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

We reveal that the origin of ferromagnetism caused by $sp$ electrons in graphene with vacancies can be traced to electrons partially filling $sp^{2*}$-antibonding and $p_z^*$-nonbonding states, which are induced by the vacancies and appear near the Fermi level. Because the spatial wavefunctions of the both states are composed of atomic orbitals in an antisymmetric configuration, their spin wavefunctions should be symmetric according to the electron exchange antisymmetric principle, leading to electrons partially filling these states in spin polarization. Since this $p_z^*$ state originates not from interactions between the atoms but from the unpaired $p_z$ orbitals due 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, and decays gradually from the vacancy as $sim$ $1/r$. According to these characteristics, we concluded that the $p_z^*$ state plays a critical role in magnetic ordering in graphene with vacancies. If the vacancy concentration in graphene is large enough to cause the decay-length regions to overlap, constraining the $p_z^*$ orbital components as little as possible on the minority sublattice atoms in the overlap regions results in the vacancy-induced $p_z^*$ states being coherent. The coherent process in the overlap region leads to the wavefunctions in all the involved regions antisymmetrized, consequently causing ferromagnetism according to the electron exchange antisymmetric principle. This unusual mechanism concerned with the origin of $sp$-electron magnetism and magnetic ordering has never before been reported and is distinctly different from conventional mechanisms.