Plasmon Resonances of Graphene-Dielectric-Metal Structures Calculated by the Method of Recurrence Relations

Graphene supports surface plasmons in the terahertz range, and compared with noble-metal plasmons, they show an extreme level of field confinement and relatively long propagation distances, with the advantage of being highly tunable via electrostatic field. Nevertheless, its interaction with light is normally rather weak. To obtain a more powerful capability of excite plasmons, a combination of graphene and artificial structures (metamaterials) present a powerful tunability for enhancing light-matter interaction. These features make graphene metamaterials a promising candidate for plasmonics and surface plasmon resonance for biological sensors. In this work, we study the plasmon spectra in a finite number of graphene layers on a metallic-dielectric substrate surrounded by materials with different dielectric constants. It is shown that using standard electromagnetic boundary conditions and solving the recurrence relation (a suitable alternative to transfer matrix method) for the coefficients of the electric potential between graphene layers, an explicit effective dielectric function of the metamaterial can be obtained giving the plasmon dispersion relations. It is found that the metal-dielectric-layered graphene structure supports both high-energy optical plasmon oscillations and out-of-phase low-energy acoustic charge density excitations. Experimentally, the Kretschmann configuration can be used to excite the surface plasmon resonances. It is based on the observation of a sharp minimum in the reflection coefficient versus angle (or wavelength) curve.