# Collective Quantum Approach to Surface Plasmon Resonance Effect

## TL;DR

This paper develops a quantum collective oscillation model for surface plasmon resonance (SPR), revealing resonance conditions through a dual length-scale damped oscillator framework, and extends the theory to localized surface plasmons in nanostructures.

## Contribution

It introduces a novel quantum dual length-scale model for SPR based on damped collective electron oscillations, extending current electrostatic theories to include polariton excitations and nanostructure applications.

## Key findings

- Resonance occurs when the driving wavenumber matches surface plasmon excitations.
- The model links surface spill-out electron oscillations to classical driven damped oscillators.
- A new equation of state for plasmon electron density limits effects in high-density plasmas.

## Abstract

In this research we present a theory of the surface plasmon resonance (SPR) effect based on the dual length-scale driven damped collective quantum oscillations of the spill-out electrons in plasmonic material surface. The metallic electron excitations are modeled using the Hermitian effective Schr\"{o}dinger-Poisson system, whereas, the spill-out electron excitations are modeled via the damped non-Hermitian effective Schr\"{o}dinger-Poisson system adapted appropriately at the metal-vacuum interface. It is shows that, when driven by external field, the system behaves like the driven damped oscillator in wavenumber domain, quite analogous to the driven damped mechanical oscillation in frequency domain, leading to the collective surface spill-out electron excitation resonance. In this model the resonance occurs when the wavenumber of the driving pseudoforce matches that of the surface plasmon excitations which can be either due to single-electrons or collective effects. Current theory of SPR is based on longitudinal electrostatic excitations of the surface electrons, instead of the polariton excitation parallel to the metal-dielectric or metal-vacuum surface. Current theory may also be extended to use for the localized surface plasmon resonance (LSPR) in nanometer sized metallic surfaces in non-planar geometry. A new equation of state (EoS) for the plasmon electron number density in quantum plasmas is obtained which limits the plasmonic effects in high-density low-temperature electron gas regime, due to small transition probability of electrons to the plasmon energy band.