Coupled electric dipole model for a Su-Schrieffer-Heeger chain of optically resonant coreshell nanoparticles

1. Problem Statement

Periodic arrays of plasmonic nanoparticles are known to exhibit topological phenomena, such as topological edge states, analogous to the Su-Schrieffer-Heeger (SSH) model in condensed matter physics. However, standard modeling approaches often treat complex nanoparticles (like core-shell structures) as single effective electric dipoles. This simplification obscures the internal physics of the particle, specifically the hybridization between surface plasmon modes at different interfaces (e.g., the core-shell interface and the shell-medium interface).

The authors identify a gap in the literature: there is no unified framework that simultaneously accounts for:

1. The complex internal mode hybridization of core-shell nanoparticles (often described by hybridization theory).
2. The collective topological behavior of periodic arrays (described by coupled dipole models).

The challenge is to develop a model that connects the microscopic resonant modes of individual core-shell particles to the macroscopic dispersion bands and topological invariants of the array, allowing for the prediction of multiple topological edge states.

2. Methodology

The authors propose a Coupled Electric Dipole (CED) formalism tailored for core-shell nanoparticles. The methodology involves three main steps:

A. Decomposition of Core-Shell Nanoparticles

Instead of modeling a core-shell nanoparticle as a single dipole, the authors decompose it into two coupled dipoles located at the same position:

• Dipole $p_a$: Represents the surface plasmon mode at the outer shell interface (radius $a$).
• Dipole $p_b$: Represents the surface plasmon mode at the inner core-shell interface (radius $b$).

The polarizability of the system is derived by considering the interaction between these two dipoles. The total polarizability is the sum of the partial polarizabilities of the sphere and the void (cavity) modes, modified by a coupling term ($g_{ab}$) that accounts for the hybridization between the inner and outer surfaces. This approach allows for a one-to-one mapping between the particle's internal resonant modes (bonding and antibonding) and the system's degrees of freedom.

B. Mapping to a Multi-Layer SSH Model

The authors arrange these core-shell nanoparticles into a 1D periodic chain with alternating distances (an SSH lattice).

• Unit Cell: Contains two nanoparticles with alternating inter-particle distances ($\beta d/2$ and $(2-\beta)d/2$).
• Hamiltonian Mapping: By applying the coupled dipole equations to the array, the system is mathematically mapped to a bilayer SSH model (or a two-leg ladder).
  • The "layers" in this analogy correspond to the two internal dipole modes ($p_a$ and $p_b$) of each nanoparticle.
  • The intra-particle coupling ($g_{ab}$) acts as the inter-layer coupling.
  • The inter-particle couplings ($G_1, G_2$) act as the intra-cell and inter-cell hoppings.

C. Topological Analysis

The authors analyze the system using:

• Dispersion Relations: Solving the eigenvalue problem for the periodic array to find band structures.
• Topological Invariants: Calculating the Zak phase and the Winding Number ($W$) to determine the topological phase of the system.
• Bulk-Boundary Correspondence: Comparing the infinite periodic spectrum with the finite array spectrum to identify edge states.
• Quasi-Static vs. Retarded Regimes: The primary analysis is in the quasi-static limit (nearest-neighbor approximation), with corrections for retardation and radiation provided in the appendices.

3. Key Contributions

1. Novel Formalism: The paper introduces a rigorous framework where a core-shell nanoparticle is treated as a system of coupled dipoles rather than a single effective dipole. This preserves the spectral complexity of the particle's internal hybridization.
2. One-to-One Correspondence: The model establishes a direct link between the number of resonant modes in a single nanoparticle and the number of dispersion bands in the array. For a core-shell particle with two resonances, the array exhibits two distinct sets of bands.
3. Multi-Gap Topology: The study demonstrates that a single SSH chain of core-shell nanoparticles hosts multiple topological gaps (one for each hybridized mode of the particle), rather than just a single gap as seen in simple metallic sphere arrays.
4. Mapping to Multilayer Lattices: The authors show that complex monolayer arrays of structured nanoparticles can be mapped to multilayer tight-binding lattices, providing a new perspective on designing topological photonic materials.

4. Results

The authors applied their model to a chain of Si@Ag (Silicon core, Silver shell) nanoparticles.

• Hybridization and Rabi Splitting:

  • The interaction between the inner (void) and outer (sphere) plasmon modes creates two distinct resonances: a bonding mode ($\omega^-_{cs}$) and an antibonding mode ($\omega^+_{cs}$).
  • The energy gap between these modes (Rabi splitting, $\Omega$) is tunable via the core-shell filling fraction ($b/a$) and the host permittivity ($\epsilon_B$).
  • For specific parameters ($b=20$ nm, $a=30$ nm), a large splitting of $\approx 1.74$ eV was observed.

• Dispersion Bands:

  • The system exhibits 12 bands (3 polarizations $\times$ 2 resonances $\times$ 2 lattice sites).
  • These bands form two distinct groups centered around the isolated particle resonances ($\omega^-_{cs} \approx 1.95$ eV and $\omega^+_{cs} \approx 3.71$ eV).
  • Within each group, alternating inter-particle distances open small topological gaps ($\Omega_{\pm, \mu}$).

• Topological Phase Transition:

  • The system undergoes a topological phase transition at the geometric parameter $\beta = 1$ (where inter-cell and intra-cell distances are equal).
  • For $\beta < 1$, the winding number $W=0$ (trivial phase).
  • For $\beta > 1$, the winding number $W=1$ (non-trivial phase).

• Edge States:

  • In the non-trivial phase ($\beta > 1$), the finite chain hosts topological edge states localized at the boundaries.
  • Crucially, because of the two internal resonances, the chain supports two pairs of edge states (one pair for the bonding mode and one for the antibonding mode).
  • These edge states are pinned to the resonant frequencies of the individual core-shell particles ($\omega^\pm_{cs}$).
  • The edge states exhibit a sextuple degeneracy (due to 2 modes $\times$ 3 polarizations $\times$ 2 ends), which can be broken by anisotropy.

5. Significance and Implications

• Tunable Multi-Frequency Topology: The ability to tune the frequencies of the edge states via the core-shell filling fraction and host permittivity allows for the design of topological devices operating at multiple, distinct frequencies simultaneously.
• Non-Linear Applications: The presence of multiple topological edge states at different frequencies makes this system highly suitable for topologically protected non-linear optics, such as the generation of higher harmonics. The edge states can be engineered to resonate at fundamental and harmonic frequencies simultaneously.
• Robustness: The study suggests that longitudinal edge states are more robust against retardation effects than transverse ones, offering a pathway to stable topological photonic circuits.
• Generalizability: The coupled-dipole decomposition method is generalizable to more complex nanostructures (e.g., nanomatryoshkas with multiple shells), allowing for the engineering of "multilayer" topological lattices with arbitrary numbers of bands and edge states.

In conclusion, this work bridges the gap between single-particle plasmon hybridization and collective topological physics, providing a powerful design tool for creating complex, multi-functional topological photonic materials.