  The incubation time preceding nucleation and growth of surface nanostructures
is interesting from a fundamental viewpoint but also of practical relevance as
it determines statistical properties of nanostructure ensembles such as size
homogeneity. Using in situ reflection high-energy electron diffraction, we
accurately deduce the incubation times for Ga-assisted GaAs nanowires grown on
unpatterned Si(111) substrates by molecular beam epitaxy under different
conditions. We develop a nucleation model that explains and fits very well the
data. We find that, for a given temperature and Ga flux, the incubation time
always increases with decreasing As flux and becomes infinite at a certain
minimum flux, which is larger for higher temperature. For given As and Ga
fluxes, the incubation time always increases with temperature and rapidly tends
to infinity above 640 {\deg}C under typical conditions. The strong temperature
dependence of the incubation time is reflected in a similar variation of the
nanowire number density with temperature. Our analysis provides understanding
and guidance for choosing appropriate growth conditions that avoid unnecessary
material consumption, long nucleation delays, and highly inhomogeneous
ensembles of nanowires. On a more general ground, the existence of a minimum
flux and maximum temperature for growing surface nanostructures should be a
general phenomenon pertaining for a wide range of material-substrate
combinations.
