Effects of size and shape on electronic states of quantum dots

A strained-modified, single-band, constant-potential three-dimensional model is formulated to study the dependence of electronic states of $\mathrm{InAs}∕\mathrm{GaAs}$ quantum dots (QDs) on shape and size variation. The QD shapes considered are (i) cuboid, (ii) cylindrical, (iii) pyramidal, (iv) conical, and (v) lens shaped. Size variations include (i) QD volume (ii) QD base length, and (iii) QD height, taking into account aspect ratio variation. Isovolume QD shapes with narrow tips were found to have higher ground-state energies than those with broad tips, and this is attributed to the smaller effective volume. The volume, base length, and height dependencies were obtained and found to tally well with both experimental results and advanced calculations. Hence, upon growth parameter variation, this can provide an alternative to confirm whether the change to the size of the uncapped QDs implies a similar change to the capped ones. Ground-state energy as function of aspect ratio does not follow a monotonic trend. Owing to the competing effect of a decrease in base length and an increase in height, the energy trend exhibits a sharp decrease to an optimum aspect ratio, followed by gentle, almost linear increase. The optimum aspect ratio varies among shapes and is predicted to be smaller for shapes with broad tips. The effective volume ratio of both shapes $(V_{\mathrm{eff},\mathrm{CUBOID}}∕V_{\mathrm{eff},\mathrm{PYRAMID}})$ was determined, and found to vary with aspect ratio. Furthermore, a “cross-over” of lens-shaped QD from “lower energy” to “higher energy” group is predicted due to significant shape transition.