In this work, we present and evaluate a (111)-rotated eight-band $mathbf{k}cdotmathbf{p}$ Hamiltonian for the zinc-blende crystal lattice to investigate the electronic properties of site-controlled InGaAs/GaAs quantum dots grown along the [111] direction. We derive the rotated Hamiltonian including strain and piezoelectric potentials. In combination with our previously formulated (111)-oriented continuum elasticity model, we employ this approach to investigate the electronic properties of a realistic site-controlled (111)-grown InGaAs quantum dot. We combine these studies with an evaluation of single-band effective mass and eight-band $mathbf{k}cdotmathbf{p}$ models, to investigate the capabilities of these models for the description of electronic properties of (111)-grown zinc-blende quantum dots. Moreover, the influence of second-order piezoelectric contributions on the polarisation potential in such systems is studied. The description of the electronic structure of nanostructures grown on (111)-oriented surfaces can now be achieved with significantly reduced computational costs in comparison to calculations performed using the conventional (001)-oriented models.
We demonstrate that cation-related localized states strongly perturb the band structure of $text{Al}_{1-x}text{In}_x$N leading to a strong band gap bowing at low In content. Our first-principles calculations show that In-related localized states are formed both in the conduction and the valence band in $text{Al}_{1-x}text{In}_x$N for low In composition, $x$, and that these localized states dominate the evolution of the band structure with increasing $x$. Therefore, the commonly used assumption of a single composition-independent bowing parameter breaks down when describing the evolution both of the conduction and of the valence band edge in $text{Al}_{1-x}text{In}_x$N.
