Near-field radiative heat transfer between general materials and metamaterials

We investigated the near-field radiative heat transfer between general materials and metamaterials. We studied the effects of metamaterial parameters on the radiative heat exchange and used three kinds of natural or artificially-constructed materials such as Al, boron-doped Si and metamaterials as examples. We calculated and analyzed the near-field radiative heat transfer processes between two semi-infinite bodies. The numerical results indicate that the radiative heat exchange between the two different materials may be less or more than the radiative heat exchange between the corresponding identical materials. It was found out to depend on the radiative properties of the materials. The work would provide a valuable reference for the selection of practical materials.

The swift advancement of micro/nanotechnology strongly encourages the development of highly-efficient miniature energy conversion and utilization devices [1][2][3][4] in which microscopic thermal transport is very complicated and needs to be considered [5]. Meanwhile, the appearance of a variety of man-made materials brings new challenges in near-field thermal radiation.
As an important part of the micro/nanoscale radiative heat transfer, various aspects of near-field radiative heat transfer has been studied by many scholars [6][7][8][9][10][11][12][13][14][15][16][17][18]. From their investigations we know that the macroscopic thermal radiation law fails to describe the radiative heat transfer processes at the subwavelength scale. Additionally, the near-field radiative heat transfer rate between two closely spaced bodies may increase by several orders of magnitude compared with that between two blackbodies at the macroscopic level because of near-field effects such as wave interference, photon tunneling and the excitation of surface waves [6][7][8][9][10][11][12][13][14][15][16][17][18]. Recently, the near-field radiative heat trans-fer between two metamaterials was studied theoretically by Joulain et al. [19] and they found that the excitation of surface waves during TE polarization and the ferromagnetic behavior of the metamaterials will result in new channels for heat transfer and will enhance the radiative heat transfer. Therefore, the near-field radiative heat transfer becomes much more complicated because of the presence of metamataterials. However, some interesting questions arise such as the type of effects that may be induced by the parameters of metamaterials on the near-field radiative heat transfer, the results of the near-field radiative heat transfer between metamaterials and nonmagnetic materials (i.e. relative permeability μ=1) and the difference in the near-field radiative heat transfer compared with that between the corresponding identical metamaterials or between the corresponding identical nonmagnetic materials. Wang et al. [12] analyzed the effects of material parameters as well as the temperature on the near-field heat transfer between the two identical normal materials using the Drude and Lorentz models. Fu and Zhang [20] comprehensively investigated the near-field radiative heat transfer between silicon surfaces with varying doping concentrations. Joulain et al. [19] studied the nearand far-field radiative heat transfer between two semi-infinite magnetodielectric materials by a combination of fluctuational electrodynamics and the fluctuation-dissipation theorem and found that the radiative heat transfer increases in the near field when surface waves are excited. Little attention has been paid to the near-field radiative heat transfer between two surfaces with dissimilar materials, especially between metamaterials and non-magnetic materials. A study into these properties will provide a valuable reference for the selection of material pairs in microscaled energy conversion devices such as thermophotovoltaic (TPV) systems.
We mainly investigated the radiative heat exchange between two semi-infinite bodies made of different materials including natural and artificial materials and we paid special attention to the effects of the metamaterial parameters on the near-field radiative heat exchange between two semiinfinite surfaces separated by a vacuum gap. The three types of materials used were Al [14,21], Si-19 (which refers to boron-doped Si at a concentration of 10 19 cm −3 [12]) and metamaterials.

Mathematical model
The structural model is shown in Figure 1. We concentrated on the radiative heat flux between two semi-infinite bodies (media 1 and 3) separated by a vacuum gap (medium 2). Media 1 and 3 are maintained at temperatures T 1 and T 3 , respectively. Because of the presence of fluctuating electrical and magnetic current sources in materials at temperatures higher than 0 K, fluctuating electromagnetic fields are present inside as well as outside the material, and thermal radiation takes place [22]. The fluctuating electromagnetic field is calculated using fluctuating-dissipation theory [23] and Maxwell's stochastic equations, which is constructed by adding extra fluctuating electrical and magnetic current sources into Maxwell's equations [22]. The spectral heat flux transferred to medium 3 because of the fluctuating current and magnetic current sources in medium 1 can be divided into two parts: One is due to the electric current source in medium 1 and the other is due to the magnetic current source in medium 1 [24]. The following expression is obtained [24]: and Figure 1 Schematic of the structure considered: the radiative heat flux is calculated between medium 1 and medium 3 with temperatures of T 1 and T 3 , respectively. The vector R represents the directionˆxx yy + wherex and ŷ respectively represent the unit vectors in the x and y directions.
Eq. (1) shows the contribution of the electric current source in medium 1 and eq. (2) shows the contribution of the magnetic current source in medium 1 [24]. In eqs. (1) and (2), ε 0 and μ 0 respectively stand for the permittivity and permeability of the vacuum while ε and μ respectively are the relative electric permittivity and relative magnetic permeability. The subscripts 1 and 3 respectively represent media 1 and 3 and the superscript * indicates a conjugated complex. Im and Re respectively represent the imaginary part and real component of the complex functions.  [27]. Θ (ω,T 1 ) is the mean energy of a Planck oscillator at the frequency ω. Details for the formulations can be found in [24]. A similar method can be applied to a calculation of the spectral heat flux that is transferred to medium 1 because of the fluctuating current and magnetic current sources in medium 3. The difference between these two radiative heat fluxes is the net spectral radiative heat transfer between the two surfaces. The total net radiative heat transfer can be obtained by integrating over the frequency ω from zero to infinite.

Relative electric permittivity and relative magnetic permeability models for metamaterials
It is well known that metamaterials belong to a new class of electromagnetic composite materials and can exhibit a host of novel properties such as negative relative dielectric permittivity (ε ), negative relative magnetic permeability (μ ) and a negative refraction index [28][29][30][31][32]. The metamaterial is both dispersive and dissipative and generally its relative dielectric and magnetic is complex and dependent on frequency [19,33,34]. For metamaterials made from an artificial array of wires and a split ring the effective relative permittivity and permeability can be expressed as [19,30,31,33,35,36] ( ) where F is the area proportionately occupied by the split ring in the unit cell, ω p is the effective plasma frequency, ω o is the resonance frequency, and γ e and γ m are the scattering rates [19,35,36]. By algebraic manipulations the real and imaginary parts of the relative permeability (μ=μ′ + iμ″) can be respectively expressed as Therefore, μ″ is always greater than 0 for ω>0 and is equal to 0 when ω=0. By setting μ′<0, one can get From eqs. (5a) and (6) it is clear that μ′<0 occurs in a frequency band above ω o or it does not occur, which is determined by the parameters F, γ m and ω o . Since eq. (3) is similar to the Drude model, a detailed analysis of the influence of γ e on ε has been given in [12,13]. Figure 2 shows the influence of γ m and γ e on the nearfield radiative heat flux between two identical semi-infinite metamaterial bodies with a vacuum gap d=10 nm for the parameters ω p =10 14 rad/s, ω o =0.4ω p , and F=0.56. Figure 2  the two surfaces were maintained at T 1 =300 K and T 3 =0 K, respectively. As shown in Figure 2, the first peak at lower frequency ω=4.717×10 13 rad/s comes from the excitation of the surface waves in TE polarization and in a narrow frequency band around ω=4.717×10 13 rad/s while there is a large number of electromagnetic states, which results in a high amount of radiative heat transfer and is referred to as the resonance of the flux [19,24,37]. Similarly, the second peak at higher frequency ω=7.075×10 13 rad/s is due to the excitation of surface waves during TM polarization and the resonance of the flux [19,24,37]. It should be noted that since μ′<0 does not occur for γ m =0.5ω p the dispersion relationship for the excitation of surface waves in TE polarization (i.e. μ 2 γ 1 + μ 1 γ 2 = 0) cannot be satisfied. Therefore, no peak is present as a result of the excitation of surface waves in TE polarization for the curve corresponding to γ m =0.5ω p in Figure 2 the corresponding heat flux peak is very sharp, which is similar to that obtained in [12]. For a small γ m the corresponding heat flux peak is also very sharp.

Near-field radiative heat transfer between two surfaces with a different materials assembly
The near-field radiative heat transfer between the two semiinfinite surfaces mode of a dissimilar materials assembly is more complicated than that discussed in the preceding section and they may exhibit quite different features. The presence of metamaterials further complicates the near-field radiative heat transfer. This section is devoted to a detailed analysis of the near-field radiative heat transfer between two semi-infinite bodies made of different materials. In the subsequent calculation the temperatures of the two semi-infinite bodies are respectively maintained at T 1 =300 K and T 3 =275 K. Although one may exchange the materials of medium 1 and medium 3, the expression for the net heat flux from medium 1 to medium 3 is formally identical, which has been shown in [19]. Figure 3 shows the spectral net radiative heat flux between the two semi-infinite bodies with the vacuum gap d=1 μm for several different combinations of metamaterials and Si-19. Note that since the vacuum gap d=1 μm is comparable to the characteristic wavelength of thermal radiation for the given temperatures the near-field effect on the radiative heat transfer between the two surfaces can be clearly identified. Meanwhile, the vacuum gap d=1 μm is not too much smaller than the characteristic wavelength for the given temperatures. On the other hand, the far-field contribution may be considered as well. Therefore, the radiative heat exchange phenomena at the vacuum gap d=1 μm may be more complicated than that for a much smaller vacuum gap such as a nanometer gap where the near-field effect is dominant in the radiative heat flux. The selected parameters for the metamaterial as described by eqs. ε (∞)=11.7, ω p =3.42×10 14 rad/s and γ =6.12×10 12 rad/s [12].
Note that to highlight the effects of the surface phonon polariton resonance, the value of parameter γ =6.12×10 12 rad/s is taken to be one tenth that in [12]. For simplicity, the material described by these parameters is still referred to as Si-19 in the following text. For the curve corresponding to case 1 (as shown in Figure 3), three peaks are present that correspond to the frequencies ω 1 =4.01×10 13 rad/s, ω 2 =4.71×10 13 rad/s and ω 3 =7.707×10 13 rad/s, respectively.
The radiative heat transfer mainly increases because of the excitation of surface waves in TE polarization around ω 2 and the excitation of surface waves in TM polarization around ω 3 [19,24]. The tiny peak at ω 1 is due to the asymptotic behavior in TM polarization around ω=ω o where the relative permeability is very large [19,24]. For the curve that corresponds to case 3, the radiative heat transfer is greatly enhanced because of the excitation of surface waves during TM polarization around ω=9.58×10 13 rad/s [12]. For the curve that corresponds to case 2 all the properties are retained for the two cases discussed above but the increases that result from these properties all decrease. The main reason is that the dissimilar materials cause the frequency band of the high radiation emission to be different from those of high absorption (i.e.

( ) ( )
, , 21 23 Im Im s p s p r r ≠ ) [37,38]. Note that the excitation of the surface waves for TE polarization only occurs at the metamaterials/vacuum interface for case 2 while excitation occurs at both the metamaterials/vacuum interfaces for case 1 and the two surface waves get coupled [24]. As shown in Table 1, it is also noteworthy that the total heat flux for case 2 is much lower than that for the other two cases.
To further understand the effect of surface waves on the radiative heat transfer between the two semi-infinite bodies, the radiative heat flux can be plotted as a function of frequency ω and the parallel wave vector component β. As shown in Figure 4, the angular frequency ω is normalized to Figure 3 The spectral net radiative heat flux between the two semi-infinite bodies with the gap d=1 μm for different assemblies of the metamaterial and Si-19.  correspond to cases 1-3, respectively, in Figure 3. Five dispersion curves are present in Figure 4(a). Curves a 2 and a 3 represent the effects of the excitation of surface waves during TE polarization. Curves a 1 , a 4 and a 5 represent the effects of the excitation of surface waves in TM polarization.
Curve a 1 becomes flatter as β increases because of the asymptotic behavior at ω=ω o resulting in a tiny heat flux peak for case 1 in Figure 3 [19,24]. The radiative heat transfer is greatly enhanced by the excitation of surface waves during the TE and TM polarizations. As shown in Figure 4(b), if one of the semi-infinite bodies is made of Si-19 instead of the metamaterial two curves, a 2 and a 5 , appear in Figure 4(a) and disappear in Figure 4(b) because of the alteration of the surface waves excitation as a result of the dissimilar material assembly. Additionally, the intensities of the radiative heat transfer for curves a 1 , a 3 and a 4 as shown in Figure 4(b) are greatly diminished, which results in a larger decrease in the radiative heat exchange between the two surfaces. Although the radiative heat transfer is enhanced by the excitation of surface waves during TM polarization because of the presence of Si-19, such an increase is not large enough to balance these decreases because of the dissimilar material assembly. Therefore, in Figure 3 the total radiative heat exchange that corresponds to case 2 is much smaller than that for case 1. As shown in Figure 4(c), for the case where both surfaces are made of Si-19, the radiative heat transfer is greatly enhanced by the excitation of the surface waves of Si-19 during TM polarization where the frequency is less than 9.97×10 13 rad/s. Furthermore, one can also see a remarkable enhancement at higher than 9.97×10 13 rad/s, which is due to the higher ( ) 21 Im s r and ( ) 21 Im p r values in these ranges. By a comparison of the curves in Figure 4(b) and (c), a curve that corresponds to the excitation of surface waves in TM polarization in Figure 4(c) at less than 9.58×10 13 rad/s vanishes in Figure 4(b). Additionally, the intensities of the increase in radiative heat transfer because of the surface waves during TM polarization and because of the higher values of ( ) 21 Im s r and ( ) 21 Im p r at higher than 9.58×10 13 rad/s in Figure 4(c) are greatly diminished in Figure 4(b), which results in a greater decrease in radiative heat transfer for the dissimilar material assembly. On the other hand, although the radiative heat transfer is also en-hanced by the excitation of surface waves during TE-and TM-polarization because of the presence of metamaterials, such an increase is not large enough to balance the decrease that results from the dissimilar material assembly. Therefore, the total heat exchange for case 2 in Figure 3 is much lower than that for the other two cases that correspond to the same material assembly.
To estimate the near-field effect, a further calculation was carried out for the situation where the vacuum gap between the two surfaces was 10 nm for the above-mentioned three material assemblies. As shown in Figure 5, with a decrease in the width of the vacuum gap, the near-field effect becomes stronger and more remarkable so that the radiative heat exchange between the two surfaces is dominated by evanescent waves (i.e. photon tunneling) [39]. A comparison between the curves in Figure 5 and those in Figure 3 reveals that the total radiative heat exchange flux for the three types of material assemblies for the 10 nm gap is much larger than that of the corresponding 1 μm gap, as shown in Table 1. The data listed in Table 1 more clearly explains the near-field effect on the radiative heat transfer between the two semi-infinite surfaces. Because of the strong near-field effect at the 10 nm gap some of the phenomena observed in Figure 3 become submerged and some peaks in the curves in Figure 3 disappear in Figure 5. The assembly of the metamaterials and Si-19 may be used as an example to further explain the near-field effect. As shown in Figure 6, the increase because of the excitation of the surface waves during TE polarization is smaller because of the excitation of the surface waves during TM polarization at the matematerial/vacuum interface when the gap was 10 nm. This was not observed in Figure 5 although it does exist, which means that for this different materials assembly with a 10 nm gap the TM-wave contribution is a dominant factor for the radiative heat transfer flux. Additionally, the enhancement because of the asymptotic behavior during TM polarization at ω=ω o becomes negligible for both the assembly Figure 5 The spectral net radiative heat flux between the two semi-infinite bodies with the gap d=10 nm for the different assemblies between the metamaterial and Si-19. cases wherein the metamaterial is involved if the vacuum gap is 10 nm since the corresponding tiny peaks cannot be found in the corresponding curves in Figure 5 [19,24]. with ω ∞ =1, ω p =2.4×10 16 rad/s and γ =1.25×10 14 rad/s [14,21] while the parameters for the metamaterials is identical to that in Figure 3. In Figure 7, case 1 was discussed above. Case 2 is where medium 1 is the metamaterial and medium 3 is Al. In this case, the effects of the excitation of the surface waves of the metamaterial during TE polarization on heat transfer are preserved but greatly diminished and during TM polarization they are almost eliminated, as shown in Figure 8. Therefore, the spectral heat flux is drastically reduced at frequencies higher than 4.71×10 13 rad/s for case 2 in Figure 7. Note that Figure 7 The spectral net radiative heat flux between two semi-infinite bodies with the gap d=1 μm for different assemblies of the metamaterial and Al. although surface plasmon polaritons also possibly excite at the Al/vacuum interface, the excitation frequency is not in the frequency domain where the contribution to the heat flux is significant for the given temperature and it has little influence on the radiative heat transfer [14]. It is emphasized that the radiative heat transfer for the metamaterialmetamaterial assembly is the highest among all three types of material assemblies. Further calculations for the three possible assemblies of Al and metamaterial at the 10 nm gap was carried out to understand the near-field effect, as illustrated in Figure 9. A similar phenomenon to that observed for the same material assembly at the 1 μm gap was obtained but the near-field effect becomes more evident. The effect of the excitation of surface polaritons during TM polarization was not observed for case 2. Note that in cases 2 and 3, both the radiative heat transfers are dominated by the TE-wave contribution. The presence of Al drastically reduces the TM-wave contribution to the radiative heat flux. The material assembly of Al and the metamaterial leads to quite a different radiative heat transfer phenomenon from Figure 9 The spectral net radiative heat flux between the two semi-infinite bodies with the vacuum gap d=10 nm for the different assemblies of metamaterials and Al.
that observed for the above-mentioned assemblies of Si-19 and the metamaterial. The radiative heat transfer for the former is dominated by a TE-wave contribution and that for the latter is dominated by a TM-wave contribution.
We will now discuss the near-field radiative heat transfer for the assembly of Al and Si-19. Figure 10 shows the spectral net radiative heat flux between the two semi-infinite bodies with the vacuum gap d=10 nm for the three different combinations of Si-19 and Al. It is clear that the radiative heat exchange for case 2 is much smaller than that for cases 1 and 3. The total radiative heat flux for cases 1 and 3 are several orders of magnitude higher than that for case 2. The main reason is that the effects of the excitation of surface polaritons of Si-19 during TM polarization decreases greatly because of the presence of Al and there are no other effects of the excitation of surface polaritons during TE or TM polarization to enhance the radiative heat transfer in the thermal frequency band for case 2.

Conclusions
We investigated the near-field radiative heat transfer between two semi-infinite surfaces made from different materials assemblies of Al, boron-doped Si and metamaterial.
The effects of the material parameters γ e and γ m on the radiative heat transfer are discussed. The results show that both the parameters γ e and γ m have little influence on the locations of the peaks. They mainly determine the sharpness of the heat flux peak because of the excitation of surface waves during TM polarization and TE polarization, respectively.
Among all the material assemblies, the radiative heat transfer for the assembly of the metamaterial-metamaterial is the highest. Compared with the radiative heat transfer of two surfaces made from the same material assembly, the radiative heat exchange between two dissimilar materials can be decreased or increased, which is dependent on the Figure 10 The spectral net radiative heat flux between two semi-infinite bodies with the gap d=10 nm for different assemblies of Si-19 and Al. material characteristics. If the material assembly is the metamaterial and Si-19, the mismatch between the spectral bands of high radiation emission and high absorptance may drastically reduce the radiative heat flux and the radiative heat transfer and it is much smaller than that for the same material assembly of the metamaterials or Si-19.
The presence of Al in a material assembly can drastically reduce the TM-wave contribution to the radiative heat flux. Therefore, the radiative heat transfer between the metamaterial and Al is much smaller than that between the two metamaterials because of the intensive reduction of the TM contribution. The radiative heat transfer between Al and the non-magnetic material Si-19 is also much smaller than that between the identical Si-19 because the assembly of Al and Si-19 suppresses the excitation of the surface polaritons in TM polarization.