  We quantify the role of Population (Pop) III core-collapse supernovae (SNe)
as the first cosmic dust polluters. Starting from a homogeneous set of stellar
progenitors with masses in the range [13 - 80] Msun, we find that the mass and
composition of newly formed dust depend on the mixing efficiency of the ejecta
and the degree of fallback experienced during the explosion. For standard Pop
III SNe, whose explosions are calibrated to reproduce the average elemental
abundances of Galactic halo stars with [Fe/H] < -2.5, between 0.18 and 3.1 Msun
(0.39 - 1.76 Msun) of dust can form in uniformly mixed (unmixed) ejecta, and
the dominant grain species are silicates. We also investigate dust formation in
the ejecta of faint Pop III SN, where the ejecta experience a strong fallback.
By examining a set of models, tailored to minimize the scatter with the
abundances of carbon-enhanced Galactic halo stars with [Fe/H ] < -4, we find
that amorphous carbon is the only grain species that forms, with masses in the
range 2.7 10^{-3} - 0.27 Msun (7.5 10^{-4} - 0.11 Msun) for uniformly mixed
(unmixed) ejecta models. Finally, for all the models we estimate the amount and
composition of dust that survives the passage of the reverse shock, and find
that, depending on circumstellar medium densities, between 3 and 50% (10 - 80%)
of dust produced by standard (faint) Pop III SNe can contribute to early dust
enrichment.
