Substrate-driven topological engineering in plasmonic Su-Schrieffer-Heeger chains

Technical Summary: Substrate-driven topological engineering in plasmonic Su-Schrieffer-Heeger chains

Problem Statement
While the Su-Schrieffer-Heeger (SSH) model serves as a cornerstone for understanding one-dimensional topological insulators, and recent work has extended these concepts to plasmonic nanoparticle chains, precise active control over topological advantages remains challenging. Traditional strategies for tuning edge modes often rely on modifying the chain geometry itself (e.g., doping dependence or photonic cavities). This paper addresses the problem of engineering the topological band structure of a plasmonic SSH chain not by altering the chain's intrinsic geometry, but by exploiting its interaction with the electromagnetic environment, specifically a planar substrate. The authors investigate how the proximity of an SSH chain to a substrate can induce topological phase transitions and alter edge mode properties.

Methodology
The study employs a theoretical framework based on the dipole approximation for a bipartite array of $N$ identical spherical nanoparticles (radius $R$, polarizability $\alpha$) positioned at a distance $z$ from a semi-infinite planar substrate. The system is modeled using the total Green's function, $G_{tot}$, which is decomposed into vacuum and scattering contributions due to the substrate.

• System Parameters: The nanoparticles and substrate are modeled as Indium Antimonide (InSb) with Drude permittivity. The system is characterized by the dimerization parameter $\beta$ (defining intra-cell vs. inter-cell spacing), the chain-substrate distance $z$, and the substrate's charge carrier density $n_{sub}$.
• Calculations: The authors solve the eigenvalue equation for the dipole moments of the coupled system. For infinite chains, Bloch's theorem is applied to derive a $2 \times 2$ block matrix equation for the dipole moments of unit cells, yielding frequency bands and Zak phases (topological invariants). For finite chains, the Inverse Participation Ratio (IPR) is calculated to quantify the spatial localization of eigenmodes and identify edge states.
• Analysis: The study analyzes the band structure and Zak phases as functions of $\beta$, $z$, and $n_{sub}$. It further investigates the robustness of these modes against structural disorder (random particle displacement) and examines the impact of these topological changes on the spectral radiative heat flux between the ends of the chain.

Key Contributions and Results
The paper identifies two distinct mechanisms driven by the substrate interaction that alter the topological band structure and Zak phases:

1. Band Hybridization (Long-range effect):

   • Mechanism: The long-range interaction between the in-plane modes of the SSH chain and the surface plasmon polaritons (SPPs) of the substrate causes the dipole-active bands to hybridize near the substrate's SPP light line.
   • Result: This hybridization leads to a band inversion and a change in the Zak phase. Crucially, this mechanism is a long-range effect mediated by the substrate's SPPs, as it disappears in nearest-neighbor approximations. It can induce edge modes for parameters that would correspond to a trivial phase in an isolated chain, but it does not necessarily open a new band gap.

2. Band Touching (Short-range effect):

   • Mechanism: The short-range interaction with the substrate induces a touching and subsequent reopening of the band gap between specific transversal bands.
   • Result: This mechanism mimics the standard SSH phase transition at $\beta=1$ but occurs at different parameter values due to the substrate. It results in a change of the Zak phase and the emergence of topologically protected edge modes. Unlike hybridization, this is identified as a short-range effect mediated by the substrate.

Dependence on Parameters:

• Dimerization ($\beta$) and Distance ($z$): Varying these parameters triggers the two mechanisms described above. The emergence and vanishing of edge mode branches are directly linked to the acquisition or loss of Zak phases via hybridization and band touching.
• Substrate Doping ($n_{sub}$): Changing the substrate's carrier concentration shifts the SPP resonance frequency relative to the localized plasmon resonance, thereby tuning the coupling strength and the specific parameter ranges where these topological transitions occur.

Robustness and Heat Transfer:

• Disorder: The topologically protected edge modes exhibit remarkable resilience against structural disorder (random particle displacements). While disorder broadens bulk bands and creates localized defect states (Anderson localization), the edge modes remain spectrally distinct and identifiable, even in the presence of 10% perturbation.
• Radiative Heat Transfer: The paper demonstrates that the signatures of these edge modes are reflected in the spectral radiative heat flux between the first and last particles of the chain. The robustness of the edge modes translates to a robust heat flux channel in the topological non-trivial phase, showing significantly less variation under disorder compared to the trivial phase.

Significance
The authors claim that their findings pave the way for engineering edge modes in plasmonic topological configurations by coupling them to a plasmonic environment, rather than solely relying on geometric modifications of the chain. This approach offers a versatile method to tune edge mode frequencies and trigger topological phase transitions by adjusting external parameters like the chain-substrate distance or substrate doping. The work suggests that fabrication tolerances for such geometries can be relaxed, as the topological protection remains largely unaffected by structural imperfections, making these systems promising for controlling radiative heat transfer and light manipulation in practical experimental setups.