Non-equilibrium Green's function formalism for radiative heat transfer

Abstract

Radiative heat transfer (RHT) at the nanoscale can vastly exceed the far-field blackbody limit due to the tunneling of evanescent waves, a phenomenon traditionally described by fluctuational electrodynamics (FE). While FE has been exceptionally successful for systems in local thermal equilibrium, its foundational assumptions break down in the growing number of scenarios involving genuine non-equilibrium conditions, such as in active devices or driven materials. This review introduces the non-equilibrium Green's function (NEGF) formalism as a powerful and versatile framework to study RHT beyond these classical limits. Rooted in quantum many-body theory, NEGF provides a unified language to describe energy transport by photons, electrons, and phonons on an equal footing. We first outline the theoretical foundations of the NEGF approach for RHT, demonstrating how it recovers the canonical results of FE in the local equilibrium limit. We then survey recent breakthroughs enabled by NEGF, including: (i) providing a quantum-accurate description of equilibrium RHT that naturally incorporates non-local and finite-size effects, resolving unphysical divergences predicted by local models; (ii) unifying heat transfer channels to reveal the non-additive synergy between radiation, electron tunneling, and phonon conduction at sub-nanometer gaps; (iii) enabling the quantum design of materials and metamaterials with tailored thermal properties through band structure and topological engineering; and (iv) describing active control of heat flow in driven systems, which allows for phenomena like isothermal heat transfer and pumping heat against a temperature gradient.

Figures (4)

• Figure 1: (a) Heat flux between two parallel graphene sheets as a function of distance. The NEGF calculation (solid line) shows a saturation of heat flux at short distances, resolving the unphysical $1/d$ divergence predicted by local theories. At larger distances, the flux follows a $\sim d^{-2}$ scaling law (red dashed line). (b) The NEGF and FE equivalence. This conceptual diagram illustrates that NFRHT, described as the tunneling of evanescent waves in fluctuational electrodynamics, is equivalently described in the NEGF formalism as energy transfer mediated by the Coulomb interactions between quantum charge fluctuations in the two bodies. The response function is obtained from first-principles DFT calculations. (c) Visualization of non-local response at a metal surface. The plot shows the real part of the normal component of the polarizability, $\rm{Re}[\Pi_{zz}(j,j')]$, as a function of lattice layer indices $j$ and $j'$. The response is highly non-diagonal, extending over $\sim 100$ atomic layers from the surface ($j = 1$), demonstrating the failure of local models which would be entirely diagonal. (d) Dimensional dependence of heat transfer. The heat flux in a line-surface (1D-2D) configuration decays with distance as $\sim d^{-5}$, a significantly steeper dependence than the $\sim d^{-2}$ law for a surface-surface (2D-2D) system. This highlights how NEGF captures the profound influence of electronic dimensionality on RHT. Panel (a) reprinted with permission from add61. Copyright 2017 American Physical Society. Panel (c) reprinted with permission from add83. Copyright 2025 American Physical Society. Panel (d) reprinted with permission from add84. Copyright 2020 Springer Nature.
• Figure 2: (a) Photon-Electron Synergy: Heat flux in a metal-vacuum-metal junction as a function of distance. The total heat flux (solid black line) is a non-trivial combination of contributions from Coulomb fluctuations (photons, blue dashed line) and electron tunneling (electrons, red dotted line). A crossover occurs at $d \approx0.92$ nm, below which electron tunneling dominates. In the strong tunneling regime ($d < 0.6$ nm), the total flux exceeds the simple sum of the two, indicating non-linear enhancement. (b) Photon-Phonon Synergy: Thermal conductance between two collinear carbyne wires. The total conductance including both radiative (RHT) and phonon (PCHT) channels (solid green line, "both") is shown along with the individual contributions. In the range $d \approx0.2-0.4$ nm, the total conductance is anomalously suppressed below either of the individual channels, demonstrating their non-additive nature. (c) Spectral Origin of Suppression: The Landauer energy transmission spectrum for the carbyne wires at $d = 0.281$ nm. The coupled spectrum ("both") is significantly suppressed in the low-frequency region compared to the pure radiative spectrum ("rad"), explaining the reduction in total conductance seen in (b). Panel (a) adapted with permission from add62. Copyright 2018 American Physical Society. Panels (b) and (c) adapted with permission from add33. Copyright 2020 American Physical Society.
• Figure 3: (a) Topological Control: Heat flux between two zigzag carbon nanotubes exhibits a non-monotonic dependence on distance, with a peak at a critical distance $d_c$. This anomalous behavior is a direct consequence of topologically protected edge states. (b) Topological Thermal Switch: By tuning a hopping parameter $\lambda$ in an SSH chain model to induce a topological phase transition (inset shows the emergence of a zero-energy edge state), the heat flux can be switched on and off. (c) Band Structure Engineering: The density of states (DOS) of twisted bilayer graphene (TBG) exhibits sharp van Hove singularities near the Fermi level when tuned to the magic angle ($\theta = 1.1^\circ$), a result of engineered flat bands. (d) Tunable Thermal Emission: The engineered flat bands in TBG lead to a sharp, intense peak in its thermal radiation spectrum. The position of this peak is highly tunable with the twist angle, allowing the material's radiative properties to be designed for specific functions. Panels (a) and (b) adapted with permission from add86. Copyright 2019 American Physical Society. Panels (c) and (d) adapted with permission from add93. Copyright 2022 Elsevier.
• Figure 4: (a) Floquet Phase Control: In a system of two bodies held at the same temperature ($T_1 = T_2$), a net energy flux can be generated by driving them with a relative phase difference $\theta = \theta_L-\theta_R$. The plot shows that the heat flux is directly controlled by this phase, enabling its magnitude and direction to be tuned externally. This provides a mechanism for thermal rectification. (b) Current-Induced Heat Flux: Heat flux between two graphene layers at the same temperature as a function of distance, induced solely by a drift current in one layer. The flux exhibits a parabolic dependence on the drift velocity $v_d$, demonstrating that the external drive is the source of the energy transfer. (c) Current-Driven Thermal Shutoff: The plot shows the spectral heat flux between two graphene layers with a temperature difference. Without a current (blue dash-dotted line), heat flows from hot to cold. An applied drift current induces a negative heat flux component via negative Landau damping (red dashed line). A sufficiently strong current (black solid line) can pump heat against the gradient, achieving a "thermal shutoff" or even active cooling. (d) Current-Induced Heat Transfer and Forces: In an asymmetrically biased graphene nanoribbon-$C_{60}$ system, the non-equilibrium current generates not only heat transfer (left panel) but also measurable forces (right panel) on the nanoribbon, with sharp resonant peaks appearing at specific bias voltages. Panel (a) adapted with permission from add24. Copyright 2024 American Physical Society. Panels (b) and (c) adapted with permission from a preprint by Peng et aladd31. Permission to reuse was requested from the authors. Panel (d) adapted with permission from add30. Copyright 2024 American Physical Society.