  The observation of intrinsic magnetic order in graphene and graphene-based
materials relies on the formation of magnetic moments and a sufficiently strong
mutual interaction. Vacancies are arguably considered the primary source of
magnetic moments. Here we present an in-depth density functional theory study
of the spin-resolved electronic structure of (monoatomic) vacancies in graphene
and bilayer graphene. We use two different methodologies: supercell
calculations with the SIESTA code and cluster-embedded calculations with the
ALACANT package. Our results are conclusive: The vacancy-induced extended $\pi$
magnetic moments, which present long-range interactions and are capable of
magnetic ordering, vanish at any experimentally relevant vacancy concentration.
This holds for $\sigma$-bond passivated and un-passivated reconstructed
vacancies, although, for the un-passivated ones, the disappearance of the $\pi$
magnetic moments is accompanied by a very large magnetic susceptibility. Only
for the unlikely case of a full $\sigma$-bond passivation, preventing the
reconstruction of the vacancy, a full value of 1$\mu_B$ for the $\pi$ extended
magnetic moment is recovered for both mono and bilayer cases. Our results put
on hold claims of vacancy-induced ferromagnetic or antiferromagnetic order in
graphene-based systems, while still leaving the door open to $\sigma$-type
paramagnetism.
