Electronic spectrum of Kekulé patterned graphene considering second neighbor-interactions

The effects of second-neighbor interactions in Kekulé-Y patterned graphene electronic properties are studied starting from a tight-binding Hamiltonian. Thereafter, a low-energy effective Hamiltonian is obtained by projecting the high energy bands at the Γ point into the subspace defined by the Kekulé wave vector. The spectrum of the low energy Hamiltonian is in excellent agreement with the one obtained from a numerical diagonalization of the full tight-binding Hamiltonian. The main effect of the second-neighbour interaction is that a set of bands gains an effective mass and a shift in energy, thus lifting the degeneracy of the conduction bands at the Dirac point. This band structure is akin to a 'pseudo spin-one Dirac cone', a result expected for honeycomb lattices with a distinction between one third of the atoms in one sublattice. Finally, we present a study of Kekulé patterned graphene nanoribbons. This shows that the previous effects are enhanced as the width decreases. Moreover, edge states become dispersive, as expected due to second neighbors interaction, but here the Kek-Y bond texture results in an hybridization of both edge states. The present study shows the importance of second neighbors in realistic models of Kekulé patterned graphene, specially at surfaces.