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Near-field radiative heat transfer between metamaterial thin films Soumyadipta Basu1,* and Mathieu Francoeur2 1

2

School for Engineering of Matter, Transport, and Energy, Arizona State University, Tempe, Arizona 85287, USA Radiative Energy Transfer Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah 84112, USA *Corresponding author: [email protected] Received December 10, 2013; accepted January 18, 2014; posted January 24, 2014 (Doc. ID 202902); published February 25, 2014 We investigate near-field radiative heat transfer between two thin films made of metamaterials. The impact of film thickness on magnetic and electric surface polaritons (ESPs) is analyzed. It is found that the strength as well as the location of magnetic resonance does not change with film thickness until the film behaves as semi-infinite for the dielectric function chosen in this study. When the film is thinner than vacuum gap, both electric and magnetic polaritons contribute evenly to near-field radiative heat transfer. At larger film thicknesses, ESPs dominate heat transfer due to excitation of a larger number of modes. Results obtained from this study will facilitate applications of metamaterials as thin-film coatings for energy systems. © 2014 Optical Society of America OCIS codes: (140.6810) Thermal effects; (160.3918) Metamaterials; (240.5420) Polaritons; (240.6690) Surface waves; (240.6680) Surface plasmons; (260.5740) Resonance. http://dx.doi.org/10.1364/OL.39.001266

The Stefan–Boltzmann law predicting radiative heat transfer between two surfaces is only applicable when the distance between them is larger than the characteristic wavelength of thermal radiation (λT ) as obtained from Wien’s displacement rule. As the distance between two surfaces becomes less than λT , near-field effects become significant due to wave interference and radiation tunneling. Consequently, thermal radiation between two bodies separated by a nanometric distance exceeds that between two blackbodies due to coupling of evanescent waves and excitation of surface polaritons (SPs) [1]. Enhanced heat transfer is useful in alternative energy systems [2–7] and can help in thermal rectification [8–13] and switching [14], as well as in radiative cooling [15]. Recently, near-field heat transfer between metamaterials has generated significant interest due to the possibility of exciting SPs in both S and P polarizations unlike ordinary, nonmagnetic materials. Heat transfer between two semi-infinite metamaterials separated by a vacuum gap was studied for different structures [16–18]. Basu and Francoeur predicted the penetration depth of evanescent waves during radiative heat transfer between metamaterials [19]. Thermal emission from metamaterials has also been calculated in order to understand the contribution of the different modes toward the total heat transfer [20,21]. Hyperbolic metamaterials [22] have been recently applied to near-field heat transfer [23–25], and have been suggested as the near-field analog of a blackbody due to the broadband enhancement of the flux [24]. However, hyperbolic metamaterials do not provide the spectral selectivity needed in an application such as thermophotovoltaic power generation. Spectral selectivity can be achieved via thin films. Near-field heat transfer between thin films made of ordinary materials has been very well analyzed. Studies have predicted that when the thickness of the films is of the same order as the vacuum gap separating them, the resulting heat transfer can slightly exceed that between two bulk materials [26–28]. Additionally, coupling of SPs within and between the films affects the resonant modes of the flux. As such, the study 0146-9592/14/051266-04$15.00/0

of near-field heat transfer between metamaterial films is of great significance because of the potential of enhancing the heat transfer over ordinary material and tuning emission/absorption spectra. Let us consider near-field radiative heat transfer between two thin films made of metamaterials as shown in Fig. 1. Medium 1, with thickness tf 1 and temperature T 1  400 K, is the emitter while medium 2, with thickness tf 2 at temperature T 2  300 K, is the receiver. The two films are separated by a vacuum gap (medium 0) of thickness d and are assumed to be in local thermal equilibrium. In this study, we consider the same structure as in [16]. The metamaterial made of thin wires and split ring resonators is chosen as an illustrative example, since the method outlined in this letter is applicable to any media of arbitrary permittivity and permeability. The relative permittivity and permeability of the metamaterial are, respectively, given by the expressions ε  ε0  iε00   1 −

ω2p ω  iωγ e 2

(1)

and

Fig. 1. Schematic of near-field radiative heat transfer between two thin films, where the emitter is at 400 K and the receiver is at 300 K. The films of thicknesses tf 1 and tf 2 are separated by a vacuum gap of thickness d. © 2014 Optical Society of America

March 1, 2014 / Vol. 39, No. 5 / OPTICS LETTERS

μ  μ0  iμ00   1 −

Fω2 ; ω − ω20  iωγ m 2

(2)

where F  0.56 is the split ring filling factor, ωp  1014 rad∕s is an equivalent plasma frequency, ω0  0.4ωp , and γ e  γ m  γ  0.01ωp [16]. The scattering rate is considered temperature independent within the range of temperature studied in this letter. Also, the effective dielectric function model is used to describe electromagnetic properties of the metamaterial. Previously, effective medium theory has been used to model the optical properties of gratings [29], porous media [30], bulk materials [16], and nanorods [31]. The near-field radiative heat flux between two thin films with arbitrary permittivity and permeability is given by [1] qtot ≈

1 π2

Z



0

Z Θω; T 1  − Θω; T 2 dω

βc ω∕c

Zβ; ωβdβ; (3)

where the contribution from propagating waves is neglected. In Eq. (3), c is the speed of light in vacuum, Θ is the mean energy of a Planck oscillator, β is the component of the wavevector parallel to the surfaces of the layer, βc is the cutoff parallel wavevector [19], and Z is the exchange function for films  2ikz0 d

Zβ; ω  e

 ImRS1 ImRS2  ImRP1 ImRP2   ; j1 − RS1 RS2 e2ikz0 d j2 j1 − RP1 RP2 e2ikz0 d j2 (4)

where kz0 is the perpendicular component of the wavevector in vacuum. Note that Eq. (4) represents contribution from evanescent waves only in the extreme near field. In Eq. (4), RSi and RPi are, respectively, the film reflection coefficients for S and P polarization; they are deS;P 2ikzi tf i S;P 2ikzi tf i  r S;P ∕1  r S;P , fined as RS;P i 0i  r i0 e 0i r i0 e where r 0i is the Fresnel reflection coefficient at the interface 0 − i, and kzi refers to the perpendicular component of the wavevector ki . The Fresnel reflection coefficients in P and S polarizations are, respectively, given by r PS  εj μj kzi − εi μi kzj ∕εj μj kzi  εi μi kzj  [27]. ij In the exchange function, contribution of S-polarized or TE waves is due to the magnetic component while that of P-polarized or TM waves is due to the electric component of the metamaterial. The exchange factor Zβ; ω is plotted as a function of β and ω in Fig. 2(a) for 5-nm-thick films and for 10-nm-thick films in Fig. 2(b). In both cases, the vacuum gap is fixed at 10 nm. The magnetic surface polariton (MSP) and electric surface polariton (ESP) are clearly seen for both film thicknesses. When the film thickness is comparable or smaller than the vacuum gap, SPs generated at the film–vacuum interfaces couple within and between the layers. This leads to a splitting of the near-field flux into symmetric (S  ) and antisymmetric (S − ) resonant modes. From Ref. [28], the real part of relative permittivity and permeability at the largest contributing parallel wavevector βmax  ln10d−1 for symmetric and antisymmetric modes during near-field heat transfer between films of identical thicknesses are

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S  :ε0opt ; μ0opt  −

1 − e− ln10D ; 1  e− ln10D

(5)

S  :ε0opt ; μ0opt  −

1  e− ln10D ; 1 − e− ln10D

(6)

where D  tf ∕d. For low scattering rates, ωe;opt ≈ ωp ∕1 − ε0opt 0.5 and ωm;opt ≈ ω0 1 − μ0opt ∕1 − F − μ0opt 0.5 . Here, ωe;opt and ωm;opt represent the frequencies at which SPs are excited. As the film thickness increases, the values of ε0opt and 0 μopt for the two different modes approach each other, implying that ωe;opt and ωm;opt also approach each other for the two modes. The calculated values of ωe;opt and ωm;opt are in good agreement with those observed in Fig. 2 for both film thicknesses. Also, from Ref. [19], for γ∕ωp  0.01, penetration depth for evanescent wave when MSPs or ESPs are excited is on the same order as the vacuum gap. This implies that, as the film thickness increases, the ωopt values approach each other and become one when tf ≥ 10 nm for a 10 nm vacuum gap. The difference between ωm;opt values for the two modes is less than that between ωe;opt values. For ESP, most of the energy transfer is due to S  modes as also seen from Fig. 2. For MSP, both modes contribute evenly to the overall heat transfer since the ωm;opt values are very close to each other. The spectral heat transfer between two metamaterial films is plotted in Fig. 3 for a 10 nm vacuum gap. For the semi-infinite case, the two peaks located at 4.62 × 1013 rad∕s and 7.07 × 1013 rad∕s correspond to MSP and ESP, respectively. Moreover,ε0opt values for different

Fig. 2. Exchange factor Zβ; ω as a function of β and ω for (a) 5 nm films and (b) 10 nm films. Horizontal axis represents angular frequency in 1013 rad∕s and the vertical color bar indicates Zβ; ω in 107 rad∕s.

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Fig. 3. Spectral heat flux for different film thicknesses when the emitter and receiver are separated by a 10 nm vacuum gap.

modes decreases as the film thickness increases. As a result, ωe;opt values approach each other as film thickness increases. For MSP, there is no significant resonance splitting for the value of the filling fraction chosen. Due to the close proximity of ωm;opt values for symmetric and antisymmetric modes, a single spectral peak is observed. However, when F  0.9, for D  0.1, ωm;opt  9.1 × 1013 rad∕s and 4.2 × 1013 rad∕s for S − and S  modes. Also, when ω0 ≈ ωp , the splitting of MSPs for the two modes can be clearly seen. Therefore, the spectral signature in Fig. 3 is highly dependent on the choice of dielectric function. Consequently, the spectral behavior and hence the near-field energy transfer can be optimized by tuning the adjustable parameters in both the permittivity and permeability model of the metamaterial. The width of the spectral heat flux curves is given by the magnitude of the imaginary part of the dielectric function [32]. From Eqs. (1) and (2) at low scattering rates, ε00 ≈ ω2p γ∕ω3 and μ00 ≈ Fω3 γ m ∕ω2 − ω20 2 . ωm;opt values do not change significantly with film thickness, as a result of which μ00opt and hence the width of the spectral heat flux curve is invariant with film thickness at magnetic resonance. On the other hand, as the film thickness increases, the value of ε00opt decreases and hence the spectral width decreases for the heat flux at electric resonance [28]. The decrease in spectral width with increasing film thickness is accompanied by a sharper peak of the spectral heat transfer curve when ESPs are excited. As discussed in previous studies for thin films [27,28], the heat transfer due to the symmetric mode is larger than that for asymmetric mode at electric resonance. Also, it appears that the heat transfer due to MSP does not change significantly with film thickness until the media becomes semi-infinite, in which case the heat flux is approximately 50% larger than that for thin films. From Fig. 2 and Ref. [19], it is seen that for the semi-infinite case, a larger number of modes can be excited at magnetic resonance compared to thin films, implying an increase in the number of channels for heat transfer. Figure 3 shows the advantage of metamaterials for energy transfer over conventional nonmagnetic materials

Fig. 4. Total near-field heat transfer between two thin films made of metamaterials as a function of vacuum gap for different film thicknesses.

in terms of spectral selectivity, due to the ability to excite MSPs at different frequencies by tuning the filling factor. The total heat transfer as a function of the vacuum gap for different film thicknesses is plotted in Fig. 4. Again, for simplicity, the emitter and receiver are assumed to have the same thickness. The heat transfer is maximal for the case of two bulk media. The energy transfer between thin films approaches the semi-infinite prediction when the film thickness equals the vacuum gap. The variation of heat flux as a function of vacuum gap for different film thicknesses has been discussed in [27]. Understandably, the trend in the total near-field heat transfer as a function of the vacuum gap for metamaterial thin films is different from that between ordinary materials due to the contribution from MSPs. Only for the semi-infinite case does the near-field heat flux due to P and S polarized waves vary as d−2 . At any given vacuum gap, the total heat flux is split evenly into contributions from S and P waves for tf ≤ d. When the film thickness becomes larger than the vacuum gap, the contribution from P waves increases, and for the semi-infinite case it is twice the contribution from S waves. As explained earlier, as the film thickness increases, ε00opt decreases while μ00opt does not vary significantly. From Ref. [19], for D > 1, βmax varies inversely as ε00opt and μ00opt when ESPs and MSPs are excited, respectively, which explains why the heat flux due to P waves dominates over S waves as film thickness increases. Note that the value of μ00 and hence contribution from S waves can be optimized by tuning the filling factor. In summary, we have calculated the near-field heat transfer between thin films made of metamaterials. The impact of film thickness on both magnetic and electric resonances was discussed in detail. Resonant frequencies corresponding to the symmetric and antisymmetric modes for magnetic resonance do not change significantly with film thickness for the permeability model chosen in this study. Contributions of both S and P waves to the total heat transfer are discussed for different film thicknesses. Results from this study will facilitate the application of metamaterials as thin film coatings in energy

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Near-field radiative heat transfer between metamaterial thin films.

We investigate near-field radiative heat transfer between two thin films made of metamaterials. The impact of film thickness on magnetic and electric ...
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