Analytical models for strained heteroepitaxial quantum dot systems have invariably assumed that the dots have a low-aspect ratio (small slopes) and that the elastic properties of the dot and the substrate are identical. In this paper, a three-dimensional analytical model for the energetics of an array of axisymmetric quantum dots is developed from physical principles. This is valid for high-aspect ratio dots (such as GeSi and InGaAs) and allows the dot and substrate to have different elastic properties. It is shown that these features are very important in determining the strain energy of both isolated dots and arrays of interacting dots. Both the elastic relaxation energy (per unit volume) of a single dot and the elastic interaction energy (per unit volume) between multiple dots are found to be greatest for tall, steep dots and for dots which are stiffer than the substrate. The equilibrium of two-facet dots is investigated and shape transition phase diagrams for small slope monoelastic theory, GeSi and InGaAs are compared. Different features of the bimodal dot size distributions in these systems are explained
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