Numerical Simulation of Grating Structures Incorporating Two-Dimensional Materials: A High-Order Perturbation of Surfaces Framework

Abstract

The plasmonics of two-dimensional materials, such as graphene, has become an important field of study for devices operating in the terahertz to midinfrared regime where such phenomena are supported. The semimetallic character of these materials permits electrostatic biasing which allows one to tune their electrical properties, unlike the noble metals (e.g., gold, silver) which also support plasmons. In the literature there are two principal approaches to modeling twodimensional materials: With a thin layer of finite thickness featuring an effective permittivity, or with a surface current. We follow this latter approach to not only derive governing equations which are valid in the case of curved interfaces, but also reformulate these volumetric equations in terms of surface quantities using Dirichlet-Neumann operators. Such operators have been used extensively in the numerical simulation of electromagnetics problems, and we use them to restate the governing equations at layer interfaces. Beyond this, we show that these surface equations can be numerically simulated in an efficient, stable, and accurate fashion using a High-Order Perturbation of Surfaces methodology. We present detailed numerical results which not only validate our simulation using the Method of Manufactured Solutions and by comparison to results in the literature, but also describe Surface Plasmon Resonances at the "wavy" (corrugated) interface of a dielectric-graphene-dielectric structure.

High-Order Spectral Simulation of Graphene Ribbons

“…We note that due to the flat interfaces present in this geometry, the DNOs are reduced to simple Fourier multipliers which can be easily computed in Fourier space. This is in stark contrast to the case of corrugated interfaces considered in [34] where a stable and accurate HOPS scheme for their computation is non-trivial to design and implement.…”

“…Following [34], the structure we consider is displayed in Fig. 1, a doubly layered, yinvariant medium with periodic interface shaped by z = g(x), g(x+d) = g(x).…”

“…In our recent contribution [34] we described an approach which overcomes both of these limitations by not only restating the frequency domain governing equations in terms of interfacial unknowns, but also describing a High-Order Spectral (HOS) algorithm which recovers solutions with remarkable accuracy (typically machine precision) with a very modest number of unknowns. A subtlety of our approach is that, in order to close the system of equations, surface integral operators must be introduced which connect interface traces of the scattered fields (Dirichlet data) to their surface normal derivatives (Neumann data).…”

“…A subtlety of our approach is that, in order to close the system of equations, surface integral operators must be introduced which connect interface traces of the scattered fields (Dirichlet data) to their surface normal derivatives (Neumann data). Such Dirichlet-Neumann Operators (DNOs) have been widely used and studied in the simulation of linear wave scattering, e.g., for enforcing far-field boundary conditions transparently [2,3,8,9,15,19,21,22,35] and interfacial formulations of scattering problems [27,29,32,34,36].…”

“…The rest of the paper is organized as follows: In Section 2 we recall the governing equations of our model [34] for the response of a two-dimensional material mounted between two dielectrics. In Section 3 we describe our surface formulation of these equations, specializing to the patterned, flat-interface configuration in Section 3.1.…”