Comments on the fractal energy spectrum of honeycomb lattice with defects

Abstract

Abstract We study the fractality of the energy spectrum of honeycomb lattice with various defects or impurities under a perpendicular magnetic field. Using a tight-binding Hamiltonian including interactions with the nearest neighbors, we investigate its energy spectrum for different choices of point defects or impurities. First, we fix a unit cell consisting of 8 lattice points and survey the energy eigenvalues in the presence of up to 2 point defects. Then it turns out that the existence of the fractal energy structure, called Hofstadter’s butterfly, highly depends on the choice of defect pairs. Next, we extend the size of a unit cell which contains a single point defect in the unit cell consisting of 18 and 32 lattice points to lower the density of the defects. In this case, the robust gapless point exists on the E = 0 eV line without depending on the size of unit cells. And we find this gapless point always exists at the center of the butterfly shape. This butterfly shape also exists for the case of no defect lattice which has the fractality.