Basic properties of nanocrystal assemblies
To investigate the effect of NC size and order, we fabricated films by drop-casting 14 nm large cubic PNCs synthesized via tip-sonication as described previously (Fig. 1a).22 Optical microscopy and scanning electron microscope (SEM) imaging reveal dense films with no high-range ordering. While there are voids around 50 nm in size, these are few and far between. Some PNCs form closed packages of roughly several tens of PNCs, but larger symmetry of such structured clusters is missing, i.e., these clusters are randomly orientated to each other. These films are in sharp contrast to superlattices (SL) visible in Fig. 1b,c via SEM imaging formed out of 8 nm and 5 nm cubes, respectively. The cubes are synthesized via the hot injection method and form SLs through solvent-drying-induced spontaneous assembly.23 The SLs are highly ordered over several microns. No voids or larger/smaller PNCs disturb the SL order. The smaller PNC cubes have lower relative deviations of their lateral size, as observed in transmission electron microscopy (TEM) images (Figure S1), resulting in a more ordered film formation. All PNCs exhibit high optical quality, as demonstrated by their intense, narrow (full width at half maximum – fwhm \(=75-93\)meV) emission spectra (Fig. 1d, Table S1). The PL peak blueshifts due to quantum confinement from 515 nm to 498 nm as the PNC size shrinks. The spacing between adjacent PNCs is determined by the organic ligands used to passivate the PNCs and amounts to approximately 2 nm in all films.24 This separation precludes charge transfer between adjacent NCs. Thus, the only possible way to transfer electron hole (e-h) pairs through the films, regardless of size and order of the NCs, is via resonant energy transfer.16
FRET processes are multiplicative and dependent on several conditions, rendering the processes non-efficient if even one of these conditions is weakly fulfilled: 1) The donor photoluminescence (PL) and acceptor absorption spectra should have a sizable overlap, 2) the orientation of the donor and the acceptor dipole moments should be aligned, i.e., as collinear as possible, and 3) the donor-acceptor distance should be small (not more than 10 nm)25. Fortunately, those conditions are highly fulfilled for PNCs, as seen in many preliminary works.9, 16, 17
Low temperature regime: Bright-dark exciton splitting
To track exciton diffusion in the PNC structures, we have realized a PL microscope inside a closed-cycle cryostat, yielding spatially and temporally resolved exciton dynamics at temperatures down to 9 K (Figure S2). Briefly, a laser is focused onto the NC assemblies, exciting electron-hole pairs that rapidly form excitons, which can emit photons and diffuse within the nanostructures. We ensure that we are in a low-power excitation regime to preclude non-linear or memory effects from playing a role.26 A single photon avalanche detector (SPAD) is scanned over the magnified focal plane of the PL, recording a PL decay trace at each spatial position with a temporal resolution of 25 ps (Figures S3).11, 12 With temperature affecting many relevant processes for exciton diffusion, such as thermally-induced hopping (Figure S4), trap formation and binding, and exciton formation and dissociation, this setup can shed light on the nature of the diffusion processes.11, 27, 28 In the measurements, the exciton concentration in an optically excited NC ensemble corresponds directly to the spatial PL emission profile, given by a Voigt function. A broadening of this function is indicative of exciton diffusion. Accordingly, we monitor the change in the PL profile variances \({\sigma }^{2}\) (from the Gaussian part of the distribution) in 25 ps steps after excitation, leading to the mean-squared displacement (MSD):
At low temperatures, we find a striking difference between the smallest and largest NCs. At 9 K, the MSD increases rapidly and nearly linearly within the first one to two nanoseconds for the 5 nm and 8 nm SLs (Fig. 2a). The MSD, however, plateaus and then begins to decline, reaching a value of \(MSD=0\frac{c{m}^{2}}{s}\) after 2 ns. This latter behavior can be naively interpreted as a negative diffusion. In contrast, for the 14 nm cubes, the MSD values continuously increase in time over the same time range as for the smaller NCs (see Figure S5).
This apparent negative diffusion is also visible at 20 K; however, the MSD does not return to 0, remaining at a constant value for higher times. At 40 K, the negative diffusion disappears, and the MSD increases continuously; yet, much faster within the first 500 ps than subsequently. This bimodal behavior becomes less prominent for higher temperatures until the MSD increase becomes monotonous for \(T>80\) K. The initial rapid increase is the same for all these temperatures, suggesting a similar diffusion source. Negative diffusion has been observed previously and generally requires multi-component systems such as singlet-triplet states in organic semiconductors or intervalley exciton–phonon scattering in 2D transition metal dichalcogenides.29, 30
To explain why negative diffusion is prevalent in the 5 nm PNCs, but not in the 14 nm PNCs, we must consider the energetic structure of the exciton. Due to the strong quantum and dielectric confinement, the exciton manifold splits into three bright levels and one dark one (Fig. 2b). While there had previously been some debate about the ordering of the levels, it has now generally been accepted that the dark exciton is the energetically lowest level.31, 32 The existence of the dark exciton can be observed in the time decay of the PL, which displays two general lifetimes, one on the order of a few nanoseconds and the other on the order of hundreds of nanoseconds or microseconds (see Figure S6). Such a slow component can also arise from delayed fluorescence from the dissociation of excitons and subsequent trapping of single charge carriers.33 We observe the power-law decay associated with such a mechanism for the 14 nm NCs and also at higher temperatures for all samples. However, at low temperatues and for long times, the 5 nm SLs exhibit the typical exponential decay one would assign to the decay of a dark exciton (see Figure S6, right). The splitting between the bright and dark levels depends strongly on size. 34, 35 We estimate the splitting energies of the NCs used here to be:
\({{\Delta }E}_{BD}^{14 nm}=1.5 meV;\) \({{\Delta }E}_{BD}^{8 nm}=9 meV;\) \({{\Delta }E}_{BD}^{5 nm}=17 meV\) (Fig. 2c).20, 35, 36 Using a Boltzmann distribution, we find that at the lowest temperatures, only the 14 nm NCs possess a significant bright-state occupation (see Figure S7). Accordingly, for the smaller NCs, excitons will quickly relax to the dark state, in which they are then fundamentally trapped, while in the larger NCs, excitons will readily switch between bright and dark states. The FRET process, which is likely responsible for the diffusion, is based on dipole-dipole coupling. With the dark states possessing nearly no dipole component (low oscillator strength compared to the bright exciton levels), the probability of a FRET hopping of excitons in the dark state from one exciton to the next is negligible. Thus, only excitons in bright states can undergo FRET processes, and excitons in dark states must first be thermally excited into an upper bright state before they can also diffuse, as indecated in the sketch in Fig. 2d by white arrows.
To confirm whether the dark exciton state can cause the observed negative diffusion, we implemented a Monte-Carlo simulation for this system (see Methods for details). In this model, the excitons are initially created nearly equally in bright/dark states. This results from the PNCs being excited far above the exciton levels within the e-h continuum and the charge carriers populating the exciton levels nearly equally upon carrier cooling.35 An exciton in its bright state can either recombine with a lifetime of approximately 1.5 ns, can hop to an adjacent NC or relax down to a dark state. Once inside the dark state, the recombination lifetime increases up to 1 µs, as previously observed,20, 21 hopping no longer occurs, and the exciton can only be thermally excited to the higher-lying bright states at higher temperatures. The simulated MSD curves match the experimental trends nicely, with a clear negative diffusion trend observable for the lowest temperatures and a bimodal increase for elevated temperatures. The bright excitons can diffuse between NCs, which initiates rapid diffusion. However, these recombine quickly or relax into dark states, which possess lifetimes in the microsecond regime and do not exhibit diffusion. Once all excitons have vacated the bright levels, the distribution returns to the original width, where the dark exciton population is also approximately equal to the size of the exciting laser spot. Accordingly, the apparent negative diffusion is not an actual diffusion process but is caused by the two different exciton species (bright and dark) with drastically different inter-NC transfer rates. At higher temperatures, thermal excitation of the dark excitons becomes progressively more likely, reducing the negative diffusion and inducing the monotonous diffusion for higher temperatures. For the largest NCs with the smallest bright-dark splitting, the trapping effect into the dark exciton state is not observable and, at most, reduces the overall diffusion.
High temperature regime: Trap states and exciton dissociation
As the temperature increases further, the MSD curves of all NC samples exhibit a steeper increase signifying an enhanced diffusion. Three effects can explain this: 1) FRET rates increase with temperature,37 2) the increased exciton lifetime allows the excitons to diffuse for a longer time than at low temperatures, and 3) thermally activated hopping is more efficient at higher T, reducing the detrimental effect of an inhomogeneous energetic landscape (see Figure S4). The observed effect progresses up to 100–150 K, upon which we again observe a contraction of the MSD (Fig. 3a,b), albeit for much longer times on the order of 10–30 ns. Moreover, the time at which the contraction begins, \({t}_{max} ,\)is highly temperature dependent, with higher temperatures leading to larger values. The bright-dark exciton state cannot explain this observation, as at these temperatures, excitons do not reside long in the dark state, even for the smallest PNCs. So far, we have neglected any contribution from trap states, which can bind excitons and charge carriers and are often detrimental to optoelectronic properties.38 Accordingly, we reworked our Monte-Carlo simulation to reproduce the observed MSD traces. In this new model, we introduced a trapping rate of free excitons into deep defects from which the excitons can still emit (albeit with a radiative rate of 1/100th that of the free exciton) but not be detrapped (see Figure S8). The result is the bottom curve in Fig. 3c, which produces a near-linear increase in MSD up to a maximum value and time (denoted \({t}_{max}\)), subsequently decaying back to zero. An additional contribution due to shallow traps from which the excitons can be detrapped, would only reduce the overall diffusion (analogous to the previous bright-dark model with thermal excitation) and not shift the \({t}_{max}\) value and is thus omitted from the new model. The increase in the initial diffusion occurring for higher temperatures can easily be reproduced by varying the inter-NC hopping rate, but this does not affect the \({t}_{max}\) value (Figure S9). The only way to shift this value to later/earlier times is through a variable trap density, which we incorporate by modifying the trapping rate of the free excitons (Fig. 3c,d). Notably, a lower trapping rate (corresponding to a lower trap density) shifts \({t}_{max}\) to higher values, which agrees nicely with the experimentally observed trends (Fig. 3e). Interestingly, we observe essentially constant \({t}_{max}\) values up to a specific temperature (14 nm: 140 K, 8 nm: 180 K, 5 nm: 220 K), after which the values increase abruptly. This result implies that the number of trap states is reduced as the temperatures increase. This can be explained by assuming that the traps in the NCs exhibit a breadth of energies that are all deep enough to preclude detrapping at lower temperatures. At higher temperatures, a certain fraction can no longer be considered deep enough and no longer bind the excitons indefinitely, reducing the effective trap density. The trap depth seems to vary with NC size, either due to the exciton Bohr radius or the binding energy, which are inversely proportional to each other, or the larger surface-to-volume ratio for smaller NCs, since traps are more likely to be located at the NC boundaries.
Temperature dependence of diffusion parameters
Understanding these processes, we evaluated the diffusion for all PNC samples and temperatures. Depending on the system and the experimental circumstances, various methods can be employed to reproduce the observed MSD curves, including a subdiffusion model39 and trapping models of varying complexity.39, 40, 41 While each model can be applied to specific combinations of PNC samples and temperature ranges, none of these models can reproduce the complete set of MSD curves of all three PNC types. However, none of the previously reported data suggests that diffusion should occur differently for the three systems. Accordingly, we numerically determine the diffusivity for all systems and temperatures with the following equation:
The average diffusivity is given by summing the differences in MSD from two subsequent time bins (separated by \({t}_{bin}= {\text{t}}_{i+1} - {\text{t}}_{\text{i}} = 25 \text{p}\text{s}\)), which are weighted by the total PL intensity at the first time bin up to the time bin corresponding to five times the PL lifetime (\(N = 5*{\tau }_{PL}/{t}_{bin}\)). The PL lifetimes differ for each NC system and temperature. The total PL intensity \({I}_{PL}\left({t}_{i}\right)\) is the sum over the PL counts of all captured PL decay traces for each time bin and the PL lifetime corresponds to the time after which total PL intensity has dropped to 1/e of the initial value. This way, we can account for the exciton population, i.e., in early times, more excitons are present. Thus those times should have a more considerable impact on the diffusivity compared to later times, where the exciton population rapidly decreases due to radiative and nonradiative recombination. If normal diffusion is present, i.e., a linear MSD behavior with time, its slope is constant, and Eq. 2 expresses the normal diffusivity: \({D}_{normal}=\frac{1}{2}\frac{\varDelta MSD}{\varDelta t}.\)Importantly, we assure that even at these very late times, when most of the exciton population has decayed, their spatial distribution can still be given by a Voigt function (see Figure SI10). With the so-obtained diffusivity and the PL lifetime measured through time-correlated single photon counting (TCSPC), we can then calculate the diffusion length \(L=\sqrt{{\tau }_{PL}{D}_{avg}}\). All three of these quantities are depicted in Fig. 4a-c.
At the lowest temperatures, both the \(5 nm\) and \(8 nm\) superlattices exhibit very low diffusivity, nearly two orders of magnitude lower than the \(14 nm\) films, which have a diffusivity of at 9 K (Fig. 4a). This is most likely due to the prevalence of the dark exciton, which severely limits diffusion in the SLs (see Fig. 2b,c). Also, considering a FRET-mediated diffusion process between equally spaced NCs (given by the ligand length), the distance covered by a single hopping process is equal to the size of the respective NCs, and so is nearly three times as large for the 14 nm NCs as for the 5 nm NCs. Accordingly, to cover the same distance in the 5 nm SLs, an exciton would need three successive hops. The diffusivities all increase with temperature, albeit at different rates, due to enhanced hopping rates (larger FRET rates and thermally activated hopping bridging energy inhomogeneities between adjacent NC (Figure S4)). Especially for the smaller NCs, a significantly increased thermal excitation of the excitons from dark to bright states can be observed, reflecting the largest exciton bright-dark splitting for these SLs. Between 80 K and 110 K, the diffusivities of the 14 nm NC films and 8 nm SLs are comparable \({D}_{avg}\approx {10}^{-1}\frac{c{m}^{2}}{s},\)peaking at 100 K and 110 K, respectively. Here, the order in the SL becomes important and can compensate for the smaller hopping distances. However, the diffusivity in the 5 nm SLs increases far slower, peaking at \({D}_{avg}\approx {4\cdot 10}^{-3}\frac{c{m}^{2}}{s}\) at 140 K\(.\) In this intermediate temperature range, the NC size and the order in the ensembles play similarly important roles in determining the diffusivities.
All samples show a steep decline in diffusivity for higher temperatures. To explain this, we again consider exciton traps in the NCs. Shallow traps likely increase in number with rising temperatures as defects are thermally ionized. Moreover, deep traps become progressively shallower with respect to thermal energy, leading to a transient binding, which reduces the overall diffusivity. The importance of bound exciton emission in perovskites was previously shown in organic/inorganic 2D and 3D halide perovskite films.42, 43 This assertion also matches the increasing \({t}_{max }\)values due to a lower deep trap density observed in the MSD measurements and Monte Carlo simulations (Fig. 3d,e).
At 200 K all three diffusivities become comparable, and for higher temperatures, the 5 nm SLs exhibit the highest values, yet, below those in our previous report on PNCs at temperatures around RT.44 This continued decline coincides with a steep rise in the PL lifetimes (Fig. 4b) and a significant drop-off in the total PL count rate at those temperatures (see Figure S11). However, as these two processes begin at higher temperatures than the diffusivity decline, the underlying effects must differ. At elevated temperatures, the thermal energy available noticeably affects the Coulomb interaction of electron and hole, in turn influencing the exciton to free charge carrier ratio in the NCs, as given by the Saha equation. This equation quantifies how the free charge carrier density increases with rising temperatures as excitons become progressively dissociated, especially for the less-confined NCs in the intermediate temperature ranges (Figure S12). Inter-NC transport of free charge carriers is prohibited, further reducing diffusivities. Additionally, the charged particles are more susceptible to binding to trap states, so in the more polarizable larger NCs, the diffusivity is even more severely reduced.
This discrepancy in the absolute diffusivity values to our previous publication is likely a result of the experimental setup. With the temperature measured just below the substrate but diffusion measurements carried out on top of the NC ensembles, which are hundreds of nanometers thick, the actual temperature in the NC samples is likely higher than indicated by the setup. We corroborate this through additional temperature-dependent PL measurements, which suggest that the temperature in the relevant part of the NC ensembles is significantly higher than anticipated (see Supporting Information, Figures S13, S14). Nevertheless, the diffusion lengths as derived from the PL decay time and diffusivity range between 10 nm and 250 nm (Fig. 4c)., which are in the order of previously determined values 12, 39, 44, 45