Spatial regularity of InAs-GaAs quantum dots: quantifying the dependence of lateral ordering on growth rate

The lateral ordering of arrays of self-assembled InAs-GaAs quantum dots (QDs) has been quantified as a function of growth rate, using the Hopkins-Skellam index (HSI). Coherent QD arrays have a spatial distribution which is neither random nor ordered, but intermediate. The lateral ordering improves as the growth rate is increased and can be explained by more spatially regular nucleation as the QD density increases. By contrast, large and irregular 3D islands are distributed randomly on the surface. This is consistent with a random selection of the mature QDs relaxing by dislocation nucleation at a later stage in the growth, independently of each QD's surroundings. In addition we explore the statistical variability of the HSI as a function of the number N of spatial points analysed, and we recommend N > 10³ to reliably distinguish random from ordered arrays.