  We study the effect of valence band spin-orbit interactions on the acoustic
phonon assisted spin relaxation of holes confined in quantum dots. Heavy
hole-light hole (hh-lh) mixing and all the spin-orbit terms arising from
zinc-blende bulk inversion asymmetry (BIA) are considered on equal footing in a
fully 3D Hamiltonian. We show that hh-lh mixing and BIA have comparable
contributions to the hole spin relaxation in self-assembled QDs, but BIA
becomes dominant in gated QDs. The dependence of the hole spin relaxation on
the QD geometry and spin splitting energy is drastically different from that of
electrons, with a non-monotonic behavior which results from the interplay
between SOI terms. Our results reconcile contradictory predictions of previous
theoretical works and are consistent with experiments.
