Near-field radiative heat transfer between isotropic, dielectric-based metamaterials is analyzed. A potassium bromide host medium comprised of silicon carbide (SiC) spheres with a volume filling fraction of 0.4 is considered for the metamaterial. The relative electric permittivity and relative magnetic permeability of the metamaterial are modeled via the Clausius-Mossotti relations linking the macroscopic response of the medium with the polarizabilities of the spheres. We show for the first time that electric and magnetic surface polariton (SP) mediated near-field radiative heat transfer occurs between dielectric-based structures. Magnetic SPs, existing in TE polarization, are physically due to strong magnetic dipole resonances of the spheres. We find that spherical inclusions with radii of 1 μm (or greater) are needed in order to induce SPs in TE polarization. On the other hand, electric SPs existing in TM polarization are generated by surface modes of the spheres, and are thus almost insensitive to the size of the inclusions. We estimate that the total heat flux around SP resonance for the metamaterial comprised of SiC spheres with radii of 1 μm is about 35% greater than the flux predicted between two bulks of SiC, where only surface phonon-polaritons in TM polarization are excited. The results presented in this work show that the near-field thermal spectrum can be engineered via dielectric-based metamaterials, which is crucial in many emerging technologies, such as in nanoscale-gap thermophotovoltaic power generation.
© 2011 OSA
Electromagnetic metamaterials are artificial materials, made of sub-wavelength functional inclusions, displaying unusual electric and magnetic properties, such as negative index of refraction [1–5]. Metamaterials research is motivated by various potential applications such as superlenses [5–9] and optical cloaking [5,10–12]. The possibility of controlling and designing the electric and magnetic responses of materials also paves the way to tailoring media with unique thermal radiative properties. So far, efforts in this area have been restricted to designing selective thermal radiation emitters and absorbers in the far-field [13–22], while, to the best of our knowledge, only three papers have investigated near-field radiative heat transfer between metamaterials [23–25].
Thermal radiation is in the near-field regime when the bodies are separated by distances of the same order of magnitude as, or less than, the dominant emission wavelength [26,27]. In that case, radiative heat transfer can exceed the blackbody predictions due to tunneling of evanescent waves. Moreover, quasi-monochromatic radiant heat exchange is possible when materials support surface polaritons (SPs) in the infrared, such as surface phonon-polaritons in silicon carbide (SiC)  and surface plasmon-polaritons in doped silicon . When dealing with non-magnetic materials, SPs can only be excited in TM polarization [30,31]. Near-field thermal radiation is an emerging area of heat transfer engineering which may find applications in imaging , energy conversion [33–37] and thermal rectification [38,39] to name only a few. In many cases, it is imperative to have some degree of control over the near-field thermal spectrum emitted and absorbed. For example, poor performance of nanoscale-gap thermophotovoltaic power generation was recently predicted due to the overheating of the cell . It was concluded that this problem could be circumvented by fine tuning the near-field thermal spectrum transferred between the radiator and the cell via thermal excitation of SPs, which can potentially be accomplished using metamaterials with negative permittivity and/or negative permeability at prescribed frequencies.
Joulain et al.  studied near-field radiative heat transfer between two hypothetical metamaterials made of an artificial array of thin wires and split ring resonators (SRRs), such that negative permittivity, negative permeability and negative refraction were obtained in some specific spectral bands. Results showed that SP mediated heat transfer in both TM and TE polarizations dominated the radiative flux. Indeed, a medium with negative permeability can support SPs in TE polarization, thus opening an additional channel through which near-field radiant energy can flow. Zheng and Xuan [24,25] provided a comprehensive procedure for solving near-field radiative heat transfer problems in one-dimensional geometry with layers of arbitrary permittivity and permeability. The method was tested for the same metamaterials discussed in reference .
Metamaterials based on metallic constitutive elements such as SRRs, thin wires, rods or fishnet structures  present several drawbacks. Fabrication of these intricate structures requires sophisticated lithography techniques. Additionally, due to the anisotropy of the metallic inclusions, the macroscopic response of the metamaterial is also anisotropic, unless complicated three-dimensional structures are employed. Finally, high losses are observed in these metal-based metamaterials at optical frequencies [40–43].
The general objective of this paper is to investigate near-field radiative heat transfer between realistic, isotropic metamaterials exhibiting negative permittivity and permeability in the infrared. This task is achieved by considering Mie resonance-based dielectric metamaterials [5,40–51]. In this work, spherical dielectric inclusions are considered due to a relatively simple fabrication process. Moreover, bulk three-dimensional dielectric sphere-based metamaterials exhibit isotropic macroscopic electric and magnetic responses since spheres are inherently isotropic . The electric and magnetic dipole resonances of a sphere can be conceptualized as the meta-atoms making up the metamaterial. A large collection of these dielectric spheres, within a dielectric host, having strong enough electric and magnetic dipole resonances can thus constitute a material with macroscopic negative permittivity and permeability for specific frequencies. Indeed, even if both the host medium and the inclusions are dielectric, negative permeability may be induced via sub-wavelength particles with very high permittivity supporting strong resonances with a large displacement current [5,42]. In this paper, SiC spheres are chosen as dielectric inclusions since the electric permittivity of this material is very large in the infrared.
The rest of the paper is structured as follows. In the next section, the problem under consideration, namely two bulk metamaterials exchanging thermal radiation and separated by a sub-wavelength vacuum gap, is described; the mathematical details required for calculating the near-field radiative heat flux, based on fluctuational electrodynamics, are provided. Subsequently, the effective permittivity and permeability models used to describe the macroscopic electric and magnetic responses of the metamaterials made of SiC spheres are discussed. In the fourth section, spectral distributions of near-field radiative heat flux are calculated and analyzed thoroughly. Concluding remarks are provided in the last section.
2. Near-field thermal radiation modeling between two bulk metamaterials
The geometry of the problem considered is schematically depicted in Fig. 1 , where two bulk materials modeled as planar half-spaces (media 0 and 2) with perfectly smooth and parallel surfaces are separated by a vacuum gap of thickness d (medium 1). The system is invariant along the x-y plane such that only the z-component of the radiative heat flux is considered.
It is assumed that the media are in local thermodynamic equilibrium, homogeneous, isotropic and described by frequency-dependent relative electric permittivity ε (= ) and relative magnetic permeability μ (= ) local in space. The problem is solved at steady-state, where media 0 and 2 are maintained at constant and uniform temperatures T 0 and T 2, respectively, via external heating or cooling devices. Such an approximation is physically realistic since the thermal resistance due to near-field radiative heat transfer in the vacuum gap is much higher than the thermal resistance due to conduction in layers 0 and 2. For example, assuming that both layers are 1 mm thick and made of SiC, we estimated that the radiative thermal resistance, in the extreme case where d = 10 nm, is about 180 times the conduction resistance for a temperature gradient of 300 K between media 0 and 2.
Near-field radiative heat transfer is predicted using fluctuational electrodynamics, where the microscopic randomly oscillating charges generating the thermal radiation field are modeled in Maxwell’s equations via macroscopic stochastic currents . When dealing with non-magnetic media, a stochastic current J r , e is added in Ampère’s law in order to model thermal radiation emission due to electric dipole oscillations [52,53]. For magnetic media, as considered in this paper, a stochastic current J r , m should also be included in Faraday’s law to model thermal emission due to fluctuating magnetic dipoles . Assuming a time-dependence of exp(-iωt), the electric and magnetic fields thermally generated can be written as follows :
The radiative heat flux along the z-direction observed at r due to a source located at r ' within an emitting medium of volume V is derived by computing the z-component of the Poynting vector , where denotes an ensemble average and * is the complex conjugate operator. The evaluation of the Poynting vector requires computation of terms , written as follows using Eqs. (1a) and (1b):Eq. (2), kv is the magnitude of the wavevector in vacuum, while α and β involve a summation over the three orthogonal components. The ensemble average of the spatial correlation function of currents can be written as a function of the local temperature of the emitting medium via the fluctuation-dissipation theorem (FDT) :
In the above, Θ (=) is the mean energy of a Planck oscillator in thermal equilibrium. Equation (3c) shows that the electric and magnetic source currents are not spatially correlated. Substitution of Eq. (2) into the z-component of the Poynting vector, after application of the FDT and assuming the ergodic hypothesis where an ensemble average is equivalent to time averaging , gives the following general expression for the near-field radiative heat flux:
The net near-field radiative heat flux between media 0 and 2 is calculated by computing the difference between the flux absorbed by medium 2 due to the emitting bulk 0 at temperature T 0 and the flux absorbed by medium 0 due to the emitting bulk 2 at T 2. An expression for the near-field radiative heat flux in terms of the Fresnel reflection coefficients can be derived provided that the DGFs for the layered system shown in Fig. 1 are known. These DGFs were reported by Joulain et al.  and are not repeated here. After some algebra, the (net) monochromatic near-field radiative heat flux between media 0 and 2 is explicitly given by:Eq. (5) has the exact same form as the near-field radiative heat flux for non-magnetic materials; the influence of the relative magnetic permeability comes into the picture in the z-component of the wavevector and in the Fresnel reflection coefficients.
Before closing this section, it should be emphasized that extra care must be taken when evaluating the z-component of the wavevector in the metamaterial. For a given kρ, the z-component of the wavevector is calculated as , where nj is the refractive index of medium j. The real part of the refractive index of a metamaterial may become negative for some frequencies. Using the complex plane representation, the relative permittivity and permeability can be written respectively as and , where ϕε and ϕμ take values between 0 and π since and are always positive. Using , it is easy to show that the real part of the refractive index becomes negative when (ϕε + ϕμ) > π. In the frequency bands where n' < 0, Skaar  showed that for absorbing materials, the signs of and for any real value of kρ should be the same as those of n' and n”, respectively. Consequently, the z-component of the wavevector in medium j should be calculated as when (ϕε + ϕμ) > π.
The only remaining step for calculating the near-field heat flux is to develop a model describing the relative permittivity and permeability of the metamaterial; this is discussed next.
3. Relative permittivity and permeability of the metamaterial
We assume a dielectric host material containing spherical dielectric inclusions of radius rs. The spheres are arranged in a periodic manner in a simple cubic lattice, where the length of the edge of a unit cubic cell is given by the lattice constant a. The number of inclusions per unit volume is given by N = 1/a 3. Alternatively, the concentration of inclusions can be characterized by the volume filling fraction f = NV, where V is the volume of a sphere. Combining these last two expressions, the volume filling fraction of the spherical inclusions can be written as . The macroscopic electric and magnetic responses of the metamaterial made of a large collection of dielectric spheres can be described via an effective electric permittivity and an effective magnetic permeability [43,44]. This is done by averaging the fields in the long-wavelength limit, roughly defined as λh >> a > 2rs, where λh is the wavelength in the host medium [40,43].
The Clausius-Mossotti model [5,43,58] is employed to describe the macroscopic effective permittivity and permeability of the metamaterial as a function of the polarizabilities of the particles. Note that this model is applicable in the long-wavelength limit and for volume filling fractions substantially smaller than unity [5,40]. The electric and magnetic polarizabilities of a sphere are given respectively by and , where a 1 and b 1 are the first order Mie coefficients (i.e., dipole terms) while kh is the magnitude of the wavevector in the host medium. In the general case of a pole of order l, the Mie coefficients are given by :
Substitution of the electric and magnetic polarizabilities of a sphere into the Clausius-Mossotti relations leads to the following effective macroscopic properties :Eqs. (7a) and (7b). Indeed, in the long-wavelength regime, infinite periodic lattices should not experience losses due to scattering. These losses, included in the polarizabilities, are corrected by performing the substitution . It is also important to note that despite the periodic arrangement of the spherical inclusions, Wheeler et al. [40,43] have shown that the effective relative permittivity and permeability given by Eqs. (7a) and (7b) can be considered as isotropic in the long-wavelength regime. From now on, for simplicity, the terminology “effective” is dropped when referring to the relative permittivity and permeability of the metamaterial.
Spheres with large magnetic dipole resonances are required to induce a macroscopic magnetic response from the metamaterial. Magnetic dipole resonance is determined by imposing b 1 → ∞. In the long-wavelength limit, the dipole term b 1 can be approximated as follows [43,44]:
Using Eq. (8), magnetic resonance arises when F( X) = -2. Substitution of this condition into Eq. (9) leads to sin( X) = 0, such that the solutions are given by X = lπ (l = 1, 2, 3, …). The fundamental resonant wavelength (l = 1) in the host medium can therefore be written as follows:
Assuming that the long-wavelength limit is about , strong magnetic dipole resonance occurs when , where εs is the relative permittivity of the sphere [43,48]. From a thermal radiation point of view, we are interested in metamaterials with magnetic response in the infrared, such that the permittivity of the spherical inclusions should be very large within this spectral band. SiC, which is a polar crystal, fulfills this requirement. The relative electric permittivity of SiC is modeled via a damped harmonic oscillator:60]. The real part of the relative permittivity of SiC reaches a value close to 300 near ωTO, such that strong magnetic dipole resonance is achievable using this type of inclusion.
For a given host medium, Eqs. (6) and (7) show that the electric and magnetic responses of the metamaterial depend upon the concentration of inclusions as well as the size and optical properties of the spheres. Engineering the macroscopic electric and magnetic responses of the metamaterial, and therefore the near-field thermal spectrum, is made possible by varying these parameters. In this paper, we restrict our attention to a host medium of potassium bromide (KBr) comprised of SiC spheres with a volume filling fraction fixed at 0.4. Around a wavelength of 10 μm, the refractive index of KBr can be considered frequency-independent with a value of 1.5 .
The real part of the refractive index n' and the sum of the phases of the relative permittivity and permeability (ϕε + ϕμ) of the metamaterial are shown in Fig. 2 for rs = 1 μm. Note that due to the fact that the relative permittivity and permeability model is valid in the long-wavelength regime, the spectral analysis is restricted to the band of from 1×1014 rad/s to 2×1014 rad/s, which corresponds to a host medium wavelength λh band of from 6.3 μm to 12.6 μm. For f = 0.4 and rs = 1 μm, the lattice constant a is about 2.2 μm, such that the minimum λh is about three times the size of a.
As discussed in section 2, the real part of the refractive index is positive within the spectral band under consideration since the condition (ϕε + ϕμ) > π is never satisfied. In fact, it can be shown that a metamaterial made of a single type of spherical inclusions cannot have a negative permittivity and a negative permeability simultaneously .
4. Spectral distributions of near-field radiative heat flux and resonance analysis
In all simulation results presented in this section, medium 0 is maintained at 300 K while medium 2 is a heat sink at 0 K. The relative permittivity and permeability of media 0 and 2 are given respectively by ε 0 = ε 2 = εeff and μ 0 = μ 2 = μeff (see Eqs. (7a) and (7b)). Spectral distributions of radiative heat flux for gaps d of 10 nm, 100 nm and 1 μm are reported in Fig. 3 for SiC spheres having a radius of 1 μm; the profiles are compared with the flux in the far-field regime and the blackbody predictions.
It can be seen in Fig. 3 that the radiative flux for small gaps d is concentrated around two resonances arising in the vicinity of 1.42×1014 rad/s and 1.69×1014 rad/s. This nearly monochromatic, marked enhancement of the near-field flux can be analyzed more closely by investigating the conditions for which SPs exist. In the problem treated here, SPs exist at both interfaces delimiting the metamaterial and the vacuum, such that coupling of these resonant surface waves occurs within the gap. For thick media, despite SP coupling in the gap, the resonance of the flux is always located at the resonant frequency of the material-vacuum interface . As such, the resonance analysis is performed hereafter by considering a single material-vacuum interface.
SP dispersion relations in TE and TM polarizations are derived by finding the poles of the Fresnel reflection coefficients and . Also, throughout the resonance analysis, losses in the relative permittivity and permeability of the metamaterial are neglected (i.e., and ). In TE polarization, → ∞ when = 0. SPs are surface waves characterized by an evanescent electromagnetic field decaying in both media 0 and 1, such that and are both pure imaginary numbers. Using , the SP dispersion relation in TE polarization can thus be written as follows:Equation (12) can be satisfied if and only if since . By combining the definition of γj with Eq. (12), the SP dispersion relation can also be written in terms of the parallel wavevector kρ:
Substitution of the dispersion relation given by Eq. (13) into γj leads to:
Specializing for the case where medium 1 is a vacuum, such that < 0 for SP to exist, Eq. (14) is modified as follows:
Equations (16a) and (16b) define the domain of existence of SPs in TE polarization [62,63]. In the special case where > 0, the numerator of Eq. (15) is always positive, such that SPs can be excited in TE polarization when and This last condition is the one prevailing for the SiC sphere-based metamaterial since the relative permittivity and permeability are never simultaneously negative for a given frequency, as discussed in section 3. Similarly, it is easy to show that SPs exist in TM polarization when and .
Resonance of the near-field radiative flux arises when kρ is very large within a narrow spectral band. For TE and TM polarizations, kρ → ∞ when = -1 and = -1, respectively. When SPs are excited by plasma oscillations or transverse optical phonons, the terminologies surface plasmon-polaritons and surface phonon-polaritons are respectively employed. Here, SPs exist due to the Mie resonances of the dielectric spherical inclusions, such that the generic terminology SP is used. Also, from now on, SPs in TE and TM polarizations will be respectively referred to as magnetic SPs and electric SPs.
The real part of the relative permittivity and permeability of the metamaterial studied in Fig. 3 are plotted in Fig. 4(a) , while the TM and TE evanescent contributions to the near-field heat flux are shown for d = 10 nm in Fig. 4(b). In both plots, the spectral bands where SPs exist are identified.
It can be seen in Fig. 4(a) that and at 1.42×1014 rad/s and 1.69×1014 rad/s, respectively. Therefore, the lowest frequency resonance of the flux observed in Fig. 3 is due to magnetic SPs, while the second at a higher frequency is due to electric SPs. Moreover, Fig. 4(b) clearly shows that the resonance around 1.42×1014 rad/s emerges only in TE polarization, while the resonance near 1.69×1014 rad/s arises in TM polarization. In the far-field regime, it can be seen in Fig. 3 that the radiative heat flux is quite low within the spectral bands supporting SPs. Indeed, Fig. 2 shows that the real part of the refractive index in these bands is low while the imaginary part () is high, thus resulting in highly reflecting zones.
The near-field radiative heat flux also exhibits secondary peaks that are not due to SPs (see Figs. 3 and 4(b)). Outside the spectral bands supporting SPs, the near-field radiative heat flux is saturated by evanescent waves generated by total internal reflection at the metamaterial-vacuum interface. These modes, contained within the region kv < kρ < n'kv, correspond to electromagnetic waves propagating within the material but evanescent within the vacuum gap . Therefore, the secondary peaks of the flux arising at some specific frequencies are due to the fact that the real part of the refractive index also exhibits peaks at these frequencies.
The influence of the size of the inclusions is studied in Figs. 5(a) and 5(b). The magnitude of the Mie coefficients a 1 and b 1 are shown in Fig. 5(a) for radii rs of 600 nm, 800 nm and 1 μm. Spectral distributions of near-field radiative heat flux at d = 10 nm are calculated for these rs values and are presented in Fig. 5(b); the profiles are compared with the predictions between two bulks of SiC.
Although not shown in Fig. 5(a), the quadrupoles a 2 and b 2 have also been calculated. For rs = 1 μm, the quadrupoles are more than an order of magnitude smaller than the dipole terms, such that it is justified to account strictly for a 1 and b 1 when modeling the relative permittivity and permeability of the metamaterial. It can be seen in Fig. 5(a) that the size of the particles affects in a non-negligible manner the strength of the magnetic dipole resonance b 1 as well as its spectral location. This observation is also true when looking at the low-frequency resonance of the flux in Fig. 5(b). The resonant wavelength of b 1 was derived in section 3 (Eq. (10)). By neglecting the losses in the relative permittivity of the spheres, the combination of Eqs. (10) and (11) leads to the following approximation for the resonant frequency of b 1:43]. Using Eq. (17), resonance of b 1 for rs of 600 nm, 800 nm and 1 μm are estimated at 1.47×1014 rad/s, 1.45×1014 rad/s and 1.43×1014 rad/s, respectively. These predictions are in good agreement with the results of Figs. 5(a) and 5(b). The resonance given by Eq. (17) is size-dependent and occurs near the transverse optical phonon frequency ωTO where the real part of the permittivity of the sphere is very large. Note that the low-frequency resonance of the flux for rs values of 600 nm and 800 nm are not due to magnetic SPs, but rather due to evanescent waves experiencing total internal reflection at the material-vacuum interface. Indeed, when rs = 600 nm, is always greater than zero, while for rs = 800 nm, the real part of the magnetic permeability becomes negative within a small spectral band but is never smaller than -1. These results therefore show that a very strong resonance of the magnetic dipole b 1 is necessary in order to induce magnetic SPs. This resonance can be enhanced by maximizing the ratio of the real part of the permittivity of the sphere over the permittivity of the host medium and by increasing the size of the inclusions.
The significant enhancement of the near-field radiative heat flux via electric SPs is physically due to surface electric dipole modes of the SiC particles . The resonant frequency of a surface mode in small spherical particles is given by [31,59]. Using Eq. (11), this resonant frequency is given by:Eq. (18), the electric dipole a 1 resonant frequency is estimated at 1.70×1014 rad/s, which is in reasonable agreement with the results of Fig. 5(a). It is also possible to estimate the near-field radiative heat flux resonance by letting . Using a procedure similar to the one described in , the resonance of the flux can be approximated as follows:
The resonant frequency of the near-field radiative heat flux is estimated to be 1.72×1014 rad/s, which is again in reasonable agreement with the results of Fig. 5(b). Note that the approximation given by Eq. (19) does not capture the small spectral shift of the resonance toward a higher frequency as the size of the inclusions decreases.
The resonant modes of the flux can also be determined by first calculating and via Eqs. (7a) and (7b), and then simply determining the frequencies for which these two values are equal to -1. The goal of the above analysis was to link the resonances of the near-field radiative heat flux with the physical mechanisms responsible for this augmentation.
Finally, it is worth noting that the radiative heat flux integrated within the spectral band from 1×1014 rad/s to 2×1014 rad/s when d = 10 nm is about 35% greater for the metamaterials made of SiC spheres with a radius of 1 μm, where both electric and magnetic SPs are excited, than the flux predicted between two bulk materials of SiC, where only surface phonon-polaritons in TM polarization are excited.
Near-field radiative heat transfer between dielectric-based metamaterials separated by a vacuum gap has been analyzed. Metamaterials made of spherical inclusions of silicon carbide (SiC) within a dielectric host medium of potassium bromide (KBr) have been considered. We have shown for the first time that electric and magnetic surface polariton (SP) mediated near-field radiative heat transfer between dielectric-based metamaterials is possible. Spectral distributions of flux for sub-wavelength vacuum gaps show a low-frequency resonance due to excitation of magnetic SPs in TE polarization and a resonance at a slightly higher frequency attributable to electric SPs in TM polarization. The strength and spectral location of the low-frequency flux resonance, due to magnetic dipole resonance of the particles, is quite sensitive to the size of the spherical inclusions. For particles with radii smaller than 1 μm, the magnetic dipole resonance of the spheres is not strong enough to induce SPs in TE polarization. Still, a high near-field radiative flux is observed at low-frequency due to evanescent waves generated by total internal reflection at the metamaterial-vacuum interface. On the other hand, the frequency and strength of the high-frequency resonance of the flux, due to electric surface modes of the spheres, are almost insensitive to the size of the inclusions.
The results presented in this paper show that it is possible to control the near-field thermal spectrum via metamaterials. Mie resonance-based metamaterials may also have significant impacts in tuning far-field emission. For example, excitation of SPs in the far-field via a grating can result in a coherent thermal source. Additionally, the presence of SPs on a flat surface induces highly reflective spectral bands, such that it becomes possible to control precisely the thermal radiative properties of materials. In future studies, it would be interesting to analyze closely the impacts of the shape and the concentration of inclusions, as well as the materials used for the host medium and the particles, on heat exchange.
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