# Plasmons in two-dimensional lattices of near-field coupled nanoparticles

## TL;DR

This paper develops an analytical framework to understand collective plasmonic modes in two-dimensional nanoparticle lattices, accounting for topological properties, environmental coupling, and retarded interactions, with implications for experimental observation.

## Contribution

It provides exact solutions for quasistatic dispersion relations in arbitrary 2D nanoparticle lattices, including non-Bravais and topologically nontrivial structures, incorporating environmental effects.

## Key findings

- Analytical expressions for plasmonic dispersion in various lattice geometries.
- Closed-form formulas for radiative and nonradiative damping rates.
- Insights into how retarded interactions modify band structures.

## Abstract

We consider plasmonic metasurfaces constituted by an arbitrary periodic arrangement of spherical metallic nanoparticles. Each nanoparticle supports three degenerate dipolar localized surface plasmon (LSP) resonances. In the regime where the interparticle distance is much smaller than the optical or near-infrared wavelength associated with the LSPs, the latter couple through the dipole-dipole interaction and form collective plasmonic modes which extend over the whole metasurface. Within a Hamiltonian model which we solve exactly, we derive general expressions which enable us to extract analytically the quasistatic plasmonic dispersion for collective modes polarized within the plane and perpendicular to the plane of the metasurface. Importantly, our approach allows us not only to consider arbitrary Bravais lattices, but also non-Bravais two-dimensional metacrystals featuring nontrivial topological properties, such as, e.g., the honeycomb or Lieb lattices. Additionally, using an open quantum system approach, we consider perturbatively the coupling of the collective plasmons to both photonic and particle-hole environments, which lead, respectively, to radiative and nonradiative frequency shifts and damping rates, for which we provide closed-form expressions. The radiative frequency shift, when added to the quasistatic dispersion relation, provides an approximate analytical description of the fully retarded band structure of the collective plasmons. While it is tempting to make a direct analogy between the various systems which we consider and their electronic tight-binding equivalents, we critically examine how the long-range retarded and anisotropic nature of the dipole--dipole interaction may quantitatively and qualitatively modify the underlying band structures and discuss their experimental observability.