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Hybridization theory for plasmon resonance in metallic nanostructures

Qi Lei, Hongyu Liu, Zhi-Qiang Miao, Guang-Hui Zheng

TL;DR

This work develops a rigorous electrostatic framework for plasmon resonances in core–shell metallic nanoshells using a Neumann-Poincaré-type operator. By modeling concentric disks with small normal perturbations and applying scale-invariant layer-potential techniques, the authors derive first-order corrections to the resonance frequencies $\tilde{\omega}_{n\pm}(\delta_1,\delta_2)$, connect solid and cavity plasmon modes, and obtain closed-form spectra for the special case of concentric disks: $\omega_{n\pm}(\delta_1,\delta_2) = (\omega_p/\sqrt{2})\sqrt{1 \pm (\delta_1/\delta_2)^{|n|} (r_1/r_2)^{|n|}}$. The analysis reveals how core/shell scaling and interface perturbations control hybridization strength and resonance bandwidth, with numerical experiments confirming the theory and demonstrating tunable plasmonic responses for sensing and nanophotonic applications. Overall, the paper provides a principled design framework linking geometry, material dispersion via the Drude model, and NP-spectrum to engineer rich, tunable plasmon resonances in nanoshells.

Abstract

In this paper, we investigate the hybridization theory of plasmon resonance in metallic nanostructures, which has been validated by the authors in [31] through a series of experiments. In an electrostatic field, we establish a mathematical framework for the Neumann-Poincaré(NP) type operators for metallic nanoparticles with general geometries related to core and shell scales. We calculate the plasmon resonance frequency of concentric disk metal nanoshells with normal perturbations at the interfaces by the asymptotic analysis and perturbation theory to reveal the intrinsic hybridization between solid and cavity plasmon modes. The theoretical finding are convincingly supported by extensive numerical experiments. Our theory corroborates and strengthens that by properly enriching the materials structures as well as the underlying geometries, one can induce much richer plasmon resonance phenomena of practical significance.

Hybridization theory for plasmon resonance in metallic nanostructures

TL;DR

This work develops a rigorous electrostatic framework for plasmon resonances in core–shell metallic nanoshells using a Neumann-Poincaré-type operator. By modeling concentric disks with small normal perturbations and applying scale-invariant layer-potential techniques, the authors derive first-order corrections to the resonance frequencies , connect solid and cavity plasmon modes, and obtain closed-form spectra for the special case of concentric disks: . The analysis reveals how core/shell scaling and interface perturbations control hybridization strength and resonance bandwidth, with numerical experiments confirming the theory and demonstrating tunable plasmonic responses for sensing and nanophotonic applications. Overall, the paper provides a principled design framework linking geometry, material dispersion via the Drude model, and NP-spectrum to engineer rich, tunable plasmon resonances in nanoshells.

Abstract

In this paper, we investigate the hybridization theory of plasmon resonance in metallic nanostructures, which has been validated by the authors in [31] through a series of experiments. In an electrostatic field, we establish a mathematical framework for the Neumann-Poincaré(NP) type operators for metallic nanoparticles with general geometries related to core and shell scales. We calculate the plasmon resonance frequency of concentric disk metal nanoshells with normal perturbations at the interfaces by the asymptotic analysis and perturbation theory to reveal the intrinsic hybridization between solid and cavity plasmon modes. The theoretical finding are convincingly supported by extensive numerical experiments. Our theory corroborates and strengthens that by properly enriching the materials structures as well as the underlying geometries, one can induce much richer plasmon resonance phenomena of practical significance.

Paper Structure

This paper contains 12 sections, 14 theorems, 100 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Mathematical setting and main results
  3. Auxiliary results
  4. Layer potentials
  5. Neumann-Poincaré-type operator
  6. Plasmon resonance
  7. Perturbations of concentric disk
  8. Rescaling and asymptotic analysis
  9. Proof of Theorem 2.1
  10. Eigenvalues of the NP-type operator K*
  11. Numerical results
  12. Conclusion

Key Result

Theorem 2.1

Let the core-shell interfaces $\partial D_i$ of the concentric disk-shaped metallic nanoshell exhibit the following $\epsilon _i$-normal perturbation then, the first-order corrected resonance frequency $\tilde{\omega} _{n\pm}\left( \delta _1 , \delta _2 \right)$ with respect to the scale factors $\delta _1$, $\delta _2$ is given by the following expression where the inner product $\left< \cdot,

Figures (7)

  • Figure 2.1: Schematic illustration of metallic nanostructures with perturbed core-shell boundaries.
  • Figure 2.2: Hybridization energy-level diagram induced by interaction between solid and cavity plasmons
  • Figure 5.1: Schematic diagrams of nanostructures defined by four different shape functions. (a)-(d) correspond to Set 1-4, respectively.
  • Figure 5.2: Hybridization behavior of metal nanoshells approaching the solid plasmon mode. (a) Spectral response with respect to $\omega$ in Set 1 ($\delta_2$ fixed, $\delta_1 = 2^{5},2^{4},2^{3}, 2^{2}, 2^{1}, 2^{0}, 2^{-1}, 2^{-2}, 2^{-3}$). (b-d) Hybridization intensity of solid-cavity plasmon as a function of $\delta_1$ for Sets 1-3, respectively.
  • Figure 5.3: Hybridization behavior of metal nanoshells approaching the cavity plasmon mode. (a) Spectral response with respect to $\omega$ in Set 1 ($\delta_1$ fixed, $\delta_2 = 2^{2}, 2^{3}, 2^{4}, 2^{5}, 2^{6}, 2^{7}, 2^{8}$). (b-d) Hybridization intensity of solid-cavity plasmon as a function of $\delta_2$ for Sets 1-3, respectively.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Definition 3.1
  • Remark 3.1
  • ...and 18 more