Hierarchical Disorder in Moir\'e Exciton Photoluminescence Probed by Spectral-Descriptor Correlations

Abstract

Hyperspectral photoluminescence (PL) maps of moir\'e transition-metal dichalcogenide heterobilayers encode rich information about the underlying disorder, but extracting it is hampered by the ambiguity of assigning individual spectral peaks. We show that the spatial correlations among a set of simple, peak-decomposition-free spectral descriptors -- such as the centroid energy, the dominant-peak energy, and a sharp-line fraction -- provide a peak-decomposition-free route to infer features of that disorder. Different descriptors act as filters that select different components of a multi-scale disorder landscape -- a smooth, micron-correlated background and a dense set of localized traps -- and therefore acquire different spatial correlation lengths. The central prediction is a correlation-length hierarchy, $\xi(E_{\rm cent}) \ge \xi(E_{\rm dom})$, which we derive by splitting the dominant-peak energy into a smooth background part and a short-range trap-switching fluctuation. The same picture explains the measured inter-descriptor correlations, including the near-perfect anticorrelation $\rho_{\rm S}(\Delta E_{\rm cd}, R_{\rm HL}) \approx -0.978$, which we show to be a robust geometric trend for spectra dominated by a common emission-envelope asymmetry. Benchmarked against phenomenological simulations, Hamiltonian diagonalization, and the measured MoSe$_2$/WSe$_2$ descriptor correlations, the framework turns descriptor maps into a quantitative, peak-decomposition-free probe of slow disorder and local traps in moir\'e excitons and, more broadly, in disordered semiconductor emitters.