Far-field heat transfer and monochromatic thermal currents in a cylindrical nonreciprocal cavity

Technical Summary: Far-field heat transfer and monochromatic thermal currents in a cylindrical nonreciprocal cavity

Problem Statement
While recent theoretical and experimental works have established that Kirchhoff's law of thermal radiation can be violated in specific directions—enabling nonreciprocal thermal emission and applications in energy harvesting and thermal circuitry—the implications of these effects in complex, non-planar far-field geometries remain poorly understood. Most existing studies on nonreciprocal radiative heat transfer are restricted to planar or near-field configurations. This work addresses the gap in understanding how nonreciprocity manifests in far-field radiative heat transfer within a hollow, infinitely long cylindrical cavity, specifically investigating whether violation of Kirchhoff's law can sustain persistent thermal currents or induce directional heat fluxes at thermal equilibrium and under non-equilibrium conditions.

Methodology
The authors model an infinitely long cylindrical cavity immersed in vacuum, with a radius $R$ significantly larger than the thermal wavelength to ensure operation in the far-field regime. The cavity surface is discretized into $n$ identical, vertically aligned elements ($\Omega_i$), each assigned either reciprocal or nonreciprocal material properties defined by their permittivity tensors.

To solve the radiative transfer problem, which resists analytical treatment in this geometry, the authors employ a custom specular ray-tracing algorithm. This approach is constrained by the generalized form of Kirchhoff's law applicable to specular surfaces: $\epsilon(\omega, \theta, \phi) = \alpha(\omega, \theta, \phi + \pi)$, where emissivity in a given direction equals absorptivity in the opposite direction. The study focuses on two primary metrics:

1. Monochromatic transmission coefficient ($S_{i \to j}$): The normalized electromagnetic energy received by element $j$ from element $i$.
2. Heat rectification coefficient ($H_{ij}$): Defined as the directional imbalance $S_{i \to j} - S_{j \to i}$.
3. Nonreciprocity factor ($\zeta$): A generalized metric quantifying the overall directional imbalance across all element pairs.
4. Monochromatic heat flux vector field ($\mathbf{v}$): The net thermal flow integrated over all directions at specific points within the cavity and on its boundary.

Simulations compare three scenarios: fully reciprocal configurations, fully nonreciprocal configurations (using an idealized "on-off" emissivity profile and a realistic Weyl semimetal model), and mixed configurations combining both material types.

Key Contributions and Results

• Thermal Equilibrium Findings:

  • Non-vanishing Rectification: In configurations containing nonreciprocal materials, the heat rectification coefficients ($H_{ij}$) are non-zero, indicating a directional imbalance in monochromatic radiative exchange between elements. For an idealized "on-off" nonreciprocal material, the magnitude of rectification reaches $|H_{ij}| = 0.26$, significantly higher than that of realistic Weyl semimetals ($|H_{ij}| = 0.028$).
  • Absence of Internal Currents: Despite the presence of non-zero rectification coefficients and closed-loop exchange channels (visualized as circulation in directed graphs), internal monochromatic thermal currents vanish at thermal equilibrium. This is attributed to the intrinsic coupling between emission and absorption at specular surfaces mandated by the generalized Kirchhoff's law. Strong emission in one direction necessitates strong absorption from the opposite direction, causing the total outgoing radiation (emitted plus reflected) to remain isotropic regardless of material reciprocity.
  • Boundary Effects: While internal currents vanish, a non-zero tangential component of the heat flux persists at the boundary, reflecting the directional imbalance in surface absorption and emission.

• Non-Equilibrium Findings:

  • Controlled Rotational Fluxes: Under non-equilibrium conditions (where specific elements are maintained at a higher temperature than others), the constraint of vanishing internal currents is lifted.
  • Reciprocal vs. Nonreciprocal: When hot elements are reciprocal, heat flux flows symmetrically from hot to cold elements. However, when hot elements are nonreciprocal, the heat flux vector field organizes into a distinct counterclockwise rotational circulation. This directional transport is driven by the asymmetric enhancement of emissivity into a specific hemisphere.
  • Mixed Configurations: The study demonstrates that even a single nonreciprocal element embedded in a reciprocal background can induce non-vanishing heat rectification coefficients across the entire system via indirect radiative coupling through reflections.

• Quantitative Metrics:

  • The nonreciprocity factor $\zeta$ reaches 0.85 for a fully nonreciprocal "on-off" configuration with 8 elements, approaching the theoretical maximum of 1 as the number of elements increases.
  • Mixed configurations (alternating reciprocal and nonreciprocal elements) yield $\zeta = 0.72$, demonstrating that substantial directional imbalance can be achieved without requiring the entire system to be nonreciprocal.

Significance and Claims
The paper claims to provide a rigorous numerical framework for analyzing nonreciprocal radiative heat transfer in cylindrical far-field geometries. Its primary significance lies in clarifying the distinction between heat rectification (directional imbalance in exchange) and persistent thermal currents (net flow). The authors demonstrate that while nonreciprocity yields maximal heat rectification coefficients even at equilibrium, the generalized Kirchhoff's law for specular surfaces prevents the formation of persistent internal thermal currents in equilibrium.

Conversely, the work establishes that nonreciprocity enables the precise control of rotational heat fluxes under non-equilibrium conditions. These findings offer design principles for nonreciprocal photonic devices and suggest new avenues for thermal management, particularly in creating directional thermal transport in the far field by strategically combining reciprocal and nonreciprocal materials. The authors note that future work will explore more complex geometries (e.g., concentric multi-cavity systems) and the interplay between near-field and far-field interactions.