From graphene to T-graphene: The electronic and topological properties of one- and two-dimensional square-octagon lattices in a magnetic field

Abstract

The discovery of graphene and its extraordinary properties, which arise from the existence of Dirac cones in the band structure, has sparked interest in other electronic lattices with different geometries. In this thesis, the electronic properties of T-graphene, or a square-octagon lattice, are studied using a tight-binding model. The band structure is calculated for different values of the intracell t1 and intercell t2 hopping parameters. A uniaxial strain is introduced and the band structure of the resulting highly anisotropic T-graphene shows the merging of Dirac-like points upon varying the hopping parameters. A finite-size T-graphene system does not show topologically protected edge states due to the hybridization of the zero-energy edge states with the zero-energy excitations in the bulk. To open a gap in the bulk, a magnetic flux is introduced in a T-graphene lattice and the band structures and topological properties are studied. However, no topological phases emerge due to the hybridization of the wave functions of the edge states. Additionally, the band structure of a one-dimensional T-graphene chain is calculated and shows two edge modes at zero energy and no closing of the gap (except for t2 = 0). The calculated Zak phases of the energy bands suggest the absence of topologically protected edge states, but the localization of the wave functions does indicate the existence of robust compact localized states. Further research is needed to determine whether the introduction of superconductivity or Rashba spin-orbit coupling can open the gap to obtain these compact localized states.