Exciton diffusion in two-dimensional metal-halide perovskites

Two-dimensional perovskites, in which inorganic layers are stabilized by organic spacer molecules, are attracting increasing attention as a more robust analogue to the conventional three-dimensional metal-halide perovskites. However, reducing the perovskite dimensionality alters their optoelectronic properties dramatically, yielding excited states that are dominated by bound electron-hole pairs known as excitons, rather than by free charge carriers common to their bulk counterparts. Despite the growing interest in two-dimensional perovskites for both light harvesting and light emitting applications, the full impact of the excitonic nature on their optoelectronic properties remains unclear, particularly regarding the spatial dynamics of the excitons within the two-dimensional (2D) plane. Here, we present direct measurements of in-plane exciton transport in single-crystalline layered perovskites. Using time-resolved fluorescence microscopy, we show that excitons undergo an initial fast, intrinsic normal diffusion through the crystalline plane, followed by a transition to a slower subdiffusive regime as excitons get trapped. Interestingly, the early intrinsic exciton diffusivity depends sensitively on the exact composition of the perovskite, such as the choice of organic spacer. We attribute these changes in exciton transport properties to strong exciton-phonon interactions and the formation of large exciton-polarons. Our findings provide a clear design strategy to optimize exciton transport in these systems.


Introduction
Metal-halide perovskites are a versatile material platform for light harvesting [1][2][3][4][5] and light emitting applications, 6,7 combining the advantages of solution processability with high ambipolar charge carrier mobilities, 8,9 high defect tolerance, [10][11][12] and tunable optical properties. [13][14][15] Currently, the main challenge in the applicability of perovskites is their poor environmental stability. [16][17][18][19] Reducing the dimensionality of the perovskite has proven to be one of the most promising strategies to yield a more stable performance. [20][21][22][23] Perovskite solar cells with mixed 2D and 3D phases, for example, have been fabricated with >22% efficiencies 24 and stable performance for more than 10.000 hours, 25 while phase pure 2D perovskite solar cells have been reported with efficiencies above 18%. 26,27 Likewise, significant stability improvements have been reported for phase pure 2D perovskites as the light emitting layer in LED technologies. [28][29][30][31][32][33][34] The improved environmental stability in 2D perovskite phases is attributed to a better moisture resistance due to the hydrophobic organic spacers that passivate the inorganic perovskite sheets, as well as an increased formation energy of the material. [20][21][22][23] However, the reduced dimensionality of 2D perovskites dramatically affects the charge carrier dynamics in the material, requiring careful consideration in their application in optoelectronic devices. 35-37 2D perovskites are composed of inorganic metal-halide layers, which are separated by long organic spacer molecules. They are described by their general chemical formula L2[ABX3]n-1BX4, where A is a small cation (e.g. methylammonium, formamidinium), B is a divalent metal cation (e.g.

Results
We prepare single crystals of n = 1 phenethylammonium lead iodine (PEA)2PbI4 2D perovskite by dropcasting a saturated precursor solution onto a glass substrate, 47-49 as confirmed by XRD analysis and photoluminescence spectroscopy (see methods section for details). Using mechanical exfoliation, we isolate single-crystalline flakes of the perovskite and transfer these to microscopy slides. The singlecrystalline flakes have typical lateral sizes of tens to hundreds of micrometers and are optically thick.
The use of thick flakes provides a form of self-passivation that prevents the typical fast degradation of the perovskite in ambient conditions.
To measure the temporal and spatial exciton dynamics, we create a near-diffraction-limited exciton population using a pulsed laser diode (ex = 405 nm) and an oil immersion objective (N.A. =  To analyze the time-dependent broadening of the emission spot in more detail, we study the temporal evolution of the mean-square-displacement (MSD) of the exciton population, given by ( ) = ( ) 2 − (0) 2 . Taking the one-dimensional diffusion equation as a simple approximation, it follows that ( ) = 2 , which allows us to extract the diffusivity and the diffusion exponent from our measurement (see SI for derivation). 46,50 In Figure 1e we plot the MSD as a function of time. background excitation of varying intensity ( Figure 2). The background excitation leads to a steady state population of excitons, which fill some of the traps and thereby reduce the effective trap density. To minimize the invasiveness of the measurement itself, the repetition rate and fluence were reduced to a minimum (see SI for details). In the absence of any background illumination, we find a strongly subdiffusive diffusion exponent of = 0.48 ± 0.02. As the background intensity is increased, an increasing is observed, indicative of trap state filling. Ultimately, a complete elimination of subdiffusion ( = 0.99 ± 0.02) is obtained at a background illumination power of 60 mW/cm 2 . For comparison, this value corresponds roughly to a 2.5 Sun illumination. Additionally, we observe that the onset of the subdiffusive regime is delayed as more and more trap states are filled, as represented by the increasing tsplit parameter (see Figure 2b, bottom panel).
To gain theoretical insights and quantitative predictions concerning the observed subdiffusive behavior of excitons and its relation with trap state densities, we performed numerical simulations based on Brownian dynamics of individual excitons diffusing in a homogeneously distributed and random trap field (see SI for details). In addition, we developed a coarse-grained theoretical model based on continuum diffusion of the exciton concentration (see SI for details). The continuum theory predicts an exponential decay of the exciton diffusion coefficient, where is the average distance between traps. The integral of this expression leads to which, as shown in Figure 2c, successfully reproduces both experimental and numerical results and allows us to accurately determine the value of the intrinsic trap state density, yielding The inset in Figure 2c shows the evolution of the effective trap state density 1 2 ⁄ with increasing illumination intensity. We note that the exponential decay of equation (1) allows for a more intuitive characterization of D(t) by relating the subdiffusion directly to the trap density 1 2 ⁄ rather than relying on the subdiffusive exponent of a power law commonly used in literature. 46 Importantly, the early diffusion dynamics are unaffected by the trap density. This gives us direct access to the intrinsic exciton diffusivity of the material and allows us to compare the exciton diffusivity between perovskites of different composition. To explore compositional variations, we substitute phenetylamonium (PEA) with butylammonium (BA) -another commonly used spacer molecule for two-dimensional perovskites. 20,28,29,32,43,[52][53][54] Figure 3a displays the MSD of the (BA)2PbI4 perovskite, again showing the distinct transition from normal diffusion to a subdiffusive regime. However, as compared to (PEA)2PbI4, excitons in (BA)2PbI4 are remarkably less mobile, displaying a diffusivity of only 0.013 ± 0.002 cm 2 /s, which is over an order of magnitude smaller than that of (PEA)2PbI4 with 0.192 ± 0.013 cm 2 /s (green curve shown in Figure 3a for comparison). Taking the exciton lifetime into account, the difference in diffusivity results in a reduction in the diffusion length from 236 ± 4 for (PEA)2PbI4 to a mere 39 ± 8 for (BA)2PbI4 (see Figure 3b). These results indicate that the choice of ligand plays a crucial role in controlling the spatial dynamics of excitons in two-dimensional perovskites. We would like to note that the reported diffusion lengths follow the literature convention of diffusion lengths in one dimension, as it is the relevant length scale for device design. The actual two-dimensional diffusion length is greater by a factor of √2.
To understand the large difference in diffusivity between (PEA)2PbI4 and (BA)2PbI4, we take a closer look at the structural differences between these two materials. Changing the organic spacer can have a significant influence on the structural and optoelectronic properties of 2D perovskites.
Specifically, increasing the cross-sectional area of the organic spacer distorts the inorganic lattice and reduces the orbital overlap between neighboring octahedra, which in turn increases the effective mass of the exciton. 55 Comparing the octahedral tilt angles of (PEA)2PbI4 and (BA)2PbI4, a larger distortion for the bulkier (PEA)2PbI4 (152.8°) as compared to (BA)2PbI4 (155.1°) is found. 56,57 The larger exciton effective mass in (PEA)2PbI4 would, however, suggest slower diffusion, meaning a simple effective mass picture for free excitons cannot explain the observed trend in the diffusivity between (PEA)2PbI4 and Rather than dealing with free excitons, a number of studies have pointed at the importance of strong exciton-phonon coupling and the formation of exciton-polarons in perovskite materials. 35,37,60,61 In the presence of an exciton, the soft inorganic lattice of the perovskite can be easily distorted through coupling with phonons, leading to the formation of polarons. As compared to a free exciton, an exciton-polaron exhibits a larger effective mass and, consequently, a lower diffusivity. The softer the lattice, the larger the distortion, and the heavier the polaron effective mass will be. 62 The correct theoretical description of the polaron in 2D perovskites is the subject of ongoing debate, though the current consensus is that the polar anharmonic lattice requires a description beyond conventional To further test the correlation between lattice softness and diffusivity, we have performed measurements on a wider range of two-dimensional perovskites with different organic spacers. In

Discussion
Given the apparent importance of exciton-phonon coupling in the spatial dynamics of excitons in 2D perovskites, structural rigidity becomes a critical design parameter in these systems. Taking into account the close correlation between diffusivity and the atomic displacement, this parameter space can be readily explored using available x-ray crystal structure data for many 2D perovskite analogues.
While the influence of the organic spacer is expected to be particularly strong in the class of n = 1 2D perovskites, we have observed consistent trends in the n = 2 analogues. Indeed, just like in n = 1, in n = 2 the use of the PEA cation yields higher diffusivities than for BA (see supporting information).
Similarly, the interstitial formamidinium (FA) cation in n = 2 yields higher diffusivity than the methylammonium (MA) cation, consistent with the trend in the atomic displacement parameters.

Sample Preparation
Chemicals: Chemicals were purchased from commercial suppliers and used as received: MX2: lead(II) iodide (PbI2) (Sigma Aldrich, 900168-5G).  The extracted lattice spacings are consistent with previously reported values of these materials. 4,5 Table S1 summarizes the XRD data of the perovskite crystals used in this study.
Binning improves the S/N ration of the data. With non-linear binning, the (S/N)k for every time-slice improves differently. As a result, every time-slice is weighted with the number of bins to correctly account for the improved (S/N)k due to binning. This multiplication with has an influence on the actual offset parameter c, that needs to be fitted to each time-slice k. After non-linear binning, every time-slice needs to be fitted with a local offset of = • , with c being the real/global offset of the data. Following this procedure, we are able to extract the variance 2 ( 2 ( ) in the previous section) for every time-slice, which are then used to generate the MSD (= 2 ( ) − 2 (0) = 2 − 0 2 ) vs. time plots, that allowed the extraction of the diffusion parameters D,  and tsplit.
We would like to note that the shortest lifetime of some perovskites is close to the width of the pulse-width of the pulsed laser diode. To minimize the influence of the laser and justify the assumption of negligible exciton generation during our measurements ( ( > 0) = 0) we only analyze the diffusion data 250 ps after the maximum of the photoluminescence lifetime data and define it as the new t = 0.
In the previous paragraphs, we derived that the MSD of normal diffusion in onedimension to be: ( ) = 2 . However, diffusion in disordered media, where the diffusivity is not constant for all times, is better described by introducing the diffusion exponent : ( ) = 2 . 9-11 With the diffusion exponent one can describe normal diffusion ( = 1), superdiffusion ( > 1, e.g. ballistic transport) and subdiffusion ( < 1, e.g. through trapping of excitons).
As described in the main text the MSD(t) of excitons in 2D perovskites shows two distinct diffusion regimes: First, a linear behavior of normal diffusion (2 , = 1), which is followed by a second subdiffusive regime (2 , < 1). We fit the two regimes simultaneously with the following fit function: with the fit parameters D, , c, and tsplit. c is generally small and is introduced to avoid overweighing the first datapoint at time t = 0. 0 = 1 1− was introduced to make MSD(t) continuous in value (excitons move continuously) and slope (speed of excitons changes continuously) at time tsplit. By fitting our experimental data with equation (S3) we were able to extract the diffusivity D, diffusion exponent , and the onset of subdiffusion tsplit from our measurements. We would like to note that fitting only the first linear regime of normal diffusion with 2 yields almost identical D values and  values of around 1.

Brownian modelling of trap states
We have performed Brownian dynamics simulations of a single exciton diffusing in a field of traps, representing ideal (non-interacting) excitons in the dilute limit carried out in experiments.
In these simulations, an exciton diffuses freely until it finds a trap, where it just stops. Free diffusion is modelled using the standard stochastic differential equation for Brownian motion in the Itô interpretation. If r(t) is the position of the exciton in the plane at time t, its displacement Δr over a time Δt is given by, where D is the free-diffusion coefficient and dW is taken from a Wiener process, such that ⟨ ⟩ = . Traps were scattered throughout the plane following a uniform random distribution. The exciton is considered to be trapped as soon its location gets closer than Rtrap = 1.2 nm to the trap center. The value was taken from estimations of the exciton Bohr radius and corresponds to a trap area of 1.44 nm 2 . 12 In any case, in the dilute regime, the diffusion is not sensitive to the trap size Rtrap, because the trap radius is much smaller than the average separation between traps, R trap ≪ . To numerically integrate the equation of motion, we used a simple second-order-accurate modification of the well-known Euler Maruyama algorithm: the BAOAB-Limit method. 13 Trajectories were computed for many independent excitons and the data was averaged to determine the MSD as a function of time.

Derivation of continuum model:
Besides studying the exciton dynamics using Brownian dynamics simulations, we derive a continuum model for the exciton diffusion in a plane having a uniform random distribution of traps. This coarser model solves the equation for the field of exciton concentration, or equivalently, for the probability field c(r,t) of finding an exciton at location r at time t. The resulting partial differential equation for c(r,t) is numerically solved using a finite difference scheme in a rectangular mesh of size h. Thus, in this description, r corresponds to the center of a given control cell of the mesh, whose area h 2 is much larger than the trap area 0 = concentration of mobile excitons will obey a reaction-diffusion equation, with a free diffusion term ( 2 ) and sink or loss rate term, s(r,t), The equation for trapped excitons is then simply, ( , ) = ( , ), and that for the total number of excitons, which is explicitly written in conservative form: note that the total exciton flux is just the diffusive flux of mobile excitons = − .
As we show below, the exciton's mean square displacement is directly given by equation (S4). But, at this stage one needs to model the loss rate s(r,t), provided that the surface fraction of trap p is fixed, and the trap area is s0. Recall that the average distance between traps is λ so, = 0 2 ⁄ and the trap density is 0 ⁄ . The loss rate has two contributions. First, over a time lapse Δt, the excitons located at the cell r (whose number is c(r,t) h 2 ) will diffuse an area

Calculation of sun-equivalent
The calculation of the sun-equivalent in Figure 2c was performed with the AM1.5 Global (ASTMG173) standard spectra. We extract that 1 sun contains 4.8 • 10 16 photons/s/cm 2 with an energy larger than the bandgap of (PEA)2PbI4 ( ℎ < ≈ 520 ). Assuming that the absorption of a photon with above-bandgap energy is wavelength independent and that every absorbed photon creates an exciton, one finds a 1 sun equivalent of 25 mW/cm 2 for a

Trap state dynamics
Influence of laser fluence: Figure S3 shows the lifetime traces of (PEA)2PbI4 recorded with a 10 MHz repetition rate and for different laser fluences of a near-diffraction limited spot. All traces show a multiexponential decay and the same early time dynamics, which shows that exciton-exciton annihilation is absent for laser fluences used in this study. On the other hand, the slow decaying component becomes slightly (logarithmic y-axis) more prominent with increasing laser fluence, which could be due to trap state filling. This is further supported by Figure S3b, where the total emission intensity (integrated lifetime data) is shown as a function of laser fluence. Exciton-exciton annihilation would result in sublinear behavior, but instead a slightly superlinear behavior is observed, which could originate from trap state filling. Influence of repetition rate: The influence or trap state filling is absent in the diffusion measurements performed with laser repetition rates of 5 MHz as shown in Figure S4. For both 50 nJ/cm 2 and 250 nJ/cm 2 , the diffusivity and diffusion exponent are the same. The finding that the laser fluence only influences the diffusion measurements for high repetition rates (40 MHz, Figure S5), but not for low ones (5 MHz, Figure S4), suggests that the traps are filled with excitons from previous pulses rather than from excitons which are generated with the same laser pulse. This can be explained by a long lifetime of trap states. With increasing laser fluence a higher percentage of laser pulses create excitons in the same inorganic layer as the following laser pulse, allowing the first exciton to fill a trap and thereby reducing the effective trap density experience by a second exciton generated by a later laser pulse. To guarantee a good signal to noise ratio most measurements in this study were performed with a 40 MHz laser repetition rate. As outlined above, this only changes the observed effective trap density (and thereby diffusion exponent ) but does not influence the intrinsic diffusivity of the material.

Determination of diffusion length.
Fluorescence lifetime measurements were performed using a laser diode of  = 405 nm (PicoQuant LDH-D-C-405, PDL 800-D, Pico-Harp 300) and an avalanche photodiode (APD, Micro Photon Devices PDM). The repetition rate was 10 MHz and the peak fluence per pulse was 50 nJ/cm 2 . Figure S7 shows the photoluminescence lifetime traces of (PEA)2PbI4 and (BA)2PbI4, and a tri-exponential fit to the data. The fit was used together with the experimentally obtained time-dependent MSD to extract the total number of surviving excitons for a given time t as presented in Figure 3b of the main text. The total number of surviving excitons at time t is given

Temperature dependent photoluminescence linewidth.
Fluorescence spectra were measured using a spectrograph with a 300 g/mm grating with a blaze of 500 nm (SpectraPro HRS-300) and an EMCCD camera (ProEM HS 1024BX3) from Princeton Instruments. The perovskites were excited with a blue LED (Thorlabs M385PLP1-C5,  = 385 nm).
The temperature of the perovskite crystal was varied with a Peltier element (Adaptive Thermal Management, ET-127-10-13-H1), using a PID temperature controller (Dwyer Instruments, Series 16C-3) connected to a type K thermocouple (Labfacility, Z2-K-1M) for feedback control and a fan for cooling. Figure S8 shows the temperature dependent emission spectrum for (BA)2PbI4 and (PEA)2PbI4. We applied the Jacobian conversion described by Mooney and Kambhampati to switch from wavelengths to energies. 15 Figure S8. Photoluminescence spectra of (BA)2PbI4 and (PEA)2PbI4 for different temperatures.
As several studies have shown before, the temperature dependent FWHM Γ( ) of photoluminescence of bulk [16][17][18][19] and 2D 20-25 perovskites can be described with the Bose−Einstein distribution, due to thermal occupation of phonon modes: where Γ is the optical phonon coupling strength, is the optical phonon energy, is the Boltzmann constant, and Γ 0 is the zero phonon linewidth. [26][27][28] For high temperatures, where the thermal energy is much greater than the phonon energy equation (S9) can be approximated through its asymptote: Literature values for (PEA)2PbI4 and (BA)2PbI4 range between around 10 to 20 meV for Γ 0 and Γ 0 = 0 and a phonon energy E LO = 30 . One can see that equation (S9) follows the linear behavior of its asymptote (equation (S10)) already for ≳ 300 . As a result, we used equation (S10) to fit our temperature dependent data and extract the optical phonon coupling strength Γ and optical phonon energy for (PEA)2PbI4 and (BA)2PbI4 (see Figure S10). The resulting Γ and values are listed in Table S2. Γ 0 was assumed to be 15 meV in accordance with literature values [20][21][22][23][24][25]29 . The exact value of Γ 0 is not critical for the fit, because Γ LO is normally significantly greater than Γ 0 . Our values, fit well with the previously reported values for (PEA)2PbI4 [20][21][22][23]29 and (BA)2PbI4. 24,25 However, we would like to note that due to the uncertainties in this technique the values listed in Table S2 should not be taken as quantitative results, but rather qualitative results to demonstrate the stronger exciton-phonon interaction in (BA)2PbI4 as compared to (PEA)2PbI4.   Figure S11 shows the diffusivity values for n = 1 perovskites (L2PbI4) with different organic spacers L. Plotting the diffusivity values as a function of average atomic displacement Ueq reveals an inverse relation and highlights the correlation of lattice softness and diffusivity. Figure S11 shows the plots for average atomic displacements (average formamidinium (FA)). We find that the organic spacer PEA yields higher diffusivities than BA, just like in the n = 1 case. As for the cation, FA yields higher diffusivities than MA. FA is a larger molecule than MA and fills out the PbI6-octahedra cage more completely. As a result, the PbI6 octahedra are tilted less for FA than for MA. Higher tilt angles were found to yield higher effective electron and hole masses in perovskites. 30,31 In addition, FA being a bulkier molecule results in a more rigid crystal as supported by the average atomic displacement values of the inorganic part (average U eq = 2 9 U eq Pb + 7 9 U eq I ) of (BA)2MAPb2I7 and (BA)2FAPb2I7 being 0.098 and 0.089 Å 2 , respectively. 8 It is important to note that Gélvez-Rueda et al. have measured an exciton to free carrier fraction of around 50 % for the n = 2 perovskite (BA)2FAPb2I7, suggesting that transport is not purely excitonic. 32 As a result, our current results serve as a qualitative scaling of the diffusivity and a more rigorous analysis of the n = 2 data would be needed for quantitative diffusivity values. Figure S12. (a) Diffusivity for n = 2 perovskites, containing two octahedra per inorganic layer, with chemical formula L2APb2I7. Organic spacers L were PEA or BA. Cation A were methylammonium (MA) or formamidinium (FA). Perovskites with PEA show higher diffusivities than perovskites with BA. Additionally, the diffusivity is higher if FA is used as cation instead of