Directive surface plasmons on tunable two-dimensional hyperbolic metasurfaces and black phosphorus: Green's function and complex plane analysis

We study the electromagnetic response of two- and quasi-two-dimensional (2-D) hyperbolic materials, on which a simple dipole source can excite a well-confined and tunable surface plasmon polariton (SPP). The analysis is based on the Green's function for an anisotropic 2-D surface, which nominally requires the evaluation of a 2-D Sommerfeld integral. We show that for the SPP contribution, this integral can be evaluated efficiently in a mixed continuous-discrete form as a continuous spectrum contribution (branch cut integral) of a residue term, in distinction to the isotropic case, where the SPP is simply given as a discrete residue term. The regime of strong SPP excitation is discussed, and the complex-plane singularities are identified, leading to physical insight into the excited SPP. We also present a stationary phase solution valid for large radial distances. Examples are presented using graphene strips to form a hyperbolic metasurface and thin-film black phosphorus. Green's function and complex-plane analysis developed allows for the exploration of hyperbolic plasmons in general 2-D materials.