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Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board Member: Fetecau, Corina / Kiefer, Claus / Röpke, Gerd / Steeb, Willi-Hans

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Volume 72, Issue 2 (Feb 2017)

Issues

Near-Field Radiative Heat Transfer under Temperature Gradients and Conductive Transfer

Weiliang Jin / Riccardo Messina
  • Corresponding author
  • Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F-34095 Montpellier, France
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/ Alejandro W. Rodriguez
Published Online: 2017-01-07 | DOI: https://doi.org/10.1515/zna-2016-0375

Abstract

We describe a recently developed formulation of coupled conductive and radiative heat transfer (RHT) between objects separated by nanometric, vacuum gaps. Our results rely on analytical formulas of RHT between planar slabs (based on the scattering-matrix method) as well as a general formulation of RHT between arbitrarily shaped bodies (based on the fluctuating–volume current method), which fully captures the existence of temperature inhomogeneities. In particular, the impact of RHT on conduction, and vice versa, is obtained via self-consistent solutions of the Fourier heat equation and Maxwell’s equations. We show that in materials with low thermal conductivities (e.g. zinc oxides and glasses), the interplay of conduction and RHT can strongly modify heat exchange, exemplified for instance by the presence of large temperature gradients and saturating flux rates at short (nanometric) distances. More generally, we show that the ability to tailor the temperature distribution of an object can modify the behaviour of RHT with respect to gap separations, e.g. qualitatively changing the asymptotic scaling at short separations from quadratic to linear or logarithmic. Our results could be relevant to the interpretation of both past and future experimental measurements of RHT at nanometric distances.

Keywords: Nanoscale Physics; Plasmonics; Radiative Heat Transfer

1 Introduction

Two bodies held at different temperatures T1 and T2 and separated by a vacuum gap exchange energy mediated by photons. A century ago, such radiative heat transfer (RHT) was thought to be limited by the famous Stefan–Boltzmann law. It sets an upper bound on the maximum amount of heat that an object can radiate into the far field [1], realised only in the ideal case of two blackbodies. Several decades ago, the pioneering works of Rytov [2], Polder and van Hove [3] showed that this law breaks down when the distance d between the two bodies is on the order or smaller than the corresponding thermal wavelength λT=ћc/kBT, close to 8 μm at ambient temperature. Essentially, at separations d<<λT, usually referred to as the near field, evanescent waves contribute flux, and thus, the cumulative amount of heat exchanged can be significantly larger (even by several orders of magnitude) than the blackbody limit [4], leading for instance to a d−2 divergence of the flux rate at short distances [5], [6]. Theoretical studies following these first results showed that such an effect is particularly pronounced (and further enhanced) if the bodies support surface resonances, e.g. plasmons in metals or phonon-polaritons in dielectrics, such that heat is primiarily mediated by collective electronic–photonic excitations propagating at material interfaces. Many experiments have since been performed – primarily in planar or nearly planar geometries, demonstrating very good agreement with theoretical predictions [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. More recently, however, experiments have begun to explore nano-metric gaps, where additional mechanisms (including phonon- and electron-mediated transfer) are expected to also contribute and have demonstrated unexpected behaviours, with either a reduction or an enahncement of the flux [18], [19], [20]. The possibility of a saturating flux has been predicted to occur at very short gaps due to the presence of nonlocal effects [21], [22], [23], phonon tunneling [24], and also due to the coupling of radiation with internal conduction [25]. These works imply that the first two effects only matter at subnanometer separations, while the conduction (the latter) proves insignificant even at such short lengthscales.

Our work focusses on the coupling between conductive and radiative effects, i.e. on the solution of the Fourier heat equation in the presence of RHT as a source term. Towards this aim, we first address the problem of calculating the heat flux between bodies having an arbitrary temperature profile. A numerical approach based on the fluctuating volume current (FVC) method allows us to tackle this problem in arbitrary geometries [26]. In this article, we exploit this powerful technique to address the conduction–radiation (CR) coupling in the case of two compact nanorods. However, we first consider a far simpler geometry, consisting of two planar slabs, by exploiting the well-established scattering-matrix formalism to derive explicit, closed-form expressions of the flux under arbitrary temperature profiles and dielectric properties. Using these analytical formulas, we are able to derive a simple analytical expression that captures the interplay of conduction and radiation as a function of the slab–slab vacuum gap and show that the temperature gradient vanishes in the limit d→0, while the heat flux saturates to a constant. In contrast to planar geometries, which support highly localised surface plasmons, we find that the contribution of bulk (delocalised) plasmonic modes in the nanorods leads to significant increase in RHT and larger as well as nonlinear temperature gradients. As we show through several numerical calculations, these effects persist in regimes already within experimental reach (e.g. silica slabs separated by tens of nanometer gaps), with even larger effects possible in heterostructured or multilayer media, e.g. a thin polaritonic film deposited on a low-conductivity substrate.

The discussions below are restricted to situations involving two bodies having linear dimensions much larger than the phonon mean free path. In this case, we are allowed to use the Fourier heat equation, which reads

[κ(x)T(x)]+d3xH(x,x)=Q(x),(1)

where κ(x) and Q(x) represent the bulk Fourier conductivity and external heat flux-rate at x, respectively, while H(x, x′) denotes the radiative power per unit volume from a point x′ to x. While the external heat flux Q will be assumed to vanish in our case, the calculation of the point-to-point radiative can be performed numerically within the FVC approach [26], described briefly below. This allows us to solve (1) in principle, for any geometry. While this method will be applied to the case of two nanorods in Section 3, we will first address the simpler geometry of two parallel slabs, demonstrating that for this geometry, an analytical solution (and additional insight) can be found.

2 Two Planar Slabs: Analytical Formulas

The geometry of two parallel slabs offers, independently of their thickness, the remarkable feature of translational invariance. For this reason, explicit expressions of the RHT between two slabs have been known for several decades. Among the possible approaches that lead to such expressions, we exploit a recently presented scattering-matrix formulation that requires only the scattering operators of the two isolated slabs, greatly simplifying the problem. This technique was recently introduced to study RHT along with interactions in planar systems out of thermal equilibrium, e.g. in the case of two [27], [28] and three [29] bodies.

In what follows, we exploit this approach and generalise it to consider the case of two bodies subject to arbitrary temperatures profiles. We begin by breaking up each body into N separate (adjacent) films of infinitesimal thickness. The field emitted by each film of thickness dz (see Fig. 1) is assumed to be in local thermal equilibrium, and its statistical properties are thus described by means of the well-known fluctuation-dissipation theorem (FDT) [30]. Note that we consider situations where the temperature gradient ∇T is small compared to some material-dependent current–current correlation lengthscale ξ (of the order of the atomic scale or phonon mean-free path), such that the charge distribution reaches local equilibrium [31]. Under this assumption, the FDT employed here can be thought of as the zeroth-order term of an expansion in powers of ξ|∇T|/T [26]. It follows that the correlation function of the counterpropagating components E0ϕ(ϕ=+,) emitted by the film is given by (see [32] for more details):

Planar slab occupying the region [z1, z2] and characterised by a temperature profile T(z). The field E0 is the one emitted by the film of infinitesimal thickness dz, while E+ and E− are the two components of the field emitted by the entire slab.
Figure 1:

Planar slab occupying the region [z1, z2] and characterised by a temperature profile T(z). The field E0 is the one emitted by the film of infinitesimal thickness dz, while E+ and E are the two components of the field emitted by the entire slab.

E0,pϕ(k,ω)E0,pϕ(k,ω)=ωdz2ε0c2N[ω,T(z)](2π)3δ(ωω)×δ(kk)δpp0,pϕϕ(k,ω),(2)

where

0,pϕϕ(k,ω)=Θ(ωck)2kzIm(kzm1+r21r2)Θ(ckω)4Im(kz)Re(kzmr1r2)e2ϕIm(kz)z,0,pϕ,ϕ(k,ω)=Θ(ωck)4kzIm(kzmr1r2)e2iϕkzz+Θ(ckω)2Im(kz)Re(kzm1+r21r2).(3)

Once the correlation functions of the fields emitted by the two slabs are known, the total field in the cavity formed by the two slabs (i.e. the vacuum gap) can be obtained as a result of the multiple reflections taking place between the planar interfaces. This finally leads to the calculation of the Poynting vector between the slabs and thus of the spectral component at frequency ω of the RHT. In the particular case of two infinitely-thick slabs at distance d made of the same material, the spectral component for propagative waves reads

φpw(ω)=12π20k0dkk(1|r|2)2kzm|1r2e2ikzd|20+dze2kzmz{N[ω,T(d2z)]N[ω,T(d2+z)]},(4)

while for evanescent waves, we have

φew(ω)=2π2k0+dkk(r)2e2kzdkzm|1r2e2kzd|20+dze2kzmz{N[ω,T(d2z)]N[ω,T(d2+z)]},(5)

where N(ω, T)=ћω/[exp(ћω/kBT)−1] denotes the Planck energy of a thermal oscillator, and c′, c″ denote the real and imaginary parts of the complex number c. Moreover, in this expression, r represents the standard Fresnel coefficient, while kz and kzm are the z components of the wavevector in vacuum and in the material, respectively.

2.1 Asymptotics

Equation 5 allows us to explore the behaviour of the heat flux under arbitrary temperature profiles and conditions. One interesting question which we now address is the degree to which temperature gradients can modify the asymptotic (short-distance) behaviour of the flux. In this case, we can restrict our attention to the evanescent, TM contribution to the flux (the only diverging term in the limit d→0), which is the only component supporting a surface mode. We first observe that the temperature profiles inside the two slabs appear in (5) through an integral calculated along the slab thickness. We start our calculation by writing the following Taylor expansion of each Planck energy:

N[ω,T(±d2±z)]=n=0+αn±(ω)n!zn.(6)

One can show [32] that only the n=0, 1, 2 terms lead to diverging flux contributions (5) in the limit d→0. A careful analysis of these three terms allows one to show (the quite intuitive statement) that only the surface behaviour of the function T(z) is relevant to the low-distance asymptotics of the RHT. In particular, if the temperatures Ta and Tb are different (see Fig. 2), the RHT exhibits the well-known d−2 divergence. If, on the contrary, we assume that the profile is such that Ta=Tb, this divergence is regularised and we are left with a flux that diverges like d−1. Assuming moreover that the first derivatives of T(z) satisfy T′(−d/2)=−T′(d/2), one can show that the flux diverges as log(αd), whereas it asymptotes to a constant as d→0 whenever T″(−d/2)=T″ (d/2) (e.g. the trivial scenario of symmetrical temperature profiles).

Two parallel slabs separated by a distance d. The two slabs are characterised by a temperature profile T(z). While the profile is arbitrary in panel (a), in panel (b), it is constant for z≤−d/2−a (where it is equal to TL) and z≥d/2+b (where it is equal to TR). Ta and Tb denote the temperatures at the left and right slab–vacuum interfaces, respectively.
Figure 2:

Two parallel slabs separated by a distance d. The two slabs are characterised by a temperature profile T(z). While the profile is arbitrary in panel (a), in panel (b), it is constant for z≤−d/2−a (where it is equal to TL) and zd/2+b (where it is equal to TR). Ta and Tb denote the temperatures at the left and right slab–vacuum interfaces, respectively.

To illustrate the various asymptotic behaviours, we calculate the RHT given by (5) in the case of two SiC slabs (whose dielectric properties are described by means of a Drude–Lorentz model [33]), for which the temperature is assumed to vary linearly over regions of thickness a=b=1 μm while the external temperatures are held at TL=600 K and TR=300 K (e.g. by means of a thermostat), as illustrated in Figure 2b. Figure 3 shows that whenever TaTb (the blue, orange, and red lines), the flux diverges under the expected d−2 power law. Nevertheless, it is evident that reducing the temperature gradient TaTb across the vacuum gap significantly reduces the overall RHT. In contrast, the case of equal surface temperatures, Ta=Tb=450 K (black line) clearly shows a drastically modified asymptotic behaviour, in agreement with our prior prediction of a d−1 divergence. The inset of Figure 3 compares the configuration (TL, Ta, Tb, TR)=(600, 451, 449, 300) K to the two uniform-temperature configurations (600, 600, 300, 300) K, and (451, 451, 449, 449) K, illustrating a transition between bulk- and surface-dominated physics. Essentially, at short distances, RHT proves to be a surface effect and thus only the temperatures near the slab–vacuum interfaces matter, whereas at larger distances, it is dominated by the averaged, bulk temperature within the slabs, and thus, the external temperatures play a dominant role.

Product of distance and RHT between two SiC slabs, assuming a=b=1μm and (TL, TR)=(600, 300) K. From top to bottom, we have (Ta, Tb)=(600, 300) K (blue), (460, 440) K (orange), (451, 449) K (red), and (450, 450) K (black). We also show are the predictions (dot-dashed lines) of the asymptotic formulas describing the d−2 and d−1 behaviours. In the inset, we compare the flux corresponding to the case (TL, Ta, Tb, TR)=(600, 451, 449, 300) K against the two configurations (600, 600, 300, 300) K (solid blue line) and (451, 451, 449, 449) K (dashed red line).
Figure 3:

Product of distance and RHT between two SiC slabs, assuming a=b=1μm and (TL, TR)=(600, 300) K. From top to bottom, we have (Ta, Tb)=(600, 300) K (blue), (460, 440) K (orange), (451, 449) K (red), and (450, 450) K (black). We also show are the predictions (dot-dashed lines) of the asymptotic formulas describing the d−2 and d−1 behaviours. In the inset, we compare the flux corresponding to the case (TL, Ta, Tb, TR)=(600, 451, 449, 300) K against the two configurations (600, 600, 300, 300) K (solid blue line) and (451, 451, 449, 449) K (dashed red line).

Thus far, we have demonstrated that with proper tailoring, the temperature profile T(z) can dramatically modify the asymptotic behaviour of RHT. In [32], we show that the former can also lead to transitions in the sign of the flux with respect to gap size, with potential applications to devices where a specific RHT power law or behaviour is desired. In the following section, we will demonstrate that the coupling between radiative and conductive transfer can result in the appearence of temperature gradients and thus strongly modify RHT, potentially becoming the main mechanism limiting RHT at nanometric distances.

2.2 Conduction–Radiation Coupling

Solutions of the heat (1) in the case of two parallel slabs (by virtue of the translation invariance) enjoy the same simplifcations as RHT calculations, allowing analytical insights. To make matters even simpler, we consider two physically motivated approximations that allow us to obtain an explicit closed-form solution both for the temperature gradient TaTb and for the RHT φ exchanged between the two slabs. In particular, the first assumption is that RHT in this system is a purely surface effect, in the sense that the heat exchanged between the two slabs is absorbed (or emitted) within a very thin region close to the slab–vacuum interface. This assumption is justified by the fact that at small separations the RHT is dominated by large-wavevector evanscent modes, strongly confined at the slab–vacuum interfaces [4]. It follows that the Fourier heat equation is actually source free in the bulk, in which case RHT can only lead to a linear temperature profile within each slab and moreover, only the temperatures Ta and Tb close to the interfaces are left as unknowns. The latter are thus determined by the boundary conditions that follow from (1). The second assumption is that, to a good approximation, near-field RHT is well described according to

φh0TaTbd2.(7)

while in general the RHT depends nonlinearly on temperature, the fact that conduction through the interior of the slabs scales linearly with TaTb and the conservation of energy (i.e. changes in conductive transfer must be offset by corresponding changes in RHT), implies that φ must also scale linearly with the temperature difference. Here, the coefficient h0 depends on the specific choice of materials and T and almost independent of the latter for sufficient high T. These two approximations allow us to write the following simple closed-form expression for the temperature gradient and the flux [34] (assuming symmetric slabs with a=b=t):

TaTbTLTR=(1+2th0κd2)1,φTLTR=h0d2(TaTbTLTR),(8)

Thus, one observes that, in general, a temperature profile is produced, with TaTb smaller than the external-temperature difference TLTR. It follows also that in the limit d→0, TaTb→0 as d2 and the behaviour of the flux is greatly modified. In particular, one can easily see that the flux tends asymptotically to the constant value, κ(TLTR)/2t, which is none other than the conductive flux in a slab of thickness 2t having externally imposed temperatures TL and TR. We note that, thanks to the linearisation with respect to temperature described by (7), it is possible to map our configuration to three resistors in series, the intermediate one (the RHT within the vacuum gap) having resistance d2/h0. Based on this parallel, the behaviour of TaTb and φ follow more intuitively as d→0.

The question we want to answer now is whether these modified behaviours are observable with everyday materials and in configurations under experimental reach. Towards this aim, we consider the specific case of two silica (SiO2) slabs subject to external temperatures TL=600 K and TR=300 K, and Figure 4 shows the resulting φ and TaTb for different thicknesses t, ranging from 100 nm to 500 μm. All the traced lines confirm the two asymptotic results discussed above, i.e. the quadratically vanishing temperature difference TaTb as well as the limiting constant value of the RHT. Moreover, as expected, changing the thickness t strongly modifies quantitatively the value of both quantities. We also remark that in the case of large thicknesses, deviations between the exact result and the one in the absence of temperature gradients are significant up to distances of the order of tens of nanometer, which have already been experimentally explored. In fact, we observe that a recent experiment measuring RHT between a silica sphere and a silica plane [18] has observed a significant reduction of the flux at distances slightly below 50 nm, suggesting the possibility of CR interplay. Note that the conduction inside the sphere differs from that of the slab, and an accurate estimation would require the general formulation provided in the following section. For more discussion of the dependence of this effect on material and geometric parameters, the reader is encouraged to read [34]. As a further remark, we observe that a configuration involving a thin highly conductive polaritonic material on top of a low-conductivity substrate, not yet explored, should also lead to RHT-mediated temperature gradients at much larger separations.

RHT φ and temperature difference Ta−Tb (inset) as a function of d between two silica slabs. We assume TL=600 K and TR=300 K. The solid lines correspond to different values of t (from top to bottom): 100 nm (black), 1μm (red), 10μm (brown), 100μm (blue), and 500 μm (green). The orange dashed line corresponds to the absence of temperature gradients.
Figure 4:

RHT φ and temperature difference TaTb (inset) as a function of d between two silica slabs. We assume TL=600 K and TR=300 K. The solid lines correspond to different values of t (from top to bottom): 100 nm (black), 1μm (red), 10μm (brown), 100μm (blue), and 500 μm (green). The orange dashed line corresponds to the absence of temperature gradients.

3 Two Nanorods: General Formulas

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 ξ=ixibi, with {bi} 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 σ=isibi can be determined via solution of a VIE equation, x+s=Ws, in terms of the unknown and known expansion coefficients {xi} and {si}, respectively, where Wi,j1=bi,(I+iωχG)bj and Gi,j=〈bi, Γ*bj〉 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 σα=sαbα at xa=bα in a body a. Such a “dipole” source induces polarisation–currents ξβ=xβbβ and EM fields ϕβ throughout space in body b (and elsewhere), such that the heat flux at xb=bβ is given (by Poynting’s theorem) by:

Φ(ω;xaxb)=12(ξβϕβ)(9)

where “〈…〉” denotes a thermodynamic ensemble average. Expressing the polarisation–currents and fields in the localised basis {bα}, 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]:

Φ(ω;xaxb)=12[Dα,αWα,β(GW)β,α](10)

where sα is a vector that is zero everywhere except at the αth element, denoted by sα, and Dα,β=sαsβ=d3xd3ybα(x)σ(x)σ(y)bβ(y) is a real, diagonal matrix encoding the thermodynamic and dissipative properties of each object [26] and described by the well-known FDT, σi(x,ω)σj(y,ω)=4πωIm[ε(x,ω)]Θ(Tx)δ(xy)δij, where Θ(T)=ћω/[exp(ћω/kBT)−1] is the Planck distribution. It follows then that the heat flux emitted or absorbed at a given position xa, the main quantity entering (1) through d3xH(x,x)=dωΦ(ω;x), is given by

Φ(ω;xa)=12[GWDbWΦaDWPbGWΦe]α,α(11)

Here, Pa(b) denotes the projection operator that selects only basis functions in a(b), such that Db=PbDPb is a diagonal matrix involving only fluctuations in object b. Furthermore, the first (second) term in (11) describes the absorbed (emitted) power in xa, 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.

(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=l2, separated by d=20 nm and held at temperatures Ta(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 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=l2, separated by d=20 nm and held at temperatures Ta(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=l2 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 Ta(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 ≈3×1014 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}×1014 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 TI=800 K while the entire nanorod b is held at Tb=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 xy 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.

(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×d2 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 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×d2 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×d2 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/d2 (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.

4 Concluding Remarks

We have studied the coupling between conduction and RHT in two different geometries. Starting from the case of two parallel planar slabs, we developed formulas based on the scattering-matrix formalism that describe the flux exchanged between the two bodies in the presence of an arbitrary temperature profile. Using these results, we have discussed the occurrence of four different asymptotic low-distance behaviour of the flux, depending on the properties of the temperature profile in proximity of the slab-vacuum interface. We have also addressed the coupled conduction–radiation problem in this setup, and obtained, by means of two approximations, explicit closed-form expressions for the temperature gradient and heat flux rates within and across the slabs, respectively. We have shown that an observable temperature gradient can indeed exist even for everyday materials and at distances within experimental reach. For these systems, the flux is significantly different from the one predicted in the absence of a temperature profile and saturates to a constant value, instead of diverging, in the limit of vanishing distance. We then considered the same coupled problem for two nanorods, for which we have calculated the RHT using a recently developed FVC approach. These structures exhibit a much higher radiative flux per unit surface due to the participation of bulk plasmon modes. As a consequence, the temperature gradients are also amplified and modified, exhibiting a nonlinear profile that is highly distance dependent. Our results could play a major role in devices aiming at manipulating and controlling the RHT at distances of some or tens of nanometers. They could aso be indeed relevant in modern experiments exploring nanometric gaps where the boundary between conduction and radiation is blurred. The idea of using heterostructures comprising low-conductivity material substrates and patterned polaritonic materials could also be promising for the observation of even larger CR effects at larger separations.

Acknowledgments

This work was supported by the National Science Foundation under Grant no. DMR-1454836 and by the Princeton Center for Complex Materials, an MRSEC supported by NSF Grant DMR 1420541.

References

  • [1]

    R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, Hemispher Publishing Corporation, Washington 1981. Google Scholar

  • [2]

    S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 3, Springer, New York 1989. Google Scholar

  • [3]

    D. Polder and M. van Hove, Phys. Rev. B 4, 3303 (1971). Google Scholar

  • [4]

    K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, Surf. Sci. Rep. 57, 59 (2005). Google Scholar

  • [5]

    J. J Loomis and H. J. Maris, Phys. Rev. B 50, 18517 (1994). Google Scholar

  • [6]

    J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, Microsc. Thermophys. Eng. 6, 209 (2002). Google Scholar

  • [7]

    A. Narayanaswamy, S. Shen, and G. Chen, Phys. Rev. B 78, 115303 (2008). Google Scholar

  • [8]

    L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, Appl. Phys. Lett. 92, 133106 (2008). Google Scholar

  • [9]

    E. Rousseau, A. Siria, G. Joudran, S. Volz, F. Comin, et al., Nature Photon. 3, 514 (2009). Google Scholar

  • [10]

    R. S. Ottens, V. Quetschke, S. Wise, A. A. Alemi, R. Lundock, et al., Phys. Rev. Lett. 107, 014301 (2011). Google Scholar

  • [11]

    T. Kralik, P. Hanzelka, V. Musilova, A. Srnka, and M. Zobac, Rev. Sci. Instrum. 82, 055106 (2011). Google Scholar

  • [12]

    T. Kralik, P. Hanzelka, M. Zobac, V. Musilova, T. Fort, et al., Phys. Rev. Lett. 109, 224302 (2012). Google Scholar

  • [13]

    P. J. van Zwol, L. Ranno, and J. Chevrier, Phys. Rev. Lett. 108, 234301 (2012). Google Scholar

  • [14]

    P. J. van Zwol, S. Thiele, C. Berger, W. A. de Heer, and J. Chevrier, Phys. Rev. Lett. 109, 264301 (2012). Google Scholar

  • [15]

    B. Song, Y. Ganjeh, S. Sadat, D. Thompson, A. Fiorino, et al., Nat. Nanotechnol. 10, 253 (2015). Google Scholar

  • [16]

    K. Kim, B. Song, V. Fernández-Hurtado, W. Lee, W. Jeong, et al., Nature 528, 387 (2015). Google Scholar

  • [17]

    R. St-Gelais, L. Zhu, S. Fan, and M. Lipson, Nat. Nanotechnol. 11, 515 (2016). Google Scholar

  • [18]

    S. Shen, A. Narayanaswamy, and G. Chen, Nano Lett. 9, 2909 (2009). Google Scholar

  • [19]

    A. Kittel, W. Müller-Hirsch, J. Parisi, S.-A. Biehs, D. Reddig, et al., Phys. Rev. Lett. 95, 224301 (2005). Google Scholar

  • [20]

    K. Kloppstech, N. Könne, S.-A. Biehs, A. W. Rodriguez, L. Worbes, et al., preprint arXiv:1510.06311 (2015). Google Scholar

  • [21]

    C. Henkel and K. Joulain, Appl. Phys. B 84, 61 (2006). Google Scholar

  • [22]

    K. Joulain, J. Quant. Spectrosc. Radiat. Transfer 109, 294 (2008). Google Scholar

  • [23]

    P.-O. Chapuis, S. Volz1, C. Henkel, K. Joulain, and J.-J. Greffet, Phys. Rev. B 77, 035431 (2008). Google Scholar

  • [24]

    V. Chiloyan, J. Garg, K. Esfarjani, and G. Chen, Nat. Commun. 6, 6755 (2015). Google Scholar

  • [25]

    B. T. Wong, M. Francoeur, and M. P. Mengüç, Int.l J. Heat Mass Transfer 54, 1825 (2011). Google Scholar

  • [26]

    A. G. Polimeridis, M. T. H. Reid, W. Jin, S. G. Johnson, J. K. White, et al., Phys. Rev. B 92,134202 (2015). Google Scholar

  • [27]

    R. Messina and M. Antezza, Europhys. Lett. 95, 61002 (2011). Google Scholar

  • [28]

    R. Messina and M. Antezza, Phys. Rev. A 84, 042102 (2011). Google Scholar

  • [29]

    R. Messina and M. Antezza, Phys. Rev. A 89, 052104 (2014). Google Scholar

  • [30]

    L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford 1963. Google Scholar

  • [31]

    A. Madrid, J. Rub, and L. Lapas, Phys. Rev. B 77, 155417 (2008). Google Scholar

  • [32]

    R. Messina, W. Jin, and A. W. Rodriguez, preprint arXiv:1609.00351 (2016). Google Scholar

  • [33]

    Handbook of Optical Constants of Solids, edited by E. Palik, Academic Press, New York 1998. Google Scholar

  • [34]

    R. Messina, W. Jin, and A. W. Rodriguez, Phys. Rev. B 94, 121410(R) (2016). Google Scholar

  • [35]

    S.-A. Biehs, M. Tschikin, and P. Ben-Abdallah, Phys. Rev. Lett. 109, 104301 (2012). Google Scholar

  • [36]

    X. Liu, R. Zhang, and Z. Zhang, Int. J. Heat Mass Transfer 73, 389 (2014). Google Scholar

  • [37]

    O. Miller, S. G. Johnson, and A. W. Rodriguez, Phys. Rev. Lett. 115, 204302 (2015). Google Scholar

  • [38]

    B. Liu and S. Shen, arXiv preprint arXiv:1509.00939 (2015). Google Scholar

  • [39]

    A. D. Phan, T.-L. Phan, and L. M. Woods, J. Appl. Phys. 114, 214306 (2013). Google Scholar

  • [40]

    Y. Yang and L. Wang, Phys. Rev. Lett. 117, 044301 (2016). Google Scholar

  • [41]

    J. Dai, S. A. Dyakov, and M. Yan, Phys. Rev. B 92, 035419 (2015). Google Scholar

  • [42]

    A. W. Rodriguez, M. H. Reid, J. Varela, J. D. Joannopoulos, F. Capasso, et al., Phys. Rev. Lett. 110, 014301 (2013). Google Scholar

  • [43]

    A. W. Rodriguez, M. H. Reid, and S. G. Johnson, Phys. Rev. B 86, 220302 (2012). Google Scholar

  • [44]

    S. Edalatpour and M. Francoeur, Phys. Rev. B 94, 045406 (2016). Google Scholar

  • [45]

    A. W. Rodriguez, O. Ilic, P. Bermel, I. Celanovic, J. D. Joannopoulos, et al., Phys. Rev. Lett. 107, 114302 (2011). Google Scholar

  • [46]

    W. Jin, R. Messina, and A. W. Rodriguez, preprint arXiv:1605.05708 (2016). Google Scholar

  • [47]

    W. H. Press, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, Cambridge 2007. Google Scholar

  • [48]

    J. Loureiro, N. Neves, R. Barros, T. Mateus, R. Santos, et al., J. Mater. Chem. A 2, 6649 (2014). Google Scholar

  • [49]

    G. V. Naik, J. Kim, and A. Boltasseva, Opt.l Mater. Exp. 1, 1090 (2011). Google Scholar

  • [50]

    S. Basu, Z. Zhang, and C. Fu, Int. J. Energy Res. 33, 1203 (2009). Google Scholar

About the article

Received: 2016-09-28

Accepted: 2016-11-27

Published Online: 2017-01-07

Published in Print: 2017-02-01


Funding Source: National Science Foundation

Award identifier / Grant number: DMR-1454836

This work was supported by the National Science Foundation under Grant no. DMR-1454836 and by the Princeton Center for Complex Materials, an MRSEC supported by NSF Grant DMR 1420541.


Citation Information: Zeitschrift für Naturforschung A, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-2016-0375.

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