We propose a method for engineering thermally excited far field electromagnetic radiation using epsilon-near-zero metamaterials and introduce a new class of artificial media: epsilon-near-pole metamaterials. We also introduce the concept of high temperature plasmonics as conventional metamaterial building blocks have relatively poor thermal stability. Using our approach, the angular nature, spectral position, and width of the thermal emission and optical absorption can be finely tuned for a variety of applications. In particular, we show that these metamaterial emitters near 1500 K can be used as part of thermophotovoltaic devices to surpass the full concentration Shockley-Queisser limit of 41%. Our work paves the way for high temperature thermal engineering applications of metamaterials.
© 2012 Optical Society of America
High temperature energy conversion processes, such as combustion, are accompanied by thermal losses which often outstrip usable produced power . As an example, for average oil and coal based energy conversion, thermal losses account for roughly 50% to 60% of the total produced power [2, 3]. For small internal combustion engines this number may climb as high as 75% . These losses can be substantially mitigated by utilizing the waste thermal energy as a heat resource. An ideal set up would convert waste thermal energy directly into usable power.
Thermal losses are also significant for photovoltaic cells. Due to the large spectral range that must be captured, over 60% of incident solar power is lost as thermal energy . Regardless of the material bandgap’s spectral position, radiation above the bandgap loses a portion of its energy to thermalization within the cell. Radiation with energy below the bandgap is essentially unused. These effects are intrinsic and place an upper limit on energy conversion for a single junction semiconductor cell . While proposals of multi-junction cells offer a potential workaround, they also require a shift to expensive materials and structures.
An alternative solution exists in the application of the thermophotovoltaic (TPV) method . In this approach, any combination of conductive, convective or radiated waste energy is concentrated to heat a structure with a spectral emission tailored to match the bandgap of a specific photovoltaic cell. Once the structure is heated, the source energy is converted to electromagnetic radiation which can be transformed into electric power with high efficiency. The method of maintaining the heated source is flexible giving the TPV approach applicability beyond that of large scale solar or solar thermal approaches. For example, TPV methods can be used to create compact devices for cogeneration of heat and electricity.
TPV devices can easily be integrated as efficiency increasing components of larger systems, or function as primary solid state energy converters . Unfortunately, the operational temperatures needed for sufficient power generation and efficient energy conversion  as well as our limited ability to tune the thermal radiation spectrum have made previous TPV approaches largely impractical. While excellent progress has been made towards designing emitters for far field TPVs through surface structuring [10–17], planar Fabry-Perot based structures [18–20] and photonic crystals [21–23], a dominant, broadly applicable set of tools for controlling thermally induced radiation has yet to emerge.
In this paper, we introduce a class of thermal effects in metamaterials for thermophotovoltaic applications. Our approach rests on engineering the poles and zeros of the dielectric constant which allow for an array of unique optical responses.We show that epsilon-near-zero (ENZ) metamaterials can behave as spectrally tunable, narrowband ultra-thin thermal emitters and introduce a new class of artificial media, epsilon-near-pole (ENP) metamaterials. This class of ENP media are ideally suited as thermal emitters in a thermophotovoltaic system where the main requirements are 1. omnidirectional thermal emission, 2. narrowband and high emissivity 3. polarization insensitivity. We also address one of the major limitations of conventional metamaterials for thermal applications: high temperature operation; by switching to plasmonic materials with high melting points. Finally, we analyze the performance of these emitters within a practical thermophotovoltaic device and show that the energy conversion efficiency can exceed the Shockley Queisser limit of single junction cells. Note that the ENZ and ENP thermal emitters introduced in this paper can function as ideal narrowband ultra-thin absorbers as well.
2. Epsilon-near-zero (ENZ) and epsilon-near-pole (ENP) narrowband absorbers and emitters
2.1. Tailored emission and optical absorption
Making use of Kirchhoff’s law of thermal radiation for a body in thermodynamic equilibrium , the engineering of thermal emission can be formulated in terms of optical absorptivity:25].
2.2. ENZ and ENP absorption
ENZ: Lossless or near lossless epsilon-near-zero resonances have recently been a topic of broad research interest, and have been shown as a plausible mechanism for creating high performance optical devices ranging from nonlinear optical switches to tailored radiation phase patterns [26, 27]. We show here that ENZ resonances have important applications in general control of thermally induced radiation (explored in Sec.III). However, the traditional Re(ε) → 0 and Im(ε) → 0 ENZ regime is not suited to the requirements of a TPV emitter.
P-polarized radiation incident on an ENZ slab shows increased absorption (non-normal incidence). This resonance arises due to the presence of a field enhancing mechanism that relies on the displacement field boundary condition: ε1E1⊥ = ε2E2⊥, where the ⊥ denotes the direction perpendicular to the slab, and either medium can be assumed to have ENZ behavior (if ε2 → 0 then E2⊥ → ∞). Kirchoff’s laws immediately reveals that this ENZ slab with enhanced absorption should show a high emissivity. However, s-polarized light which does not have a component of the field perpendicular to the slab does not show this field enhancement or the ENZ resonance. By this constraint, no s-polarized light can be thermally excited, and p-polarized radiation cannot be efficiently emitted at low polar angles. Since the emission of an ideal blackbody shows no angular or polarization preference, the maximal averaged emitted spectral power in an ENZ region is less than half of what can be achieved theoretically.
Nevertheless, polarization averaged emissivity near that of a blackbody can still be attained if the Im(ε) → 0 condition is relaxed. In moving away from true ENZ behavior by the addition of extra loss, two separate, but connected, absorption improving effects occur. First, in the Re(ε) → 0 region, the high impedance mismatch between an ENZ material and free space is greatly reduced as the added loss acts to decrease the impedance of the material. Since this also dictates a general relaxation of ENZ resonance characteristics, polarization sensitivity is greatly diminished. Second, at wavelengths shorter than the Re(ε) → 0 crossing where material impedance is similar to that of free space (Re(ε) ≈ 1), the addition of material losses begins to allow for significant absorptivity even if the material film is thin. Both effects push this pseudo ENZ resonance towards near-omnidirectional and high absorptivity for both electromagnetic polarizations. At higher losses, they combine to create a single highly absorptive spectral region (Fig. 1(a)).
Yet, improving absorptivity in this manner comes at the cost of an increased spectral width. Due to the natural dispersion limitations of a region where Re(ε) → 0, the spectral width over which the additional loss achieves impedance matching is comparatively broad. As a direct result, high emissivity occurs over a much wider range than that ideal for high efficiency TPVs. The onset of this behavior is depicted in Fig. 1(a), where the emissivity of a 100 nm thick film moving away from the true ENZ characteristics, following a Drude model, on a perfectly reflecting backing is calculated using (1) and the Poynting vector relation . Again, as will be discussed in Sec.V, while broader emissivity may be useful for certain TPV applications, it does not match the ideal narrowband criterion. In light of these results, we introduce the concept of ENP resonances for achieving the thermal emission characteristics necessary for high efficiency TPVs. Note that both the ENZ and ENP resonances can be engineered using nanostructured metamaterials.
ENP: The primary advantage of operating at an ENP resonance is the extremely dispersive nature of these regions. This characteristic allows for tight spectral control even with moderate material losses. Yet, beyond this most important feature, several other benefits for TPV type emitters and absorbers exist. At an ideal pole of the dielectric constant we have, Re(ε) → (±)∞. The addition of losses regularizes the singularity, reduces impedance mismatch and leads to enhanced absorption (high emissivity) in a narrow spectral region (Fig. 1(b)). The ENP resonance associated with such a pole shows no polarization sensitivity and achieves omnidirectional high emissivity in isotropic media.
We now compare the critical role of losses in the ENZ and ENP resonance. In contrast to the fixed Re(ε) → 0 condition of an ENZ resonance, which forces any impedance matching to be accomplished only through the addition of loss, the behavior of Re(ε) itself in an ENP region is highly loss dependent. When loss is added, the real part of the dielectric function can no longer achieve large absolute values, opening the possibility of impedance matching even with relatively lower losses as compared to the ENZ resonance. This leads to a narrower spectral window of high emission and absorption than the previously mentioned ENZ approach. A comparative visualization of the characteristic spectral emissivity of a 100 nm thick material film in the ENP regime with moderate losses, based on a Lorentz model, on a perfectly reflecting backing is shown in Fig. 1(b). Noting that ENP type resonances match all the requirements of a high efficiency TPV emitter, we now focus on practical realizations of ENZ and ENP materials.
3. Practical ENZ and ENP with metamaterials
3.1. Natural ENZ and ENP resonances
In natural materials, ENZ regions occur at bulk plasmon as well as longitudinal optical phonon resonances, while ENP characteristics are related to transverse optical phonons. Yet, despite the ubiquity of these features in optical responses, few materials exhibit ENZ or ENP characteristics in the 0.5 eV to 1.0 eV range, crucial for TPV devices. The bulk plasmon energy, proportional to ωp ∝ (N/me)1/2, is generally pushed to much higher energies due to the small effective electron mass, me, and the high electron concentration, N ≈ 1022cm−3, of typical metals . The energy of material phonon resonances, proportional to ωLO ∝ ωTO ∝ (1/M)1/2, occurs at significantly lower energies due to the relatively large reduced ionic mass, M . The prospect of natural ENP or ENZ infrared emitters is limited to a small collection of highly lossy materials such as osmium or molybdenum which are not capable of creating the spectrally narrow emission required for high efficiencies.
To engineer ENP and ENZ responses we consider two approaches based on metamaterial crystals. The following subsections show how ENP and ENZ features beyond those found naturally can be created and tuned through the use of either planar material stacks or embedded nanowires. Schematics of the multilayer and nanowire structures are depicted in Fig. 2.
3.2. One-dimensional multilayer structure
The simplest structure for creating the ENZ/ENP metamaterials consists of alternating layers of metal and dielectric forming a multilayer structure (Fig. 2(a)). The effective medium parameters are given by
The physical characteristics of these effective parameters can be understood in terms of the restrictions placed on the material’s electrons. Since the motion of electrons is nearly free within each metal plane (Fig. 2(a)) the parallel effective medium parameter follows the frequency dispersion relation of an effective metal.In turn, this creates ENZ behaviour at the effective plasmon resonance of the parallel direction. Conversely, in the perpendicular direction nearly free electrons are confined to the thickness of their particular metal plane. On account of this confinement, the frequency dispersion of the perpendicular permittivity mirrors that of an effective Lorentz model dielectric with an accompanying effective ENP resonance. The performance of these effective medium parameters, for calculating emissivity, is compared with the exact theoretical transfer matrix technique for a model tantalum/titanium dioxide system in Fig. 3. Note that throughout the paper, we verify the predictions of effective medium theory (EMT) using numerical simulations taking into account the finite size of the unit cell, absorption in the constituent materials, dispersion as well as effects due to the substrate.
Taking the dispersion characteristics into account, the limitations of the planar structure for thermal applications become apparent. As ENP behavior occurs along the optic axis, its interaction with light is limited to the p-polarization. For realistic materials, this limited domain of interaction leads only to a small lobe of enhanced emissivity at high polar angles. In the parallel direction, the ENZ resonance displays the same characteristics as an isotropic ENZ material. Since the parallel optical response parameter interacts strongly with both polarizations, ENZ behavior dominates the overall emissivity features of the multilayer structure. Correspondingly, the same broad emissivity that earlier caused us to turn toward ENP behavior is also a general feature of the multilayer structure. A visualization of the emissivity of a realizable multilayer system using model tantalum and titanium dioxide is shown in Fig. 3. Possible application of this structure as an emitter for TPVs will be explored in Sec.V.
3.3. Two-dimensional nanowire structure
Seeking spectral emissivity control beyond what can be accomplished using the multilayer structure, we switch to the nanowire metamaterial system (Fig. 2(b)) [30, 31]. In contrast to the multilayer structure, embedded nanowires allow nearly free electron propagation along the perpendicular direction (the optic axis), but not in the parallel plane. As the confinement of electrons is flipped from the multilayer structure, so too is the functional frequency dispersion. The perpendicular direction now follows the frequency dispersion characteristics of a metal, while the parallel mimics that of an effective Lorentz model dielectric. Employing the same generalized Maxwell-Garnett approach used for the planar metamaterial, the effective medium parameters for metallic nanowires with a square lattice embedded in a dielectric matrix  are defined as:Fig. 4(b), the ENP and ENZ conditions are now properly aligned for TPV applications. The ENP resonance which results in high narrowband emissivity is located along the parallel direction where it interacts with both s- and p-polarized light. As such, the general emissivity pattern of the nanowire structure is very similar to that of an isotropic ENP material, and nearly ideal for creating a TPV emitter. This is shown in Fig. 4, where titanium nitride nanowires embedded in silicon have been used to create a spectrally narrow emissivity spike at a wavelength just below the bandgap of a gallium antimonide (GaSb) photovoltaic cell. The spectral location of GaSb’s bandgap, 1.75 μm, makes it well suited for use in a TPV device at realistic operating temperatures (inset Fig. 3(b)).
3.4. Metamaterial thermal antenna
We now show how the ENZ region of the nanowire array can lead to highly directional thermal emission. Primarily, this occurs as the optical dispersion relation, relating the energy and momentum of an electromagnetic wave in a medium, is altered from the isotropic form of (where we have used the previously mentioned direction convention, k denotes the spatial frequency of the wave and ω its frequency) to the anisotropic form of:
As the perpendicular direction nears low loss ENZ behavior, the isofrequency surface of the anisotropic crystal, Eq. (4), with Re(ε||) > 0 becomes very narrow (Fig. 5). Accordingly, very small changes in incident angle (parallel wave momentum) result in extreme variation of the material impedance. The electromagnetic wave begins to probe the perpendicular ENZ component at non-normal incidence and material impedance becomes very large leading to high reflections . This allows a small angular region (for p-polarized light) over which the wave can penetrate the medium and be absorbed (Fig. 5). Thus the metamaterial can show high emissivity only over this narrow angular region at the ENZ wavelength. Concurrently, the closer Re(ε⊥) is to 0, the narrower this region becomes, eventually forming a needle-like angular peak. Since spatial coherence is directly connected to the angular spread of an electromagnetic wave, the heating of an anisotropic crystal in the region of a perpendicular ENZ resonance presents a grating-free method of thermally exciting coherent radiation. The angular and spectral location of this antenna-like behavior is tunable based on the fraction of metal, ρ, filling the underlying unit cell. It is worth mentioning that in the extremely low loss ENZ case a second, purely longitudinal, electric field solution is possible [30, 35]. Regardless, the general behavior of the low loss ENZ results shown in Fig. 5(c) is still possible with the inclusion of this second solution. For TPV applications, omnidirectional emission achievable at the ENP wavelength is preferable and a more detailed treatment of the ENZ thermal antenna will be presented elsewhere.
4. High temperature plasmonic metamaterials
To this present time, optical metamaterial applications have been confined to temperatures near or significantly below 300 K. Yet, many of the most interesting applications of thermal engineering require temperatures in the vicinity of 1500 K. It follows directly that the scope of traditional optical metamaterials for thermal engineering applications is quite restricted. However, this limitation is strictly material based. In this section, we show that by switching away from the conventional low melting point metals, such as silver and gold, optical metamaterial design principles can be employed for high temperature thermal engineering.
Following the results of the above table, several thermally robust materials display weakly metallic optical characteristics and relatively low loss (Fig. 6(a)) through the 1 to 2 μm spectral range. These characteristics prove crucial for thermal engineering. Building on the results of inset Fig. 3(b), energy based applications of thermal engineering are most likely in the approximate temperature range of 1500 K. At significantly cooler temperatures, the spectrally broad and relatively small power density emitted by a blackbody (see Sec.V) strongly limits the energy conversion performance of any possible far field TPV device. Returning to the inset of Fig. 3(b), since the bulk of the emitted power for a blackbody in this temperature range is positioned at wavelengths slightly longer than 1 μm, this becomes the most important spectral region for emissivity control.
According to the effective medium theory presented in previous sections, both ENP and ENZ resonances require the metallic and dielectric components of the unit cell to have |Re(ε)| of the same order. As Re(ε) is at most on the order of 10 for a dielectric in the near infrared spectral region, metamaterial based ENP and ENZ behavior in this region of high spectral power density is only possible if the metallic permittivity component satisfies −10 ≤ Re(ε) ≤ −1. This condition makes the replacement of silver or gold with more thermally robust interstitial nitrides such as titanium nitride , transition metals such as tantalum, or transparent conductive oxide semiconductors such as aluminum zinc oxide , all the more important for metamaterial thermal engineering. Fine-tuning of the ENP resonance using high temperature plasmonic materials is shown in Fig. 6(b).
5. Energy conversion efficiency of TPV devices
5.1. Characteristics of ENZ and ENP emitters for TPVs
We now turn our attention to characterizing the performance of the metamaterial emitters made of high temperature plasmonic materials for TPV systems. We follow the theoretical energy conversion argument developed by Shockley and Queisser . For completeness, we provide a detailed comparison with a blackbody emitter of the various efficiencies that need to be accounted for in a practical device.
The model begins with the primary assumption that the photovoltaic cell, of the larger TPV device, operates with unity quantum efficiency perfectly cutoff at the bandgap energy. Under this assumption, photons with energy greater than the material bandgap create an electron hole carrier pair with energy equal to that of the bandgap, and any additional energy is lost to thermalization within the cell. Photons with energy below the bandgap of the photovoltaic cell are lost to free space. This perfect cutoff model defines an ultimate performance limitation on the TPV device following the usable power created by the emitter:41,42].
To understand the consequences of this relation, the efficiencies ηult resulting from the angularly averaged emissivity profiles of an ideal blackbody and the titanium nitride/silicon nanowire system, described in Fig. 4, are plotted for a gallium antimonide photovoltaic cell with temperatures varying from 500 to 2500 K in Fig. 7(a). Gallium antimonide is chosen as the photovoltaic cell as its material bandgap, 0.71 eV, is well matched to the spectral power distribution of the blackbody near 1500 K. At low temperatures, since the blackbody spectral power distribution is broad and flat, the power emitted near the photovoltaic cell bandgap is not an appreciable percentage of the total emitted power. As the temperature increases, the black-body spectral radiance (Eq. (6)) increases and shifts towards shorter wavelengths. This leads to higher percentages of the emitted radiation being usable by the photovoltaic cell and, consequently, increased ultimate efficiency. This effect peaks as the efficiency gained by shifting a larger portion of the spectral power density to lower wavelengths is countered by mounting thermal losses due to the cell only producing electron hole pairs at the bandgap energy. A visualization of the spectral power distributions of an ideal black body, the titanium nitride nanowire design, and the emitter offering peak device efficiency for a gallium antimonide photovoltaic cell are shown in Fig. 7(b) at a temperature of 1500 K.
5.2. Detailed balance efficiency
To determine the total efficiency of our emitter designs as part of a TPV device, we take into account the extra efficiency terms corresponding to 1) the additional ways in which charge carriers may be removed from the cell and 2) specific current voltage relations of a standard pn-junction.
In addition to the desired external current, both radiative and nonradiative recombinations reduce the total number of charge carriers within a photovoltaic cell. The characteristics of these recombination mechanisms can be understood through the equilibrium condition of the photo-voltaic cell with and without the presence of emitter photons. With no emitter contribution, and the cell in thermodynamic equilibrium, the net number of carriers within the cell is constant, and can be described by:Eq. (10), we can see that a final efficiency reduction arises due to the interrelation of current and voltage in the pn-junction cell. As current is allowed to flow the number of free carriers within the cell drops, reducing the voltage from its open circuit value. As current must be drawn to create usable power, an efficiency factor relating the operational voltage, VP, to the open circuit voltage, VOC, must be included by finding the maximal power-point (VPI(VP)) of the photovoltaic cell. Using (10) we find:
5.3. Energy conversion results
From the ultimate efficiency (Eq. (5) and Fig. 7) we can begin to see that the principal characteristic of a good emitter for TPVs is the suppression of subbandgap photons. Near unity ultimate efficiency, ηult, requires the spectral emissivity of the emitter, ζE (λ, θ), to become needle-like at a wavelength slightly shorter than that of the material bandgap. Yet, this ultra-sharp spectral behavior is pragmatically undesirable. As the spectral range of emission is narrowed, the power produced at a fixed temperature is decreased. This reduction not only limits the amount of power that can be extracted from the TPV device, but also acts to reduce the total energy conversion efficiency. Reviewing Eq. (13), if the number of carriers in the pn-junction is near the unilluminated thermal equilibrium amount, a large percentage of the incident emitter energy will be either reradiated or lost within the cell. Consequently, the efficiency of converting the power radiating from the emitter will also be reduced. Since the temperature of the emitter, and thus the amount of usable power it can produce, is intrinsically limited by the materials used, whether a spectrally wider or narrower emission is preferable depends on the particular application. At low temperatures, broader emitters are more efficient and provide greater power densities; however, as temperature is increased their efficiency falls below that of the narrow emitter.
Clearly, this tradeoff has direct consequence for our TPV emitter designs. For applications in which it is not possible to reach emitter temperatures of ≈ 1000 K, broad ENZ type emitters will create better TPV devices. Above this approximate temperature, spectrally thinner ENP type absorption peaks will provide superior net efficiency. Here, even lower emitted power begins to become a benefit of the spectrally thin design. As the emitter is in thermodynamic equilibrium, all emitted energy must be replaced. This becomes increasingly difficult with a spectrally broad emitter at high temperatures. As an alternative view, given a fixed input power, a spectrally thinner emitter will operate at a higher temperature and thus higher efficiency (Fig. 8(b)).
Taking all three efficiency considerations into account, the theoretical metrics for TPV devices using metamaterial emitters, neglecting any loss associated with an absorber stage, are shown in Fig. 8. The full concentration limit for solar based single junction photovoltaics, following the assumptions of Shockley and Queisser, is surpassed for emitter temperatures near 1500 K in the AZO nanowire system. The overall performance of these designs show either greater high temperature efficiency [11, 23], tighter spectral emissivity, or similar efficiency behavior with a higher emitter power  to other contemporary designs.
6. Concluding remarks
In this article we have introduced a class of artificial media called epsilon-near-pole metamaterials and developed a general means of exerting control over thermally excited far field electromagnetic radiation through the use of ENP and ENZ resonances. Thermal radiation control using these generalized bulk material resonances shows great potential for creating a new class of realizable thermal devices. In particular, we have shown specific metamaterial implementations for creating emitters for TPVs and coherent thermal sources in the near infrared range crucial for energy applications. We have also introduced the concept of metamaterials based on high temperature plasmonic materials. This switch away from typical plasmonic materials is crucial for many far field thermal applications and more generally for furthering the scope of optical metamaterial designs. This work paves the way for future use of metamaterials for the control of thermally excited radiation.
We would like to acknowledge funding from the Canadian National Science and Engineering Research Council (NSERC), Canadian School of Energy and Environment (CSEE) and Alberta Nanobridge.
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