Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials

Abstract

By an integral equation approach to the time-harmonic classical Maxwell equations, we describe the dispersion in the nonretarded frequency regime of the edge plasmon-polariton (EPP) on a semi-infinite flat sheet. The sheet has an arbitrary, physically admissible, tensor valued and spatially homogeneous conductivity, and serves as a model for a family of two-dimensional conducting materials. We formulate a system of integral equations for the electric field tangential to the sheet in a homogeneous and isotropic ambient medium. We show how this system is simplified via a length scale separation. This view entails the quasi-electrostatic approximation, by which the tangential electric field is replaced by the gradient of a scalar potential, ϕ. By the Wiener-Hopf method, we solve an integral equation for ϕ in some generality. The EPP dispersion relation comes from the elimination of a divergent limiting Fourier integral for ϕ at the edge. We connect the existence, or lack thereof, of the EPP dispersion relation to the index for Wiener-Hopf integral equations, an integer of topological character. We indicate that the values of this index may express an asymmetry due to the material anisotropy in the number of wave modes propagating on the sheet away from the edge with respect to the EPP direction of propagation. We discuss extensions such as the setting of two semi-infinite, coplanar sheets. Our theory forms a generalization of the treatment by Volkov and Mikhailov (1988 Sov. Phys.

Edge plasmon-polaritons on isotropic semi-infinite conducting sheets

“…In the nonretarded frequency regime, when |ωµσ(ω)/k 0 | 1 with σ(ω) > 0, 9,10 the EP dispersion relation can be derived by the quasi-electrostatic approach. 19,23,25 By carrying out an asymptotic expansion for exact result (2a), we derive the formula…”

“…Our formulation includes nonhomogeneous and anisotropic sheets with local surface conductivities; a generalization is provided in Ref. 25.…”

“…More generally, J can be a linear functional of E at z = 0. 25 We alert the reader that (3) and ( 4) introduce the volume current density, J e , as distinct from the induced surface current density, J, on Σ. This distinction is justified if the domain of Maxwell's equations is R 3 \ Σ; and highlights the different physical origins of the two current densities.…”

“…For a generalization of (8a) to 2D materials in which the induced surface current density is a linear functional of (u(x), v(x)), see Ref. 25. The problem is to find q so that matrix equation (8a) admits nontrivial solutions.…”

“…[20][21][22][23][24] To our knowledge, many studies of the EP are restricted to the nonretarded frequency regime, in which the electric field is approximated by the gradient of a scalar potential ("quasi-electrostatic approach"). 5,25 In this paper, we use the time harmonic classical Maxwell equations in three spatial dimensions (3D) in order to formally derive the dispersion relation of the EP on a semiinfinite, flat sheet with a straight edge and a homogeneous and isotropic surface conductivity.…”