Localized surface plasmons in a continuous and flat graphene sheet

TL;DR

This paper derives an exact integral equation for localized surface plasmons in a flat graphene sheet with a dielectric protrusion, revealing their dependence on geometry and supporting multiple modes.

Contribution

It introduces a precise mathematical model for surface plasmons in graphene with geometric features, solving the Fredholm equation exactly and analyzing mode properties.

Findings

Dispersion relation depends solely on protrusion geometry

Supports both even and odd localized modes

Electrostatic potential illustrates localized plasmon fields

Abstract

We derive an integral equation describing surface-plasmon polaritons in graphene deposited on a substrate with a planar surface and a dielectric protrusion in the opposite surface of the dielectric slab. We show that the problem is mathematically equivalent to the solution of a Fredholm equation, which we solve exactly. In addition, we show that the dispersion relation of the localized surface plasmons is determined by the geometric parameters of the protrusion alone. We also show that such system supports both even and odd modes. We give the electrostatic potential and the stream plot of the electrostatic field, which clearly show the localized nature of the surface plasmons in a continuous and flat graphene sheet.