Analytical solutions for strain in pyramidal quantum dots

Analytical solutions for strain in buried pyramidal quantum dots (QDs) of arbitrary truncation, assuming isotropy of the elastic constants, are presented for the first time. The expressions allow the strain to be evaluated extremely rapidly compared to atomistic, finite-element, or other numerical techniques. As a demonstration of the new results, calculations are performed for InAs QDs buried within a GaAs matrix for a range of truncations for the cases of an isolated QD, an isolated QD with a graded composition profile, and a QD array. Results show that strain magnitudes and trends are highly dependent upon geometry and composition. For example, the biaxial strain at the apex of a pyramid is shown to be negative, but becomes positive as the truncation increases.