Surface-modes mediated long-range radiative heat transfer through a plasmonic Su-Schrieffer-Heeger chain

This study demonstrates that coupling a plasmonic Su-Schrieffer-Heeger chain of InSb nanoparticles to an InSb substrate enables long-range radiative heat transfer mediated by surface modes, where topologically protected edge modes in the non-trivial phase significantly enhance heat transport compared to the trivial phase.

The Gist

Imagine you have a long line of tiny, glowing marbles (nanoparticles) floating in space. You heat up the first marble, and you want to know how much heat reaches the very last marble at the other end of the line.

In the world of physics, this is usually a slow, difficult process. Heat tends to leak out into the empty space around the marbles before it can travel all the way down the line. But what if you could build a "super-highway" for heat to travel on?

That is exactly what this paper explores. The researchers are studying a special chain of marbles made of a material called Indium Antimonide (InSb) placed just above a flat floor made of the same material. They discovered that this setup creates a "topological" effect—a kind of invisible, protected path that allows heat to zip from one end of the chain to the other much faster than usual.

Here is the breakdown of their discovery using simple analogies:

1. The "Su-Schrieffer-Heeger" (SSH) Chain: A Rhythm Game

Think of the chain of marbles as a rhythm game.

• The Trivial Phase (Boring Rhythm): If the marbles are spaced out evenly (like clap, clap, clap, clap), the heat has to hop from one to the next. It's a steady but slow walk.
• The Non-Trivial Phase (The Special Rhythm): If you change the spacing so the marbles are in tight pairs with gaps between them (like clap-clap... pause... clap-clap... pause), something magical happens. This specific pattern creates "edge states."

The Analogy: Imagine a line of people passing a bucket of water.

• In the even spacing (trivial) line, everyone passes the bucket to their neighbor. If someone drops it, the line stops.
• In the paired spacing (non-trivial) line, the two people at the very ends of the line (the edges) develop a secret, super-fast connection. They can pass the bucket directly to each other, skipping the middle of the line entirely. This is the "topological edge mode."

2. The Substrate: The "Trampoline Floor"

Usually, if you put a chain of marbles in a vacuum, heat just radiates away into the void. But in this experiment, the chain is hovering just above a flat floor (the substrate).

The Analogy: Imagine the floor is a giant trampoline.
When the marbles vibrate (heat up), they don't just send energy into the air; they also bounce energy onto the trampoline floor. The floor has its own "waves" (surface waves) that can carry that energy.

• The floor acts like a conveyor belt. It catches the heat from the marbles, carries it along the floor, and drops it back off at the other end.
• This allows the heat to travel much further than it could in empty space. It's like sending a letter via a high-speed train (the floor) instead of walking it (through the air).

3. The "Sweet Spot" Distance

The researchers found something very interesting about how close the marbles need to be to the floor.

• Too far: The marbles can't "feel" the trampoline floor, so the conveyor belt doesn't work.
• Too close: The marbles get "stuck" to the floor, and the energy transfer gets messy and inefficient.
• Just right: There is a specific distance where the connection is perfect. It's like tuning a radio; you have to find the exact frequency (or distance) to get the clearest signal.

4. The Big Discovery: Topology Wins

The main result of the paper is a comparison between the "boring" even spacing and the "special" paired spacing.

• The Result: When the chain is in the "special" (topological) pattern, the heat transfer is significantly stronger than in the "boring" pattern.
• Why? Because in the special pattern, the "edge modes" (the secret connection between the first and last marble) are active. They act like a superhighway that bypasses the traffic jams in the middle of the chain.
• The Catch: This superhighway only works well if the "conveyor belt" (the floor's surface waves) is long enough to carry the heat. If the floor's waves die out too quickly, the advantage disappears.

Summary

Think of this research as designing a heat delivery service.

1. The Problem: Heat usually gets lost in transit.
2. The Solution: Use a special arrangement of particles (the SSH chain) sitting on a special floor (the substrate).
3. The Magic: By arranging the particles in a specific "paired" pattern, you unlock a topological superhighway at the ends of the chain.
4. The Benefit: Heat travels from one end to the other much faster and more efficiently, especially when the floor below helps carry the load.

This isn't just about marbles; it could lead to better ways to manage heat in tiny computer chips, making them run cooler and faster by using these "topological" shortcuts to move heat away from sensitive areas.

Technical Summary

1. Problem Statement

The paper addresses the challenge of controlling and enhancing long-range radiative heat transfer between nanoparticles. While near-field heat transfer is known to be enhanced by surface waves (such as Surface Plasmon Polaritons, SPPs, or Surface Phonon Polaritons, SPhPs) and can be mediated by a third body (like a substrate), the interplay between topological edge states in nanoparticle chains and substrate-mediated surface modes remains largely unexplored.

Specifically, the authors investigate a bipartite chain of plasmonic nanoparticles (an analogue of the Su-Schrieffer-Heeger or SSH model) placed in close proximity to a plasmonic substrate. The goal is to determine how the coupling between the chain's topological edge modes and the substrate's surface waves affects radiative heat transport, particularly whether topological protection can lead to enhanced long-range heat flux compared to the trivial phase.

2. Methodology

The study employs the framework of fluctuational electrodynamics within the dipole approximation.

• System Configuration: A chain of $N$ spherical Indium Antimonide (InSb) nanoparticles is arranged in a bipartite lattice (sublattices A and B) above a semi-infinite InSb substrate. The system is characterized by a lattice constant $d$ and an intra-cell separation $t$, defined by a parameter $\beta = 2t/d$.
  • $\beta < 1$: Topological trivial phase.
  • $\beta > 1$: Topological non-trivial phase (supporting edge modes).
• Material Properties: Both the nanoparticles and the substrate are modeled as InSb with Drude permittivity. The carrier concentration ($n$) is varied to tune the surface plasmon resonance frequency ($\omega_{SPP}$) of the substrate relative to the localized plasmon resonance of the particles ($\omega_{LP}$).
• Theoretical Framework:
  • Green's Function Approach: The total Green's function is decomposed into vacuum and scattering (substrate) parts to calculate the electromagnetic interaction between dipoles.
  • Band Structure Calculation: The eigenvalue equation for an infinite chain is solved to determine frequency bands for In-plane (IP) and Out-of-plane (OP) modes.
  • Topological Invariants: The Zak phase is calculated to identify topological phase transitions and predict the existence of edge modes in finite chains.
  • Heat Flux Calculation: The radiative heat flux is computed using the fluctuational electrodynamics formalism, calculating the power absorbed by the last particle in the chain when the first particle is heated.
  • Local Density of States (LDOS): The photonic LDOS is calculated to visualize the spatial distribution of edge modes and their interaction with the substrate.

3. Key Contributions

• Coupling of Topology and Substrate Modes: The paper demonstrates that the presence of a plasmonic substrate significantly deforms the frequency bands of the SSH chain. It shows that while the substrate introduces hybridization between longitudinal and transversal modes, the topological phase transition (determined by the Zak phase) persists.
• Mechanism of Long-Range Transport: The authors establish that long-range heat transfer is mediated by the substrate's surface waves. The efficiency of this transport is governed by the propagation length of these surface modes ($\Lambda_{SPP}$).
• Edge Mode Enhancement: The study proves that in the topological non-trivial phase, topologically protected edge modes provide an additional channel for heat transfer, leading to enhanced flux compared to the trivial phase, provided the chain length is comparable to the surface mode propagation length.
• Non-Monotonic Distance Dependence: A novel finding is the non-monotonic dependence of heat transfer on the chain-substrate distance ($z$). The coupling between edge modes and surface waves is maximized at a specific distance where momentum and energy conservation are optimally satisfied, rather than simply increasing as the gap closes.

4. Key Results

A. Band Structure and Topological Phases

• Band Deformation: Coupling to the substrate causes dips in the dipole-active bands. For low substrate carrier concentrations ($n_{sub}$), the coupling is strong, leading to band hybridization and the closing of band gaps for IP modes.
• Zak Phase: The Zak phase calculation confirms a topological phase transition at $\beta = 1$. For $\beta = 1.3$ (non-trivial), the Zak phase is $\pi$, predicting edge modes. For $\beta = 0.7$ (trivial), it is $0$.
• Edge Modes: In finite chains ($N=60$), distinct edge modes appear within the band gap for the non-trivial phase. These modes are highly localized at the first and last particles, confirmed by a high Inverse Participation Ratio (IPR $\approx 0.5$).

B. Radiative Heat Flux

• Substrate Enhancement: The presence of the substrate increases heat transfer to the last particle by up to a factor of 100 compared to the vacuum case, due to the surface wave channel.
• Optimal Chain Length: The heat flux exhibits a non-monotonic behavior with chain length. It peaks when the total chain length ($L$) is approximately equal to the surface plasmon propagation length ($\Lambda_{SPP}$).
• Topological vs. Trivial:
  • In the non-trivial phase ($\beta = 1.3$), heat flux is generally higher than in the trivial phase ($\beta = 0.7$) due to the contribution of edge modes.
  • This enhancement is most pronounced when the surface mode propagation length is much larger than the lattice constant.
  • For very short chains or low $n_{sub}$ (where $\Lambda_{SPP}$ is small), the edge mode advantage diminishes.
• Spectral Analysis: Spectral power analysis reveals that in the non-trivial phase, heat transfer is dominated by the narrow edge mode frequencies, whereas the trivial phase relies on broader dipole-active bands.

C. Distance Dependence ($z$)

• Non-Monotonic Behavior: For the non-trivial phase, the heat flux increases as the chain approaches the substrate but then decreases at very small distances ($z < 400$ nm).
• Explanation: This is attributed to the momentum matching condition. Perfect coupling between the localized edge mode and the surface wave occurs when $2k_{SPP} z \approx 1$. Deviating from this optimal distance reduces the coupling efficiency. In contrast, the trivial phase shows a monotonic increase as $z$ decreases.

D. Local Density of States (LDOS)

• The LDOS analysis confirms that the non-trivial phase exhibits enhanced LDOS at the edge particles compared to the trivial phase.
• The interaction term of the LDOS shows positive lobes above and below the edge particles in the non-trivial phase, visually confirming the enhanced coupling to surface modes.

5. Significance

• Fundamental Physics: The work bridges the fields of topological photonics and near-field thermodynamics, proving that topological protection can be utilized to control thermal transport in the presence of complex environments (substrates).
• Thermal Management: The findings suggest a pathway for designing thermal diodes or thermal switches where heat flow can be toggled or enhanced by tuning the topological phase (via geometry $\beta$) or the substrate properties (via doping $n_{sub}$).
• Experimental Guidance: The paper provides specific predictions for experimental verification, such as the non-monotonic distance dependence and the spectral signatures of edge modes. It suggests that scattering-type scanning near-field optical microscopy (s-SNOM) could be used to directly image the enhanced LDOS at the chain edges, offering a direct method to detect topological thermal states.
• Long-Range Transport: It validates the concept of using surface waves to mediate heat over distances significantly larger than the near-field limit, with topological edge states acting as efficient "bridges" for this transport.