  A semiconductor quantum dot (QD) can generate highly indistinguishable
single-photons at a high rate. For application in quantum communication and
integration in hybrid systems, control of the QD optical properties is
essential. Understanding the connection between the optical properties of a QD
and the growth process is therefore important. Here, we show for GaAs QDs,
grown by infilling droplet-etched nano-holes, that the emission wavelength, the
neutral-to-charged exciton splitting, and the diamagnetic shift are strongly
correlated with the capture zone-area, an important concept from nucleation
theory. We show that the capture-zone model applies to the growth of this
system even in the limit of a low QD-density in which atoms diffuse over
$\mu$m-distances. The strong correlations between the various QD parameters
facilitate preselection of QDs for applications with specific requirements on
the QD properties; they also suggest that a spectrally narrowed QD distribution
will result if QD growth on a regular lattice can be achieved.
