c) shows the exciton profile c(r) (cyan line) that results from Eq. (S3) (this profile is not expected to match the experimental profile, because the latter involves convolution with the optics’ point-spread function). nm; a laser source intensity low enough that no exciton- 100 exciton interactions occur (we are in the linear regime); a point-spread function for received light that is an Airy profile of FWHM 270 nm; and excitons of lifetime ≈ 1 10−1 ns. In Supplemental Fig. S13 we show the experimentally measured photoluminescence profile on a sparse nanopar- 10−2 ticle substrate (black). Here we expect the rate of exci- ton hopping to be effectively zero, and so we can use this case as a baseline to isolate the effect of our optics. Also 10−3 shown in the figure are an Airy function of FWHM 240 −2 −1 0 1 2 nm (blue), which is the profile of the laser source on the substrate, and (in green) the convolution of this function r(µm) with an Airy function of FWHM 270 nm. The latter is the point-spread function of the optics at the received FIG. S15: As suggested by Supplemental Fig. S14, photolumi- wavelength. The convolution matches the observed pro- nescence profiles resulting from the diffusion equation do not describe the shape of the experimental profile (black dashed file, even into the tails, indicating that our optics func- line). The green lines show profiles I(r) = (c?S)(r), resulting tions as expected. Knowing this baseline is important, from Eq. (S3) and Eq. (S18), for three different values of D. because it allows us to attribute the broadening of the profile seen on the dense nanoparticle substrate to ex- citon hopping. We show the experimentally measured photoluminescence profile on a dense nanoparticle sub- terms of the approximate width of the profile. Taking−1 2 strate (black) in Supplemental Fig. S14. We also show µ ∼ 1 (ns) gives D ∼ 1(cm) /s, consistent with the es- 2 the profile I(r) expected for diffusive excitonic motion timate of the exciton diffusion constant (D ≈ 0.5(cm) /s) (green). This profile results from the exciton profile c(r), made in the main text. In detail, however, the profiles do calculated from Eq. (S3) in the low-power regime ρ = 0 not match: small discrepancies can be seen in the tails – for D/µ = 0.095 (µm)2, convolved, per Eq. (S18), with evident in the logarithmic plot of panel (b) – indicating the point-spread function of the optics. In that equa- that Eq. (S3) does not perfectly describe exciton motion tion, S(r) is an Airy function of FWHM 270 nm. The on the dense substrate. In Supplemental Fig. S15 we exciton profile itself is shown in cyan in panel (c). The show calculated profiles (green) for three different values source profile in Supplemental Fig. S14 is the same as of D atop experimental data: the comparison indicates that shown in Supplemental Fig. S13, providing a mea- that the shape of the experimental profile is not perfectly sure of the extent to which exciton hopping broadens described by exciton diffusion with a single diffusion con- the profile. The calculated profile shown in Supplemen- stant. tal Fig. S14 is consistent with the experimental result in The iso-energetic nature of our nanoparticles suggests I/I0 I/I0 I/I0 I/I0 12 time 92400000, var = 0 time 92800000, var = 0 these simulations is k/µ = 103. Taking µ ∼ 1 (ns)−1 and the nanoparticle size a ∼ 10 nm gives an estimate for the diffusion constant (on a pristine substrate) of D ∼ 1 3210 µa2 ∼ 12 (cm)2s−1. This value is consistent with the estimate derived from experimental data. The comparison of experimental and calculated profiles shown in Supplemental Fig. S17 suggests that energy transport in our experiments results from iso-energetic hopping, with a diffusion constant of order (cm)2s−1, on a spa- tially imperfect nanoparticle substrate. The middle curve in panel (a) of Supplemental Fig. S17, produced using a FIG. S16: Snapshots of time-averaged photon emission statis- tics from continuous-time Monte Carlo simulations of exciton vacancy fraction of 20%, is the green line in Panel (g) of creation and self-destruction on a square nanoparticle lattice. the main text. In the right-hand panel we also simulate exciton hopping. The laser source Φ(r) is an Airy function of FHWM 240 nm. that exciton subdiffusion of the type seen in previous work [3] and in Section S5 B does not occur in our ex- periments. Instead, the presence of imperfections in the nanoparticle substrate, and the practice of averaging ex- perimental profiles over different spatial locations, may lead to non-diffusive profiles. To investigate this possi- bility we turned to microscopic simulations of the kind described in Section S5 A. On a square nanoparticle lat- tice we simulated the processes of creation (S19), self- destruction (S20), and hopping ∅ k+ A←→ A + ∅, (S28) using continuous-time Monte Carlo [4]. Hops were con- sidered to any vacant nearest-neighbor nanoparticle. We assume the process of self-destruction to give rise to a photon at the same spatial location, and so the photon emission profile is the exciton profile c(r) in the long- time limit. In Supplemental Fig. S16 we show example snapshots of the time-averaged photon emission statistics that result from these processes in the absence (left) and presence (right) of hopping. The source Φ(r) is again an Airy function of FWHM 240 nm; the left-hand panel is essentially an image of this function. In Supplemental Fig. S17 we show the experimentally measured photoluminescence profile on a close-packed nanoparticle monolayer (black), together with profiles I(r) (green) obtained by convolving, via Eq. (S18), the exciton profile c(r) obtained from microscopic simula- tions with a point-spread Airy function S(r) of FWHM 270 nm. We work in the low-power regime, with Φ0/µ = 10−2 (see Supplemental Fig. S9). Simulations done on pristine substrates match the diffusive profiles obtained using Eq. (S3). To mimic substrate imperfections we did microscopic simulations with a fraction fV of nanoparti- cle vacancies. No excitons can be created on, or hop to, a vacancy. We created vacancies in a spatially uncorrelated way, which is probably not true of vacancies produced by the nanoparticle self-assembly process: there, vacan- cies appear to cluster as gaps. However, the effect leads to profiles with shapes similar to those seen in experi- ment. The value of the hopping rate k used to produce 13 (a) (b) 100 1 100 disorder 10−1 10−1 0.5 10−2 10−2 10−3 − 10 −3 1 0 1 −1 0 1 −1 0 1 r(µm) r(µm) r(µm) FIG. S17l:eEfxtp: e0ri.m1e,n0ta.2l p,0ho.t3o ludmisinoesrcdenecerprofile (black dashed line) compared with simulated profiles I(r) = (c?S)(r) (green).The profile c(r) results from continuous-time Monte Carlo simulations of exciton creation, self-destruction, and hopping; this function is convolved with S(r), an Airy function of FWHM 270 nm, via Eq. (S18). In panel (a) we show results for parameters Φ /µ = 10−2 30 and k/µ = 10 , with three different mean nanoparticle vacancy fractions fV of 0.1, 0.2, and 0.3 (from the outside in). In panels (b) and (c) we show results for vacancy fraction 0.2 on linear-linear and linear-log plots, respectively: its shape issaimbetptelromt_astchh iffotr(tph2e ,e"xcperreimaetnitoalnp_ro0fi.le01th_andearse tprruoficletsiofronm_dth_e1d_iffaunsionniheqiulaattioion n(s_ee0F_idgsi.ffSu14 and S15). Tlines are the exciton profiles c(r). sion_1000 he_dcyiasnorder_ 0.2_iteration_0/report_pl_cut_convolved.dat"," ",pen_green_solid); S7. ADDITIONAL EXPERIMENTAL FIGURES • The excitation laser intensity profile is shown in Supplemental Fig. S18. • The schematics of the optical setups are shown in Supplemental Fig. S19. • Additional time-resolved PL spectroscopy data are provided in Supplemental Fig. S20. • Additional lifetime measurements are provided in Supplemental Fig. S21. • A time-resolved PL microscopy map is shown in Supplemental Fig. S22. • The normalized PL intensity as a function of the ex- citation laser power is shown in Supplemental Fig. S23. I/I0 I/I0 I/I0 14 Laser – 450 nm 500 nm Gaussian fit HWHM 120 nm FIG. S18: (a) Diffraction limited CW laser spot (wavelength 450 nm) imaged with a CCD camera after 530X magnification. (b) Laser spot cross-section (blue) and Gaussian fit (FWHM 240 nm). 15 a) b) Sample 100x Pulsed laser 465 nm Dichroic filter Photodiode Picoharp Monochromator + APD c) FIG. S19: Setup for (a) steady-state PL microscopy, (b) time-resolved PL spectroscopy, and (c) time-resolved PL microscopy. 16 a) a) b) d) c) e) FIG. S20: Time-resolved PL spectra on (a) an ordered area made of uniformly sized PNCs and on (b) a disordered area made of PNCs of different sizes. (c) PL spectra from data in (a) integrated between 0 ns and 0.2 ns (blue) and between 3.4 ns and 5.4 ns (red). The two spectra overlap. (d) PL spectra from data in (b) integrated between 0 ns and 0.2 ns (blue) and between 3.4 ns and 5.4 ns (red). The spectrum at later time is slightly red shifted. (e) Integrated PL spectrum; the blue and red vertical lines show the PL wavelengths displayed in Figure 3-b and 3-c. 17 a) 𝜏 = 1.98 ns b) 𝜏 𝜏 = 1.14 ns FIG. S21: Time-resolved PL of (a) PNCs assembled in a sparse monolayer; (b) PNCs assembled in a close-packed monolayer (integrated over the entire collection area). The lifetime of the system, measured as the time to reach a 37% or 1/e decay, is 1.94 ns, and 1.14 ns respectively. 18 a) b) FIG. S22: (a) Time-integrated PL intensity map. (b) Lifetime map calculated from. FIG. S23: Normalized PL intensity as a function of the excitation laser power. The upward scan is shown in red dots, the downward scan is shown in blue dots, and the average. 19 [1] Yang, J., Choi, M. K., Kim, D.-H., and Hyeon, T., Designed [5] Zwanzig, R., Diffusion in a Rough Potential, Proc. Natl. Acad. Assembly and Integration of Colloidal Nanocrystals for Device Sci. USA 1988, 85, 2029-2030. Application, Adv. Mater. 2016, 28, 1176-1207. 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