Quantifying non-Gaussian diffusion in transient microscopy using excess kurtosis

TL;DR

This work uses transient scattering microscopy to visualize exciton transport in bulk $WSe_2$ and reveals pronounced non-Gaussian diffusion as quantified by excess kurtosis $EK$. Through repetition-rate and fluence studies, it distinguishes early, surviving-population effects (positive $EK$) from late-time dynamics dominated by shallow traps (negative $EK$), supported by numerical simulations that include Meitner-Auger recombination and trap states. The authors show Gaussian fits can misestimate diffusivity under non-Gaussian conditions, proposing a robust discrete variable method to extract $D$ from 2D distributions, with a measured diffusivity around $4~\mathrm{cm^2/s}$. Overall, kurtosis emerges as a powerful diagnostic for identifying anomalous diffusion in transient microscopy data and guiding analysis beyond Gaussian assumptions.

Abstract

Research on energy transport has advanced in recent years with the emergence of transient microscopy techniques that allow for imaging of carriers with high spatial and temporal resolution. In this context, transient scattering microscopy (TScM), has emerged as an alternative to traditional techniques. However, the sensitivity of TScM to different carriers can complicate the interpretation of results, highlighting the need to develop models tailored to TScM. Here, TScM is used to visualize exciton transport in bulk TMDCs. We show that exciton populations exhibit non-Gaussian profiles by analyzing the their excess kurtosis. Numerical simulations incorporating anomalous diffusion -- such as Auger recombination and trap states -- reproduce these experimental observations. Furthermore, by tuning the injected carrier density, we demonstrate that the temporal signature of the kurtosis is distinct for Auger-dominated and trap-dominated regimes. Additionally, we find that traditional Gaussian-fitting methods can yield inconsistent results for the extracted diffusivities. As an alternative, we implement a discrete variable calculation which yields robust, consistent diffusivity values. Our results establish kurtosis as a vital diagnostic parameter for identifying anomolous diffusion and demonstrate the necessity of moving beyond Gaussian approximations for accurate analysis of TScM data.

Figures (14)

• Figure 1: a. Schematic of the transient scattering microscopy setup. b. Transient scattering differential images taken at different pump-probe time delays. c. Corresponding azimuthally averaged profiles of the pictures shown in b., dark points are the resulting averaged data points while dashed red lines show the best gaussian fit for the provided data. d. Excess kurtosis as a function of time for the complete dataset shown in b., kurtosis is calculated using the full 2D image for each time delay to increase SNR.
• Figure 2: a. Excess kurtosis as a function of time for selected datasets acquired with different laser repetition rates. b. Numerical simulations for different dynamics representing the corresponding evolution of the excess kurtosis as a function of time. Black corresponds to pure Gaussian diffusion with no other effects, the red line represents the dynamics when Meitner-Auger recombination is present, the blue line shows the evolution of the kurtosis for dynamics dominated by trap states and the green line the calculation for a combination of non-interacting heat and exciton populations.
• Figure 3: a. Excess kurtosis as a function of time for different injected carrier densities. b. Total calculated intensity contrast for the measured laser fluences.
• Figure 4: a. Extracted values for $\Delta \sigma^2$ using one-dimensional gaussian fits from linecuts for selected injected energy densities. b. Variances extracted from the same datasets as a. using the discrete variable approach on the two-dimensional distributions. c. Comparison of the calculated diffusivities for each method at different incident fluences, all diffusivities are obtained by performing a linear fit on the first 1.5 ns, before the sublinear diffusion regime.
• Figure S1: a. Slow component of the double Gaussian fit. b. Fast component of the Gaussian fit c. Variance extracted from the azimuthally averaged profiles using the discrete variable method d. Ratio of fast (excitons) vs slow (heat) components, the values are extracted from the double Gaussian fit