Homogenization of plasmonic crystals: seeking the epsilon-near-zero effect

Abstract

By using an asymptotic analysis and numerical simulations, we derive and investigate a system of homogenized Maxwell's equations for conducting material sheets that are periodically arranged and embedded in a heterogeneous and anisotropic dielectric host. This structure is motivated by the need to design plasmonic crystals that enable the propagation of electromagnetic waves with no phase delay (epsilon-near-zero effect). Our microscopic model incorporates the surface conductivity of the two-dimensional (2D) material of each sheet and a corresponding line charge density through a line conductivity along possible edges of the sheets. Our analysis generalizes averaging principles inherent in previous Bloch-wave approaches. We investigate physical implications of our findings. In particular, we emphasize the role of the vector-valued corrector field, which expresses microscopic modes of surface waves on the 2D material. We demonstrate how our homogenization procedure may set the foundation for computational investigations of: effective optical responses of reasonably general geometries, and complicated design problems in the plasmonics of 2D materials.

Keywords: Maxwell's equations; asymptotic analysis; graphene; homogenization; plasmonic crystals; surface plasmon-polariton.

Figure 1.

Schematic of geometry. (a) The unit cell, Y = [0, 1]³, with microstructure Σ, a conducting sheet. (b) Computational domain Ω with rescaled periodic layers Σ^d and spatially dependent surface conductivity σ^d(x). The ambient medium has a heterogeneous permittivity, ε^d(x).

Figure 2.

Prototypical examples of microscopic geometries with conducting sheets for cell problem (3.2). (a) Infinite planar sheet, with no edges. (b) Planar strip (nanoribbon). (c) Sheet forming circular cylinder (nanotube). The corrector, χ, can be characterized as follows: (a) χ≡0; (b) χ₂ = χ₃≡0 while χ₁ is non-trivial; and (c) χ₃≡0 while χ₁ and χ₂ are non-trivial.

Figure 3.

Real and imaginary parts of corrector component χ₁ for geometries of figure 2b,c in y₁y₂-plane. It is evident that internal edges in nanoribbons and curvature in nanotubes create SPPs. (a) Re χ₁, ribbons, (b) Im χ₁, ribbons, (c) Re χ₁, tubes and (d) Im χ₁, tubes. (Online version in colour.)

Figure 4.

(a) Geometry of a plasmonic-crystal slab of height 1 with corrugated layers of 2D material, where the corrugation is sinusoidal and the period is equal to the spacing, d (in the schematic d = 2⁻³). (b) Real part of electric field in the y-direction in the corresponding homogenization limit (as d → 0), via solution of cell problem (3.2) and computation of the homogenized solution given by (3.3). (c–e) Real part of electric field in the y-direction, based on direct numerical simulations of (2.1)–(2.3) (see [41]), for decreasing spacing , d (i.e. increasing degree of scale separation): (c) d = 2⁻⁴; (d) d = 2⁻⁵; (e) d = 2⁻⁶. (Online version in colour.)

Figure 5.

Plots of real and imaginary parts of matrix elements of ε^eff as a function of frequency by (5.2) for the geometries of figure 2b,c. (a) ε^R₁₁ for nanoribbons; and (b) ε^T₁₁ or ε^T₂₂ (ε^T₁₁ = ε^T₂₂) for nanotubes. The shaded area indicates the frequency regime for negative real part in each case. (Online version in colour.)