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One intriguing finding in graphene is the vacancy-induced magnetism thathighlights the interesting interaction between local magnetic moments andconduction electrons. Within density functional theory, the currentunderstanding of the ground state is that a Stoner instability gives rise toferromagnetism of $\pi$ electrons aligned with the localized moment of a$\sigma$ dangling bond and the induced $\pi$ magnetic moments vanish at lowvacancy concentrations. However, the observed Kondo effect suggests that $\pi$electrons around the vacancy should antiferromagnetically couple to the localmoment and carry non-vanishing moments. Here we propose that a phase possessingboth significant out-of-plane displacements and $\pi$ bands withantiferromagnetic coupling to the localized $\sigma$ moment is the groundstate. With the features we provide, it is possible for spin-resolved STM, STS,and ARPES measurements to verify the newly proposed phase.

Competing magnetism in $\pi$ electrons in graphene with a single carbon vacancy