Electronic structure of self-organized InAs/GaAs quantum dots bounded by $\{136\}$ facets

Recent experiments indicate that the shape of self-organized InAs quantum dots grown on GaAs [001] is an elongated pyramid with bounding facets corresponding to a family of four $\{136\}$ planes. This structure, which possesses $C_{2v}$ symmetry, is quite different from square-base pyramidal or lens geometries, which have been assumed in previous electronic structure calculations for this system. In this paper, we consider theoretically the influence of the $\{136\}$ shape on the electronic structure and optical properties of the quantum dots. We present a valence force-field calculation of the inhomogeneous strain and incorporate the results into an eight band $\overrightarrow{k}\cdot \overrightarrow{p}$ electronic structure calculation. The size dependence of the electronic structure is calculated and compared to experimental photoluminescence spectra. The effects of perturbations on the $\{136\}$ shape are also considered. Calculations based on the $\{136\}$ shape give good agreement with the observed level structure and optical polarization properties of self-organized InAs/GaAs quantum dots.