Dependence of the electronic structure of self-assembled $(In,Ga)As∕GaAs$ quantum dots on height and composition

While electronic and spectroscopic properties of self-assembled $In1−xGaxAs∕GaAs$ dots depend on their shape, height, and alloy compositions, these characteristics are often not known accurately from experiment. This creates a difficulty in comparing measured electronic and spectroscopic properties with calculated ones. Since simplified theoretical models (effective mass, $k∙p$⁠, parabolic models) do not fully convey the effects of shape, size, and composition on the electronic and spectroscopic properties, we offer to bridge the gap by providing accurately calculated results as a function of the dot height and composition. Prominent features of our results are the following: (i) Regardless of height and composition, the confined electron energy levels form shells of nearly degenerate states with a predominant $s,p,…$ orbital character. On the contrary, the confined hole energy levels form shells only in flat dots and near the highest hole level (HOMO). (ii) In alloy dots, the electrons $s-p$ splitting depends weakly on height, while the $p-p$ splitting depends nonmonotonically due to alloy fluctuations. In pure, nonalloyed $InAs∕GaAs$ dots, both these splittings depend weakly on height. Furthermore, the $s-p$ splitting is larger, while the $p-p$ has nearly the same magnitude. For hole levels in alloy dots, the $s-p$ splitting decreases with increasing height (the splitting in tall dots being about four times smaller than in flat dots), whereas the $p-p$ splitting remains nearly unchanged. Shallow, pure, nonalloyed dots have a $s-p$ splitting of nearly the same magnitude, whereas the $p-p$ splitting is about three times larger. (iii) As height increases, the $s$ and $p$ characters of the wave function of the HOMO becomes mixed, and so does its heavy-hole and light-hole characters. (iv) In alloy dots, regardless of height, the wave function of low-lying hole states are localized inside the dot. Remarkably, in nonalloyed $InAs∕GaAs$ dots these states become localized at the interface as height increases. The localized states are nearly degenerate and polarized along $[11¯0]$ and [110]. This localization is driven by the peculiarities of the biaxial strain present in the nanostructure.