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We reveal that the origin of ferromagnetism caused by $sp$ electrons ingraphene with vacancies can be traced to electrons partially filling$sp^{2*}$-antibonding and $p_z^*$-nonbonding states, which are induced by thevacancies and appear near the Fermi level. Because the spatial wavefunctions ofthe both states are composed of atomic orbitals in an antisymmetricconfiguration, their spin wavefunctions should be symmetric according to theelectron exchange antisymmetric principle, leading to electrons partiallyfilling these states in spin polarization. Since this $p_z^*$ state originatesnot from interactions between the atoms but from the unpaired $p_z$ orbitalsdue to the removal of $p_z$ orbitals on the minority sublattice, the $p_z^*$state is constrained, distributed on the atoms of the majority sublattice, anddecays gradually from the vacancy as $\sim$ $1/r$. According to thesecharacteristics, we concluded that the $p_z^*$ state plays a critical role inmagnetic ordering in graphene with vacancies. If the vacancy concentration ingraphene is large enough to cause the decay-length regions to overlap,constraining the $p_z^*$ orbital components as little as possible on theminority sublattice atoms in the overlap regions results in the vacancy-induced$p_z^*$ states being coherent. The coherent process in the overlap region leadsto the wavefunctions in all the involved regions antisymmetrized, consequentlycausing ferromagnetism according to the electron exchange antisymmetricprinciple. This unusual mechanism concerned with the origin of $sp$-electronmagnetism and magnetic ordering has never before been reported and isdistinctly different from conventional mechanisms.

Ferromagnetism in graphene traced to an antisymmetric orbital combination of involved electronic states