Statistical analysis of time-resolved emission from ensembles of semiconductor quantum dots: interpretation of exponential decay models

We present a statistical analysis of time-resolved spontaneous emission decay curves from ensembles of emitters, such as semiconductor quantum dots, with the aim to interpret ubiquitous non-single-exponential decay. Contrary to what is widely assumed, the density of excited emitters and the intensity in an emission decay curve are not proportional, but the density is a time-integral of the intensity. The integral relation is crucial to correctly interpret non-single-exponential decay. We derive the proper normalization for both a discrete, and a continuous distribution of rates, where every decay component is multiplied with its radiative decay rate. A central result of our paper is the derivation of the emission decay curve in case that both radiative and non-radiative decays are independently distributed. In this case, the well-known emission quantum efficiency can not be expressed by a single number anymore, but it is also distributed. We derive a practical description of non-single-exponential emission decay curves in terms of a single distribution of decay rates; the resulting distribution is identified as the distribution of total decay rates weighted with the radiative rates. We apply our analysis to recent examples of colloidal quantum dot emission in suspensions and in photonic crystals, and we find that this important class of emitters is well described by a log-normal distribution of decay rates with a narrow and a broad distribution, respectively. Finally, we briefly discuss the Kohlrausch stretched-exponential model, and find that its normalization is ill-defined for emitters with a realistic quantum efficiency of less than 100 %.