Using our recently developed method we analyze the radiative heat transfer in micron-thick multilayer stacks of metamaterials with hyperbolic dispersion. The metamaterials are especially designed for prospective thermophotovoltaic systems. We show that the huge transfer of near-infrared thermal radiation across micron layers of metamaterials is achievable and can be optimized. We suggest an approach to the optimal design of such metamaterials taking into account high temperatures of the emitting medium and the heating of the photovoltaic medium by the low-frequency part of the radiation spectrum. We show that both huge values and frequency selectivity are achievable for the radiative heat transfer in hyperbolic multilayer stacks.
© 2013 Optical Society of America
The purpose of the present study is to show the way to a significant improvement of so-called micron-gap thermo-photovoltaic system (MTPVS) [1–8] through the huge and frequency-selective enhancement of thermal radiation which dramatically exceeds the black-body limit.
The key components of MTPVS are emitter and photovoltaic (PV) cell substrated by a heat sink. The micron thick vacuum gap allows the PV material to capture the whole radiation of the emitter. The emitter is a layer of a material possessing high emissivity and radiating under high temperatures (T ≥ 1000°K) i.e. mainly in the range of near-infrared (NIR) waves. The PV cell comprises a semiconductor layer with a p-n junction operating at NIR. The frontal electrode of the cell is usually implemented as metal strips of deeply submicron thickness. In the important theoretical works [6–8] devoted to MTPVS one concentrates on the radiative heat transfer (RHT) between two surfaces, that of the emitter and that of the PV layer, which are separated by a micron-thick vacuum gap. Though the overall geometry of MTPVS is often cylindrical the radius of curvature is very large compared to the gap thickness h, and both emitting and PV surfaces can be modeled as flat ones. Since the electromagnetic absorption in these media is high the emitting and the PV layers can be replaced by two half-spaces. The first systematic analysis of such a problem was given in . The gap between these two half-spaces cannot be completely filled with a continuous medium because the non-radiative thermal conductance through the gap would obviously suppress the PV conversion  (see also in [5, 6, 8]). Special mechanical systems [10, 11] maintain the micron vacuum gap over a sufficient area with needed precision (50–100 nm). The thermal conductance through the supporting elements is reduced to the safe level [3–5, 7, 8].
The main drawback of conventional thermophotovoltaic (TPV) systems which results in their rather low efficiency originates from the ultra-broad frequency spectrum of thermal radiation. In solar cells, the bandgap edge λg of the PV material can be chosen so that the solar light spectrum concentrated mainly in the visible range nearly coincides with the operational band of the PV cell, which is determined by the PV spectral response of a semiconductor. In this case, it is possible to locate λg near the upper edge of the solar spectrum. The same design approach where the sun would be substituted by the thermal emitter is not acceptable for MTPVS. Even at high temperatures of the emitter the spectrum of the thermal radiation is much broader than that of the visible sunlight. The irradiance of a black body is described by the well-known Planck formula:Eq. (1) the spectrum of the thermal radiation of a surface with the temperature T = 2000°K is effectively located between λmin ≈ 0.6 μm and λmax ≈ 5 μm. It is not reasonable to choose for the PV cell of a TPV system the semiconductor with λg = 5 μm. First, there are no PV materials operating in the whole range [λmin, λmax]. If λg = 5 μm a PV cell could operate, for example, in the range λ = 2.5 − 5 μm. Then 90% of the radiation impinging the PV cell will be concentrated in the band 0.6 μm < λ < 2.5 μm, where its spectral response vanishes. This radiation would mainly result in the harmful heating of the PV cell. Moreover, all known PV materials operating in the range λ > 2 μm are very inefficient beyond cryogenic temperatures as compared to PV materials operating at room temperatures below 2 μm, such as CuInSe2, CdTe, Ge, etc. and used in known MPTVS. For a realistic MTPVS the bandgap of the PV medum can be, for example, equal to λg ≈ 1.5 μm. Then, the emitter operating at the temperature T1 = 2000°K produces in the useful interval of wavelengths [λmin, λg] nearly one half of the radiated power. This part of the radiation will be converted into photocurrent, but the low-frequency part (λg, λmax) will produce only the harmful heating. This heating is the main reason of the low efficiency of conventional TPV systems (see e.g. in [5, 12]) which is not much higher than that of existing thermoelectric systems.
In modern TPV systems such as MTPVS this heating is reduced to negligible values. First, one inserts an electromagnetic filter into the gap between the hot and the PV surfaces . Such filter is often realized as a finite-thickness photonic crystal whose stop-band covers the harmful range [λg, λmax] [5, 13–20]. The photonic crystal layer with 5–7 unit cells over its thickness is sufficient to achieve the high filtering quality . Since it is necessary to prevent the electron and phonon thermal conductance, an empty gap of micron thickness h = 1 − 3 μm is left between the filtering multilayer stack and the hot surface [19, 20]. Second, the further improvement is done by a modification of the emitter using the nanopatterning/nanotexturing or fabricating the arrays of metal nanoantennas on its surface [8, 16–20]. This way the emissivity of the hot surface Eω can be engineered maximal (i.e. close to unity) in the useful range [λmin, λg], and minimal (close to zero) – in the harmful range. Since the thermal irradiance of the hot surface is equal to Iω = EωBω and the radiation impinging the PV layer has the energetic spectrum Sω = Iωτ2(ω), where τ(ω) is the amplitude transmittance of the filter, the frequency selectivity of the emission and that of the photonic crystal layer are multiplicative effects. Combining them, the harmful impact of the low-frequency part of thermal radiation can be practically removed. This modification results in the maximally possible increase of the overall efficiency defined as the ratio of its output electric power to the power of thermal radiation produced by the emitter. The achieved gain of a modern MTPVS compared to a conventional TPV system is 27-fold. The overall efficiency of MTPVS attains 30–35% . Moreover, in TPVS of solar-thermal type the low-frequency part of the radiative heat is fully involved into the energy conversion since being reflected back to the emitter determines its operation temperature. For these TPVS the 45-fold enhancement of the overall efficiency has been achieved , overcomming the Shottky-Queisser efficiency limit.
However, besides the efficiency enhancement there is a way to enhance the power output of MTPVS once more. This further enhancement is possible through overcoming the black-body (Planckian) limit for the thermal radiation. The PV cell of the known MTPVS only harvests the wave fields radiated by the thermal emitter and does not extract the radiative energy from it . In the known MTPVS the advantage of the presence of the PV material in the near vicinity of the emitter is used only for capturing the far-field radiation, and sometimes for the optimization of the frequency filtering [5–8, 15]. However, the electromagnetic coupling between the PV cell and the emitter of a MTPVS can be engineered so that the transferred thermal radiation may dramatically exceed the black-body limit and become the super-Planckian one.
The super-Planckian emission enhancement has been known and experimentally observed since long ago in connection with the so-called near-field TPV systems. In these systems the vacuum gap (2) between the hot medium (1) and the PV medium (3) should be very narrow. Practically, its thickness is within h = 10 − 50 nm depending on the frequency range [21–30]. The super-Planckian RHT across such a tiny gap arises due to the so-called photon tunneling, which was first considered in connection to thermal radiation by S. M. Rytov  whose approach was later developed in the seminal work . In this phenomenon, evanescent waves, i.e. spatial harmonics whose tangential wave number q exceeds k0, form wave packets that carry energy and thus are involved in RHT. A detailed overview of this phenomenon allowing the RHT to significantly exceed the black-body limit can be found in books [21, 22] and in paper . For a black body the emissivity for complex radiation angles i.e. for q > k0 is, by definition, equal to zero, and packages of evanescent waves between two black bodies do not carry the power. Therefore the photon tunneling of radiative heat across a nanogap is an advantage of real media compared to two black bodies.
The gain of the RHT compared to the black-body limit becomes huge in those near-field TPV systems which explore either the so-called surface-phonon-polariton or surface-plasmon-polariton enhancement mechanisms [23–30, 33–35]. Then the photon tunneling is enhanced by the excitation of paired surface waves at the boundaries of media 1 and 3, and the gain of RHT compared to the black-body limit attains 3–4 orders. However, this huge gain does not mean the high power output of near-field TPV systems. First, it is difficult to combine the requirement of the nanometer gap between the hot and PV media and the frequency selectivity of RHT. Second, such a tiny vacuum gap is possible to maintain only on a very small area. Therefore near-field TPV systems can practically operate as sensing and imaging devices (see e.g. in [30, 36–38]) at low temperatures of the emitter, i.e. in the mid-IR range where thermal fluxes are weak and the cryogenic cooling of a PV cell can be used. Unlike MTPVS they cannot be used as sources of electric energy converted from the heat [5, 7, 8, 33]. As to MTPVS, when the vacuum gap is as thick as h ≥ 1 μm the photon tunneling and coupling of surface waves become negligibly small [27, 34, 35]. Therefore, the RHT in available MTPVS does not exceed the black-body limit.
As to prospective MTPVS, the possibilities of the photon tunneling in them using the partial filling of the micron gap with a metamaterial were considered in several works. In [39–41] one suggested to fill one half of the gap with the so-called double-negative (or Veselago’s) metamaterial possessing negative material parameters ε ≈ μ ≈ −1. Then the metamaterial layer inserted into the gap together with the empty part of the gap would form the so-called perfect lens which was earlier suggested by J.B. Pendry for sub-diffraction imaging . Imaging properties of the Pendry lens are hardly needed for MTPVS, however the structure suggested by Pendry has another unique feature – the total photon tunneling, This effect would occur in the lossless structure without internal granularity regardless of the gap thickness. It means the unit transmittance τ(q) = 1 for all spatial harmonics including evanescent waves. The hopes of the authors of [39–41] were related with the creation of double-negative metamaterials whose optical losses will be sufficiently small. Then the huge enhancement of RHT could be achieved for micron-thick gaps . Unfortunately, known realizations of the Veselago metamaterial obtained during a rather long period after the publication of  suffer of high optical losses (see e.g. in ) which definitely not allow the gain of RHT. Besides losses, there is a restriction related to the granularity of these metamaterials. In our opinion, double-negative metamaterials will hardly lead to a breakthrough in MTPVS.
2. Main ideas and findings
In our work  a design approach to MTPVS has been suggested which should allow the broadband photon tunneling of spatial harmonics q > k0 across micrometer thick gaps completely filled with a metamaterial possessing the hyperbolic dispersion. The idea is based on the fact that spatial harmonics q > k0 in hyperbolic metamaterials are propagating  i.e. photon tunneling is due to the conversion of evanescent waves into propagating ones. This kind of the photon tunneling is much more advantageous than that of evanescent wave packets because the last one is possible only for nanometer gaps and is a strongly resonant phenomenon. The key feature of the photon tunneling in a hyperbolic layer is its broad band over both frequency axis and the axis of spatial frequencies q. The thickness h of a hyperbolic metamaterial layer for which the photon tunneling is practically significant is restricted only by the optical losses and in a realistic metamaterial can be as large as several microns. How to implement such a metamaterial gap preventing the non-radiative thermal conductance through it? This becomes clear from Fig. 1(a). Arrays of aligned metal nanowires possess in the visible and NIR ranges the properties of a hyperbolic medium (see e.g. in [46–51]). These arrays can be grown in the medium of emitter (medium 1) and in the PV medium (medium 3) with free-standing parts. This technology is described e.g. in [51, 52]. Then two layers (medium 1 and medium 3) should be stacked so that the distance h between them is larger than the lengths of the free-standing parts of nanowires (L1 = h1 + h2 and L2 = h2 + h3, respectively), but L1 + L2 ≥ h. Here the value of h is determined by the supporting system (e.g. ). Then nanowires will form an interdigital structure. The interdigital arrangement corresponds to the complete filling of the vacuum gap with a hyperbolic mematerial without mechanical contacts between its hot and cold parts. In Fig. 1(a) some nanowires grown in media 1 and 3 have no free-standing parts and therefore the averaged inter-wire distances in these media (c and d, respectively) are not equal to those in the gap. The necessity of these additional nanowires will be explained below. The arrays of free-standing nanowires have the same averaged period a in the top and bottom parts of the gap, and in the central part of the gap the averaged period is equal .
In this paper we do not calculate the exact RHT in the original structure shown in Fig. 1(a) since it is too challenging for the full-wave simulations. Our study is done within the framework of the homogenized model illustrated by Fig. 1(b). The gap h is assumed to be filled with three layers of hyperbolic metamaterials. The effective media 2.1 and 2.3 of thicknesses h1 and h3, respectively, are described by the same effective permittivity tensor. The internal layer 2.2 is characterized by a smaller period of the nanowires. Since the whole gap is filled with hyperbolic media the energy is carried by all spatial frequencies up to the certain limit qmax. The question on this bound for spatial frequencies is discussed below.
In  the huge enhancement of RHT in the mid IR range was theoretically obtained across a hyperbolic medium layer formed by single-wall metal-state carbon nanotubes. Similar results for metal nanowires and NIR were obtained in . In the case of RHT at NIR the presence of metal nanowires in media 1 and 3 turns out to be important. If only the vacuum gap is filled with a hyperbolic metamaterial (as in ) the NIR radiation experiences strong reflections at the interfaces. If all media of the structure are hyperbolic (as in in ) these reflections are reduced. It is clear now why the density of nanowires in media 1 and 3 is suggested in the present work to be higher than that in medium 2. The material of nanowires in our calculations is gold. The higher volume fraction of gold (negative permittivity in the NIR range) helps to compensate the optical contrast of the host dielectric materials with free space and to make the optical properties of effective media 1 and 3 closer to those of effective medium 2.
In the present paper we consider the RHT in a structure similar to that analyzed in  in connection to its possible TPV applications. In  and  we have neglected the thermal processes in the hyperbolic metamaterial of the gap. Here we study the contribution of medium 2 into RHT taking into account the finite temperature T2 of this metamaterial. Since our goal is the proof of the concept and thus qualitative estimations are sufficient, we use the simplest model of the temperature distribution in the structure, i.e. assume that T2 is the average value of two temperatures – that of the emitter (T1) and that of the PV medium (T3). The PV material is assumed to be maintained at room temperature T3 = 300°K. Since T1 corresponds to the NIR range i.e. T1 > 1000°K the value T2 turns out to be much higher than T3. Under such temperatures as T1 and T2 = (T1 + T3)/2 gold becomes a very lossy material – its optical absorption coefficient κ attains values on the order of 100. To achieve the strong RHT with such lossy nanowires seems to be problematic. However, we have found that such high values of the optical absorption of gold are useful to obtain rather low optical losses in the metamaterial formed by golden nanowires.
The possibility to achieve the high broadband enhancement of RHT in hyperbolic metamaterials even at very high temperatures is the first important finding of the present paper. The second important result is the possibility of the frequency filtering of RHT in multilayer stacks of hyperbolic metamaterials. It will be shown that the harmful radiative heat flux in the range [λg, λmax] in a 7-layer structure with 5 hyperbolic layers between two isotropic half-spaces can be made much smaller than the flux within the useful spectrum [λmin, λg].
In order to estimate the operation of the suggested structure and to optimize its parameters we use the so-called circuit model of radiative heat transfer developed in . The model is rigorously based on the fluctuation-dissipation theorem as well as the classical model by Rytov (see e.g. in [21, 22, 31, 32]). For a 1D problem our model is fully strict and generalizes the classical approach based on the Green’s function of a multilayer structure to the case when the materials of the layers possess anisotropic and spatially dispersive properties. The relative simplicity and quite broad bounds of applicability of our method make it advantageous for the case when the layers are nanostructured and can be modeled as metamaterial ones.
3. Modeling of a three-layer structure
For simplicity, in this section we consider the model which corresponds to the special case when the central layer (2.2) in Fig. 1(b) is absent. This situation holds by letting h2 = 0 in the original structure of Fig. 1(a). Since both top and bottom layers (2.1) and (2.3) have the same inter-wire distance a they can be described by the same tensor of effective permittivity ε2 with axial ε2|| and transverse ε2⊥ components and unified into one effective medium layer 2.
3.1. Circuit model formulas
All formulas derived in this subsection were presented in a more general form in our work . However, the density of information in our paper  is very high, and its reading is probably not so easy. Therefore, for the convenience of readers we reproduce below some derivations which are simplified for the special case under study.
The circuit model  is based on the expansion of radiative heat (thermal noise) into both frequency and spatial Fourier spectra. The circuit scheme of the heat transfer in the system under study presented in Fig. 2(a) refers to one spatial harmonic determined by the tangential wave number q and frequency ω. The stochastic electromotive force (EMF) e1(t) with the spectral density ℰ1(ω) describes a spatial harmonic of the thermal noise produced by effective medium 1. This EMF is connected in series with the surface impedance of medium 1. The surface impedance of a half-space is equal to the wave impedance of the medium depending on both q and ω (see e.g. ). Similarly, the wave impedance Z3 of medium 3 is connected to the fluctuating EMF e3(t) with the spectral density ℰ3(ω) which describes the thermal noise generated by medium 3. Two mutually correlated EMFs e2−(t) and e2+(t) with the spectral densities ℰ2−(ω) and ℰ2+(ω) connected, respectively, to the input and output of the 4-pole modeling the medium 2, describe the radiative heat production in medium 2. The scattering properties of this 4-pole can be described by the 2 × 2 matrix of Z-parameters (see e.g. ):44] in this paper we use the time dependence exp(jωt) adopted in the circuit theory. The equivalence Z12 = Z21 expresses the reciprocity of medium 2, and the equivalence Z11 = Z22 means the absence of bianisotropy in it .
The mean squares of the fluctuating EMFs (per unit of frequency) are proportional to the real parts of the corresponding input impedances for media 1 and 3, and to the real parts of Z11 and Z22 for medium 2 :44]:
It is easy to express the effective current I+ in the impedance Z3 through the equivalent impedance Zh referred to the plane z = h as shown in Fig. 2(b). This is the output impedance of the 4-pole (modeling the medium 2) loaded by the lumped load Z1 (modeling the medium 1):53]): ℰ′1,2− = ℰ1,2−Z12/(Z11 + Z1). Taking into account Eqs. (3) and (4) we obtain
Calculating the mean square of I+ using Eq. (6) we take into account that the thermal noise in media 1, 2 and 3 is not mutually correlated. From here we obtain:Eq. (8), the effective power absorbed in medium 3: , splits into the following terms:
The term P33 has nothing to do with RHT. It corresponds to the radiative heat produced by the thermal fluctuations in medium 3 and absorbed in it. This absorption process in medium 3 is mediated by the series connection of Z3 and Zh because the radiation produced by medium 3 is absorbed in it after being reflected from the interface z = h, and media 1 and 2 both influence this reflection. The term P23 corresponds to the contribution of medium 2 into RHT. Using formulas (2) it is possible to prove that for both TE-polarized and TM-polarized waves the value P23 is either positive or zero (for lossless medium 2 the expression on the right-hand side of Eq. (11) vanishes). Finally, the term P13 in (9) corresponds to the heat transferred from medium 1 to medium 3. In the classical theory of RHT only this term is analyzed.
Let us also note that in the considered model the current I+ is the conventional circuit form of the magnetic field tangential to the interface z = h. This magnetic field refers to a spatial harmonic q of either TM or TE polarization. The product Z3I+ expresses the tangential electric field, and Re(Z3|I+|2) = |I+|2R3 is the z-component of the Poynting vector of a spatial harmonic (ω and q) at the interface z = h. Since all the power flux calculated at this interface and directed along z is absorbed, the value expresses the power flux density absorbed in medium 3 per unit intervals of q and ω. The value describes the backward (antiparallel to the z-axis) power flux at the same interface. This power flux is absorbed in media 1 and 2, where the impedance Zh describes the cascade connection of media 1 and 2. In this flux we can also share out separate contributions of media 1, 2 and 3. The power flux absorbed in medium 1 can be expressed through the current I− flowing in Z1 which can be also easily found. If we want to find the radiative heat absorbed in medium 2 we may either subtract P1 from P1+2 or find the power P2 separately. For it one can present the 4-pole modeling the layer of medium 2 as a T-circuit (also a Π-circuit representation is possible) formed by three lumped elements (two series impedances Z11 + Z12 and Z22 + Z12, and one shunting impedance Z12), find three effective currents flowing through these three elements and, finally, find the total power absorbed in these three impedances. The same result can be obtained by the integration of the radiative heat flux over the gap [0, h], however, for it we have to find the flux explicitly. To do it in the closed form is not so easy for the stack of anisotropic and spatially dispersive media. Procedures corresponding to the circuit model are much simpler.
The total heat flux absorbed by medium 3 per unit time and unit area (power flux density) is expressed through P3 as follows :(2) and (5) into (12) we can write it in the following form: Eq. (15) to formula (46) of : 44] for a gap filled with an arbitrary medium 2. This representation is suitable for both evanescent and propagating spatial spectra. If medium 2 is free space, then Im(β2) = 0 for the propagating waves (q < k0) and for the evanescent waves (q > k0). Since for micron vacuum gaps exp (−k0h) ≪ 1, the spatial harmonics with q > k0 practically do not contribute to RHT because of the exponential factor |exp−2jβ2h| in (17). Filling the gap with a hyperbolic medium results in the relation , and the factor |exp−2jβ2h| may approach unity for all spatial harmonics which are within the range of applicability of the continuous medium model. This is the main idea of our work . In an ideal continuous hyperbolic medium all spatial harmonics 0 < q < ∞ are propagating and S3 turns out to be theoretically infinite even in presence of optical losses in the medium. This problem has been discussed in  and was resolved by the introduction of the integration limit qmax = π/a in Eq. (13) which takes into account the granularity of the original structure.
To conclude this subsection we make several instructive remarks. First, it is easy to show thatEq. (16), and substituting q = k0 sinθ, Z2 = R2 for plane waves incident from medium 2 to medium 1 under the angle θ, one can easily derive from Eq. (19) a known formula for the hemispherical spectral emissivity of medium 1 seen from medium 2 (see e.g. in [21, 22]): Eq. (20) is valid either for TM-waves or for TE-waves, whereas the total emissivity of the black body is the sum of and .
3.2. Maximal enhancement of radiative heat transfer
Hyperbolic metamaterials are special case of uniaxially anisotropic media. Wave impedances and axial propagation factors for TM-polarized and TE-polarized waves for uniaxial media can be found e.g. in :
If media 1 and 3 are also hyperbolic the impedance matching at the interfaces z = 0 and z = h is improved compared to the case when media 1 and 3 are isotropic dielectrics. Since media 1, 2 and 3 are different, the dependence of Zi(q) on q in expression (21) is different for these media. At first glance it appears that such spatial dispersion forbids the perfect impedance match. However, a more accurate analysis of (21) shows that the dependence of Zi(q) on q can be greatly suppressed in the interval of spatial frequencies if |εi||| ≫ 1. The interval of spatial frequencies is practically responsible for the RHT, because only within this interval one has . Spatial harmonics with experience stronger decay due to non-vanishing optical losses in medium 2, and also because the decay factor typically grows versus q.
If all three media are hyperbolic, their optical losses are sufficiently low, and their longitudinal permittivity components are large enough in absolute value, the wave impedances of the three media are all nearly real and weakly dependent on q in the important interval of spatial frequencies. In this case a good matching for thermal radiation is achievable at both interfaces within the structure.
Equating the imaginary part of the expression (Z1 + Z2)(Z2 + Z3) + (Z1 − Z2)(Z2 − Z3) exp (−2 jβ2h) in the right-hand side of Eq. (15) to zero we obtain an equation which after some algebra can be reduced to the form:Eq. (23) is equivalent to the requirement of perfect matching at the plane z = 0 i.e. to equation , where Z0 is the input impedance of the 4-pole modeling the gap (medium 2) loaded by the impedance Z3. The perfect matching condition is satisfied for every β2 if and only if media 1 and 3 are conjugate matched i.e. when R1 = R3 and X1 = −X3. Such an ideal conjugate matching of two media is, however, impossible in practice since the reactive part of the wave impedance of the TM-waves is inductive in any passive uniaxial medium. This conclusion follows from Eq. (21) if one takes into account that Im(β) < 0, and Im(εi⊥) < 0. Even if medium 3 has optical gain the conjugate matching with passive medium 1 is not achievable since R1 > 0 for a passive material and R3 < 0 for an active one. We can only approach the perfect matching by reducing X1 and X3 (i.e. decreasing optical losses in media 1 and 3) so that they are small compared to R1 and R2.
Assuming that R1 ≈ R2 ≈ R3 and replacing Ri in Eq. (15) by R we can represent this relation in the form
When optical losses in media 1 and 3 are negligible i.e. X1,3 ≈ 0 formula (24) reduces to P13 = (2/π)(Θ1/4) or N = 1/4. This is the well-known result obtained in  for the global maximum of the spatial spectrum N(ω, q) = (π/2)[P13(ω, q, T)/Θ1(ω, T)] of the heat transfer function. Ways to approach this target maximum at some points within a given spatial frequency range using the SPP enhancement are discussed in [27, 34]. However, such a resonant enhancement of the heat transfer provides for high values of N only in a finite range of ω and q. The larger is the gap h between media 1 and 3 the narrower is the [ω, q] band within which N is close to 1/4. When h ≥ 100 nm the band within which N(ω, q) ≈ 0.25 becomes so narrow (even if the photon tunneling is enhanced by surface phonon polaritons) that the photon tunneling does not lead to a gain in RHT [27, 34, 45]. Although the enhancement of RHT by surface states allows the values of N to approach the global maximum, this technique is less advantageous for the RHT than the conversion of the evanescent waves into the propagating waves in hyperbolic layers, especially when their thickness is larger than 100 nm. In  a very high gain (3 orders of magnitude) for RHT was obtained due to the photon tunneling through a hyperbolic medium layer with the thickness h = 1 μm. However, this result was obtained in the mid-IR range asuumng a hyperbolic material with very low optical losses, e.g. formed by aligned single-wall metal-state carbon nanotubes. For a more lossy medium of metal nanowires in the near-IR range (where the same gap h = 1 μm is wider in terms of λ) the photon tunneling through the hyperbolic layer must be complemented by impedance matching .
The strategy of maximizing N in a wide band [ω, q] can be formulated as follows. First, the gap h should be filled with a hyperbolic material. This grants conversion of the evanescent waves, i.e. ensures that the exponential factor e−2Im(β2)h ≈ 1 is irrelevant. Second, we have to impedance match all the media. In order to achieve this, media 1 and 3 must be as well hyperbolic, so that Xi ≪ Ri and R1 ≈ R2 ≈ R3. The last condition after substitution of Eq. (21) reads
For any q the condition (25) is satisfied strictly if and only if ε1⊥ = ε2⊥ = ε3⊥ and ε1|| = ε2|| = ε3|| i.e. when all media are equivalent. However, the approximate equivalence of the left-and right-hand sides of Eq. (25) can be achieved e.g. when |εi||| ≪ 1 for i = 1, 2, 3. If this requirement is fullfiled, the approximate matching condition applicable within a broad interval of spatial frequencies is expressed by the following system of equations:(26), (27) together with |εi||| ≫ 1 are the practical conditions for maximizing the RHT within the spatial spectrum band and at each frequency ω for which these conditions hold. The key idea here is the high absolute value of the axial permittivity in the three hyperbolic media which allows their wave impedances to become almost independent of the spatial frequency q. From these observations it follows that the fraction of nanowires in media 1 and 3 should be larger than that in medium 2, in order to compensate for the nonzero host dielectric susceptibility of media 1 and 3.
We would like to note also that although it formally follows from our theory that the maximal RHT is achieved for lossless media 1 and 3, it is meaningless to try to reduce these losses too much. If optical loss in medium 3 is very low, the significant part of radiative heat will be transmitted through this medium, because the thickness of medium 3 is in fact finite and is determined by the practical design of the PV cell. If the radiative heat transmission through the overall structure is significant it will dramatically change the whole optimization approach. Thus, we assume that optical losses in media 1 and 3 are sufficient to justify their approximation as two semi-infinite media.
3.3. Radiative heat transfer in hyperbolic materials at low temperatures
Although in this paper we restrict our analysis by homogeneous hyperbolic materials it is instructive to discuss the practical range of parameters of the original structure shown in Fig. 1 (a) when it can be replaced by a stack of the effective medium layers shown in Fig. 1 (b).
In accordance with  a reliable analytical model of a nanowire medium suitable for the whole IR range is the one suggested in work  (see also ). Following it, the axial component of the effective permittivity of an aligned array of metal nanowires with the relative permittivity εm embedded into a host material with the relative permittivity εh is calculated as follows:48]: (28) has to be found using the second formula of (21) as a function of q. This allows to find ε|| as a function of both q and ω. Since the effective medium formed by golden nanowires is spatially dispersive its isofrequency contours are not exactly hyperbolic even at the room temperature. For golden nanowires, the shape of the isofrequency curves varies versus λ from hyperbolas in the visible range to nearly straight infinite lines in the mid IR. The spatial dispersion does not deteriorate the photon tunneling .
The negative values of Re(ε||) in the NIR range follow from the representation of Eq. (28) in the equivalent form:Eq. (30) has a negative real part.
For medium 2 formed by gold nanowires the reasonable value for the averaged lattice constant a is on the order of 100 nm. It is one order of magnitude smaller then the wavelength in free space for the range λ = 0.7−1.5 μm in which many TPV systems operate. Reducing a further is useless since the wire thickness b of a free-standing nanowire cannot be too small. In our numerical examples we have fixed a = 100 nm. Practically important values of the volume fraction of gold in medium 2 p2 = 0.1 − 0.35 correspond to nanowires of thicknesses b = 40 − 70 nm. Lattice constants c and d in media 1 and 3 were chosen within the limits (1.6 − 2.8)b so that p1,3 ≤ 0.4. The latter value of the volume fraction is nearly at the applicability limit of the homogenization model . For an array of aligned golden nanowires with the aforementioned design parameters operating at room temperatures in the band λ = 0.5 − 2 μm the dependence ε||(q) turns out to be negligible over the wide range of spatial frequencies q = (0...30)k0.
The aim of the following analytical calculations is to maximize the integral gain for the radiative heat absorbed by medium 3 (the gain is achieved due to the presence of nanowires). As discussed above, we approximate the original nanostructures forming the layers 1–3 by continuous hyperbolic media with feasible material parameters. The integral gain is defined as , where S3 is calculated for the structure with nanowires, and corresponds to the case without nanowires (when media 1–3 are the uniform host materials). The evanescent waves for wavelengths λ < 2 μm do not contribute into radiative heat transfer across a 1-μm thick vacuum gap. Therefore the difference between and the spectrum of radiative heat flux between two black bodies is only due to the emissivity of the hot (SiC) and PV (CIGS) media which vary in the operation band of the PV cell based on CIGS within the range 0.89 − 0.91 and 0.61 − 0.63, respectively.
First, we calculate this gain neglecting the radiative heat flux produced by medium 2. Here we also neglect the heating of medium 3 by its own radiation reflected from the interface z = h. In other words, instead of we calculate in this subsection the value . Following , S13 and are expressed through P13 and as follows:Eq. (31) are proportional to Θ1 this gain does not depend on the temperature and is equal to:
The results of calculations are represented in Figs. 3 and 4, where the data for the infrared complex permittivity of gold at temperatures T3 = 300°, T2 = 350°, and T1 = 400° K have been taken from [60, 61]. Under the temperature T = 300° K the complex refraction index of gold for h̄ω = 1 eV (λ = 1.24 μm) equals n − jκ ≈ 0.37 − j8.77 [60, 61] that corresponds to the nearly real and negative complex permittivity. In the range 0 < T < 400° K, Im(εAu) grows weakly versus T and gold has properties typical for a negative-ε low-loss metal [60–62]. The host material of medium 1 is assumed to be polycrystal SiC [56, 57]. This material is often used in TPV systems operating at NIR due to its high temperature stability and weak dispersion of its complex refraction index, which is approximately equal to n − jκ ≈ 2.4 − j0.02 over the range 0.5 − 2.5 μm . The host material of medium 3 is doped CuInSe2 with minor carrier density 3·1018 cm−3. Its complex permittivity in the NIR range can be found in works [58, 59]. For this material λg ≈ 1.7 μm and the PV cell can operate in the range 0.8 − 1.7 μm.
The thickness b of the nanowires and their average periods c and d in media 1 and 3 were optimized in order to achieve the highest gain for the gap h = 1 μm. This value for the optimized structure with h = 1 μm turned out to be nearly equal to 1.1 · 103. The optimization was performed by varying three values b, c and d within the allowed ranges (see above) with the 5 nm step. Optimal parameters of nanowire arrays were found as follows: a = 100 nm, b = 45 nm, c = 75 nm, d = 85 nm.
Unfortunately, due to the frequency dispersion of the optical constants of both Au and CuInSe2 it is hardly possible to satisfy both conditions (26) and (27) simultaneously. The analysis of the matching condition (25) in the range λ = 1−2 μm has shown that integral matching over all q is more sensitive to the mismatch of ε|| than to the mismatch of ε⊥. This observation has been confirmed by calculations of . The dispersion of longitudinal and transverse components of effective permittivities of the three media ε1,2,3 for this optimized structure is presented in Fig. 3. As we can see in Fig. 3, in the optimized structure the axial permittivities of all three media are close one to another, whereas the transverse ones are quite different.
In Fig. 4(a) we depict the spatial spectrum of the heat transfer function N(q) = (π/2)(P13/Θ1) calculated at λ = 1.25 μm. N(q) has been calculated for four structures: one without nanowires (p1,2,3 = 0), one with the same amount of nanowires in all three media p1 = p2 = p3 = 0.3, one without nanowires in media 1 and 3 (only medium 2 contains nanowires with p = 0.16), and, one with optimized volume fractions of nanowires p1 = 0.22, p2 = 0.16, and p3 = 0.28. In the case p1,2,3 = 0 there is no RHT in the evanescent range i.e. N ≈ 0 at q > k0. In this case we observe a Fabry-Perot resonance the interval 0 < q < k0. The structure in which nanowires are absent in media 1 and 3 but are present in medium 2 (p1,3 = 0, p2 = 0.16) is hardly feasible. However, it is instructive to consider this case and the case when p1,2,3 = 0.3 in order to better distinguish the impacts of the photon tunneling from the impact of the impedance matching.
The result for the case p1,3 = 0, p2 = 0.16 qualitatively repeats the one obtained in  for carbon nanotubes. It illustrates the photon tunneling effect when the matching is poor. Over the whole interval k0 < q < 30k0 we observe periodic Fabry-Perot resonances. It is seen that at the lowest resonance the function N nearly reaches the global maximum. However, in between the resonances, the value of N is small due to the impedance mismatch. If p1,2,3 = 0.3 the matching is better than in the case p1,3 = 0, p2 = 0.16 since all three media are now hyperbolic, and the averaged level of N(q) increases. However, the Fabry-Perot resonances are weaker due to the increased fraction of lossy materials in the structure, and the matching is not as good as needed to obtain a huge gain due to the nanowires. For the optimized design (p1 = 0.22, p2 = 0.16, p3 = 0.28) the matching is good, and the Fabry-Perot resonances are sufficiently pronounced. In this case the averaged value of N(q) is maximal. In Fig. 4(b) we depict the spectral gain in the RHT G13(λ) due to the presence of nanowires. The value G13 is defined as the spectrum of Gint:27] the radiative heat transfer function. The value M13(ω) is the RHT function in presence of nanowires. We have compared for the present case (medium 1 is SiC, medium 3 is CIGS) with that in the case of two black bodies. Function turns out to be practically constant in the range λ < 2 μm and equals g ≈ 0.5. Therefore the gain G13(ω) presented in Fig. 4(b) in dB (10log10G13) is nearly twice (3 dB) larger than that between two black bodies. The gain G13(ω) is weakly dispersive and keeps three orders of magnitude in the range 0.7 − 1.8 μm. This gain does not take into account the contribution of medium 2 into RHT. The total spectral gain due to the presence of nanowires G123 is defined as: Eq. (11). In the low-temperature regime for h = 1 μm the value G23 is much smaller than G13. In this regime the integral gain attaining in the optimized structure does not immediately correspond to the huge absolute value of RHT. In the presented example, the emitter temperature is rather low. The emitted radiation (compared to which this gain has been calculated) in this case has its maximum in the mid-IR range, and thus is very weak at λ < 2 μm. The power output of an MTPVS operating at emitter temperatures as low as T1 = 400°K will be anyway very low, even in presence of a huge gain in the thermal emission.
3.4. Radiative heat transfer in hyperbolic materials at high temperatures
When T exceeds 400° K the optical absorption coefficient κ of gold starts to grow fast versus T. When T ≈ 800° K in accordance with [61, 62]κ at h̄ω = 1 eV attains the value κ ≈ 69.4. We have not found any available data for the complex permittivity of gold at temperatures higher than 800° K. However, following , the complex permittivity εAu can be approximately extrapolated until the melting point (for gold Tm = 1337° K) in the following way. The real part of the complex permittivity of any metal weakly changes when T varies from the room temperature to the melting point. The imaginary part changes strongly and its temperature growth in the vicinity of the melting point is nearly linear . Then we can write for λ = 1.24μm the following estimations εAu ≈ −77 − jIm(εAu), where Im[εAu(T = 300° K)] ≈ −6.5, Im[εAu(T = 800° K)] ≈ −4816 (these data correspond to [60–62]) and Im[εAu(T = 1300° K)] ≈ −9688 (this value is obtained by extrapolation). Similar extrapolations for T = 1300° K were done at other wavelengths.
Under temperatures T ≥ 770° K the absolute value of the complex permittivity |εAu| in the range 0.8−1.7 μm turns out to be higher than 3·103. Then the skin-depth of gold in this range is as small as δ < 15 nm and is much smaller than the nanowire thickness. In the array of such nanowires the electromagnetic waves of the TM-type become quasi-TEM waves . The field of these waves is concentrated in between the nanowires and weakly penetrates into the metal. As a result, in spite of huge dissipative losses of the metal the whole array has rather low optical losses and operates as a nearly hyperbolic metamaterial (with strong spatial dispersion, since the dependence ε||(q) in this regime is significant). The rather low level of optical losses of the metamaterial is granted by the very high level of optical losses in the material of its constitutive elements. Note that in the range of temperatures T = 450 − 600° K at the same wavelengths 0.8 < λ < 1.7 μm gold is also a very lossy material, however in this range |εAu| < 500 and the skin-depth is not sufficiently small compared to b. At these temperatures, golden nanowires form an effective medium with high optical losses (and still strong spatial dispersion) which cannot be used for the enhancement of RHT.
In Fig. 5 we present the results analogous to those depicted in Fig. 4, however, here T1 = 1300° K, T2 = 800° K, T3 = 300° K, and the optimal values of p1,2,3 are different. Although the optical losses in the media 1–3 are acceptable, they are quite high compared to the case of the room temperatures, and the numerous narrow-band Fabry-Perot resonances of N(q) are replaced by a single broadband resonance. Here, the impact of impedance matching becomes dominating over that of the photon tunneling, and the result for p1,2,3 = 0.4 is much worse than that obtained for the optimized structure when the matching is good. Anyway, the integral gain due to nanowires is not as huge in this case as in the previous example: .
However, this modest gain does not take into account the contribution of medium 2 into RHT. In the high-temperature regime the value P23 turns out to be larger than P13 for h = 1 μm by one order of magnitude. The ratio P23/P13 grows versus h. The contribution of medium 2 allows the giant enhancement of RHT even if the layer of medium 2 is as thick as h = 5 μm. In Fig. 6 we depict the spectral gain G123 in the radiative heat transferred to medium 3 from both media 1 and 2 in comparison with G13 for h = 5 μm. Fig. 6(a) corresponds to T1 = 400° K, T2 = 350° K, T3 = 300° K. Neglecting the contribution of medium 2 the gain G13 is nearly 30-fold. The contribution of medium 2 grants the 800-fold gain. So, for h = 5 μm the contribution of medium 2 into RHT dominates also in the low-temperature regime (unlike the case h = 1 μm). It is not surprising since medium 2 has a common boundary with medium 3, whereas medium 1 is now very distanced. At high temperatures the impact of medium 2 becomes very spectacular. Fig. 6(b) corresponds to T1 = 1300° K, T2 = 800° K, T3 = 300° K. In this case the gain granted by the nanowires calculated without the contribution of medium 2 is almost absent: G13 ≈ 1. Rather high optical losses in the effective medium suppress the photon tunneling across such a thick hyperbolic layer. However, the contribution of the five-micron thick layer of the hot medium 2 dramatically changes the situation. It results in the increase of RHT by four orders of magnitude. In the band 0.7 − 1.5 μm the value S13 + S23 for the gap with h = 5 μm filled with hyperbolic media has the value on the same order as for the vacuum gap as thin as h ≈ 300 nm. In the range 1.5 − 2.3 μm the value S13 + S23 expressed in dB is negative (except a narrow band 1.95 − 2 μm) since the heat transfer function in presence of nanowires turns out to be smaller than that in their absence. In the range 2.3 − 6.1 μm the radiative heat transfer is also enhanced by nanowires, however this enhancement is 3 order of magnitude smaller than that in the photovoltaic region. Also, the black-body radiation (to which the function G is normalized) decreases in this long-wave region. One can conclude that the claimed frequency selectivity of the enhancement of radiative heat transfer is achieved.
4. Modeling of a seven-layer structure
Our last result concerns the 7-layer structure possessing the strong frequency selectivity for RHT. The structure is illustrated by Fig. 7. In the present section nanowires are assumed to be submerged into the hot and PV media by finite depths H1 and H3, respectively. The nanowires form an interdigital structure with the interval of overlapping equal to h2. In the seven-layer effective-medium model the two half-spaces (isotropic hot medium 1 and isotropic PV medium 3) are separated by five layers of effective hyperbolic media, as is shown in Fig. 7(b). This is already a non-periodic multilayer structure which can also possess filtering properties, which may be even better that those of photonic crystals with same amount of unit layers . In  it was shown that using a stack of nanolayers of dielectrics and refractory metals one can achieve an excellent filtering quality, whereas both reflected and transmitted spectra are very broad. However, in  the filtering was referred to a normally incident plane wave. In our case the filtering problem is much more difficult. First, our filter is a stack of hyperbolic media, otherwise we cannot achieve the huge RHT. Filtering in such multilayers is loosely studied. Second, the effective permittivity of the central layer h2 is related to that of the effective medium filling the intervals h1 and h3, because the nanowire period in the layer h2 equals . Third, the host material for these three layers is defined as free space. Finally (and this is the most important difficulty), the filtering of RHT implies the integral filtering over all the effective spatial spectrum q = [0, qmax]. Our aim is to maximize the integral transmission of all spatial harmonics at every desirable wavelength between λmin and λg, and to minimize this integral transmission at every non-desirable wavelength λg < λ < λmax, where λmin is the lower bound of the wavelength range in which the PV cell operates, and λmax is the effective maximal wavelength of the thermal emission.
Such filtering, to our knowledge, has not yet been studied, and represents a complex scientific problem whose general solution would be important not just for the thermo-photovoltaic applications. In the this paper we present the results of an initial study. They are depicted in Fig. 8. We have obtained them by varying the following parameters: 1) thicknesses of all five layers H1,3 and h1,2,3 where H1,3 and the thickness of the vacuum gap h = h1 + h2 + h3 varied within the interval 0.8 − 2 μm with the step 10 nm; 2) period of the nanowire array in the vacuum gap (a) and 3) periods of nanowires in layers H1 and H3 (c and d, respectively). Periods a, b, c were varied within the interval (1.5 − 4)b with the step 5 nm. The nanowire thickness was fixed as b = 40 nm.
The parameters of gold were taken corresponding to T2 = 800° K. In this study we optimized only the gain in the direct RHT, i.e. G13, where index 1 refers to the hot medium and 3 – to the PV medium. This gain in accordance with relation (33) is determined by the heat transfer function M13. It is enhanced by the presence of the nanowires as compared to the heat transfer function M13 calculated in their absence, and does not depend on the temperature. The heat transfer function M13 is defined in the same manner as the integral of the spatial spectrum N(ω, q), with the latter one being calculated using the general formula (32) of , which is valid for an arbitrary multilayer stack. For the special case under study, when the radiative heat is produced in one half-space (here, in the hot medium) and is dissipated in another half-space (here, in the PV medium), the formula (32) of  coincides with the result of the transfer matrix approach . The value N(ω, q) = (π/2)[P13(ω, q, T)/Θ1(ω, T)] is related to the transmission coefficient τ(ω, q) of a spatial harmonic q (see ) by the relation (18).
In Fig. 8(a) we present the results of optimization of the frequency spectrum of the gain G13 obtained in the presence of nanowires, where a rather high gain G13 in the wavelength range 1.2 μm < λ < λg is combined with relatively low values of G13 for larger wavelength. This gain is calculated with respect to RHT in the structure without nanowires. The latter RHT does not coincide with the RHT between two semi-infinite black bodies separated by the same gap h = 0.92 μm due to the following reasons. First, multilayer media 1 and 3 have the emissivity smaller than unity (see above). Second, the vacuum gap in the present example is submicron, and the RHT due to the photon tunneling may not be negligible (for black bodies it is negligible by definition). Therefore, in the same plot we show the ratio of to the . Again, it turns out that the value g is almost dispersion-less and now is close to 0.6. The spectrum of the black-body radiation is shown in this Figure in arbitrary units. These three plots allow one to visually estimate the ratio between the useful (high-frequency) and harmful (low-frequency) parts of the thermal radiation received by the PV medium. This ratio is nearly equal to 4. Design parameters of the optimized structure are as follows: H1 = 0.7 μm, c = 75 nm, H3 = 0.5 μm, d = 95 nm, h1 = h3 = 220 nm, a = 90 nm, h2 = 480 nm. In Fig. 8 one can see stopbands where the gain G13(λ) is absent. In these regions the RHT in presence of nanowires is smaller than in their absence. Being expressed in dB G13(λ) is then negative and therefore is not shown.
Comparing Figs. 8(a) and 5(b) we see that in the seven-layer structure the frequency selectivity is achieved together with a lower integral enhancement of RHT over the range 0.8−1.7 μm than the one obtained for the three-layer structure. The latter structure is not frequency selective and the gain is achieved mostly because of the broadband impedance matching. In the present case the enhancement is very resonant and though attains 38 (in absolute values of magnitude) at λ = 1.35 μm, the integral gain in the useful range turns out to be as small as 3.
This result can be significantly improved if we assume that the materials that host the nanowires below and above the vacuum gap are different from host materials of effective media 1 and 3 (i.e. different from SiC and CuInSe2, respectively). This variety allows for a better frequency selectivity and a higher integral gain. We have considered a structure in which the materials hosting the nanowires below and above the vacuum gap (i.e. forming the layers H1 and H3) are equivalent and we have chosen amorphous silicon. Its complex permittivity in the range λ = 0.5 − 1.24 μm has been taken from . At 1.24 < λ < 6 μm the dispersion of a-Si is weak and one can use the approximation εSi ≈ εSi(λ = 1.24μm) ≈ 12.8 − j0.003. The array of aligned metal nanowires in the a-Si matrix is feasible as well (see e.g. in ). This structure demonstrates a very high sensitivity of G13 to the design parameters of the original structure. In Fig. 8(b) we show the results analogous to those shown in Fig. 8(a). Here the integral gain granted by the nanowires in the useful spectrum 0.8 − 1.7 μm has three orders of magnitude. The harmful part of the RHT also experiences an enhancement, however the integral gain over the harmful range 1.7−6 μm is one order of magnitude smaller than the integral gain in the useful range. The optimal parameters of the structure are as follows: H1 = 1.43 μm, c = 130 nm, H3 = 1.31 μm, d = 140 nm, h1 = h3 = 110 nm, a = 155 nm, h2 = 890 nm (i.e. h = 1.11 μm).
5. Discussion and conclusions
In this work we have developed the concept of giant radiative heat transfer in a layered structure of hyperbolic metamaterials initially suggested in our previous works  and . Here we have analyzed the possibilities granted by this concept for the breakthrough in prospective micron-gap thermo-photovoltaic systems operating in the near infrared region. We have shown that the gain of RHT in the stack of hyperbolic layers due the replacement of the vacuum gap by the hyperbolic metamaterial layer is possible to improve by the broadband impedance matching. Accordingly to our model, the impact of high temperatures does not result in critically high optical losses in the effective medium though the material of nanowires (in the present example – gold) acquires huge optical losses due to its strong heating. We have explained this seeming paradox: the metamaterial keeps properties of a hyperbolic medium for the TM-waves because they weakly penetrate into lossy nanowires. Finally, we have shown that the huge enhancement can be combined with the frequency selectivity. This way one can dramatically reduce the harmful impact of the low-frequency part of the emission. Hopefully, this work will serve to the development of a new direction – MTPVS enhanced by hyperbolic metamaterials.
We have to stress that within the framework of the effective-medium model the circuit theory is fully strict . For a structure of carbon nanotubes the effective-medium model (and, consequently, the circuit model) was validated numerically (see in [44, 45]). For metal nanowires the effective-medium model has been validated for spatial frequencies q < k, corresponding to the propagation in free space and partially validated for the region q > k through the operation of so-called wire-medium super-lenses and endoscopes for which the predictions based on the effective-medium model have been confirmed by full-wave simulations and experiments (see e.g. in ).
For the structure of carbon nanotubes from  the hyperbolic effective-medium model becomes inadequate for spatial frequencies q > π/a, where a is the array period. This is the highest limit of validity for any homogenization model. However, it does not mean that the effect of giant RHT for the structure of carbon nanotubes holds only for spatial harmonics with q < π/a. Beyond the hyperbolic effective-medium approximation the guiding performance of the array keeps for q > π/a on the nearly same level. Though the shape of isofrequency contours of the original array of nanotubes in the region q > π/a differs from a hyperbola the main property of the wire medium – to support propagation of spatial harmonics with q > k – still pertains. It keeps until the limit value qmax restricted by losses. In accordance to the theory of wire media [48–50], losses lead to a more strong distortion of hyperbolic isofrequencies than the distortion due to the spatial dispersion. Due to losses the tails of both hyperbola branches at a certain values of q (which can be higher than π/a) bend and approach to one another forming two closed contours instead of two infinite branches of a hyperbola. For the array of carbon nanotubes from  the bending of the hyperbola happens at q ≈ 50k (this limit weakly depends on the wavelength in the range λ = 7 − 9 μm studied in that work). The strict modeling in  gave the close result to that obtained by the integration of the transmittance τ over the interval q = [0, 50k] within the framework of the effective-medium model. In other words, though the homogenization within the interval π/a < q < 50k is physically not adequate, the order of magnitude for the RHT is still predicted correctly.
We assume that the same situation holds for metal nanowires. In the present study we fix the upper limit qmax of the integration of N(ω, q) in formula (33) is fixed in the same way as it was done for nanotubes in . The calculation of isofreqiencies for the medium of metal nanowires has been done using the accurate analytical model of the infrared wire medium . For low-temperature case (T2 = 350°) the bend of the isofrequency contour occurs at qmax = (25 − 30)k depending on the frequency within the range under study. For the high-temperature case (T2 = 800°) this bend occurs at qmax = (10 − 12)k. If instead of these values we would substitute qmax = π/a into Eq. (33) the claimed gain for the low-temperature cases decrease by one order of magnitude (G13 ∼ 102 − 103 instead of G13 ∼ 103 − 104). However, for the high-temperature case the claimed gain does not change since π/a > 12k in the major part of the frequency band.
The second important approximation of our model is the uniform temperature T2 of all nanowires. In fact free-standing parts of nanowires connected to medium 1 will have the temperature closer to T1 than to T2, whereas free-standing parts of nanowires connected to medium 3 will have the temperature closer to T3. Since, the arrays are overlapping it implies the temperature variation in the horizontal plane (x − y) for the interval h1 < z < h1 + h2. Such a variation is not be taken into account by our model. Moreover, a so strong step-wise temperature variation with the period 100 nm would be probably also inadequate due to the strong photon tunneling between adjacent hot and cold nanowires. Vertical gradients of temperature in the non-overlapping parts of the nanwire array are also possible. On physical grounds, we estimate that our approximation is adequate to predict at least the order of magnitude for RHT.
Briefly, we cannot claim that our results are exact for the original structures of nanowires depicted in Figs. 1(a) and 7(a). However, the purpose of this paper is only the proof of concept. Full-wave simulations for these original structures are not doable with available commercial software and represent a difficult problem (perhaps solvable for leading scientific groups). These simulations, experimental investigations, and the final optimization of the nanostructures, as well as practical design issues, are presently under consideration in the context of our potential collaboration with best scientific groups working in the field.
Let us note that the photon tunneling due to the conversion of spatial harmonics with q > k into propagating waves had been numerically revealed before the publication of our first work . Namely, in  while studying the filtering in 1D photonic crystals the authors observed the strong enhancement of simulated RHT for a structure of alternating metal and dielectric nanolayers. This structure was referred by the authors to a class of 1D photonic crystals, however, in fact, the effect was revealed in the low frequency region where the stacked structure in accordance to existing homogenization models [72–75]) behaves as an effectively homogeneous hyperbolic metamaterial. It is not only a theory since experimental realizations of hyperbolic metamaterails of alternating metal and dielectric nanolayers are known (see e.g. in [65, 66]).
In  the effect of photon tunneling in such structures has been studied in more details. Though the authors of  do not use the terminology of an effective hyperbolic medium, the photon tunneling is explained namely as the conversion of evanescent waves into propagating ones. Partial oscillations in the stacked structure form the spatial spectrum of propagating Bloch waves in a certain frequency range. The range of this spatial spectrum exceeds the limit q = k that corresponds to the photon tunneling. Notice, that in the theory of 1D lattices the formation of the spatial spectrum of propagating Bloch waves is the same as the homogenization of the stack (see e.g. in [77–79]). Moreover, authors of  have separated the contribution of the coupled surface phonon polaritons (induced at the interfaces between the metal and dielectric nanolayers) into RHT from radiative heat carried by the Bloch wave and the dominating role of the last one in the enhancement of RHT compared to the vacuum gap has been clearly shown. In their next paper  same authors analyzed the near-field emissivity of a hyperbolic metamaterial and established an analogy between it and a black body. In works [81–83] three research groups revealed nearly on the same time the effect of the huge concentration of electromagnetic energy near the surface of a hot hyperbolic metamaterial. The results of these works made a significant contribution into electromagnetics of hyperbolic metamaterials which is becoming nowadays an important scientific direction.
An important contribution of our group is an original geometry of the hyperbolic metamaterial. Although the photon tunneling in a hyperbolic metamaterial performed as a stack of metal-dielectric nanolayers is possible and also promising for TPV applications, this design approach is hardly compatible with the concept of MTPVS. Solid dielectric nanolayers would imply the mechanical contact between media 1 and 3 which is prohibited for TPV systems. If we assume that in the hyperbolic metamaterial performed as a stack of metal-dielectric nanolayers some dielectric nanolayers are replaced by vacuum (it was suggested in works [35, 64]), the TPV operation becomes in principle possible. However, such a TPV system would refer to the class of near-field TPVS whose disadvantages have already been discussed. Our design approach to a hyperbolic metamaterial seems to be more promising namely for MTPVS. Also, the contribution of the present paper is the analysis of the matching issues and the frequency selectivity of RHT in a stack of hyperbolic layers. Without this last result it would be difficult to claim TPV applications for our variants of hyperbolic media.
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