Strongly coupled near-field radiative and conductive heat transfer between planar objects
SStrongly coupled near-field radiative and conductive heat transfer between planar objects
Riccardo Messina, ∗ Weiliang Jin, ∗ and Alejandro W. Rodriguez Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Universit´e de Montpellier, F- 34095 Montpellier, France Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
We study the interplay of conductive and radiative heat transfer (RHT) in planar geometries and predict thattemperature gradients induced by radiation can play a significant role on the behavior of RHT with respect to gapsizes, depending largely on geometric and material parameters and not so crucially on operating temperatures.Our findings exploit rigorous calculations based on a closed-form expression for the heat flux between twoplates separated by vacuum gaps d and subject to arbitrary temperature profiles, along with an approximate butaccurate analytical treatment of coupled conduction–radiation in this geometry. We find that these effects canbe prominent in typical materials (e.g. silica and sapphire) at separations of tens of nanometers, and can playan even larger role in metal oxides, which exhibit moderate conductivities and enhanced radiative properties.Broadly speaking, these predictions suggest that the impact of RHT on thermal conduction, and vice versa, couldmanifest itself as a limit on the possible magnitude of RHT at the nanoscale, which asymptotes to a constant(the conductive transfer rate when the gap is closed) instead of diverging at short separations. Two non-touching bodies held at different temperatures canexchange heat through photons. In the far field, i.e. separa-tions d (cid:29) thermal wavelength λ T = (cid:126) c/k B T (of the order of8 µ m at room temperature), the maximum radiative heat trans-fer (RHT) between them is limited by the well-known Stefan–Boltzmann law [1]. In the near field, i.e. d (cid:28) λ T , evanescentwaves can tunnel and contribute flux, enabling RHT to ex-ceed this limit by several orders of magnitude [2–4]. Such en-hancements can be larger in nano-structured surfaces [5, 6],but only a handful of these have been studied thus far [7–10], leaving much room for improvements [5]. A more com-monly studied heat-transport mechanism is thermal conduc-tion, involving transfer of energy through phonons or elec-trons. Although conduction is typically more efficient thanRHT [11–13], there are ongoing theoretical and experimen-tal efforts aimed at discovering novel materials and structuresleading to larger RHT, with recent work suggesting the possi-bility of orders of magnitude enhancements [5], in which caseRHT could not only compete but even exceed conduction insituations typically encountered in everyday experiments (e.g.under ambient conditions).In this paper, we present an approach for studying coupledconduction–radiation (CR) problems between planar objectsseparated by gaps that captures the full interplay of near-fieldRHT and thermal conduction at the nanoscale. Using an ex-act, closed-form expression of the slab–slab RHT in the pres-ence of arbitrary temperature distributions, we show that CRinterplay can give rise to significant temperature gradients andthereby greatly modify RHT, causing the latter to asymptote toa constant, the conductive flux when the gap is closed, whichsets a fundamental limit to radiative heat exchange at shortseparations. We provide evidence of the validity of a simplebut useful surface–sink approximation that treats the impactof RHT on conduction as arising purely at the vacuum–slabinterfaces, yielding analytical expressions with which one canstudy the scaling behavior and impact of these effects withrespect to relevant parameters. For instance, we find thattheir prominence is largely independent of operating temper-atures but strongly tied to the choice of materials (e.g. glasses and oxides versus highly conductive metals) and geometries(e.g. thin versus macroscopic films). Furthermore, thesephenomena lie within the reach of current-generation exper-iments [14], leading to significant changes in RHT betweentypical materials like silica and saphire at relatively large gapsizes ∼ several tens of nanometers, and potentially playingan even greater role in metal oxides, which exhibit low-lossinfrared polaritons [15, 16] and therefore enhance RHT.Coupled photonic and phononic diffusion processes innanostructures are becoming increasingly important [17, 18].While recent works have primarily focused on the interplaybetween thermal diffusion and external optical illumination,e.g. laser-induced, localized heating of plasmonic struc-tures [19–23], the thermal radiation emitted by a heated bodyand absorbed by nearby objects can also be a great sourceof heating or cooling. To date, however, the impact of RHTon conduction remains largely unexplored, with the consen-sus view being that radiation is insufficiently large to result inappreciable temperature gradients [12, 13, 24]. On the otherhand, modern experiments measuring RHT between planarsurfaces are beginning to probe the nanometer regime [14, 25–37], and in certain cases offer evidence of deviations fromthe typical /d behavior associated with near-field enhance-ment [38, 39] at nanometric distances, often attributed to non-local [17, 40] or phonon-tunneling [18] effects. Here, wefind that depending on geometric configuration and materials,radiation-induced temperature gradients can play a significantrole on transport above the nanometer regime, requiring a fulltreatment of the coupling between conduction and radiation. Exact formulation of coupled conduction–radiation.—
Inwhat follows, we present a formulation of coupled CR ap-plicable to the typical situation of two planar bodies (the sameframework can be extended to multiple bodies), labelled a and b , separated by a gap of size d . We assume that the slabs ex-hibit arbitrary temperature profiles and exchange heat amongone other. Neglecting convection and considering bodies withlengthscales larger or of the order of their phonon mean-freepath, in which case Fourier conduction is valid, the stationarytemperature distribution satisfies the one-dimensional coupled a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p CR equation: ∂∂z (cid:20) κ ( z ) ∂∂z T ( z ) (cid:21) + (cid:90) d z (cid:48) ϕ ( z (cid:48) , z ) = Q ( z ) , (1)where κ ( z ) and Q ( z ) represent the bulk Fourier conductivityand external heat-flux rate at z , respectively, while ϕ ( z (cid:48) , z ) denotes the radiative power per unit volume from a point z (cid:48) to z . Previous studies of (1) considered only radiative en-ergy escaping into vacuum through the surfaces of the ob-jects, exploiting simple, albeit inaccurate ray-optical approx-imations that are inapplicable for sub-wavelength objects orin the near field [11, 41]. The novelty of our approach to(1) is that ϕ , as written above and computed below, is fun-damentally tied to accurate and modern descriptions of RHTbased on macroscopic fluctuational EM [2, 42], allowing usto explore regimes (e.g. distances (cid:28) λ T ) where near-field ef-fects dominate RHT among different objects. In particular, aswe show below, in some regimes RHT can lead to observabletemperature distributions. Although we only consider the im-pact of external radiation on the temperature profile and viceversa, under large temperature gradients, RHT could poten-tially modify the intrinsic thermal conductivity of these ob-jects [17, 43, 44], a situation that we leave to future work. Wealso ignore far-field radiation since it is negligible comparedto conduction or RHT at the distances considered in this work.We focus on the scenario illustrated on the inset of Fig. 1,in which the temperature of slab a ( b ) slab is fixed at T L ( T R )by means of a thermostat, except for a region of thickness t a ( t b ). To study this problem, we calculate the RHT via aFourier expansion of the slabs’ scattering matrices [45]. Suchtechniques were recently employed to obtain close-formed an-alytical expressions of RHT between plates of uniform tem-perature [46–49]. Here, we extend these results to considerthe more general problem of slabs under arbitrary tempera-ture distributions. Toward this aim, we generalize prior meth-ods [47, 49] by dividing each slab into films of infinitesi-mal thicknesses, each having a fixed temperature. Describ-ing the associated EM fields at each point by means of thefluctuation-dissipation theorem (from which the RHT can bededuced), and considering multiple reflections associated withthe various interfaces, we find that the evanescent RHT perunit volume from a point z a in slab a to a point z b in slab b , ϕ ( z a , z b ) = (cid:82) ∞ d ω (cid:82) ∞ ω/c d β ϕ a ( ω, β ; z a , z b ) , can be ex-pressed analytically in the closed form [50]: ϕ ( ω, β ; z a , z b ) = 4 βπ ( r (cid:48)(cid:48) k (cid:48)(cid:48) zm ) e − k (cid:48)(cid:48) z d e − k (cid:48)(cid:48) zm ( z b − d/ | − r e − k (cid:48)(cid:48) z d | × (cid:16) N [ ω, T ( z a )] − N [ ω, T ( z b )] (cid:17) , (2)where β denotes the conserved, parallel ( x – y ) wavevector k z = (cid:112) ω /c − β and k zm = (cid:112) εω /c − β the perpen-dicular wavevectors in vacuum and the interior of the slabs, re-spectively, and where we introduced the transverse-magnetic(dominant) polarization Fresnel reflection coefficient of a pla-nar ε –vacuum interface, r = ( εk z − k zm ) / ( εk z + k zm ) . Note FIG. 1. The left inset shows a schematic of two parallel slabs sepa-rated by a distance d along the z direction ( z = 0 denotes the middleof the gap). The two slabs exhibit a temperature profile T ( z ) : thetemperature is constant and has values T L and T R in the regions z ≤ − d/ − t a and z ≥ d/ t b , respectively, and variable inthe regions − d/ − t a < z < − d/ and d/ < z < d/ t b ,with T a and T b denoting the temperatures at the slab–vacuum inter-faces. In the main part of the figure, temperature profile along thetemperature-varying region of the hot slab in a configuration involv-ing two silica slabs with t a = t b = 100 µ m held at external temper-atures T L = 600 K and T R = 300 K at the outer ends, and separatedby vacuum gaps d . The points z = 0 , µ m represent the boundaryof the thermostat and the interface with the vacuum gap. The threelines correspond to d = 10 nm (black), 20 nm (red) and 50 nm (blue).The right inset shows the z -dependent radiative flux-rate ϕ ( z a ) (seetext) along slab a at d = 100 nm. that we restrict our analysis to the transverse-magnetic polar-ization because only it supports a surface phonon-polaritonresonance. Equation (2) permits fast solutions of coupled CRproblems in this geometry for a wide range of parameters. Surface–sink approximation:
At small separations, near-field RHT is dominated by large- β surface modes that areexponentially confined to the slab–vacuum interfaces [4].Hence, it is sufficient (as discussed below and confirmedthrough exact results in Fig. 1) to treat its impact on conduc-tion as a purely surface effect, in which case the entire prob-lem can be described through the surface temperatures. Inparticular, under this assumption, given a distance d , identicalconductivities κ and temperature-varying regions t = t a = t b ,and external temperatures T L and T R , the only unknowns arethe interface temperatures T a and T b , which satisfy the fol-lowing boundary conditions: − κ T a − T L t = − κ T R − T b t = ϕ, (3)Here, ϕ denotes the net heat exchanged between the two slabs(assumed to take place at the surfaces), whose spectral ω and β components are given by [50]: ϕ ( ω, β ) = 2 βπ ( r (cid:48)(cid:48) ) k (cid:48)(cid:48) zm e − k (cid:48)(cid:48) z d | − r e − k (cid:48)(cid:48) z d | (cid:90) + ∞ dz e − k (cid:48)(cid:48) zm z × (cid:16) N [ ω, T ( − d/ − z )] − N [ ω, T ( d/ z )] (cid:17) . (4)Despite the complex dependence of the heat flux on separationand temperature profile, we find that it is possible to approxi-mate the former using a simple, power-law expression of theform, ϕ (cid:39) h ( T a − T b ) /d [39] (valid as long as the radiationis primarily coming from the surface of the slabs), with thecoefficient h calculated as the near-field heat flux betweentwo uniform-temperature slabs held at T L and T R , dividedby T L − T R . Essentially, while the dependence of RHT onabsolute temperature is generally nonlinear, the fact that con-duction through the interior of the slabs scales linearly with T a − T b and that energy must be conserved (i.e. changes inconductive transfer must be offset by corresponding changesin RHT), implies that ϕ must also scale linearly with T a − T b ,with the precise value of the coefficient h determined fromthe radiative conductivity at large values of d where radia-tion does not impact conduction. Given these simplifications,Eq. (3) can be solved to yield: T a − T b T L − T R = (cid:18) th κd (cid:19) − , ϕT L − T R = h d (cid:18) T a − T b T L − T R (cid:19) . (5)These formulas reveal that the interplay of conduction and ra-diation causes T a − T b → quadratically with d , producing acontinuous temperature profile and leading to a finite value of ϕ → κ ( T L − T R ) / t as d → , the conductive flux througha gapless slab of thickness t subject to a linear temperaturegradient T L − T R (as it must, from energy conservation). Be-low, we show that the existence of such temperature gradientsalong with deviations from the typical /d RHT power laware within the reach of present experimental detection.
Numerical predictions.—
To begin with, we first addressthe validity of the surface–sink approximation above. In or-der to do so, we of course need to consider the full coupledCR problem described by (1), requiring numerical evaluationof the spatial heat transfer ϕ ( z a , z b ) in (2). For concreteness,we consider a practical situation typical of RHT experiments,involving two silica (SiO ) slabs subject to external tempera-tures T L = 600 K and T R = 300 K by a thermostat at distance t = t a = t b = 100 µ m away from the slab–vacuum interfaces.Silica not only has relatively low κ ≈ . W/m · K but also sup-ports polaritonic resonances at mid-infrared wavelengths andhas well-tabulated optical properties [51]. Figure 1 illustratesthe increasing, linear temperature gradient present in slab a with decreasing separations d , a consequence of the exponen-tial decay of the spatial heat transfer, ϕ ( z a ) = (cid:82) dz b ϕ ( z b , z a ) ,illustrated on the inset at a fixed d = 100 nm. Results obtainedthrough (5), with h = 5 . × − W/K, are in almost per-fect (essentially indistiguishable) agreement with those of thefull CR treatment and are therefore not shown. The same is - FIG. 2. Total flux ϕ and temperature difference T a − T b (inset) asa function of distance d between two silica slabs (shown schemati-cally on the left inset of Fig. 1) that are being held at T L = 600 Kand T R = 300 K. The various solid lines correspond to differenttemperature-varying regions t (from top to bottom): 100 nm (black), µ m (red), µ m (brown), µ m (blue) and µ m (green). Theorange dashed line shows ϕ in the absence of temperature gradients. true at smaller values of t , down to tens of nanometers, belowwhich the surface–sink approximation begins to fail.Figure 2 shows ϕ and T a − T b (inset), normalized by theexternal temperature difference T L − T R , as a function of d and for the same slab configuration but considering multiple t = { , . , , , , } µ m, with decreasing values of t leading to smaller temperature gradients and larger ϕ . Here, t = 0 (dashed line) corresponds to the typical scenario whereconduction dominates and hence there are no temperature gra-dients, in which case ϕ = h ( T L − T R ) /d exhibits the ex-pected divergence. Quite interestingly, we find that at typi-cal values of t = 100 µ m, the flux decreases by ≈ atdistances d ≈ nm, well within the reach of current experi-ments [14, 25, 34–36]. This result may be particularly relevantto recent experiments [14] investigating RHT between largesilica objects, which indicate deviations from the /d scal-ing behavior (along with flux saturation) at similar distances.We now explore the degree to which these saturation effectsdepend on the choice of material and operating conditions,quantified via the separation regime at which they become sig-nificant. In particular, inspection of (5) alllows us to define thedistance ˜ d = (cid:112) th /κ at which T a − T b = ( T L − T R ) and ϕ = h ( T L − T R ) / ˜ d , corresponding to half the value of theRHT obtained when conduction and radiation do not influenceone another. Figure 3 shows ˜ d as a function of the material-dependent ratio h /κ for the particular choice of T L = 600 K, T R = 300 K, and t = 100 µ m, highlighting the square-rootdependence of the former on the latter. Superimposed are theexpected ˜ d associated with various materials of possible ex-perimental interest (solid circles), obtained by employing ap-propiate values of κ and h , which depend primarily on thechoice of external temperature. Within the surface–sink ap-proximation (valid here), the latter do not influence the scal-ing of either ϕ or T a − T b with respect to separation, as evi-dent from (5). The inset of Fig. 3 shows h as a function T L for SiC, SiO , and aluminum zinc oxide (AZO), identified bytheir increasing values of h , illustrating the near constancyof the coefficient over a wide range of acceptable temperaturedifferences. Note that we consider unrealistically large valuesof T L only to illustrate asymptotic behavior.Noticeably, despite small differences in the value of h between various materials, there are striking variations in ˜ d ,which can range anywhere from a few nanometers in the caseof SiC and GaAs, up to several tens of nanometers for SiO and AZO, respectively. Such variations are almost entirelydue to differences in thermal conductivities, which naturallyplay a major role in this problem, with the conductivities ofSiC, SiO , and AZO taken to be κ (cid:39) W/m · K, . W/m · K,and . W/m · K, respectively. Note that, generally, zinc ox-ides exhibit moderate values of thermal conductivities at hightemperatures, depending on their fabrication method, with thevalue here taken from Ref. 52. The open circle in Fig. 3 in-dicates the expected ˜ d ∼ hundreds of nanometers associatedwith ultra-low conductivity ( κ (cid:46) . W/m · K) nanocompos-ite oxides that can now be engineered [15, 52, 53] and whichare likely to play a more prominent role in future thermal de-vices [16]. We stress that our predictions are consistent withthe lack of gradient effects observed in recent experiments in-volving materials such as silicon and Au, which exhibit lowand high values of h and κ , respectively. The case of silica isparticularly interesting, however, since it is typically used inRHT experiments, yet the possibility of temperature gradientshas never been considered. These results along with (5) canserve as a reference for future experiments, allowing estimatesof the regimes under which these effects become relevant.While our analysis above is based on the assumption of vac-uum gaps, it is straightforward to generalize Eq. (5) to includethe possibility of finite intervening conductivities, κ > ,requiring only that h be replaced with h + κ d in the firstexpression of Eq. (5). We find, however, that similar conclu-sions follow for small but finite κ (cid:46) − W/m · K (typical ofRHT experiments).In conclusion, we have presented a study of coupledconduction–radiation heat transfer between planar objects atshort distances. We have expressed the resulting temperaturegradients and radiative-flux modifications in terms of simple,analytical expressions involving geometric and material pa-rameters, showing that in systems well within experimentalreach or already considered in experiments [14], both tem-perature gradients and flux saturation should be observed. Asimilar saturation phenomenon has been predicted to occurdue to non-local damping [17, 40] and/or phonon-tunnelingbelow the nanometer scale [18] (note that at atomistic scaleswhere continuum electrodynamics fails, the boundary be-tween phonon and radiative conduction is blurred). Our worksuggests that even at and above nano-meter gaps, and depend-ing on material and geometric conditions, CR interplay could - - - - ��� ��� [ ��� ] ��� [ ���� ] ��� � �� � � � ���� ��� FIG. 3. Typical distance scale ˜ d relevant to conduction–radiationproblems (see text) as a function of the material-dependent ratio h /κ for two slabs with t a = t b = 100 µ m. Solid circles de-note corresponding values for specific material choices (abbrevia-tions), with AZO[1.2] and AZO[0.05] denoting aluminum zinc ox-ides of different conductivities κ = 1 . W/m · K [52] and potential κ = 0 . W/m · K [15], respectively. The inset shows the dependenceof the radiative-heat transfer coefficient h on the external tempera-ture gradient ∆ T = T L − K, for three cases, AZO (red), silica(blue) and SiC (black), with decreasing values. instead become the dominant mechanism limiting RHT. Fur-thermore, there are significant efforts underway aimed at ex-ploring regimes, e.g. smaller gap sizes or materials and struc-tures leading to larger RHT (for applications in nanoscalecooling [54] and other thermal devices [55]), where these ef-fects may be observed at even at larger separations. Our on-going work generalizing the coupled CR formulation to ar-bitrary geometries reveals even larger interplay in structuredsurfaces [56, 57]. Arguably, advances in either or both direc-tions will make such analyses necessary.
Acknowledgements
This work was supported by the Na-tional Science Foundation under Grant no. DMR-1454836and by the Princeton Center for Complex Materials, a MR-SEC supported by NSF Grant DMR 1420541. ∗ These authors contributed equally to this work.[1] J. R. Howell, M. P. Meng¨uc¸, and R. Siegel,
Thermal radiationheat transfer (CRC press, 2010).[2] S. Basu, Z. Zhang, and C. Fu, International Journal of EnergyResearch , 1203 (2009).[3] A. Volokitin and B. N. Persson, Rev. Mod. Phys. , 1291(2007).[4] K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Gr-effet, Surf. Sci. Rep. , 59 (2005).[5] O. D. Miller, S. G. Johnson, and A. W. Rodriguez, Phys. Rev.Lett. , 204302 (2015).[6] C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Ro-driguez, preprint arXiv:1511.04492 (2015). [7] A. Narayanaswamy and G. Chen, Phys. Rev. B , 075125(2008).[8] M. Kruger, T. Emig, and M. Kardar, Phys. Rev. Lett. ,210404 (2011).[9] A. P. McCauley, M. H. Reid, M. Kruger, and S. G. Johnson,Phys. Rev. B , 165104 (2012).[10] A. W. Rodriguez, M. H. Reid, and S. G. Johnson, Phys. Rev. B , 054305 (2013).[11] J. Holman, Heat transfer (McGraw-Hill Inc, 2010).[12] B. T. Wong, M. Francoeur, and M. P. Meng¨uc¸, InternationalJournal of Heat and Mass Transfer , 1825 (2011).[13] J. Z.-J. Lau, V. N.-S. Bong, and B. T. Wong, J. Quant. Spectrosc.Radiat. Transfer , 39 (2016).[14] S. Shen, A. Narayanaswamy, and G. Chen, Nano Letters ,2909 (2009).[15] C. Chiritescu, D. G. Cahill, N. Nguyen, D. Johnson, A. Bodap-ati, P. Keblinski, P. Zschack, Science , 351 (2007).[16] D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E.Goodson, P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar,Applied Physics Reviews , 011305 (2014).[17] K. Joulain, J. Quant. Spectrosc. Radiat. Transfer , 294(2008).[18] V. Chiloyan, J. Garg, K. Esfarjani, and G. Chen, Nature Com-munications , 6775 (2015).[19] G. Baffou, C. Girard, and R. Quidant, Phys. Rev. Lett. ,136805 (2010).[20] G. Baffou, E. B. Urea, P. Berto, S. Monneret, R. Quidant, andH. Rigneault, Nanoscale , 8984 (2014).[21] H. Ma, P. Tian, J. Pello, P. M. Bendix, and L. B. Oddershede,Nano Letters , 612 (2014).[22] C. L. Baldwin, N. W. Bigelow, and D. J. Masiello, The journalof physical chemistry letters , 1347 (2014).[23] R. Biswas and M. L. Povinelli, ACS Photonics , 1681 (2015).[24] B. T. Wong, M. Francoeur, V. N.-S. Bong, and M. P. Meng¨uc¸,J. Quant. Spectrosc. Radiat. Transfer , 46 (2014).[25] A. Kittel, W. M¨uller-Hirsch, J. Parisi, S.-A. Biehs, D. Reddig,and M. Holthaus, Phys. Rev. Lett. , 224301 (2005).[26] A. Narayanaswamy, S. Shen, and G. Chen, Phys. Rev. B ,115303 (2008).[27] L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, Appl. Phys.Lett. , 133106 (2008).[28] E. Rousseau, A. Siria, G. Joudran, S. Volz, F. Comin, J.Chevrier, and J.-J. Greffet, Nature Photon. , 514 (2009).[29] R. S. Ottens, V. Quetschke, S. Wise, A. A. Alemi, R. Lundock,G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, Phys.Rev. Lett. , 014301 (2011). [30] T. Kralik, P. Hanzelka, V. Musilova, A. Srnka, and M. Zobac,Rev. Sci. Instrum. , 055106 (2011).[31] T. Kralik, P. Hanzelka, M. Zobac, V. Musilova, T. Fort, and M.Horak, Phys. Rev. Lett. , 224302 (2012).[32] P. J. van Zwol, L. Ranno, and J. Chevrier, Phys. Rev. Lett. ,234301 (2012).[33] P. J. van Zwol, S. Thiele, C. Berger, W. A. de Heer, and J.Chevrier, Phys. Rev. Lett. , 264301 (2012).[34] B. Song et al. , Nature Nanotechnology , 253 (2015).[35] K. Kim et al. , Nature , 387 (2015).[36] K. Kloppstech et al. , preprint arXiv:1510.06311 (2015).[37] R. St-Gelais, L. Zhu, S. Fan, and M. Lipson, Nature Nanotech-nology , 515 (2016).[38] P.-O. Chapuis, S. Volz, C. Henkel, K. Joulain, and J.-J. Greffet,Phys. Rev. B , 035431 (2008).[39] J.-P. Mulet, K. Joulain, R. Carminati, and J.-J.Greffet, Mi-croscale Thermophysical Engineering , 209 (2002).[40] C. Henkel and K. Joulain, Appl. Phys. B 84, 61 (2006).[41] R. M. S. da Gama, Applied Mathematical Modelling , 795(2004).[42] S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles ofStatistical Radiophysics (Springer, Berlin, 1988).[43] D.-Z. A. Chen, A. Narayanaswamy, and G. Chen, Phys. Rev. B , 155435 (2005).[44] S. Volz, et al. , The Eur. Phys. J. B , 1 (2016).[45] M. Francoeur, M. P. Meng¨uc¸, and. R. Vaillon, J. Quant. Spec-trosc. Radiat. Transfer , 2002 (2009).[46] P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues,J. Appl. Phys. , 044306 (2009).[47] R. Messina and M. Antezza, Phys. Rev. A , 042102 (2011).[48] R. Messina, M. Antezza, and P. Ben-Abdallah, Phys. Rev. Lett. , 244302 (2012).[49] R. Messina and M. Antezza, Phys. Rev. A , 052104 (2014).[50] R. Messina, W. Jin, and A. W. Rodriguez, in preparation.[51] Handbook of Optical Constants of Solids , edited by E. Palik(Academic Press, New York, 1998).[52] J. Loureiro et al. , J. Mater. Chem. A , 6649 (2014).[53] P. Jood, R. J. Mehta, Y. Zhang, G. Peleckis, X. Wang, R. W.Siegel, T. Borca-Tasciuc, S. X. Dou, and G. Ramanath, NanoLetters , 4337 (2011).[54] B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, NanoLetters , 4546 (2012).[55] P. Ben-Abdallah and S.-A. Biehs, AIP Advances , 053502,(2015).[56] A. G. Polimeridis, M. T. H. Reid, W. Jin, S. G. Johnson, J. K.White, and A. W. Rodriguez, Phys. Rev. B92