  A microscopic explanation for the room temperature ferromagnetism in diluted
ZnO:Co is at present rather elusive. Although standard secondary phases can
usually be ruled out, it is less clear whether regions with high Co
concentration coexist with undoped portions of the film, i.e. whether some form
of CoO polymorph can be responsible for the magnetic signal. Since X-ray
usually excludes the presence of the native rock-salt phase, the study of CoO
polymorphs becomes particularly interesting. In this work we investigate
theoretically the magnetism of CoO in both the wurtzite and zincblende phases.
By using a combination of density functional theory with the LDA+$U$
approximation and Monte Carlo simulations we demonstrate that wurtzite and
zincblende CoO have a complex frustrated anti-ferromagnetic ground state with
no net magnetic moment in the bulk. Most importantly the estimated critical
temperatures are well below room temperature for both cases. This suggests that
bulk CoO polymorphs are not responsible for the room temperature magnetism
observed for ZnO:Co, although the role of clusters with uncompensated spins or
arranged in a spinodal decomposed structure still remains an open question.
