A theoretical model study on interplay between Coulomb potential and lattice energy in graphene-on-substrate

Abstract

The graphene-on-substrates breaks the sub-lattice symmetry leading to the opening of a small gap. The small band gaps can be enhanced by electron–phonon interactions by keeping strongly polarized superstrate on graphene. To describe the band gap opening in graphene, we propose a tight-binding model Hamiltonian taking into account of third-nearest-neighbor electron-hoppings. We introduce repulsive Coulomb interaction at two sub-lattices of graphene. Further, we consider phonon coupling to the electron densities centered at two sub-lattices in the presence of phonon vibration with a single frequency. For high frequency phonons, the present interaction represents the Holstein interaction. Applying Lang–Firsov canonical transformation in the high phonon-frequency limit, we calculate the modified Coulomb interaction and the effective hopping integral which are functions of electron–phonon coupling, phonon-frequency and nearest-neighbor electron-hopping integral. The electron Green’s functions are calculated by Zubarevs technique. The electron occupancies at two sub-lattices for up and down spins are calculated and computed self-consistently. Finally, we calculate the modulated substrate induced gap of graphene-on-substrate, which is computed numerically for [Formula: see text] grid points for electron momentum. We have studied the interplay of Coulomb interaction, electron–phonon interaction in high phonon-frequency limit. The maximum band gap achieved due to the interplay is nearly 67% more than the substrate induced gap. To achieve this condition, one requires low Coulomb energy for low frequency phonon, while one needs high Coulomb interaction and high electron–phonon interaction of a given lattice vibration frequency. For given electron–phonon interaction and phonon-frequency, the modified gap is enhanced throughout the temperature range with increase of Coulomb interaction.