LAUSR

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: ${C}_{2\mathrm{v}}$, ${C}_{3\mathrm{v}}$, and ${D}_{2\mathrm{d}}$. 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) $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ 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.