Dispersion of electromagnetic waves in layered graphene-dielectric metamaterials

Abstract

 In this paper possibility of creating a medium based on graphene-dielectric (quartz) structures with a hyperbolic shape of the dispersion relation is considered. Using the Bloch theorem and the transfer matrix method, dispersion relations are found for hyperbolic media, consisting of periodically located layers of graphene and a dielectric. The transfer matrix method gives the possibility to relate the amplitudes of the incident and reflected electromagnetic waves at the input to the layered structure with the amplitudes at the output by means of propagation and transfer matrixes. The frequency dependent optical conductivity of graphene sheet is calculated by Kubo formula which takes into account both interband and intraband transitions. It is shown that when the chemical potential of graphene is increased, the elliptic form of the dispersion curve turns into to hyperbolic. This result is due to the fact that the perpendicular component of the permittivity changes its sign when the chemical potential is changed.