  We theoretically investigate the influence of the fluctuating Overhauser
field on the spin of an electron confined to a quantum dot (QD). The
fluctuations arise from nuclear angular momentum being exchanged between
different nuclei via the nuclear magnetic dipole coupling. We focus on the role
of the nuclear electric quadrupole moments (QPMs), which generally cause a
reduction in internuclear spin transfer efficiency in the presence of electric
field gradients. The effects on the electron spin coherence time are studied by
modeling an electron spin echo experiment. We find that the QPMs cause an
increase in the electron spin coherence time and that an inhomogeneous
distribution of the quadrupolar shift, where different nuclei have different
shifts in energy, causes an even larger increase in the electron coherence time
than a homogeneous distribution. Furthermore, a partial polarization of the
nuclear spin ensemble amplifies the effect of the inhomogeneous quadrupolar
shifts, causing an additional increase in electron coherence time, and provides
an alternative to the experimentally challenging suggestion of full dynamic
nuclear spin polarization.
