  In this work, we present and evaluate a (111)-rotated eight-band
$\mathbf{k}\cdot\mathbf{p}$ 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
$\mathbf{k}\cdot\mathbf{p}$ 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.
