  The properties of an exciton in a type II quantum dot are studied under the
influence of a perpendicular applied magnetic field. The dot is modelled by a
quantum disk with radius $R$, thickness $d$ and the electron is confined in the
disk, whereas the hole is located in the barrier. The exciton energy and
wavefunctions are calculated using a Hartree-Fock mesh method. We distinguish
two different regimes, namely $d<<2R$ (the hole is located at the radial
boundary of the disk) and $d>>2R$ (the hole is located above and below the
disk), for which angular momentum $(l)$ transitions are predicted with
increasing magnetic field. We also considered a system of two vertically
coupled dots where now an extra parameter is introduced, namely the interdot
distance $d_{z}$. For each $l_{h}$ and for a sufficient large magnetic field,
the ground state becomes spontaneous symmetry broken in which the electron and
the hole move towards one of the dots. This transition is induced by the
Coulomb interaction and leads to a magnetic field induced dipole moment. No
such symmetry broken ground states are found for a single dot (and for three
vertically coupled symmetric quantum disks). For a system of two vertically
coupled truncated cones, which is asymmetric from the start, we still find
angular momentum transitions. For a symmetric system of three vertically
coupled quantum disks, the system resembles for small $d_{z}$ the pillar-like
regime of a single dot, where the hole tends to stay at the radial boundary,
which induces angular momentum transitions with increasing magnetic field. For
larger $d_{z}$ the hole can sit between the disks and the $l_{h}=0$ state
remains the groundstate for the whole $B$-region.
