Radiative Spin Caloritronics

Technical Summary: Radiative Spin Caloritronics

Problem Statement
Recent advances in many-body radiative heat transfer have revealed phenomena analogous to condensed-matter physics, such as thermal Hall effects and topological heat flow. In magneto-optical systems, external magnetic fields break Lorentz reciprocity, coupling the orbital and spin degrees of freedom of thermal photons. While the inverse spin thermal Hall effect (ISTHE)—where a longitudinal gradient of photon spin angular momentum generates a transverse radiative heat flux—was recently predicted, the reciprocal question remained open: Does a longitudinal radiative heat flux generate a transverse accumulation of photon spin angular momentum? Furthermore, it was unclear whether these two conversion mechanisms are connected by Onsager-Casimir reciprocity relations despite the explicit breaking of time-reversal symmetry by the external magnetic field.

Methodology
The authors investigate a nonreciprocal many-body system consisting of four identical Indium Antimonide (InSb) nanoparticles ($R=50$ nm) arranged in a square with $C_4$ symmetry, immersed in a thermal bath at $T_b = 300$ K. The system is subjected to an external magnetic field $\mathbf{H}_{ext}$ along the $z$-axis.

• Theoretical Framework: The study utilizes fluctuational electrodynamics within the electric-dipole approximation. The dielectric permittivity of the nanoparticles is modeled using a gyrotropic Drude–Lorentz model, introducing off-diagonal components that break Lorentz reciprocity.
• Transport Calculation: Net radiative power exchange between particles and the bath is calculated using a many-body Landauer formalism. The transmission coefficients $T_{j,i}(\omega, H_{ext})$ are shown to satisfy the Onsager-Casimir symmetry $T_{j,i}(\omega, H_{ext}) = T_{i,j}(\omega, -H_{ext})$.
• Spin Definition: The local spin angular momentum (SAM) density is defined via the correlation functions of electric and magnetic fields. The study focuses on the out-of-plane component $S_z$, which is non-zero due to the magnetic field lifting the degeneracy of circular polarizations.
• Linear Response: The system is analyzed in the linear-response regime, where a longitudinal temperature bias ($\Delta T$) drives a heat current, and the resulting transverse spin accumulation ($\Delta S_z$) is measured.

Key Contributions and Results

1. Prediction of the Spin Thermal Hall Effect (STHE): The paper predicts that a longitudinal radiative heat current ($J_Q^x$) driven by a temperature gradient induces a transverse accumulation of photon spin angular momentum ($\Delta S_z$) in the network. This is characterized by a spin contrast between the upper and lower particles of the square.
2. Onsager-Casimir Reciprocity: The authors demonstrate that the STHE and the previously known ISTHE constitute an Onsager-Casimir reciprocal pair. By defining thermodynamic forces ($X_T$ for temperature gradient, $X_S$ for spin gradient) and fluxes ($J_Q$ for heat, $J_S$ for spin), they derive the transport matrix. They show that the coupling coefficients satisfy $G_{TS}(H_{ext}) = G_{ST}(-H_{ext})$, confirming that heat and photon spin are coupled transport channels in nonreciprocal photonic systems.
3. Symmetry Properties: Numerical results for the InSb network reveal specific symmetry signatures:
   • The transverse spin accumulation $\Delta S_z$ is an odd function of the temperature bias ($\Delta T$).
   • $\Delta S_z$ is an even function of the magnetic field magnitude ($H_{ext}$), despite the underlying magneto-optical response being odd. This arises from the combined action of the gyrotropic response and the $C_4$ network symmetry.
   • Reversing the magnetic field reverses the sign of the transverse spin accumulation.
4. Microscopic Origin and Spectral Analysis: The spin-heat coupling coefficient ($G_{ST}$) is derived from the fluctuational-electrodynamic description. Spectral decomposition shows the response is strongly resonant, dominated by a narrow high-frequency resonance associated with the free-carrier dipolar mode of the nanoparticles. The magnetic field enhances this contribution through mode hybridization.
5. Thermodynamic Bounds and Figure of Merit: Applying the second law of thermodynamics to the transport matrix imposes a fundamental bound on the spin-heat coupling strength. The authors introduce a dimensionless parameter $\kappa$ and a thermal-spin figure of merit $Z_{TS} = \kappa / (1-\kappa)$, analogous to the thermoelectric figure of merit $ZT$. This metric quantifies the efficiency of radiative spin-heat conversion, bounded by $0 \le \kappa \le 1$.

Significance
The paper establishes a complete thermodynamic framework for "photon spin caloritronics." By identifying photon spin as a genuine thermodynamic transport variable capable of driving energy transport (and vice versa) in nonreciprocal systems, the work completes the thermodynamic description of spin-resolved radiative transport. The authors claim these results lay the conceptual foundations for spin-controlled thermal radiation and the development of nonreciprocal photonic thermal devices, providing a universal metric ($Z_{TS}$) to quantify the efficiency of such spin-heat conversion processes.