Based on first principles calculations, 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 pz orbitals due to the removal of pz 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 ∼ 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. Consequently, we can explain how such a weak magnetization with such a high critical temperature can be experimentally observed in proton-irradiated graphene.
To elucidate the physics underling magnetism observed in nominally nonmagnetic materials with only sp-electrons, we built an extreme model to simulate H-adsorption (in a straight-line form) on graphene. Our first principles calculations for the model produce a ferromagnetic ground state with a magnetic moment of one Bohr magneton per H atom and a high Curie temperature. The removal of the pz-orbitals from sublattice B of graphene introduces pz-vacancies. The pz-vacancyinduced states are not created from changes in interatomic interactions but are created because of a pz-orbital imbalance between two sublattices (A and B) of a conjugated pz-orbital network. Therefore, there are critical requirements for the creation of these states (denoted as p imbalance z ) to avoid further imbalances and minimize the effects on the conjugated pz-orbital network. The requirements on the creation of p imbalance z are as follows: 1) p imbalance z consists of pz-orbitals of only the atoms in sublattice A, 2) the spatial wavefunction of p imbalance z is antisymmetric, and 3) in principle, p imbalance z extends over the entire crystal without decaying, unless other pz-vacancies are crossed. Both the origin of spin polarization and the magnetic ordering of the model arise from the aforementioned requirements.
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