1. Introduction

Thermophotovoltaic (TPV) cells are a promising way to efficiently convert thermal energy into electricity, providing an environmentally-friendly energy source from waste heat or other thermal sources. As just a few examples, TPV cells have been used in solar applications using optical concentration and selective absorbers [1, 2], portable microgenerators through the incorporation of photonic crystals [3], and hybrid waste-heat converters, combining a TPV cell with solid oxide fuel cells [4]. The design of TPV cells for a given application is a multi-faceted optimization problem; prior studies include study of the details associated with near-field thermal radiation [5] and global optimization of the system [6], as well as experimental development and characterization of TPV devices [7–9]. Detailed reviews of the design and optimization of the key components of TPVs can be found in [10, 11] and references therein. TPV cell design relies on two main components: an emitter, which emits thermal radiation at high temperatures, and a photovoltaic (PV) cell that converts this radiation into electricity. Selective emitters are designed to reshape the thermal radiation spectrum, shifting the maximum emittance to a wavelength that matches the absorption characteristics of the PV cell. Since PV cell technology based on semiconductor junctions is mature and well understood, the primary opportunity for further improvement of TPV cell conversion efficiency is the development and optimization of efficient selective emitters.

For conventional TPV cells based on blackbody emission, the emitters must operate at high temperatures. For example, from Wien’s displacement law one can see that a blackbody emitter would need to be at 1680 K in order for the peak emittance to match the band gap of a GaSb PV cell (E_g = 0.72 eV). For this reason, high melting point materials are required. Consequently, structures built of refractory metals, such as tungsten [12], tantalum [13], and molybdenum [14] have been proposed and analyzed previously. On the other hand, the use of periodic structures such as metamaterials [15, 16] and photonic crystals [17, 18] has been shown to modify the emittance spectrum of selective emitters. Due to their simplicity and ease of fabrication, gratings are a promising practical option. Chen and Zhang [19] studied complex tungsten gratings, and showed that the excitation of surface plasmon polaritons (SPP) can greatly improve the thermal emittance. Nguyen-Huu et al. [20] studied tungsten pyramidal nanogratings that combine structures with low and high filling ratios. This combination was shown to improve the emittance as well as to reduce the sensitivity to the angle of incidence for transverse magnetic (TM) polarization. Song et al. [21] proposed trilayer metal-insulator-metal (MIM) gratings made of W-SiO₂-W, and Shuai et al. [22] studied more general multilayer metal-dielectric gratings. In both [21] and [22], perfect emittance at the design wavelength was achieved for TM polarization, through the excitation of SPPs and magnetic polaritons (MP). To achieve improved polarization insensitivity, Nguyen-Huu et al. [23] evaluated a deep grating that takes advantage of Wood’s anomalies (WA) [24], cavity resonances (CR), and SPPs. Recently, multilayer W-HfO₂ metamaterial structures have been shown to enable thermal emission control through topological transitions [25]. In addition, Yeng et al. [26] filled a 2D tantalum photonic crystal with SiO₂ or HfO₂, and obtained omnidirectional and polarization insensitive thermal emitters by exploiting CRs. The HfO₂ dielectric in these structures appears promising for TPV applications due to its high melting point (3031 K).

As noted above, both SPPs and MPs have been used to manage and improve the thermal emittance. However, these improvements are only obtained for TM–polarized incident radiation. For general TPV applications, the incident radiation is randomly polarized, and thus both the transverse electric (TE) and TM response are important. However, the strategies used for TM emittance control are not effective for TE; instead, for TE polarization, CRs are required to enhance the thermal emittance. Unfortunately, for gratings able to excite both SPPs and CRs [23], the wavelength associated with the emittance peak is typically close to the visible range (0.4–0.8 µm), which translates to very high operating temperatures. Therefore, in order to improve thermal emitters and make them suitable for infrared TPV applications, it is necessary to develop strategies to enable efficient TPV operation at lower emitter temperatures.

In this work, we explore simple structures combining the use of a thin HfO₂ spacer and deep W gratings to achieve high emittance at technologically relevant temperatures. We show that a grating with an embedded HfO₂ layer supports both SPPs and MPs, while the deep grating grooves allow the excitation of CRs. By using rigorous coupled wave analysis (RCWA), two kinds of gratings have been studied. First, a shallow grating designed to excite MPs is evaluated. A peak emittance of 99.9% for λ = 1.73 µm is obtained for the case of TM polarization, but the performance with TE radiation is appreciable lower. To improve on this performance, a deeper grating that supports CRs is evaluated for TE waves. It is found that by the simultaneous excitation of SPPs, MPs, and CRs it is possible to reach normal thermal emittances of 97.7% and 99.7% for TE and TM waves, respectively, at wavelengths close to λ = 1.65 µm. These gratings are promising for use with low bandgap PV cells, such as GaSb (0.72 eV) and In_0.53Ga_0.47As (0.75 eV).

2. Emitter design

Hafnia is a high permittivity oxide that has favorable properties for use in selective emitters. HfO₂ has a very high refractive index (n ∼ 4) for short wavelengths, and remains almost constant, close to n = 2, for wavelengths λ > 0.4 µm. Moreover, the absorption coefficient—which is related to the imaginary part of the index of refraction—is negligible in the optical and near-IR where TPV systems can be expected to operate [27]. In addition, hafnia has a large energy bandgap (E_g ∼ 6 eV) and, critically for thermal emitters, possesses a very high melting point, 3031 K, which translates to good thermal stability [28]. Moreover, its thermal expansion coefficient is compatible with W [29]—a key factor when operating at high temperatures. Recently, hafnia layers have been deposited onto photonic crystals to prevent degradation when operated at high temperatures [30–32], providing experimental validation of its promise for high temperature applications.

The gratings evaluated here consist of a tungsten substrate, a hafnia spacer layer, and a tungsten grating similar to the structure proposed by Wang and Zhang [33]. In addition, to help clarify the physics of operation, a W grating without a HfO₂ spacer layer has also been simulated for comparison. To simulate the selective emitter explored here, Helmholtz’s equation is solved by using the RCWA method [34, 35] in order to find the reflectance (ρ) and transmittance (τ). The absorptance (α) is calculated indirectly by energy conservation using α = 1 − ρ − τ. Finally, the emittance is obtained through Kirchhoff’s law, which states that in thermal equilibrium the absorptance is equal to the emittance (α = ε) [34]. To obtain more realistic estimates of thermal emittance for selective emitters operating at high temperatures, we considered a temperature dependent Drude model for tungsten [36, 37]. The properties of HfO₂ are taken from [27]. In addition, both the electric and magnetic fields inside the structures are obtained by using the finite element method [38]. Two gratings with the same geometry except for their height have been evaluated. First, a shallow grating with a period Λ = 1.1 µm, thickness h = 0.1 µm, and a filling factor f = w/Λ = 0.2 (with a distance between ridges a = 0.88 µm) above a hafnia layer of thickness d = 0.15 µm was considered (inset Fig. 1(a)). As shown in Fig. 1(a), simulations of this structure result in nearly perfect normal emittance, 99.9%, for TM incident radiation. The incorporation of a hafnia spacer significantly improves the emittance, generating a peak at λ = 1.73 µm, due to MP excitation as explained by the concentration of the magnetic field (Fig. 1(b)). However, the emittance in response to TE radiation is modest, with a peak emittance that is both blueshifted to λ = 1.255 µm, as well as being appreciable lower, close to 88.13%. However, as can be seen in the comparison in Fig. 1(a), the TE and TM responses of the W/HfO₂/W structure are much better than that of a plane W grating.

 

Fig. 1 Simulated normal emittance for the studied shallow grating. (a) Emittance for both TE and TM polarizations for W gratings with and without hafnia spacer layers. The inset shows the geometry of the structure. (b) Magnetic field strength for λ = 1.73 µm.

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To develop a polarization–insensitive selective emitter, a deep grating with Λ = 1.1 µm, h = 1.0 µm, f = 0.2, a = 0.88 µm, and d = 0.15 µm, is considered (inset Fig. 2(a)). This deep grating produces an emittance peak of 97.7% for TE radiation at λ = 1.614 µm and 99.7% for TM waves at λ = 1.674 µm, thereby demonstrating that the polarization sensitivity can be greatly reduced by using the deep grating design in combination with the HfO₂ spacer.

 

Fig. 2 Simulated normal emittance for the studied deep grating. (a) Emittance for both TE and TM polarizations for W gratings with and without hafnia spacer layers. The inset shows the geometry of the structure. (b) Electric field magnitude for λ = 1.614 µm and the magnetic field magnitude for λ = 1.674 µm.

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To understand the physical origins of these results, we note that it is well-known that metallic gratings can excite SPPs for TM polarization. This excitation is given by the collective oscillation of electrons and photons, due to the perpendicular component of the electric field. On the other hand, for TE incident radiation, the tangential component of the electric field must vanish on the surface [39], and therefore no surface polarization can be produced. For this reason, design of a polarization-insensitive selective emitter requires the use of different mechanisms to control the thermal radiation in each polarization. For TE radiation, CRs in deep gratings are an attractive option. These resonances can be predicted by ${\lambda }_{mn}=2/\sqrt{{(m/h)}^2+{(n/a)}^2}$, where by choosing m = 0 and n = 1 we obtain a maximum wavelength λ₀₁ = 2a [41]. The SPP excitation can be predicted by using the dispersion relation for a metal-dielectric interface, together with momentum conservation k_x = k_spp(ω) + jK, where k_x is the horizontal component of the wavevector, k_x = (2π/λ) sin θ_i, $k_{spp}=(2\pi /\lambda )\sqrt{{\epsilon }_m{\epsilon }_d/({\epsilon }_m+{\epsilon }_d)}$ is the wavenumber of the surface wave, K = 2π/Λ is the grating momentum, and j is an integer [23]. By including a dielectric spacer below the grating, MPs are generated as a result of the coupling between internal and external SPPs [40].

As noted above, the incorporation of a dielectric layer enables the excitation of MPs. Moreover, by using an LC circuit model [33] to predict the excitation wavelength of the MPs (λ_MP), it can be shown that when the dielectric constant of this layer is high, as is the case of HfO₂, MPs can be excited at longer wavelengths which allows for lower working temperatures. Figure 3 illustrates these trends for deep and shallow gratings as a function of the spacer permittivity.

 

Fig. 3 Excitation wavelength for MPs (λ_MP) as a function of spacer dielectric constant for both shallow (blue) and deep (red) gratings, for several spacer thicknesses.

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Regarding the polarization-insensitive deep gratings, emittance enhancement results from the simultaneous excitation of CRs, SPPs, and MPs. By combining these three effects, emittance is improved for wavelengths larger than 1.6 µm for both TE and TM polarizations (Fig. 2(a)). The peak for TE polarization at λ = 1.614 µm (highlighted with the blue dot in Fig. 2(a)) is due to CRs as predicted by the slab waveguide cut-off for the TE₁₀ mode, λ₁₀ [41]; as shown in Fig. 2 (b), the waveguide mode concentration of the electric field at resonance produces an increase in the absorptance. On the other hand, to understand the phenomena that enhance the emittance in the near IR frequency range, we find that the emittance peak at λ = 1.14 µm for TM polarization is generated by the excitation of SPPs, according to momentum conservation for radiation in free space above the grating. The emittance peak at λ = 1.674 µm is due to the excitation of MPs as can be inferred from the long-wavelength TM peak (highlighted by the orange dot in Fig. 2(a)). As shown in Fig. 2(b), the magnetic field at that wavelength is concentrated in the hafnia layer.

The angular dependence of the directional emittance is presented for both polarizations in Fig. 4. For TE incident radiation, the emittance is close to one up to an angle of θ = 30°, but decreases abruptly at larger angles. For the TM case, the emittance peak redshifts from 1.674 µm to 2.47 µm and it is reduced from 99.7% to 58.9% as the angle increases to θ = 60°. The dependence on the angle of incidence is detrimental to the efficiency of a TPV cell, due to the fact that thermal emission is an hemispherical property. Ideally, the structure should present high emittance over as large a range of angles as possible, as well as to maximize the peaks for both TE and TM polarizations.

 

Fig. 4 Directional emittance of the deep grating as a function of the angle of incidence. (a) and (b) show the emittance for different angles of incidence (θ = 0°, 30°, 45°, and 60°) for TE and TM polarizations, respectively.

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To evaluate the efficiency performance of the selective emitters, we consider the normal in-band emittance (ϵ_in) [32] given by:

 

Fig. 5 Thermal radiation for (a) shallow and (b) deep gratings at T = 1680 K and T = 1750 K, respectively. The blackbody thermal radiation is shown for comparison. An in-band emittance of 88.25% and 85.17% is achieved for the deep gratings for TE and TM polarization, respectively.

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The in–band polarization–averaged hemispherical emittance was also calculated. For the shallow grating a modest in–band hemispherical emittance of ϵ_in,_h = 57.63% was obtained; the value is low because the performance of the TE radiation is poor. In the case of the deep grating an in–band hemispherical emittance of ϵ_in,_h = 67.66% was achieved. This value is lower than the normal in–band emittance due to the fact that the CRs start disappearing for θ > 45° and, in the case of TM radiation, the peak is red shifted for wavelengths larger than the bandgap of the PV cell.

3. Conclusion

Gratings based on W with and without HfO₂ spacer layers have been explored through numerical simulations. We see that use of an HfO₂ layer can improve the performance of tungsten grating-based selective thermal emitters. The use of shallow gratings results in a maximum emittance of 99.9% at λ = 1.73 µm, but with appreciable lower TE performance. To improve the polarization insensitivity, a deep grating was investigated. The deep grating exhibits emittance peaks of 97.7% and 99.7% for TE and TM radiation, respectively. The mechanisms responsible for the improvement in the emittance have been explored; it was found that for TM radiation SPPs and MPs play a crucial role, while for the case of TE polarization, CRs are responsible for the high emittance. The peak emittance wavelength for both polarizations, close to λ = 1.65 µm, requires a working temperature of approximately 1750 K. Both tungsten and hafnia can operate up to this temperature due to their high melting points and material compatibility. The performance of the emitters was evaluated by the normal in-band emittance, obtaining ϵ_in,_TM = 85.17% and ϵ_in,_TE = 82.25% for the deep grating case. In addition, we estimated the hemispherical in–band emittance. The shallow and deep gratings exhibit hemispherical in–band emittances of 57.63% and 67.66%, respectively. The results show that deep gratings with a high dielectric constant spacer are promising alternatives for TPV applications, whose performance can be improved by further optimization.