  Phonons strongly influence the optical control of semiconductor quantum dots.
When modeling the electron-phonon interaction in several theoretical approaches
the quantum dot geometry is approximated by a spherical structure, though
typical self-assembled quantum dots are strongly lens-shaped. By explicitly
comparing simulations of a spherical and a lens-shaped dot using a
well-established correlation expansion approach we show that indeed lens-shaped
dots can be exactly mapped to a spherical geometry when studying the phonon
influence on the electronic system. We also give a recipe to reproduce spectral
densities from more involved dots by rather simple spherical models. On the
other hand, breaking the spherical symmetry has a pronounced impact on the
spatio-temporal properties of the phonon dynamics. As an example we show that
for a lens-shaped quantum dot the phonon emission is strongly concentrated
along the direction of the smallest axis of the dot which is important for the
use of phonons for the communication between different dots.
