We present a statistical analysis of time-resolved spontaneous emission decay curves from ensembles of emitters, such as semiconductor quantum dots, with the aim of interpreting 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 by its radiative decay rate. A central result of our paper is the derivation of the emission decay curve when both radiative and nonradiative decays are independently distributed. In this case, the well-known emission quantum efficiency can no longer be expressed by a single number, but 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%.
A real-space study is presented on the occurrence of stacking faults in crystals of silica colloids with diameters of about 1 and 1.4 m formed through sedimentation. The softness of the interaction potential is varied from slightly repulsive to hard-sphere like, both intrinsically by variation of the diameter, as well as through the addition of salt, which screens the surface charges. Our results indicate that the equilibrium crystal structure for these colloids is an fcc-crystal, with the number of stacking faults determined by the interplay between sedimentation and crystallization kinetics, irrespective of the softness of the interaction potential. For spheres with a certain diameter the number of stacking faults decreases with decreasing initial volume fractions. These results provide a way to grow fcc-crystals of hard-sphere particles by slow sedimentation. The relative number of stacking faults in the first few layers above the bottom wall can be as much as a factor of 10 higher than deeper into the crystal. This effect is due to the crystallization kinetics on a plain wall in a gravitational field. A patterned bottom wall that favors a specific hexagonal orientation was found to drastically reduce the number of stacking faults in the crystal.
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