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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163

http://d0022-40

n CorrE-m

journal homepage: www.elsevier.com/locate/jqsrt

Metamaterial-based perfect absorbers for efficientlyenhancing near field radiative heat transfer

Nan Zhou, Xianfan Xu n

School of Mechanical Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA

a r t i c l e i n f o

Article history:Received 9 June 2015Received in revised form31 July 2015Accepted 25 August 2015Available online 3 September 2015

Keywords:Radiative heat transferMetamaterialsNear fieldSurface polaritons

x.doi.org/10.1016/j.jqsrt.2015.08.01573/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: +17654945639, fax:ail address: xxu@ecn.purdue.edu (X. Xu).

a b s t r a c t

The fascinating capability of manipulating light using metamaterials (MMs) has inspired asignificant amount of studies of using MMs for energy related applications. In this workwe investigate MM-based perfect absorbers for enhancing near field radiative heattransfer, which is described by the fluctuation dissipation theorem. MM structuresdesigned at two wavelengths are analyzed, corresponding to two working temperatures.Both electric and magnetic surface polaritons are found to contribute to heat transfer,while natural materials support only electric polaritons. The near-perfect absorption isdemonstrated to be related to the modification of effective optical properties, which isimportant for enhancing radiative heat transfer efficiently. By comparing different designs,the bandwidth of the heat flux spectrum is found to increase with the absorption band-width, which is originated from the spatial field distributions. This study will contribute tothe understanding of surface polaritons in near field radiative heat transfer and facilitatethe optimization of MMs for near field heat transfer applications.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The demonstration of negative refractive index mate-rials [1–3] has generated significant amount of interests inresearches of metamaterials (MMs) [4–7]. The fascinatinglight manipulation capability of MMs has many potentialapplications related to energy utilization. For example,MMs can be designed for radiation selective surfaces tai-loring the solar absorption and emission of light at a par-ticular wavelength, i.e., selective absorbers or emitters[8–13]. This holds the potential in the field of thermalphotovotaics, where the conversion efficiency could besignificantly enhanced if the wavelength is matched to thebandgap of photovoltaic materials [14,15].

+17654940539.

Nano-structured surfaces or meta-surfaces can also bedesigned to enhance near field radiative heat transfer(NRHT). Polder and Van Hove [16] found a heat fluxincrease between two parallel plates, separated by a dis-tance smaller than λT due to the contribution from eva-nescent waves, where λT is the peak wavelength of thethermal radiation spectrum given by Wien's displacementlaw: λTTE2898 mm K. Recent work has shown that it ispossible to enhance spectral heat flux by several orders ofmagnitude through resonantly excited surface phononpolaritons (SPhPs) in polar materials such as SiC and glass[17–19]. The intrinsic optical properties of polar materialslimit the SPhPs to around 10 mm, corresponding to anoperating temperature close to the room temperature.Other materials investigated in the infrared range includedoped Si [20,21] and graphene [22,23], which enhanceNRHT through exciting surface plasmon polaritons (SPP).

For noble metals that also support SPP, the resonancetypically lies in the visible or UV wavelengths, which is

Fig. 1. Schematic of the nanoscale thermal radiation between two par-allel plates.

N. Zhou, X. Xu / Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163 157

significantly above the thermal frequency range. This cor-responds to an operating temperature higher than 2000 Kwhich could melt the metal, thus limiting the directapplication of metallic materials in NRHT. However, byapplying nanostructures, the response can be stronglymanipulated through the interaction between incidentelectromagnetic waves and the properly designed struc-tures. The incident light can be coupled to not only electricsurface polaritons, but also magnetic polaritons [11,13].This method provides flexibility in tailoring the opticalproperties in a wide frequency range and thus provides apotential to effectively enhance NRHT.

Analysis of the NRHT usually employs fluctuational elec-trodynamics [24], which has been extended to MMs witharbitrary permittivity ε and permeability m by Joulain et al.[25]. In classical electromagnetics, the Drude–Lorentz modelis applied to ε (¼ε'þ iε") and m (¼m'þ im"), derived from theoscillation of charges or fictitious magnetic charges. For adilute metal design with extremely low plasma frequency[26], an analytical expression was derived for the effectivepermittivity εeff, which still obeys the Drude–Lorentz model.Similarly, the average response of split ring resonator (SRR)MMs was also analytically characterized [27]. More generally,analytical solutions are not available for the collective opticalresponse of inhomogeneous structures. Instead, the effectivemedium theory (EMT) is widely applied to homogenize thestructure with the aid of the geometric filling ratio [28–30] orthe S-parameters from numerical simulation [15,31,32]. It isaccepted that EMT is generally applicable if the incidentwavelength is much larger than size of the unit cell, and thedistance to the structure surface is not smaller than the size ofthe unit cell [29–31]. Many researchers have developedalternative approaches for evaluating the MM-based NRHTwith improved accuracy at small plate separations. A scat-tering approach [33] based on rigorous coupled-wave analysis(RCWA) has been applied to compute the NRHT betweenmetallic [34] and dielectric [35] gratings. Wen [36] conducteddirect simulations for NRHT between two parallel plates usingWiener chaos expansion, which has no explicit constraint onthe geometry and relies on finding eigenmodes of the thermalfluctuating current. Liu and Shen [37] extended this methodto hyperbolic MM consisting of metallic wire array. Othermethods for solving NRHT problems in arbitrary configura-tions include direct finite-difference time-domain (FDTD)computations [38,39], boundary-element method (BEM) [40],and thermal discrete dipole approximation (T-DDA) [41].

In this study we focus on MMs that have a top metallicantenna array and a bottom metal ground layer, separatedby a dielectric spacer. This type of MMs has been demon-strated both numerically and experimentally to providenear perfect absorption and functions as a blackbody at thedesired wavelength [11,15]. We extend the study of theseMMs to the field of near field radiative heat transferbetween parallel plates, through enhancing the strength ofsurface polaritons and thus enhancing the NRHT. We alsoinvestigate the effect of temperature and compare differentMM designs.

2. Methods and geometry

The parallel plates geometry under study is illustrated inFig. 1, where the medium 3 between two plates is typicallyvacuum with ε3¼1. By applying the fluctuation dissipationtheorem, the net radiative heat flux is given as [42]

⎡⎣ ⎤⎦⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

P T T d I T I T

KdKc

r r

r r e

KdKc

r r

r r ee

,

/

1 1

1

/

4 Im Im

1,

1

s p

c

i d

Pr opagating

c dd

Evanescent

1 20

01

02

, 0

/

2

312

322

31 322 2

/ 231 32

31 322 2

2

3

3

3

( )( )

∫

∫

∫ ( ) ( )

∑

π ω

ω

ω

( ) = ↓ ( ) − ( )

↓×( )

− −

−⏟

+( ) −

⏟ ( )

ω ω

α

ωα α

α α γ

ω

α α

α α γγ

∞

=

∞

− ‵‵− ‵‵

where K is the in-plane wave vector andI T c e/4 / 1k T0 3 3 2 / Bω π( ) = ℏ ( − )ω

ωℏ is the intensity of blackbodyradiation, which serves as a ω-domain filter that cuts offmodes with k T/Bω > > ℏ [42]. Both propagating (Koω/c)and evanescent (K4ω/c) waves are accounted in Eq. (1). Thedenominator involving Fresnel reflection coefficients corre-sponds to multiple reflections. When the surface plasmon/phonon polaritons are excited, Im(rs) is negligible whileIm(rp)E2Im(ε)/|εþ1|2 [17]. As a result, the p-polarized eva-nescent channel peaks at the same condition as the surfacemodes that originated from plasmon or phonon polaritons(ε¼�1), significantly enhancing the NRHT. Similar analysiscan be applied to the s-polarized channel if the plate has anegative permeability m', indicating the existence of surfacemagnetic polaritons. Thus, instead of applying naturallyexisted bulk materials, we design nanostructures or MMs tomanipulate both ε and m. The regions with both negative ε' orm' will considerably enhance the NRHT.

According to the analysis in [31], the effective refractiveindex (n¼n'þ in") and impedance (z¼z'þ iz") are related tothe complex reflection coefficient r and transmissioncoefficient t by

nk Dr t

tcos

12 2a0

2 2( ) = − +

( )

N. Zhou, X. Xu / Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163158

zr tr t

11 2b

2 2

2 2= ± ( + ) −( − ) − ( )

Here, D is the total thickness of the structure, and k0 isthe wave vector in vacuum. The passive mediumassumption requires that z'40 and n"40. We simulatedthe unit cell of the structure and calculated r and t coef-ficients using ANSYS HFSS, a commercial frequency-domain finite-element method solver. With n and z, theeffective optical properties can be readily computed using

n z/ 3aε = ( )

n z. 3bμ = ⋅ ( )

MM-based perfect absorbers/emitters are typicallymade of a top metallic antenna array and a bottom metalground layer, separated by a dielectric spacer. The uniquenear-perfect absorption/emission properties take advan-tages of the electric resonance in the top nano-structure,as well as the magnetic resonance which induces anti-parallel currents in the two metallic layers [9,11]. Electricand magnetic resonances can be separately tuned in dif-ferent elements and simultaneously achieved [8]. Effec-tively, the structure provides an impedance matching tothe free space with ε¼m, which makes the perfectabsorption possible [8,32]. Various MM patterns have beenstudied, including cross [10,15,43], patch [12,32,44], disk[9], and hole [45]. In addition to the strong manipulationof absorption spectrum, MM-based absorbers can bedesigned to be angular independent or omnidirectionalunder both TE and TM illuminations, as has beendemonstrated experimentally [12,9,44].

In this work we study two resonant wavelengths: 10 mmand 4 mm. They correspond to working temperatures ofabout 300 K and 700 K, correspondingly. For the wave-length of 10 μm, we investigate two cross patterns, calledcross A and cross B here, to illustrate the relation betweenfar field perfect absorption and near field enhancement(Section 3.1). For the wavelength of 4 μm, we considerseveral different top metallic patterns as illustrated in Fig. 2.The cross pattern is similar to the fishnet structure, whichhas been investigated for near field radiation at near-IRwavelengths [46]. The complementary bowtie aperture(CBA) in Fig. 2(d) has been demonstrated in our earlier workto generate enhanced and confined magnetic field around

Fig. 2. Unit cell of MM patterns made of (a) patch, (b) cross, (c) hole and (d) compparameters are shown in (d), which is applicable to (a–c).

the metal strip [47]. Table 1 summarizes geometry para-meters that have been optimized for each design to achievenear-perfect absorption at the resonance. The pattern size land period p scale with the resonant wavelength. As hasbeen discussed in literature [15,32,44,45,47], the absorptionspectrum is generally most sensitive to l, p and the spacerthickness ts. They determine the response of a single pat-tern, the interaction between adjacent patterns and thecoupling between two metallic layers, respectively. Gold isused as the metallic material. For wavelengths λo2 mm, itsproperties were taken from [48]. For long wavelengths, theDrude model i1 /p

2 2ε ω ω ω γω( ) = − + is applied withωp andγ taken from [49]. The spacer material is typically SiO2 orAl2O3. Here, SiO2 is applied for designs at 4 mm and itsoptical properties from [48] are used in the simulation. SiO2

is known to have a strong dispersion in the mid-infraredrange, thus Al2O3 is used instead for designs at 10 mm andits frequency-dependent optical constant is taken from [50].

3. Results and discussion

3.1. Importance of perfect absorption for near fieldenhancement

In this section, we investigate two cross patterns Aand B that work around the wavelength of 10 mm, aslisted in Table 1. The importance of strong absorption ofMMs on NRHT will be discussed in details by comparingthe two designs.

From Table 1 we can see that the only difference betweencross A and cross B is the spacer thickness. Fig. 3(a) showsthat both designs produce a far field absorption peak at9.42 mm (2�1014 rad/s) under normal incidence. But onlycross A, with a spacer thickness of 300 nm, achieves near-perfect absorption at resonance. Fig. 3(b–d) presents theretrieved optical properties of cross A and crosss B, whichshow effective Lorentz oscillations in the vicinity of2�1014 rad/s. Far away from the oscillation, the absolutevalue of ε generally decreases with increasing ω, same asthat observed in the Drude model. Thus the effective prop-erties, which are very different from those of bulk or thinfilm gold, can be well described by the Drude–Lorentz modelas follows:

lementary bowtie aperture. Top view is used for (a–c). The cross-sectional

Table 1Geometric parameters for the metamaterial patterns studied in this work.

Wavelength Pattern ts [nm] h [nm] tg [nm] p [mm] l [mm] w [mm] d [nm]

10 mm Cross – A 300 100 100 4.4 3.8 0.4Cross – B 100

4 mm Patch 25 40 100 1.3 0.85Cross 25 40 100 1.3 0.85 0.4Hole 150 40 100 2.7 2.6CBA 150 50 100 2.4 1.95 0.1

N. Zhou, X. Xu / Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163 159

i i 4aD

D

p e

e e

2

2,

2

20,2

ε ω ε ωω γ ω

ω

ω ω γ ω( ) = −

+−

− + ( )∞

i.

4b

p m

m m

,2

20,2

μ ω μω

ω ω γ ω( ) = −

− + ( )∞

Here subscript D stands for the Drude term. The cor-responding fitting parameters are summarized in Table 2.

In Fig. 3(b–d), we plot fitting results using parameters inTable 2 together with the directly retrieved properties. Forcross A, its near-perfect absorption indicates an effectivemagnetic Lorentz oscillator that is strong enough to inducenegative m' over a wavelength range. On the other hand, itspermittivity is also manipulated from extremely negativevalues, as observed in bulk gold, to moderate negative ε'. Thishelps enhance the NRHT more efficiently, as indicated by theexpression Im(rp)E2Im(ε)/|εþ1|2. Compared with cross A,the magnetic oscillator resonance produced by cross B ismuch weaker. As a result, m' stays positive in the simulatedspectrum. In addition, ε' is very negative for most of thespectrum. It should be mentioned that ε' will graduallyapproach �1 on the short wavelength/large frequency side,which will make additional contribution to the p-evanescentchannel. We ignore this short wavelength resonance becauseit is outside of the simulation range and it corresponds to amuch higher temperature.

The corresponding NRHT calculations of cross A andcross B are shown in Fig. 4. The temperatures are fixed atT1¼350 K and T2¼300 K. Fig. 4(a) presents the net radia-tive heat flux as a function of the distance d. The heat fluxfor cross A is broken down to three channels, the propa-gating wave, and the s- and p-polarized evanescent waves.The red and green dash lines correspond to s- and p-eva-nescent modes of cross B, respectively. We can see thatboth s- and p-evanescent channels of cross A outperformthat of B as expected. As the distance increases, the con-tribution of s-evanescent mode decreases rapidly becauseof the very narrow bandwidth of negative m' for effectivelyexciting magnetic polaritons, and the total heat flux isdominated by the p counterpart and the propagatingmode. At d¼4 mm, close to the period of the designs, theevanescent channels (mainly p-evanescent) contributeabout 33% to the total flux. For the shaded area of do4 mmin Fig. 4(a), the EMT is not well-suited and the results willlikely produce errors for the contribution from evanescentmodes. However, we present the corresponding results toexplore the effect of optical properties on NRHT, and toobtain a rough idea of the NRHT behavior of the designed

structure. It is expected that both electric and magneticsurface polaritons will gradually dominate the heat flux atsmaller separations. However, a quantitative analysis is notavailable with the EMT method.

Fig. 4(b) compares the total heat flux of cross A and crossB with two parallel Au and SiC plates. To show a fair com-parison, the radiative heat flux is integrated within thesame spectral range from 1.2�1014 rad/s to 3.8�1014 rad/sfor all cases presented in Fig. 4(b). The surface phononresonance near 10 mm (within the integration range) in SiCis naturally strong, thus the heat flux between two parallelSiC plates is larger than that of MMs and bulk gold plates. Atd¼4 mm, the heat flux of cross A is thrice that of B andremains about twice at larger distances. It is also seen thatthe radiative heat flux between two bulk Au plates is atleast one order of magnitude smaller than that of theproperly design MMs. Fig. 4(b) indicates that one shouldstill choose SiC over MM-based materials for near-roomtemperature applications. However, the advantage of usingMM-based materials is that their design can be easilymodified for application at other wavelength/temperature,or even for multi-/broad-band applications, while theintrinsic surface phonon resonance in SiC is monochromaticnear the 10 mm wavelength.

3.2. Near-field enhancement using different geometricdesigns

As was mentioned, MM-based perfect absorbers/emit-ters can have various geometric patterns. In this section,we compare four different patterns shown in Fig. 2 andlisted in Table 1. They are all designed around the wave-length of 4 mmwith a near-perfect absorption at resonancein the far field, as shown in Fig. 5.

Fig. 6(a) and (b) show the near field heat flux spectra forsquare patch, cross, hole, and complementary bowtieaperture (CBA) patterns at d¼100 nm and 2 mm, respec-tively. The temperatures are fixed at T1¼750 K and T2¼700 K. The corresponding total heat flux integrated in thespectral range from 4�1014 rad/s to 5.4�1014 rad/s ispresented in Fig. 6(c). Although the results at d¼100 nm,which is much smaller than the period, could overestimatethe heat flux, they are still used to estimate the geometriceffect on NRHT. We see that, firstly, for all designs, twopeaks can be observed. The stronger one on the large fre-quency/short wavelength side corresponds to the s-polar-ized surface wave, which outperforms the p counterpart atsmall distances. Secondly, we see from Fig. 6(a) that thepeak values are slightly different among four designs and

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 3 3.5 4

Cross-A

Cross-B

6.28μm

9.42 μm

ω [1014 rad/s]

Abso

rptio

n

-300

-200

-100

0

100

200

300

400

500

1.5 2 2.5 3 3.5

ε'ε"ε' fittedε" fitted

ω [1014 rad/s]

Perm

ittiv

ity

Cross-A

-8-4

0

4

8

12

16

20

24

1.5 2 2.5 3 3.5

μ'μ"μ' fittedμ" fitted

Perm

eabi

lity

ω [1014 rad/s]

Cross-A

-600

-400

-200

0

200

400

1.5 2 2.5 3 3.5

ε'ε"ε' fittedε" fitted

ω [1014 rad/s]

Perm

ittiv

ity

Cross-B

-2

0

2

4

6

8

1.5 2 2.5 3 3.5

μ'μ"μ' fittedμ" fittedPe

rmea

bilit

y

ω [1014 rad/s]

Cross-B

Fig. 3. (a) Far field absorption spectra for cross A and cross B that resonate at 10 mm. The retrieved and fitted optical properties for (b, c) cross A and (d, e)cross B.

N. Zhou, X. Xu / Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163160

the CBA absorber produces the strongest peak (at4.73�1014 rad/s). It has been noted that a large loss term ofε or m will deteriorate the contribution of surface modes[30]. Thus, at d¼100 nm, the strong s-evanescent modefrom the CBA absorber can be attributed to the relativelysmall imaginary part in mE�1þ i0.4 (at 4.73�1014 rad/s).For other designs, mE�1þ i0.5 (at 5.08�1014 rad/s) for the

square, mE�1þ i0.84 (at 4.76�1014 rad/s) for the cross,and mE�1þ i1.11 (at 4.8�1014 rad/s) for the hole. Thirdly,the full width at half maximum (FWHM) bandwidths Δωare very different, which were found to roughly scale withthe absorption bandwidth shown in Fig. 5. For square, cross,hole and CBA patterns, the absorption bandwidths Δω are2.93�1013, 2.43�1013, 1.55�1013, and 1.45�1013 rad/s,

Table 2Drude–Lorentz fitting parameters for cross – A and cross – B designs.

ε1 ωD [rad/s] γD [rad/s] ωp,e [rad/s] ω0,e [rad/s] γe [rad/s]

Cross – A 14 2�1015 2.8�1013 6.05�1014 2.09�1014 4.25�1012

Cross – B 14 3.2�1015 2.1�1013 7.25�1013 2.025�1014 6�1012

m1 ωp,m [rad/s] ω0,m [rad/s] γm [rad/s]Cross – A 4.72 1.43�1014 1.985�1014 5�1012

Cross – B 4.16 0.83�1014 1.978�1014 5.5�1012

10-4

10-3

10-2

10-1

100

101

102

103

10-7 10-6 10-5 10-4

prop (A)s evan (A)p evan (A)total (A)s evan (B)p evan (B)

d [m]

Net

Hea

t Flu

x [W

m-2

]

10-1

100

101

102

103

104

10-7 10-6 10-5 10-4

Cross-ACross-BAuSiC

Net

Hea

t Flu

x [W

m-2

]

d [m]Fig. 4. (a) The net heat flux of different channels as a function of plateseparation for the mid-infrared cross design A. The red and green dashlines correspond to s- and p-evanescent modes of cross B. (b) Comparethe total heat flux of cross A and cross B with that from two parallel Auand SiC plates. For all cases in this figure, the temperatures are fixed at T1¼350 K and T2¼300 K. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this article.)

0

0.2

0.4

0.6

0.8

1

4 4.2 4.4 4.6 4.8 5 5.2 5.4

squarecrossholeCBA

ω [1014 rad/s]

Abso

rptio

n

3.49μm

3.77μm

4.09μm

4.48μm

Fig. 5. Far field absorption spectra for four designs that resonate near4 mm.

N. Zhou, X. Xu / Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163 161

respectively. The geometric effects on resonance char-acteristics have also been studied [51], which alsoreported that the square patch design has a largerabsorption bandwidth than cross pattern because ofsmaller effective mutual and kinetic inductances. At a

larger distance of 2 mm, the peaks from evanescentchannels significantly decrease, resulting in similarpeak values. However, the difference in bandwidth stillexists, as shown in Fig. 6(b). The net heat flux at dif-ferent distances is computed from the spectralresponse, therefore depending on both the peak valueand the bandwidth. Fig. 6(c) shows that at d¼2 mm, theintegrated heat flux roughly scales with the band-width, while at d¼100 nm, the CBA absorber producesa relatively large heat transfer because of the strongs-evanescent peak shown in Fig. 6(a).

Lastly, we compare the heat flux of SiC to the four MMpatterns analyzed above, considering temperatures at T1¼750 Kand T2¼700 K. Fig. 6(d) shows the heat flux spectra from1.2�1014 to 5.4�1014 rad/s between two SiC plates atd¼100 nm and 2 mm. The shaded area corresponds to thefrequency range used for MM patterns, namely4�1014–5.4�1014 rad/s. There is no surprise that SiC stillproduces strong surface phonon resonance near 1.8�1014 rad/s.At d¼2 mm, the integrated heat flux is 2.21�103W/m2, about10 times that of MM patterns. At a smaller distance d¼100 nm,the heat flux integrated in the range from 1.2�1014 to5.4�1014 rad/s is 2.94�104W/m2, which is only slightlyhigher than the 2.35�104W/m2 produced by the patch pat-tern. For temperatures higher than 800 K or distances less than70 nm, the MM-based designs provide a higher NRHT than SiC.

4. Conclusion

The MM-based perfect absorbers are applied forenhancing nanoscale radiative heat transfer, using the

102

103

104

square cross hole CBA

d = 10-7 m

d = 2x10-6 m

Pattern

Net

Hea

t Flu

x [W

m-2

]

Fig. 6. The heat flux spectra at (a) d¼100 nm and (b) 2 mm for fourdesigns for 4 mm wavelength. The legend is the same for (a) and (b).(c) The integrated net heat flux at two different distances for four pat-terns. (d) The heat flux spectra between two SiC plates at two separa-tions. The shaded area indicates the frequency range used for the fourMM patterns. For all cases, the temperatures are fixed at T1¼750 K and T2¼700 K.

N. Zhou, X. Xu / Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163162

effective optical properties retrieved from EMT and thefluctuation electrodynamics for parallel plate geometry.The near-perfect absorption by MM, due to the enhancedstrength of surface polaritons, is thus related to the effi-cient enhancement of the near field heat flux. At smallplate separations, both electric and magnetic surfacemodes can contribute to the heat transfer, while the latterdoes not exist in natural materials. MMs offer great flex-ibility for tuning resonance in different frequency rangesfor enhancing heat flux at different target temperatures.The comparison among different designs reveals that thebandwidth of heat flux roughly scales with the absorptionbandwidth, which originates from the spatial field dis-tributions in the unit cell. Therefore, MM absorbers withperfect absorption as well as a broad resonance peak arepreferable for NRHT.

Acknowledgments

The authors gratefully acknowledge the support of theNational Science Foundation (Grant No. CMMI-1120577).

References

[1] Pendry JB. Negative refraction makes a perfect lens. Phys Rev Lett2000;85:3966–9.

[2] Zhang S, Fan WJ, Panoiu NC, Malloy KJ, Osgood RM, SRJ Brueck.Experimental demonstration of near-infrared negative-index meta-materials. Phys Rev Lett 2005;95:137404.

[3] Dolling G, Enkrich C, Wegener M, Soukoulis CM, Linden S. Low-lossnegative-index metamaterial at telecommunication wavelengths.Opt Lett 2006;31:1800–2.

[4] Smith DR, Pendry JB, Wiltshire MCK. Metamaterials and negativerefractive index. Science 2004;305:788–92.

[5] Shalaev VM. Optical negative-index metamaterials. Nat Photonics2007;1:41–8.

[6] Engheta N. Circuits with light at nanoscales: optical nanocircuitsinspired by metamaterials. Science 2007;317:1698–702.

[7] Liu YM, Zhang X. Metamaterials: a new frontier of science andtechnology. Chem Soc Rev 2011;40:2494–507.

[8] Landy NI, Sajuyigbe S, Mock JJ, Smith DR, Padilla WJ. Perfect meta-material absorber. Phys Rev Lett 2008;100:207402.

[9] Liu N, Mesch M, Weiss T, Hentschel M, Giessen H. Infrared perfectabsorber and its application as plasmonic sensor. Nano Lett 2010;10:2342–8.

[10] Liu XL, Starr T, Starr AF, Padilla WJ. Infrared spatial and frequencyselective metamaterial with near-unity absorbance. Phys Rev Lett2010;104:20.

[11] Watts CM, Liu XL, Padilla WJ. Metamaterial electromagnetic waveabsorbers. Adv Mater 2012;24:OP98–120.

[12] Wu CH, Neuner B, Shvets G, John J, Milder A, Zollars B, et al. Large-area wide-angle spectrally selective plasmonic absorber. Phys Rev B2011;84:075102.

[13] Zhao B, Wang LP, Shuai Y, Zhang ZM. Thermophotovoltaic emittersbased on a two-dimensional grating/thin-film nanostructure. Int JHeat Mass Transf 2013;67:637–45.

[14] Narayanaswamy A, Chen G. Surface modes for near field thermo-photovoltaics. Appl Phys Lett 2003;82:3544–6.

[15] Liu XL, Tyler T, Starr T, Starr AF, Jokerst NM, Padilla WJ. Taming theblackbody with infrared metamaterials as selective thermal emit-ters. Phys Rev Lett 2011;107:045901.

[16] Polder D, Vanhove M. Theory of radiative heat transfer betweenclosely spaced bodies. Phys Rev B 1971;4(10):3303–14.

[17] Mulet JP, Joulain K, Carminati R, Greffet JJ. Enhanced radiative heattransfer at nanometric distances. Microscale Therm Eng 2002;6:209–22.

[18] Shen S, Narayanaswamy A, Chen G. Surface phonon polaritonsmediated energy transfer between nanoscale gaps. Nano Lett2009;9:2909–13.

N. Zhou, X. Xu / Journal of Quantitative Spectroscopy & Radiative Transfer 167 (2015) 156–163 163

[19] Rousseau E, Siria A, Jourdan G, Volz S, Comin F, Chevrier J, et al.Radiative heat transfer at the nanoscale. Nat Photonics 2009;3:514–7.

[20] Fu CJ, Zhang ZM. Nanoscale radiation heat transfer for silicon atdifferent doping levels. Int J Heat Mass Transf 2006;49:1703–18.

[21] Basu S, Lee BJ, Zhang ZM. Infrared radiative properties of heavilydoped silicon at room temperature. J Heat Trans—Trans ASME, 132;023301.

[22] van Zwol PJ, Thiele S, Berger C, de Heer WA, Chevrier J. Nanoscaleradiative heat flow due to surface plasmons in graphene and dopedsilicon. Phys Rev Lett 2012;109:264301.

[23] Svetovoy VB, van Zwol PJ, Chevrier J. Plasmon enhanced near-fieldradiative heat transfer for graphene covered dielectrics. Phys Rev B2012;85:155418.

[24] Rytov SM, Krastov YA, Tatarskii VI. Principles of statistical radio-physics. Berlin: Springer-Verlag; 1989.

[25] Joulain K, Drevillon J, Ben-Abdallah P. Noncontact heat transferbetween two metamaterials. Phys Rev B 2010;81:165119.

[26] Pendry JB, Holden AJ, Stewart WJ, Youngs I. Extremely low frequencyplasmons in metallic mesostructures. Phys Rev Lett 1996;76:4773–6.

[27] Pendry JB, Holden AJ, Robbins DJ, Stewart WJ. Magnetism fromconductors and enhanced nonlinear phenomena. IEEE T MicrowTheory 1999;47:2075–84.

[28] Francoeur M, Basu S, Petersen SJ. Electric and magnetic surfacepolariton mediated near-field radiative heat transfer betweenmetamaterials made of silicon carbide particles. Opt Express2011;19:18774–88.

[29] Biehs SA, Ben-Abdallah P, Rosa FSS, Joulain K, Greffet JJ. Nanoscaleheat flux between nanoporous materials. Opt Express 2011;19:A1088–103.

[30] Liu XL, Bright TJ, Zhang ZM. Application conditions of effectivemedium theory in near-field radiative heat transfer between mul-tilayered metamaterials. J Heat Trans—Trans ASME, 136; 092703.

[31] Smith DR, Schultz S, Markos P, Soukoulis CM. Determination ofeffective permittivity and permeability of metamaterials fromreflection and transmission coefficients. Phys Rev B 2002;65:195104.

[32] Hao JM, Zhou L, Qiu M. Nearly total absorption of light and heatgeneration by plasmonic metamaterials. Phys Rev B 2011;83:165107.

[33] Lambrecht A, Marachevsky VN. Casimir Interaction of DielectricGratings. Phys Rev Lett 2008;101:160403.

[34] Guerout R, Lussange J, Rosa FSS, Hugonin JP, Dalvit DAR, Greffet JJ,et al. Enhanced radiative heat transfer between nanostructuredgold plates. Phys Rev B 2012;85:180301.

[35] Lussange J, Guerout R, Rosa FSS, Greffet JJ, Lambrecht A, Reynaud S.Radiative heat transfer between two dielectric nanogratings in thescattering approach. Phys Rev B 2012;86:085432.

[36] Wen SB. Direct numerical simulation of near field thermal radiationbased on wiener chaos expansion of thermal fluctuating current.J Heat Trans—Trans ASME, 132; 072704.

[37] Liu BA, Shen S. Broadband near-field radiative thermal emitter/absorber based on hyperbolic metamaterials: direct numericalsimulation by the Wiener chaos expansion method. Phys Rev B2013;87:115403.

[38] Rodriguez AW, Ilic O, Bermel P, Celanovic I, Joannopoulos JD, Sol-jačić M, Johnson SG. Frequency-selective near-field radiative heattransfer between photonic crystal slabs: a computational approachfor arbitrary geometries and materials. Phys Rev Lett 2011;107:114302.

[39] Didari A, Mengüç MP. Analysis of near-field radiation transfer withinnano-gaps using FDTD method. J Quant Spectrosc Radiat Transf2014;146:214–6.

[40] McCauley AP, Reid MTH, Krüger M, Johnson SG. Modeling near-fieldradiative heat transfer from sharp objects using a general three-dimensional numerical scattering technique. Phys Rev B 2012;85:165104.

[41] Edalatpour S, Čuma M, Trueax T, Backman R, Francoeur M. Con-vergence analysis of the thermal discrete dipole approximation.Phys Rev E 2015;91:063307.

[42] Chapuis PO, Volz S, Henkel C, Joulain K, Greffet JJ. Effects of spatialdispersion in near-field radiative heat transfer between two parallelmetallic surfaces. Phys Rev B 2008;77:035431.

[43] Ye YQ, Jin Y, He SL. Omnidirectional, polarization-insensitive andbroadband thin absorber in the terahertz regime. J Opt Soc Am B2010;27:498–504.

[44] Hao JM, Wang J, Liu XL, Padilla WJ, Zhou L, Qiu M. High performanceoptical absorber based on a plasmonic metamaterial. Appl Phys Lett2010;96:251104.

[45] Hu CG, Zhao ZY, Chen XN, Luo XG. Realizing near-perfect absorptionat visible frequencies. Opt Express 2009;17:11039–44.

[46] Bai Y, Jiang YY, Liu LH. Multi-band near-field radiative heat transferbetween two anisotropic fishnet metamaterials. J Quant SpectroscRadiat Transf 2015;158:36–42.

[47] Zhou N, Kinzel EC, Xu X. Complementary bowtie aperture for loca-lizing and enhancing optical magnetic field. Opt Lett 2011;36:2764–6.

[48] Johnson PB, Christy RW. Optical constants of the noble metals. PhysRev B 1972;9:5056–70.

[49] Ordal MA, Long LL, Bell RJ, Bell SE, Bell RR, Alexander RW, et al.Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag,Ti, and W in the infrared and far infrared. Appl Opt 1983;22:1099–119.

[50] Palik ED. Handbook of optical constants of solid. San Diego: Aca-demic; 1998.

[51] Sakurai A, Zhao B, Zhang ZM. Resonant frequency and bandwidth ofmetamaterial emitters and absorbers predicted by an RLC circuitmodel. J Quant Spectrosc Radiat Transf 2014;149:33–40.