Bulk and surface plasmons in graphene finite superlattices

In recent years, graphene plasmonics, has attracted significant interest motivated by graphene's unique high carrier mobility, electrically or chemically tunable carrier density, long-lived and strong plasmon excitation confinement. In the context of optoelectronics and nanophotonics, graphene is considered as promising plasmonic material working in the mid-infrared and terahertz spectral windows. However, in a single layer graphene, the electromagnetic energy has a limited capability to plasmon excitations. To obtain a more powerful capacity of excite plasmon resonance, one can employ multilayer graphene structures as possible solution. In this work, we find an analytical expression for the dielectric function of the N-layer graphene structure solving a recurrence relation between the electric potential at different graphene layers. We also find that the highest energy of the plasmon oscillations supported by the multilayer graphene stacks at long wavelength correspond to electrons in each plane oscillating in phase. In addition, a surface plasmon mode emerges at certain finite wave vector outside the bulk plasmon band and it is Landau damping in the continuous interband single particle excitation with increasing wave vector.