# Localized Surface Plasmon Resonances of Simple Tunable Plasmonic Nanostructures

## TL;DR

This paper derives systematic analytical formulas for localized surface plasmon resonances in simple plasmonic nanostructures, enabling easier calculation and tuning of their optical properties, with validation against hybridization theory.

## Contribution

It introduces new formulas and a tree diagram framework that relate core-shell plasmons to solid and cavity plasmons, simplifying the analysis of complex nanostructures.

## Key findings

- Formulas agree with plasmon hybridization theory
- Dielectric effects enable LSPR tuning
- Complex structures' LSPR reduced to simpler cases

## Abstract

We derive and present systematic relationships between the analytical formulas for calculation of the localized surface plasmon resonances (LSPR) of some plasmonic nano-structures which we have categorized as simple. These relationships, including some new formulas, are summarized in a tree diagram which highlights the core-shell plasmons as the generators of solid and cavity plasmons. In addition, we show that the LSPR of complex structures can be reduced to that of simpler ones, using the LSPR of a nanorice as a case study, in the dipole limit. All the formulas were derived using a combination of the Drude model, the Rayleigh approximation, and the Fr\"ohlich condition. The formulas are handy and they are in good agreement with the results of the plasmon hybridization theory. The formulas also account for dielectric effects, which provide versatility in the tuning of the LSPR of the nanostructures. A simplified model of plasmon hybridization is presented, allowing us to investigate the weak-coupling regimes of solid and cavity plasmons in the core-shell nanostructures we have studied.