Epitaxial self-assembled quantum dots (SAQDs) are of interest for nanostructured optoelectronic and electronic devices such as lasers, photodetectors and nanoscale logic. Spatial order and size order of SAQDs are important to the development of usable devices. It is likely that these two types of order are strongly linked; thus, a study of spatial order will also have strong implications for size order. Here a study of spatial order is undertaken using a linear analysis of a commonly used model of SAQD formation based on surface diffusion. Analytic formulas for film-height correlation functions are found that characterize quantum dot spatial order and corresponding correlation lengths that quantify order. Initial atomic-scale random fluctuations result in relatively small correlation lengths (about two dots) when the effect of a wetting potential is negligible; however, the correlation lengths diverge when SAQDs are allowed to form at a near-critical film height. The present work reinforces previous findings about anisotropy and SAQD order and presents as explicit and transparent mechanism for ordering with corresponding analytic equations. In addition, SAQD formation is by its nature a stochastic process, and various mathematical aspects regarding statistical analysis of SAQD formation and order are presented.
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