  We study the influence of spin-orbit coupling on the hole states in InAs/GaAs
quantum dots grown on [001]- and [111]-oriented substrates belonging to
symmetry point groups: C2v, C3v and D2d. We investigate the impact of various
spin-orbit mechanisms on the strength of coupling between s- and p-shell
states, which is a significant spin-flip channel in quantum dots. We calculate
spin relaxation rates between the states of lowest Zeeman doublet and show that
the [111]-oriented structure offers one order of magnitude slower relaxation
compared to the usual [001]-oriented self-assembled QD. The magnetic-field
dependence of the hole states is calculated using multiband (up to 14 bands)
k.p model. We identify the irreducible representations linked to the states and
discuss the selection rules, which govern the avoided-crossing pattern in
magnetic-field dependence of the energy levels. We show that dominant
contribution to the coupling between some of these states comes from the shear
strain. On the other hand, we demonstrate no coupling between s- and p-shell
states in the [111]-oriented structure.
