Even larger conduction–radiation interplays are expected to arise in structured materials, e.g. hyperbolic metamaterials [35], [36] or lattices of metallic antennas [37], [38], where RHT can be further enhanced [35], [37], [38], [39], [40], [41] and modified [42], [43], [44], [45] compared to planar structures [37]. However, our ability to solve the coupled conduction–radiation problem (1) in arbitrary geometries hinges on our ability to compute *H*(**x**, **x**′) in full generality, which is possible thanks to a recently introduced FVC method that exploits powerful EM scattering techniques [26] to enable fast calculations of RHT between arbitrarily shaped objects subject to arbitrary temperature distributions.

The starting point of the FVC method is the volume-integral equation (VIE) formulation of EM, in which the scattering unknowns are 6-component polarisation currents *ξ* in the interior of the bodies coupled via the homogeneous 6×6 Green’s function Γ of the intervening medium [26]. Given two objects described by a susceptibility tensor *χ*(**x**) and a Galerkin decomposition of the induced currents $\xi ={\displaystyle {\sum}_{i}{x}_{i}{b}_{i}},$ with {*b*_{i}} denoting localised basis functions throughout the objects (*i* is the global index for all bodies), the scattering of an incident field due to some fluctuating current–source $\sigma ={\displaystyle {\sum}_{i}{s}_{i}{b}_{i}}$ can be determined via solution of a VIE equation, *x*+*s*=*Ws*, in terms of the unknown and known expansion coefficients {*x*_{i}} and {*s*_{i}}, respectively, where ${W}_{i\mathrm{,}j}^{-\text{\hspace{0.17em}}1}=\u3008{b}_{i}\mathrm{,}\text{\hspace{0.17em}}\mathrm{(}I+i\omega \chi G\mathrm{)}{b}_{j}\u3009$ and *G*_{i,j}=〈*b*_{i}, Γ**b*_{j}〉 are known as VIE and Green matrices [26]. Previously, we exploited this formalism to propose an efficient method for computing the total heat transfer between any two compact bodies [26], based on a simple voxel basis expansion (uniform discretisation). The solution of (1) requires an extension of the FVC method to include the spatially resolved heat transfer between any two voxels, which we describe below. Consider a fluctuating current–source *σ*_{α}=*s*_{α}b_{α} at **x**_{a}=*b*_{α} in a body *a*. Such a “dipole” source induces polarisation–currents *ξ*_{β}=*x*_{β}b_{β} and EM fields *ϕ*_{β} throughout space in body *b* (and elsewhere), such that the heat flux at **x**_{b}=*b*_{β} is given (by Poynting’s theorem) by:

$$\Phi \mathrm{(}\omega \mathrm{;}\text{\hspace{0.17em}}{x}_{a}\to {x}_{b}\mathrm{)}=\frac{1}{2}\u3008\Re \mathrm{(}{\xi}_{\beta}^{\ast}{\varphi}_{\beta}\mathrm{)}\u3009$$(9)

where “〈…〉” denotes a thermodynamic ensemble average. Expressing the polarisation–currents and fields in the localised basis {*b*_{α}}, and exploiting the volume equivalence principle to express the field as a convolution of the incident and induced currents with the vacuum Green’s function (GF), *ϕ*=Γ*(*ξ*+*σ*), one finds that (9) can be expressed in a compact, algebraic form involving VIE matrices [46]:

$$\Phi \mathrm{(}\omega \mathrm{;}\text{\hspace{0.17em}}{x}_{a}\to {x}_{b}\mathrm{)}=\frac{1}{2}\Re [{D}_{\alpha \mathrm{,}\alpha}{W}_{\alpha \mathrm{,}\beta}^{\u2020}{\mathrm{(}GW\mathrm{)}}_{\beta \mathrm{,}\alpha}]$$(10)

where *s*^{α} is a vector that is zero everywhere except at the *α*th element, denoted by *s*_{α}, and ${D}_{\alpha \mathrm{,}\beta}=\u3008{s}_{\alpha}^{\ast}{s}_{\beta}\u3009={\displaystyle \iint {d}^{3}\overrightarrow{x}{d}^{3}\overrightarrow{y}{b}_{\alpha}^{\ast}\mathrm{(}\overrightarrow{x}\mathrm{)}\u3008\sigma \mathrm{(}\overrightarrow{x}\mathrm{)}{\sigma}^{\ast}\mathrm{(}\overrightarrow{y}\mathrm{)}\u3009{b}_{\beta}\mathrm{(}\overrightarrow{y}\mathrm{)}}$ is a real, diagonal matrix encoding the thermodynamic and dissipative properties of each object [26] and described by the well-known FDT, $\u3008{\sigma}_{i}\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}\omega \mathrm{)}{\sigma}_{j}^{\ast}\mathrm{(}y\mathrm{,}\text{\hspace{0.17em}}\omega \mathrm{)}\u3009=\frac{4}{\pi}\omega \text{Im}\mathrm{[}\epsilon \mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}\omega \mathrm{)}]\Theta \mathrm{(}{T}_{x}\mathrm{)}\delta \mathrm{(}x-y\mathrm{)}{\delta}_{ij},$ where Θ(*T*)=*ћω*/[exp(*ћω*/*k*_{B}T)−1] is the Planck distribution. It follows then that the heat flux emitted or absorbed at a given position **x**_{a}, the main quantity entering (1) through $\int {\text{d}}^{3}{x}^{\prime}H\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}{x}^{\prime}\mathrm{)}}={\displaystyle \int \text{d}\omega \Phi \mathrm{(}\omega \mathrm{;}\text{\hspace{0.17em}}x\mathrm{)}},$ is given by

$$\Phi \mathrm{(}\omega \mathrm{;}\text{\hspace{0.17em}}{x}_{a}\mathrm{)}=\frac{1}{2}\Re {\left[\underset{{\Phi}_{\text{a}}}{\underbrace{GW{D}^{b}{W}^{\u2020}}}-\underset{{\Phi}_{\text{e}}}{\underbrace{D{W}^{\u2020}{P}^{b}GW}}\right]}_{\alpha \mathrm{,}\alpha}$$(11)

Here, *P*^{a}^{(b)} denotes the projection operator that selects only basis functions in *a*(*b*), such that *D*^{b}=*P*^{b}DP^{b} is a diagonal matrix involving only fluctuations in object *b*. Furthermore, the first (second) term in (11) describes the absorbed (emitted) power in **x**_{a}, henceforth denoted via the subscript “a(e).” Equation (11) is a generalisation of our previous expression for the total heat transfer between two arbitrary inhomogeneous objects [26] in that it includes both the spatially resolved absorbed and emitted power throughout the entire geometry. In Ref. FVC, we showed that the low-rank nature of the GF operator enables truncated, randomised SVD factorisations and therefore efficient evaluations of the corresponding matrix operations. We find, however, that in this case, the inclusion of the absorption term does not permit such a factorisation, except in special circumstances [46].

Given (11), one can solve the coupled CR equation in any number of ways [47]. Here, we exploit a fixed-point iteration procedure based on repeated and independent evaluations of (10) and (1), converging once both quantities approach a set of self-consistent steady-state values. Equation (1) is solved via a commercial, finite-element heat solver, whereas (10) is solved through a free, in-house implementation of our FVC method [26]. While the above formulation is general, below we explore the computationally convenient situation in which object *b* is kept at a constant, uniform temperature by means of a carefully chosen thermal reservoir, such that the absorbed power in object *a* can be computed efficiently. Furthermore, whenever one of the objects is heated to a much larger temperature than the other (as is the case below), it is also reasonable to ignore the impact of temperature gradients in the cooler body on the corresponding profile of the hotter body. The power emitted by *a* (the heated object) turns out to be much more convenient to compute, since the time-consuming part of the scattering calculation can be precomputed independently from the temperature distribution in object *a* and hence stored for repeated and subsequent evaluations of (1) under different temperature profiles.

As a proof of principle, we now apply the above-mentioned method to a simple geometry consisting of two metallic nanorods of cross-sectional widths *l* and thickness *t*; in practice, both for easy of fabrication and to obtain even larger RHT [17], such a structure could be realised as a lattice or grating, shown schematically in Figure 5a. However, for computational convenience and conceptual simplicity, we restrict our analysis to the regime of large grating periods, in which case it suffices to consider only the transfer between nearby objects. In the following, we take AZO as an illustrative example [48], [49]. To begin with, we show that even in the absence of CR interplay, the RHT spectrum and spatial RHT distribution inside the nanorods differ significantly from those of AZO slabs of the same thickness.

Figure 5: (a) Schematic illustration of two square lattices of nanorods (labelled *a* and *b*) of thickess *t*, period Λ, cross-sectional area *l*×*l*, and separation *d*, whose temperature distribution and energy exchange is mediated by both conductive ∇·[*κ*(**x**)*T*(**x**)] and radiative *H*(**x**, **x**′) heat transfer. (b) Total radiative heat transfer spectrum Φ(*ω*) between two AZO nanorods (solid lines) of thickness *t*=500 nm and cross-sectional area *A*=*l*^{2}, separated by *d*=20 nm and held at temperatures *T*_{a}_{(b)}=800(300) K. The spectrum is shown for different cross sections *l*={10, 20} nm (blue and red lines) and in the limit *l*=∞, corresponding to two planar slabs. (c) Spatial radiative heat flux in nanorod *a* for the case *l*=20 nm, corresponding to the (i) first, (ii) second, and (iii) SPP plasmon resonances, respectively, annotated in (b).

Figure 5b shows the RHT spectrum Φ(*ω*) per unit area *A*=*l*^{2} between two AZO nanorods (with doping concentration 11 wt% [49]) of length *t*=500 nm and varying widths *l*={10, 20, ∞} nm (blue solid, red solid, and black dashed lines), held at temperatures *T*_{a}_{(b)}=800(300) K and vacuum gap *d*=20 nm. The limit *l*→∞ corresponds to the slab–slab geometry explored in Section 2, in which case the Φ(*ω*) exhibits a single peak occuring at the SPP frequency ≈3×10^{14} rad/s. The finite nature of the nanorods results in additional peaks at lower frequencies, corresponding to bulk/geometric plasmon resonances (red and blue solid lines) that provide additional channels of heat exchange, albeit at the expense of weaker SPP peaks, leading to a roughly 5-fold enhancement in RHT compared to slabs. More importantly and well known, such structured antennas allow tuning and creation of bulk plasmon resonances in the near- and far-infrared spectra (much lower than many planar materials) that can more effectively transfer thermal radiation. The contour plots in Figure 5(i–iii) reveal the spatial RHT distribution Φ(*ω*, **x**) (in arbitrary units) at three separate frequencies *ω*={0.4, 0.8, 2.3}×10^{14} rad/s, corresponding to the first, second, and SPP resonances, respectively. As expected, the highest frequency resonance is primarily confined to the corners of the nanorod surface (becoming the well-known SPP resonance in the limit *l*→∞), with the fundamental and intermediate resonances have flux contributions stemming primarily form the bulk. As we now show, such an enhancement results not only results in larger temperature gradients but also changes the resulting qualitative temperature distribution.

Figure 6a shows the temperature profile along the *z* direction for the nanorod geometry of Figure 5a, with width *l*=20 nm and gap size *d*=20 nm, obtained via solution of (1). For the purpose of generality, we show results under various doping concentrations {2, 6, 11}wt% (green, red, and black solid lines), corresponding to different SPP frequencies and bandwidths [49]. In particular, we consider a situation in which the boundary I of nanorod *a* is kept at *T*_{I}=800 K while the entire nanorod *b* is held at *T*_{b}=300 K (through contact with a room-temperature reservoir), and assume an AZO thermal conductivity of *κ*=1 W/m·K [48]. The temperature along the *x*–*y* cross section is nearly uniform and therefore only shown in the case of 11 wt% (inset), a consequence of the faster heat diffusion associated with *l*<<*t*. In all scenarios, the temperature gradient is significantly larger in the case of nanorods (solid lines) than for slabs (*t*→∞, dashed lines), becoming nearly an order of magnitude larger in the case of 6 wt%, whose SPP frequency is much higher than the peak Planck wavelength at *T*=800 *K*. More interestingly, we find that while slabs exhibit linear temperature profiles (since RHT is dominated by surface emission [50]), the bulk and delocalised nature of RHT in the case of nanorods leads to visibly nonlinear temperature distributions.

Figure 6: (a) Temperature profile along the *z* coordinate of a nanorod (solid lines) when it is heated from one side to a temperature of 800 K and is separated from an identical, constant- and uniform-temperature nanrod held at *T*=300 K on the other side, by a gap size *d*=20 nm. The nanorods have cross-sectional width *l*=20 nm and thicknesses *t*=500 nm and are made up of AZO with results shown for multiple values of the doping concentration {2, 6, 11}wt% (blue, red, and black lines). Also shown are the temperature profiles of slabs (dashed lines) of the same thickness (corresponding to the limit *l*→∞). Inset: Temperature distribution throughout the nanorod in the case of 11 wt%. (b) Temperature profiles of nanorods of width *l*=10 nm under various separations *d*={5, 10, 20, 30} nm (black, blue, red, and green lines). (c) The radiative flux *H*×*d*^{2} for nanorods of width *l*=10 nm (red lines) and the slabs (*blue lines*) as a function of *d* in the presence (solid lines) or absence (dashed lines) of temperature gradients induced by conduction and radiation interplay. (Inset:) The ratio of radiative heat flux from the nanorods to slabs in the presence (red dots) and absence (black dots) of conduction and radiation interplay.

Figure 6b shows the temperature profile at various separations *d*={5, 10, 20, 30} nm (black, blue, red, and green lines) in the case of nanorods of width *l*=10 nm and 11 wt%, illustrating the sensitive relationship between the degree of CR interplay and magnitude of the gradients and the gap size. Notably, while the RHT and therefore temperature gradients increase as *d* decreases, the profile becomes increasingly linear as the geometry approaches the slab–slab configuration. The transition from bulk- to SPP-dominated RHT and the increasing impact of the latter on conduction and vice versa is also evident from inspection of the dependence of the RHT rate on gap size. In particular, Figure 6c shows the asymptotics of RHT flux *H*×*d*^{2} as a function of *d* in the presence (solid lines) and absence (dashed lines) of CR interplay (with the latter involving uniform temperatures) and for both slabs (black lines) of thickness *t*=500 nm and nanorods (red lines) of equal thickness and width *l*=10 nm. For the well-known uniform temperature scenarios, the RHT for both nanorods and slabs behaves as 1/*d*^{2} (dashed lines); however, the temperature gradients induced by CR interplay significantly changes the RHT asymptotic behaviour for nanorods at a separation of *d*=15 nm, while not obvious for the slabs down to *d*=5 nm. This effect is further illustrated by the ratio of RHT flux as a function of *d*, as shown in the insets. While the ratio remains almost a constant for the uniform temperature case (black dots), it decreases visibly when considering CR interplay (red dots), implying that temperature gradients are larger in the nanorods. As in the case of plates, in the limit *d*→0 (not shown), RHT will asymptotic to a constant rather than a diverge.

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