Determination of the particle size distribution of cube-shaped colloidal perovskite quantum dots from photoluminescence spectra: A combined theoretical–experimental approach

A theoretical-experimental approach is proposed to convert the photoluminescence spectra of colloidal perovskite quantum dot ensembles into accurate estimates for their intrinsic particle size distribution functions. Two main problems were addressed and properly correlated: the size dependence of the first excitonic transition in a single cube-shaped quantum dot and the inhomogeneous broadening of the fluorescence line shape due to the size nonuniformity of the chemically prepared quantum dot suspension in addition to the single-dot homogeneous broadening. By applying the reported methodology to CsPbBr3 quantum dot samples belonging to the strong and intermediate confinement regimes, the calculated size distributions exhibited close agreement with those obtained from transmission electron microscopy, with precise estimates for the average particle size and standard deviation. Specifically for strongly confined ultrasmall CsPbBr3 quantum dots, the presented spectroscopic model for size distribution computation is based on a new analytical expression for the size-dependent bandgap, which was developed within the framework of the finite-depth square-well effective mass approximation accounting for band nonparabolicity effects. Such a quantum mechanical approach correctly predicts the expected transition to the intermediate confinement regime in sufficiently large quantum dots, which are traditionally described by the well-known bandgap equation in the infinite potential barrier limit with a spatially correlated electron-hole wavefunction and nonparabolic carrier effective masses. The proposed calculation scheme originates from general theoretical considerations so that it can be readily adapted to semiconductor quantum dots of many other systems, from all inorganic metal halides to hybrid perovskite materials, regardless of the adopted chemical synthesis route.