Persistent enhancement of exciton diffusivity in CsPbBr3 nanocrystal solids

In semiconductors, exciton or charge carrier diffusivity is typically described as an inherent material property. Here, we show that the transport of excitons among CsPbBr3 perovskite nanocrystals (NCs) depends markedly on how recently those NCs were occupied by a previous exciton. Using transient photoluminescence microscopy, we observe a striking dependence of the apparent exciton diffusivity on excitation laser power that does not arise from nonlinear exciton-exciton interactions or thermal heating. We interpret our observations with a model in which excitons cause NCs to transition to a long-lived metastable configuration that markedly increases exciton transport. The exciton diffusivity observed here (>0.15 square centimeters per second) is considerably higher than that observed in other NC systems, revealing unusually strong excitonic coupling between NCs. The finding of a persistent enhancement in excitonic coupling may help explain other photophysical behaviors observed in CsPbBr3 NCs, such as superfluorescence, and inform the design of optoelectronic devices.


Abstract:
In semiconductors, exciton or charge carrier diffusivity is typically described as an inherent material property.Here, we show that the transport of excitons (i.e., bound electron-hole pairs) in CsPbBr3 perovskite nanocrystals (NCs) depends markedly on how recently those NCs were occupied by a previous exciton.Using fluence-and repetition-rate-dependent transient photoluminescence microscopy, we visualize the effect of excitation frequency on exciton transport in CsPbBr3 NC solids.Surprisingly, we observe a striking dependence of the apparent exciton diffusivity on excitation laser power that does not arise from nonlinear exciton-exciton interactions nor from thermal heating of the sample.We interpret our observations with a model in which excitons cause NCs to undergo a transition to a metastable configuration that admits faster exciton transport by roughly an order of magnitude.This metastable configuration persists for ~microseconds at room temperature, and does not depend on the identity of surface ligands or presence of an oxide shell, suggesting that it is an intrinsic response of the perovskite lattice to electronic excitation.The exciton diffusivity observed here (>0.15 cm 2 /s) is considerably higher than that observed in other NC systems on similar timescales, revealing unusually strong excitonic coupling in a NC material.The finding of a persistent enhancement in excitonic coupling between NCs may help explain other extraordinary photophysical behaviors observed in CsPbBr3 NC arrays, such as superfluorescence.Additionally, faster exciton diffusivity under higher photoexcitation intensity is likely to provide practical insights for optoelectronic device engineering.
In semiconducting materials, the diffusion of charge carriers or excitonswhich are bound electron-hole pairsis central to the operation of electrical devices, generating energy in the form of electricity, light or heat.][3][4][5][6][7][8][9][10][11] In particular, experimental access to nonequilibrium regimes of exciton/carrier transport has revealed new insights into mesoscale dynamics.3] At later times, the effect of energetic disorder manifests in subdiffusive transport phenomena, wherein the ensemble average diffusivity decreases over time. 14Here, we report observation of a novel nonequilibrium transport modality in which the diffusivity of excitons in a nanocrystal array depends on how recently those nanocrystals were previously in the excited state.
To characterize exciton dynamics in CsPbBr3 NC solids, we employed transient photoluminescence microscopy (TPLM) to track radiative recombination of photogenerated excitons with both temporal and spatial resolution, as illustrated in Fig. 1a & S1.Briefly, a variable repetition rate pulsed laser (405 nm, ~50 ps, FWHM 500 nm, 0.5-40 MHz) generated an initial population of excitons in a near-diffraction-limited spot.Epifluorescence was collected by a microscope objective lens and magnified by ~500x using a telescope, then an avalanche photodiode (APD, 50 µm × 50 µm active area) was raster scanned across the magnified imaging plane.Transient photoluminescence data were collected at each spatial position, allowing the time-dependent spatial distribution to be re-constructed and analyzed.The overall temporal resolution (~80 ps) is limited by the excitation laser pulse width and the APD response time.As a superresolution optical technique, the spatial resolution is ultimately limited by the total photon counts and other ancillary factors impacting the signal-to-noise ratio of the measurement, meaning that exciton diffusion lengths much smaller than the focused laser beam waist can be reliably determined for bright emitters. 2A diagram of the instrument and further experimental details (Supplementary Note 3, Fig. S1-2) can be found in the Supporting Information.
We first investigated CsPbBr3 NCs capped with a mixture of oleylamine and oleic acid (OLA/OA) surface ligands, which were spun-cast into thin solid films supported on quartz glass substrates.The quasi-cubic CsPbBr3 NCs were 8.3 nm in size, and the solid film exhibited a photoluminescence quantum yield of 68%.The NC film absorption and emission spectra are shown in Fig. 1b.The transmission electron micrograph (TEM) image shows the quasi-cubic shape of the NCs (Fig. 1b inset).The thin film samples were 30-40 nm in thickness and relatively uniform, as confirmed by atomic force microscopy (AFM) (Fig. S4).
To ensure that TPLM experiments were conducted in the linear regime, we performed power-dependent photoluminescence spectroscopy to verify the absence of nonlinear multiexciton interactions.Fig. 1c shows normalized photoluminescence spectra collected with nominal laser power ranging from 0.5 to 20 nW at a constant repetition rate of 2.5 MHz focused to a ~500 nm diameter laser spot, corresponding to 0.004 to 0.16 photons absorbed per NC per laser pulsesee Table S1.Across this power regime, no peak shifting or broadening is observed; more importantly, there is a linear relationship between integrated photoluminescence intensity and excitation laser power, indicating that additional non-radiative multi-exciton (i.e.Auger) processes are not introduced at higher power.The same linear trend was also observed when excitation laser power was increased under constant laser fluence and varying repetition rate (Fig. S5).
As observed by others, 15 the transient photoluminescence decay of CsPbBr3 NCs exhibits both prompt and delayed emission characteristics (Fig. 1d, Fig. S6).The prompt emission, covering the first ~0-3 ns following photoexcitation, is well fit by a single-exponential decay curve (solid black lines on the log-log plot in Fig. 1d), while the delayed emission (≥3 ns) can be described by a power-law (dashed line).While the origin of these complex emission dynamics is still debated, single-NC studies suggest that the prompt emission results from direct exciton recombination, whereas the delayed power-law emission arises from excitons that have undergone multiple trapping-detrapping processes. 39With increasing laser power, the prompt emission lifetime slightly decreased from ~2.3 ns to ~1.5 ns, while the power-law exponent for the delayed emission increased from ~1.5 to ~1.7 (Fig. S6).
The results of one TPLM measurement are shown in Fig. 1e.These data were collected with a nominal laser power of 10 nW at 2.5 MHz repetition rate and 500 nm laser spot size, corresponding to a laser pulse fluence of ~0.76 µJ/cm 2 , or 0.08 photogenerated excitons per NC per pulse (see Supplementary Note 6).To quantitatively extract exciton diffusivity from the TPLM data shown in Fig. 1e, the instantaneous spatial profile was fitted to a Gaussian shape at each delay time and the mean square displacement (MSD), equal to the change in variance of the Gaussian distribution () 2 −  0 2 , was extracted (Supplementary Notes 3, Fig. S2). 2 The diffusivity of the exciton population is then extracted from the slope of the MSD curve.For normal diffusion, the MSD follows a linear relationship with time: where  is the diffusivity.
The MSD curve generated from the TPLM data in Fig. 1e is shown in Fig. 1f.Interestingly, the MSD grows linearly with time during the first ~3 ns then becomes sublinear afterwards, matching the transition from prompt to delayed emission in the transient photoluminescence data (Fig. 1d).The MSD behavior is consistent with the transient photoluminescence decay interpretation: at early times excitons diffuse freely within the NC solid while spontaneously undergoing radiative recombination; at later times, exciton trapping begins to dominate the spatiotemporal dynamics leading to a dramatic decrease in exciton diffusivity.For the remainder of this study, we focus exclusively on the early-time dynamics (<3~4 ns), corresponding to free exciton diffusion.
TPLM experiments were performed while independently varying the laser repetition rate, laser pulse fluence (proportional to the excitation density, 〈〉; see Supplementary Note 6), and, accordingly, the time-averaged laser power (Fig. 2).We observed that increasing either the fluence or repetition rate while holding the other variable constant always led to a greater measured value of the exciton diffusivity (Fig. 2a-d).Surprisingly, if the fluence and repetition rate were varied simultaneously such that the time-average power remained constant, the measured value of the exciton diffusivity did not change (Fig. 2e, Fig. S3).TPLM experiments performed under a variety of laser excitation conditions are aggregated and plotted together in Fig. 2f-h.While there is no correlation of diffusivity with either laser pulse fluence or laser repetition rate alone (Fig. 2g,h), there is a strong positive correlation between diffusivity and time-averaged laser power (Fig. 2f).The grey oval is a guide to the eye.The x-axis is in linear scale when power is less than 10 nW and logarithm scale when power is above 10 nW.(g,h) Exciton diffusivity measured under varying experimental conditions plotted vs laser repetition rate or <N>.Data points are colorcoded according to time-average power, indicated to the right.
The observation of power-dependent diffusivity in the low-excitation-density regime is surprising.Typically, exciton transport in semiconductor NC solids is understood in the framework of incoherent hopping mediated through dipole-dipole interactions, i.e.[42] In this picture, excitons act effectively as point dipoles that undergo discrete stochastic hopping transitions to neighboring NCs; varying the excitation laser power is expected to have no effect on the rate at which excitons move.We note that, at sufficiently high excitation density, 〈〉 , exciton-exciton interactions will introduce additional nonlinear recombination pathways that can interfere with the interpretation of time-resolved microscopy measurements; 2-3 however, no dependence of the measured diffusivity on laser pulse fluence alone was observed (Fig. 2h).The correlation between exciton diffusivity and time-averaged laser power (Fig. 2f)but not repetition rate or laser pulse fluence alonecannot be explained within the traditional exciton random walk model.Consequently, we explored multiple experimental and materials factors that might contribute to this anomalous observation, including 1) sample degradation, 2) instrumentation/data analysis artifacts, 3) laser heating, and 4) surface effects.
We first sought to verify that the power-dependent trend shown in Fig. 2f is repeatable and reversible.A series of TPLM measurements was performed at the same spot on the sample while non-monotonically varying laser power (Fig. 3a).We established a low-power baseline diffusivity by starting the measurement at 1.5 nW, and then raised the power to 6 nW.The TPLM scan was repeated again at 6 nW, then the power was lowered back to 1.5 nW, confirming that the lowpower diffusivity value could be recovered after the same sample spot was exposed to higher laser power.The last two runs further confirmed the general reproducibility of the measurement, while also revealing some of the scatter in the data evident in Fig. 2f.Next, we investigated whether varying signal-to-noise ratio of the TPLM measurement at different laser power could lead to systematic errors in data analysis.In Fig. 3b we compare a typical TPLM measurement series (open circles) to one in which variable neutral density filters were placed in the detection pathway to achieve a constant signal-to-noise ratio across all measurements (crosses).The two data sets overlap completely (albeit with some scatter in the data), demonstrating that differences in signal intensity or APD count rate are not responsible for the power-dependent trend.Moreover, we tested our data analysis procedure on simulated data with varying shot noise and background noise and found that diffusivities below 0.01 cm 2 /s could be reliably extracted from experimental data of the quality presented here (Supplementary Note 7).
Perhaps the biggest concern in any power-dependent spectroscopy trend is possible effects of laser heating.When a laser pulse is absorbed by the sample, heat is generated as the excess photon energy is dissipated over varying timescales.Typically, carrier-carrier scattering first leads to formation of a Boltzmann distribution over the electronic degrees of freedom within ~100 fs, 43 followed by hot carrier relaxation to the band edge via phonon emission on a picosecond timescale. 44As time-averaged power rises, more heat is generated, increasing the lattice temperature.The temperature rise is ultimately limited by thermal transport away from the laser excitation spot, a process whose timescale depends on the geometry of the measurement and the thermal conductivity of the sample.Thermal transport simulations of our TPLM experiment showed that, under typical laser fluence used here (~0.1 µJ/cm 2 ), each absorbed laser pulse increases the local sample temperature by less than 0.01 Kwith the majority of that heat dissipating within the first few nanoseconds following photoexcitation (Supplementary Note 8).
These findings are consistent with experimental results obtained using time-resolved X-ray diffraction by Kirschner et al., who found that excitation fluences on the order of a few mJ/cm 2 (~10,000x larger than that used in our TPLM experiments) were required to increase the temperature of CsPbBr3 NCs by ~100 K, and that the heat fully dissipated within ~10 ns. 45 further investigate the potential consequences of sample heating, we directly measured the exciton diffusivity as a function of sample temperature (Fig. 3c).Temperature-dependent TPLM measurements were performed inside a closed-cycle liquid helium cryostat under vacuum (Montana Instruments Cryostation).Two CsPbBr3 NC samples having similar sizebut different surface chemistrywere investigated.One sample was terminated with a mixture of oleylammonium and oleate ligands (OLA/OA), while the other sample was terminated with a zwitterionic ligand, 3-(N,N-dimethyloctadecylammonio)propanesulfonate. 46In both samples, the diffusivity exhibited a similar non-monotonic trendfirst increasing as the temperature decreased below room temperature, then eventually decreasing again as the sample was further cooled below ~150 K.The full temperature-dependent behavior is the subject of ongoing investigation, but the salient observation here is the trend of decreasing diffusivity with increasing sample temperature near room temperature.If laser heating were responsible for the power-dependent trend shown in Fig. 2f and Fig. 3e, we would expect the opposite behavior.Consequently, we conclude that laser heating is not responsible for the anomalous power-dependent trend in the data.
Unusual behavior in semiconductor nanocrystals is sometimes associated with their unique surface properties.Next, we examined whether varying surface treatments led to the same powerdependent phenomenon.Four different types of samples were investigated: 1) the OLA/OAcapped NCs shown in Fig. 1, 2) NCs synthesized with a zwitterionic ligand, [3-(N,Ndimethyloctadecylammonio)-propanesulfonate], 3) spun-cast NC solids coated with AlOx using standard atomic layer deposition (ALD), 47 and 4) NCs coated with an AlOx shell grown via colloidal ALD prior to spin-casting, 48 as illustrated in Fig. 3d.Samples #1, 3, and 4 were prepared at EPFL (Lausanne), while sample #2 was prepared at ETH (Zurich), before being shipped to MIT for transient microscopy measurements.Optical characterization of all samples is reported in Fig. S9, and details of the sample preparation methods used are included in the Supplementary Information.
In Fig. 3e we compare the results of TPLM measurements made on the four samples with different surface treatments.All four samples show the same monotonic trend of increasing exciton diffusivity with increasing laser power.Similar to the OLA/OA samples shown in Fig. 2, there was no correlation between exciton diffusivity and laser pulse fluence or repetition rate alone (Fig. S10).Importantly, the two samples with AlOx coatings exhibited lower absolute diffusivity than the uncoated NCs, which is consistent with previous studies demonstrating slower exciton transfer rates between NCs with thicker surface shells. 14Moreover, when the colloidal AlOx shell thickness increased from 4 layers to 8 or 12 layers, the diffusivity became too small to measure (Fig. S11).
These findings build confidence in the measurement technique and its connection to the physics of exciton transport.Meanwhile, the repeated observation of power-dependent diffusivity in multiple CsPbBr3 NC samples synthesized by different labs and terminated with different chemistries suggests that this unexpected correlation is intrinsic to the CsPbBr3 perovskite lattice and not derived from surface-related phenomena.
To understand the observation of power-dependent diffusivity, we consider the microscopic meaning of time-averaged laser power in the context of this experiment.In the low excitation density regime (i.e.<N> less than 1), time-averaged power informs on the frequency of NC photoexcitation events; specifically, the inverse of time-averaged power is proportional to the average waiting time between NC excitation events.In Fig. 4a, we plot the measured exciton diffusivity vs. time between NC excitations (proportional to 1/power).When plotted in this way, the data exhibit a characteristic exponential relaxation curve, with an extracted relaxation time constant of ~6 μs.As the waiting time between individual excitation events becomes longer, the observed ensemble exciton diffusivity decreases.The limiting diffusivity (as time between excitation events tends toward infinity) is ~0.01 cm 2 /s, while the "enhanced" diffusivity measured under frequent photoexcitation conditions is up to 15x larger.The observation that exciton diffusivity changes as the exciton generation frequency on any given NC changes is surprising.
Significantly, the relaxation time constantthe characteristic time it takes for a NC to relax to the low-diffusivity stateis on the microsecond timescale, which is 2-3 orders of magnitude longer than the exciton lifetime (Fig. 1).This striking observation suggests that CsPbBr3 NCs retain some persistent memory of previous exciton occupation events.The exponential relaxation behavior shown in Fig. 4a suggests that NCs can adopt two different states: a stable "slow state" associated with the low-diffusivity exciton transport regime, and a metastable "fast state" that permits dramatically increased exciton hopping rates.These states are illustrated schematically in Fig. 4b,c.To test our understanding, we built a phenomenological model of exciton transport that includes persistent excitation memory effects and studied model dynamics using kinetic Monte Carlo (KMC) simulations (Fig. 4d,e).In the model, the NC solid is represented as a two-dimensional square lattice -each lattice site representing a single NCwith lattice spacing,  ≈ 10, roughly corresponding to the neareast neighbor spacing.
In the model, NCs can adopt either a slow or fast state, depicted in Fig. 4d as green or purple squares, respectively.Excitons, depicted in red in Fig. 4d, occupy individual NCs and can hop between nearest neighbor NCs with a rate  hop that depends on the state of the neighboring NCs.Excitons hop to slow or fast state NCs with rates  hop (slow) = 0.006ps −1 or  hop (fast) = 0.7ps −1 , respectively.The rates were empirically chosen to match the experiments.Excitons decay stochastically with a probability determined by their lifetime,  ex = 2.5ns.We model the creation of excitons via laser pulse by randomly exciting sites with a Gaussian spatial distribution whose width,  pulse , and amplitude,  pulse , are selected to match experimental estimates.Subsequent pulses create new excitons at regular intervals whose spacing in time far exceeds the exciton lifetime, so that no excitons from previous pulses remain.
The state of a NC (i.e., slow or fast) depends on the history of its exciton occupancy.Upon excitationvia laser pulse or hoppingan NC immediately transitions into the fast state and can remain in this state even if the exciton decays or hops away.Unoccupied fast state NCs decay stochastically back to the slow state with a rate of  relax = (8.1 ) −1 .Due to the difference between exciton (~ns) and fast state (~µs) lifetimes, all NC relaxation is assumed to occur in the time between laser pulses, after all excitons have decayed.Fig. 4d illustrates a snapshot of the simulation a few nanoseconds after one of the laser pulses.
We compute exciton diffusivity from the simulation in a manner analogous to the experiment.Specifically, we compute the mean radial exciton density as a function of time following the most recent laser pulse,  ex (, ), assuming the center of the laser pulse is located at the origin.We extract the time-dependent width of this distribution,  2 () = 〈 2 ()〉, where the angle brackets imply an average over  ex , and a simulated diffusivity as  sim = ( 2 () −  2 (0))/2 , evaluated at  = 1ns .Fig. 4e plots the simulated values of  sim for a range of simulations with differing pulse frequency and pulse intensity.We find that  sim is primarily a function of the average time between NC photoexcitation as evaluated at the center of the Gaussian laser pulse.This result is in near quantitative agreement with experimental observation.
0][51][52] Photocharging in CsPbBr3 NCs has been observed at higher laser pulse fluences at cryogenic temperatures, 39 and is typically associated with electron (or hole) transfer to the NC surface.To investigate this possibility, we compared the diffusivity relaxation curves for OLA/OA-capped NCs (shown in Fig. 4a) to the colloidal AlOx-coated CsPbBr3 NCs, shown in Fig. S12.Despite the presence of the insulating oxide layer on the AlOx-coated NC surface, the relaxation time constant was 10.8 μsonly slightly larger than the 6 μs value measured for the uncoated NCs.Moreover, photocharging is typically a nonlinear effect, requiring interaction of one exciton with another exciton or photon, which we have shown is not the case in this study (Fig. 2h).Finally, it is not clear why a charged NC should exhibit dramatically enhanced excitonic coupling to its neighbors.
A more provocative explanation for the excitation memory effect illustrated in Fig. 4b,c is the presence of a lattice polarization that persists long after exciton recombination.4][55][56][57][58] Particularly intriguing is the possibility of microscopic ferroelectric domains, which could act to collapse the oscillator strength along a particular NC axis.In this picture, the transition back to the relaxed/disordered state requires thermal energy from the surrounding which results in a slow relaxation time, as is shown in Fig. 4b,c.However, the presence of ferroelectric-like behavior in halide perovskites is debated. 59gardless of the microscopic mechanism, the phenomenological observation of a persistent enhancement in excitonic coupling in CsPbBr3 NC solids is significant.1][62][63][64] However, under the highest laser powers used in this study, the diffusivity reached 0.15 cm 2 /s (we note that Penzo et al. reported a diffusivity of 0.5 cm 2 /s in similar CsPbBr3 NCs). 7For reference, carrier diffusivity in bulk halide perovskite single crystals at room temperature has been estimated between 0.3-0.5 cm 2 /s, while diffusivity in perovskite thin films is typically an order of magnitude smaller, ~0.01-0.0566] The observation of an exciton diffusivity in CsPbBr3 NC solids that exceeds the carrier diffusivity in perovskite thin filmsdespite the presence of long-chain organic ligands separating individual NCsis striking.Our findings may also inform other observations of strong excitonic coupling in CsPbBr3 NCs.5] Different from excitonic coherences sometimes observed in molecular systems, 67 the term superfluorescence implies the presence of strong excitonic coupling in the excited statebut not the ground state.Temporary transition to a transient lattice configuration within the excited state manifold in which excitonic coupling is enhanced could help explain this behavior.Finally, the observation of power-dependent diffusivity has implications for the design of high-brightness LEDs 21, 68-75 and lasers [76][77][78] featuring CsPbBr3 NCs.
Nanosocope IIIa (Veeco, USA), operated in tapping-mode, with Nanosensors PP-NCSTR AFM probes.Thin lines were scratched on the samples to reveal the Si substrate.The mapping was carried-out at the edge of the lines.
Transmission Electron Microscopy (TEM): TEM images were acquired on an Analytical JEOL-2100F FETEM equipped with a Gatan camera, using a beam energy of 120 kV.NC were dropcasted on Cu TEM grids (Ted Pella Inc.) prior to the imaging.Size measurements were performed using the software Image J and counting 200 particles per sample.

Steady-State Absorption:
Steady-state UV−Visible absorption measurements were performed in transmission mode using a PerkinElmer Lambda 950 spectrophotometer equipped with deuterium and tungsten halide lamps for UV and Vis-IR ranges, respectively.A PMT and Peltier-controlled PbS were used for detection.
Steady-State Photoluminescence (PL) Spectroscopy: Steady-state emission and quantum yield (QY) PL measurements were recorded by a Horiba Jobin Yvon Fluorolog-3 spectrometer equipped with a PMT detector.All PL spectra were collected at an excitation wavelength of 370 nm.Absolute QY measurements were performed in a Spectralon® coated integrating sphere.For each sample, four measurements were performed: (i) sample emission (Sem); (ii) blank glass emission (Bem) (iii) sample excitation (Sexc) and (iv) blank glass excitation (Bexc).The absolute QY was then calculated as follows: The reported QY values is the average of three measurements.

Model dynamics:
Model dynamics were simulated with a simple Monte Carlo algorithm.Initially, there are no excitons and every NC is in the gnd configuration.Whenever an exciton occupies a NC, via laser pulse or hopping, the NC immediately transitions into the mod configuration.Exciton hopping dynamics are then simulated with a 1ps timestep until all excitons have decayed.At each timestep, each exciton attempts a hop to a randomly selected neighbor and with probability  hop (mod) = 0.7 or  hop (gnd) = 6 × 10 −3 if the neighbor is in the modified or ground configuration, respectively.
Following this, each exciton randomly decays (is eliminated from the simulation) with probability  decay = 4 × 10 −4 (selected to yield an exciton lifetime of 2.5ns).Excitons are constrained to single occupancy, so if an exciton attempts to hop to an occupied site, its hopping probability is zero.
After all excitons have decayed, the NC configurations are relaxed over the time increment to the next pulse.To do this we select a random relaxation time, Δ relax , for each NC in the mod configuration.Selecting Δ relax = − τ relax ln(1 − ), where  is a random number between 0 and 1 yields a proper exponential distribution of configurational lifetimes.For a given NC, if Δ relax ≤  pulse , then the NC is returned to the ground state for the next simulated laser pulse.If Δ relax >  pulse , the NC remains in the mod state.Thus, as the pulse frequency nears  relax , excitons can hop to NCs that transitioned into a mod state during the previous pulse.
For each set of  pulse and  pulse , we simulated a sequence of 1000 laser pulses, recording exciton positions every 50 timesteps.Time-dependent exciton density,  ex (, , ) was computed by averaging over the occupancy state of site (, ) at time  after the most recent laser pulse for all 1000 pulses.Exciton diffusivity was computed based on the shape of this distribution, as described in the main text.
The diffusion data were taken as a function of time delay and spatial position.The general diffusion equation in one dimension is as follows: (, )  = ()  2 (, )  2 − ()(, ) where (, ) is the exciton density distribution as a function of time and location, () is the time-dependent exciton diffusivity, and () is the time-dependent exciton decay rate.For CsPbBr3 NCs, we assume that the diffusivity and decay rate are time-independent during the first 3 ns.Additionally, we treat the laser pulse as an instantaneous source, ( ̃, 0).Then, the general solution to Eqn. ( 2) could be written as: As we normalize the PL emission at any given time delay, Eqn. ( 3) can be re-written into Eqn.(3) as the following: where (, ) =   ) and an exciton diffusivity of 0.01 cm2 /s, similar to the experimental conditions.An experimentally obtained normalized lifetime trace () was added to simulate the data decay as a function of time.Therefore, the spatial profile at the laser arrival time could be expressed as:  where  is a pre-factor modulating the intensity of the signal.Here, we arbitrarily set  = 2 * 10 5 to match the experimental signal intensity.At later times , the variance and spatial distribution could be written as:  Both ambient background noise and shot noise were considered and added to the simulated PL emission data (Fig. S7a).Specifically, ambient background noise refers to both the dark current of the detector and the photons from the experiment surroundings that reached the detector.To recreate the noise, we created a matrix of uniformly distributed random numbers with the intensity similar to experimentally observed dark counts.In addition, shot noise describes the fluctuation of number of photons detected as photon detections are individual events.Shot noise is proportional to the square root of the experimental signal as:  ℎ ∝ √ (11)  For each data point, the shot noise is recreated as:  ℎ ∝ (0,1) * √ (12)  where (0,1) is a random number generated between 0 and 1.The same data analysis and fitting procedure were then applied to the simulated data set, and a diffusivity of 0.01 cm 2 /s was recovered (Fig. 7b-c).It should be noted that only when the full spatial profile (when there were negligible PL counts towards both ends) was fitted and when the background counts were properly accounted for, could the diffusivity be accurately recovered; otherwise, a value significantly larger than 0.01 cm 2 /s was extracted.

Fig. 1 .
Fig. 1.Transient photoluminescence microscopy (TPLM) of CsPbBr3 NC solids.(a) Schematic showing a near diffraction-limited laser pulse generating a population of excitons, which subsequently diffuse within the film, leading to spatial broadening of the photoluminescence signal over time (laser excitation spot not to scale).(b) Normalized absorbance (solid line) and photoluminescence (dashed line) spectra of the CsPbBr3 (OLA-OA) NCs in toluene.Arrow indicates the laser excitation wavelength for all of the TPLM measurements in this study.Inset shows a transmission electron micrograph (TEM) image of the CsPbBr3 NC sample.(c) Normalized power-dependent emission spectra showing no peak shift as a function of excitation laser power (0.5~20 nW, 2.5 MHz, 500 nm diameter laser spot).Inset shows the integrated photoluminescence intensity as a function of laser excitation power.Dashed line is a linear fit of the data with fitted parameters labeled above.(d) Power-dependent transient photoluminescence plotted on log-log scale (5 MHz repetition rate).The solid lines are singleexponential fits to the first 3 ns, and dashed lines are power law fits to 3~20 ns.(e) Sample normalized TPLM color map plotted as a function of spatial position (x-axis) and time (y-axis) at 10 nW (2.5 MHz repetition rate).(f) Mean square displacement as a function of time extracted from the data shown in (e).See TableS1-2 for detailed measurement parameters.

Fig. 2 .
Fig. 2. Dependence of measured exciton diffusivity on laser pulse fluence, repetition rate, and time-averaged power.(a,b) TPLM data with varying repetition rate, while holding pulse fluence constant at <N> = 0.02 absorbed photons per NC per laser pulse.(c,d) TPLM data with varying pulse fluence, while holding repetition rate constant (5 MHz).(e) TPLM data while varying pulse fluence and repetition rate together to achieve constant time-averaged power.(f) Exciton diffusivity measured under varying experimental conditions plotted vs time-averaged power.The grey oval is a guide to the eye.The x-axis is in linear scale when power is less than 10 nW and logarithm scale when power is above 10 nW.(g,h) Exciton diffusivity measured under varying experimental conditions plotted vs laser repetition rate or <N>.Data points are colorcoded according to time-average power, indicated to the right.

Fig. 3 .
Fig.3.Measurement repeatability, and effect of temperature, signal-to-noise ratio, and surface chemistry.(a) Demonstration that power-dependent diffusivity is a repeatable and reversible observation.TPLM was performed at the same sample spot while varying the excitation power non-monotonically.(b) Invariance of the power-dependent trend with measurement signal-tonoise ratio.Open circles correspond to the typical TPLM measurement, in which signal count rate varies naturally with the laser power used.Crosses correspond to a power series in which variable neutral density filters were placed in the signal path to achieve a constant signal-tonoise ratio across all powers.(c) Temperature-dependent exciton diffusivity of CsPbBr3 NCs capped with different ligands, as measured at 1 nW and 2.5 MHz.(d) Illustration of four different surface treatments investigated.Top left shows the CsPbBr3 NCs capped with oleylammonium and oleate ligands (OLA/OA).Bottom left shows the NC samples capped with zwitterionic ligands, 3-(N,N-dimethyloctadecylammonio)-propanesulfonate.Top right shows the NC film coated with AlOx using standard atomic layer deposition (ALD).Bottom right shows the NC sample coated with AlOx using a colloidal ALD growth.(e) Correlation of exciton diffusivity with time-averaged laser power for CsPbBr3 NC solids with different surface treatments.The xaxis is in linear scale when power is less than 10 nW and logarithm scale when power is above 10 nW.The grey ovals are a guide to the eye, separately grouping the uncoated and AlOx-coated samples.

Fig. 4 .
Fig. 4. Persistent enhancement of exciton diffusivity.(a) Experimentally measured diffusivity relaxation curve.Exciton diffusivity of CsPbBr3 NCs with OLA/OA ligands as a function of time between NC excitation events.Mirror x-axis shows the corresponding time-averaged power.The red dashed line is a single exponential fit to the data, and the best fit parameters are annotated within the figure.Inset is a magnification of the early-time data points.(b) Schematic illustration of the excitation memory effect leading to persistent enhancement of exciton diffusivity.(c) Potential energy surface description of the phenomenon illustrated in panel (b).Black curves indicate the electronic excited state and electronic ground state of the NCs.(d) Snapshot of a kinetic Monte Carlo (KMC) simulation of exciton transport in a 2D NC array, which includes excitation memory effects.NCs in the relaxed state are shown in green and NCs in the metastable state are shown in purple.Red spots indicate current location of excitons within the time-dependent simulation.(e) KMC simulation results plotted as a function of time between NC excitation events, showing consistency with the experimentally measured phenomenon.Different colors represent different simulated excitation fluences.Inset shows simulated diffusivity as a function of time between laser pulses (reciprocal of laser repetition rate).The red dashed line is a single exponential fit to the data.

1 √4𝜋𝐷𝑡𝑒 − 𝑥 2 4𝐷𝑡Fig. S2 .
Fig. S2.Sample time slices of the TPLM measurements in CsPbBr3 NC solids at 25 nW.Open circles are data points and solid lines are the fitted Gaussian shapes.

Fig. S3 .
Fig. S3.Mean square displacement (MSD) curves while holding the time-averaged power relatively constant and changing the repetition rates and fluences.The TPLM data are shown in Fig. 2e.

Fig. S7 .
Fig. S7.Simulated data and analysis.(a) 2D color maps of unnormalized (left) and normalized (right) simulated transient photoluminescence microscopy data considering both dark counts and shot noise, assuming exciton diffusivity  = 0.01  2 / and initial spot size (FWHM) of

Table S1 .
Experimental conditions for the power-dependent photoluminescence spectra shown in Fig.1c: excitation at 405 nm, 2.5 MHz, spot size FWHM ~500 nm.

Table S2 .
Experimental conditions for the transient photoluminescence data shown in Fig.1d: excitation at 405 nm, 5 MHz, spot size FWHM ~500 nm.