Relativistic Tight-Binding Model for Hexagonal Lattice: Application to Graphene

A non-perturbative relativistic tight-binding (TB) approximation method applicable to crystalline material immersed in a magnetic field was developed and tested for crystalline silicon in 2015. To apply this method to any material in a magnetic field, the electronic structure of the material in absence of the magnetic field must be calculated. In this study, we present the relativistic TB approximation method for graphene in a zero magnetic field. The Hamiltonian and overlap matrices are constructed considering the nearest neighboring atomic interactions between the s and p valence orbitals, where the relativistic hopping and overlap integrals are calculated using the relativistic version of the Slater-Koster table. The method of constructing the Hamiltonian and overlap matrices and the resulting energy-band structure of graphene in the first Brillouin zone is presented in this paper. The appearance of a small band-gap at the k-points (also known as the spin-orbit gap) due to the relativistic effect seen at low temperature whose magnitude is 25 μeV , have also been shown by the theory.