Dirac equation description of the electronic states and magnetic properties of a square graphene quantum dot

Electronic eigenstates of a square graphene quantum dot (GQD) terminated by both zigzag and armchair edges are derived in the theoretical framework of the Dirac equation. We find that the Dirac equation can determine the eigenenergy spectrum of a GQD with high accuracy even if its size is reduced to a few nanometers. More importantly, from the Dirac equation description we can readily work out the number and energy gap of the conjugate surface states, which are intimately associated with the magnetic properties of the GQD. By using the Hartree-Fock mean field approach, we study the size dependence of the magnetic ordering formation in this square GQD. We find that there exists a critical size of the width between the two zigzag edges to indicate the onset of the stable magnetic ordering. On the other hand, when such a width increases further, the magnetic ground state energy of a charge neutral GQD tends to a saturated value. These results coincide with the previous results obtained from the first-principles calculation. Then, based on the Dirac equation solution about the surface state, we establish a simple two-state model which can quantitatively explain the size dependence of the magnetic ordering in the square GQD.