  In self assembled III-V semiconductor quantum dots, valence holes have longer
spin coherence times than the conduction electrons, due to their weaker
coupling to nuclear spin bath fluctuations. Prolonging hole spin stability
relies on a better understanding of the hole to nuclear spin hyperfine coupling
which we address both in experiment and theory in the symmetric (111)
GaAs/AlGaAs droplet dots. In magnetic fields applied along the growth axis, we
create a strong nuclear spin polarization detected through the positively
charged trion X$^+$ Zeeman and Overhauser splittings. The observation of four
clearly resolved photoluminescence lines - a unique property of the (111)
nanosystems - allows us to measure separately the electron and hole
contribution to the Overhauser shift. The hyperfine interaction for holes is
found to be about five times weaker than that for electrons. Our theory shows
that this ratio depends not only on intrinsic material properties but also on
the dot shape and carrier confinement through the heavy-hole mixing, an
opportunity for engineering the hole-nuclear spin interaction by tuning dot
size and shape.
