## Abstract

We theoretically demonstrate a novel, efficient and cost effective thermal emitter using a Mie-resonance metamaterial for thermophotovoltaic (TPV) applications. We propose for the first time the design of a thermal emitter which is based on nanoparticle-embedded thin film. The emitter consists of a thin film of SiO_{2} on the top of tungsten layer deposited on a substrate. The thin film is embedded with tungsten nanoparticles which alter the refractive index of the film. This gives rise to desired emissive properties in the wavelength range of 0.4 *μ*m to 2 *μ*m suitable for GaSb and InGaAs based photovoltaics. Effective dielectric properties are calculated using Maxwell-Garnett-Mie theory. Our calculations indicate this would significantly improve the efficiency of TPV cells. We introduce a new parameter to gauge the efficacy of thermal emitters and use it to compare different designs.

© 2016 Optical Society of America

## 1. Introduction

Thermophotovoltaics (TPV) is a promising technology for heat recovery and an attractive alternative for existing electricity generation technologies [1–3]. A TPV system consists of a thermal emitter which operates at high temperatures (∼1500 K) [4, 5] and a photovoltaic cell and can directly convert thermal energy into electricity [6]. In principle, TPV systems can achieve an efficiency of Carnot engine for monochromatic radiation [7] and such TPV systems with efficiencies approaching the limit have been discussed in the literature [8–10]. In reality, efficiencies of TPV systems suffer from the mismatch between emission spectrum of emitter and absorption spectrum of PV cell [11]. External quantum efficiency (EQE) of a PV cell is defined as the fraction of incident photons converted into electron-hole pairs [12] and is indicative of the amount of current generated for a given incident photon wavelength. A PV cell has non-zero EQE above the band-gap, i.e. a PV cell can generate electric current only when incident photons have higher energies than the bandgap of semiconductor material [13]. Moreover, absorption of incident photons having wavelength longer than bandgap wavelength is undesirable as it causes thermal leakage and reduces the efficiency [14]. So, in order to achieve high efficiency of a TPV system, goal is to develop a thermal emitter with high emissivity in the region of high external quantum efficiency (EQE) of PV cell and low emissivity in rest of the spectrum. Several studies have focused on using photonics crystals [15, 16], plasmonic metamaterials [17], single micro/nano sized spheres [18], doped materials [19, 20], surface gratings in order to obtain spectrally selective emission [21]. 1-D, 2-D and 3-D photonic crystals and surface gratings have also been developed [22–24]. Numerous works involve use of intermediate filters which reflect the low energy photons back to emitter [25–28].

Metamaterials is another class of nanomaterials/nanostructures which has been the topic of many articles focused on selective thermal emitters. Liu et al [29] have demonstrated a narrow-band mid infrared thermal emitter based on a metamaterial. Plasmonic metamaterials have also been explored for solar TPV applications as in [30, 31]. Woolf et al [32] have presented and also experimentally demonstrated a heterogeneous metasurface for selective emitters and predicted 22% conversion efficiency for InGaAs based TPV system at the operating temperature of 1300 K for the emitter. Use of refractory materials such as tungsten is common in the literature. These studies and several others [33–38], although not specifically focused on TPV applications, involve complex surface patterns which are difficult to fabricate. In this paper, we propose for the first time the use of Mie-metamaterial thin films, i.e., nanoparticle-embedded thin films to design a novel, efficient and low cost thermal emitter suitable for GaSb and InGaAs based TPVs. Mie-metamaterials or Mie-resonance metamaterials are artificial materials which utilize Mie resonances of inclusions for the shaping of emission spectra. Several theoretical studies of nanoparticle inclusions into host materials have been performed [39–41]. However, literature pertaining to optical and emissive properties of nanoparticle embedded thin films is rare. Authors recently published a broad study on the role of polar and metallic nanoparticles on far-field and near-field thermal radiation from thin films by considering different cases of hypothetical Mie-metamaterial thin films [42]. In present study we exploit the idea by using thin films of SiO_{2} embedded with tungsten (W) nanoparticles to design a novel thermal emitter specifically for GaSb and InGaAs based TPVs.

Schematic of a typical TPV system is shown in Fig. 1(a). Thermal radiation from the heat source is absorbed by the blackbody absorber. This heat is emitted by the selective thermal emitter only in the desired band of wavelengths in which the external quantum efficiency of the PV cell is high. Figure 1(b) depicts an example of 1-D grating structure that could be used to realize wavelength selective radiative properties of thermal emitter. It comprises gratings of SiC and W of period Λ = 50 nm and filling ratio *ϕ* = 0.55. Grating layer is 50 nm thick. It’s emissive properties along with the emittance of 2-D grating structure considered in Zhao et al [13] will be compared to the Mie-metamaterial based thermal emitter of present study. The prime focus of this paper is the proposed design of thermal emitter shown in Fig. 1(c). It consists of a 0.4 *μ*m thick film of SiO_{2} on the top of 1 *μ*m thick film of tungsten deposited on a substrate. Thickness of the layer of W makes sure that all the radiation from the substrate is reflected back and the radiative properties of the emitter are dictated by the top two layers alone. The SiO_{2} film is doped with W nanoparticles of 20 nm radius. Inclusion of W nanoparticles changes the optical properties of the film and results in desired emission spectrum. Volume fraction of W nanoparticles was adjusted to 30% to achieve the best result. While the mixture of SiO_{2} and W gives rise to spectrally selective emission for the PV, the choice of materials also makes sure that it will withstand high temperatures.

## 2. Theoretical fundamentals

The hemispherical emissivity of the thermal emitter can be expressed as [42]

*ω*is the angular frequency and

*k*is the magnitude of inplane wave vector. ${\tilde{R}}_{h}^{(\mu )}$ and ${\tilde{T}}_{h}^{(\mu )}$ are the polarization dependent effective reflection and transmission coefficients which can be calculated using the recursive relations of Fresnel coefficients of each interface [43]. The dielectric functions can be related to real (

_{ρ}*n*) and imaginary (

*κ*) parts of refractive index as $\sqrt{\epsilon}=n+j\kappa $. Dielectric functions of the materials (SiO

_{2}, W and SiC) considered in this paper are taken from literature [44–46]. Having very low temperature coeffcients, room temperature values of dielectric function are used for SiO

_{2}and SiC [47, 48]. Dielectric properties of tungsten were also assumed to be unchaged as the operating temeprature is much less than the melting point. The top layer of the thermal emitter considered in our calculations is either 1-D grating structure or nanoparticle embedded thin-film. In both cases, it can be approximated as a homogeneous layer of an effective dielectric function.

If the period of grating (Λ) is much smaller than the wavelength of interest such as the case considered here, dielectric function of the grating layer can be approximated using effective medium theory [49]. We use second order effective medium approximation to calculate the effective dielectric function which is given by [49]

*ε*and

_{A}*ε*are the dielectric functions of the two materials in surface gratings, Λ is the period and

_{B}*ϕ*is the filling ratio. The expressions for zeroth order effective dielectric functions

*ε*

_{TE,0}and

*ε*

_{TM,0}are given by

For calculating the effective dielectric function the Mie-metamaterial, we use Clausius-Mossotti equation. [50, 51].

*ε*is the dielectric function of the matrix,

_{m}*α*is the electric dipole polarizability,

_{r}*r*and

*f*are the radius and volume fraction of nanoparticles respectively. To consider the size effects of nanoparticle inclusions, we use the Maxwell Garnett formula which employs the expression for electric dipole polarizability using Mie theory [52], ${\alpha}_{r}=3j{c}^{3}{a}_{1,r}/2{\omega}^{3}{\epsilon}_{m}^{3/2}$, where

*a*

_{1,r}is the first electric Mie coefficient given by

*ψ*

_{1}and

*ξ*

_{1}are Riccati-Bessel functions of the first order given by

*ψ*

_{1}(

*x*) =

*xj*

_{1}(

*x*) and ${\xi}_{1}(x)=x{h}_{1}^{(1)}(x)$ where

*j*

_{1}and ${h}_{1}^{(1)}$ are first order spherical Bessel functions and spherical Hankel functions of the first kind, respectively. Here, ‘′’ indicates the first derivative. ${x}_{m}=\omega r\sqrt{{\epsilon}_{m}}/c$ and ${x}_{\mathit{np}}=\omega r\sqrt{{\epsilon}_{\mathit{np}}}/c$ are the size parameters of the matrix and the nanoparticles, respectively;

*ε*being the dielectric function of nanoparticles.

_{np}It is worth mentioning that Maxwell-Garnett-Mie theory is applicable when average distance between inclusions is much smaller than the wavelength of interest [53]. This criteria is satisfied in the calculated presented. Also noteworthy is the fact that nanoparticle diameter (40 nm) is much smaller than the thickness of the thin film (0.4 *μ*m) considered. Thus, effective medium theory holds true for the calculations presented in this study.

## 3. Results

Figure 2(a) illustrates the effect of W nanoparticle inclusions on the refractive indices of SiO_{2} host. Pure SiO_{2} has a near constant refractive index (n) ∼ 1.55 and a negligible extinction coefficient (*κ*). Nano-sized W spheres (2*r* = 40 nm) much smaller than the operating wavelength are introduced into the SiO_{2} matrix. This brings about an increased refractive index and extinction coefficient. Although real and imaginary parts of the refractive index have different implications, it is imperative that real and imaginary parts of the mixture depend on both the indices of its constituents. It is essential to point out that change in refractive indices of the mixture is a result of addition of new material as well as the Mie-scattering of electromagnetic waves by spherical nanoparticles. Consequently, refractive indices of the mixture depend on volume fraction and particle size. The resulting changes in the emission spectrum are displayed in Fig. 3(a). Owing to its metallic nature, W is moderately absorptive only at lower wavelengths and highly reflective beyond 2 *μ*m. Although SiO_{2} bulk has a high absorptivity/emissivity, 0.4 *μ*m thick layer of SiO_{2} is practically transparent for the entire spectrum. Therefore, both the bare tungsten and the combination of SiO_{2} layer on tungsten base have low emissivity in the range of 0.4 to 2 *μ*m and negligible emissivity for the majority of spectrum beyond 2 *μ*m. Upon inclusion of W nanoparticles, emissivity is enhanced for spectral region of interest. The geometric parameters available for spectral shaping of the emissivity are thickness of SiO_{2} film, volume fraction of W nanoparticles and particle radius. If size distribution of nanoparticles is available, it can be incorporated into the extended Maxwell-Garnett formula as in [55]. For the present study, the particle radius is fixed to 20 nm. Figure 3(a) also highlights the effect of volume fraction of W nanoparticles on emission spectra. An increase in volume fraction from 10% to 30% results in higher emissivity and the broadening of the emission spectrum. The final proposed configuration has the SiO_{2} film thickness of 0.4 *μ*m and 30% volume fraction of W nanoparticles. Note that calculated emissivity of W (black curve in Fig. 3(a)) matches well with high-temperature (1600 K) emissivivty reported in [56]. Also refractive index of SiO_{2} has a low temeprature coefficient [57]. Therefore these materials are well suited for high temeprature applications.

The thermal emitter presented here is suitable for a typical TPV system consisting GaSb or InGaAs based photovoltaics. The EQE of the two PV cells is shown in Fig. 3(b). Both the GaSb and InGaAs based PV cells efficiently generate electricity if the wavelength of incident photons falls in the range of 0.4 *μ*m to 2 *μ*m. Configuration of the proposed design i.e. thickness of the SiO_{2} layer, radius and volume fraction of W nanoparticles was chosen to achieve best match between the emission spectrum and the EQE curve of the PV cells. Our design exhibits high emissivity (> 0.8) in the wavelength range 0.4 *μ*m to 1.7 *μ*m and very low emissivity (< 0.2) beyond 2 *μ*m. The close match between the emission spectrum of the emitter and EQE curve PV cell ensures high conversion efficiency and minimizes the thermal leakage of long wavelength photons. For comparison, the emission spectrum of 1-D structure (Fig. 1(b)) and that of a 2-D grating structure discussed in Zhao et al [13] is also shown. It is apparent that the emission spectrum of the design presented here is comparable to that of 1-D structure and the 2-D structure presented in [13] for wavelengths below 1.7 *μ*m. Although a substantial part of blackbody radiation at high temepraturs (1000 to 1500 K) exists between 2 to 6 *μ*m, very little energy is emitted by proposed emitter in this region as emissivty is very low. Moreover, the proposed design has lower emissivity beyond 1.7 *μ*m when compared to the 2-D design. This indicates much lower losses due to thermal leakage. It is worth mentioning that, while choice of materials in the present study is same as that used in [13], fabrication of thermal emitter based on tungsten embedded SiO_{2} thin film is relatively simple. This guarantees low cost of large scale fabrication of such emitters. As refractory materials such as platinum and aluminum oxide (Al_{2}O_{3}) have been tested before for thermal emitter [32], they can also be explored for this new class of emitters.

## 4. Discussion

While the match between emission spectrum of the emitter and the EQE curve of the PV cell can be a good indicator of TPV system’s performance, the emitter temperature has an equally important role as it decides the dominant wavelength and can dictate the overall efficiency. Therefore, discussion of conversion efficiency cannot be left out in this work. Ideal emitter has an *e*(*ω*) = 1 above bandgap and *e*(*ω*) = 0 below the bandgap. The conversion efficiency of the TPV system could be defined as the ratio of the power output of the PV cell to the power incident on the PV cell given by

*P*is the power radiated by the emitter [58]

_{rad}*P*is proportional to the number density of the photons above bandgap and the EQE of PV cell defined as [32]

_{out}*q*,

*V*,

_{OC}*FF*and

*η*are electronic charge, open circuit voltage, fill factor and EQE of PV cell, respectively.

_{EQE}*n̄*(

*ω*,

*T*) is the number density of incident photons

*e*(

*ω*) = 1. Such a system would have lower power output than the one with ideal thermal emitter despite having maximum efficiency with no losses. It is apparent that conversion efficiency of the TPV system alone cannot be regarded as a criterion to gauge the efficacy of a thermal emitter. Therefore we propose a new parameter to calculate the effectiveness of the thermal emitter for a given PV cell and emitter temperature. The parameter termed as effectiveness index is defined as the ratio of the efficiency of the TPV system to the efficiency of the TPV system with ideal thermal emitter at the same temperature and for the given PV cell,

*β*=

_{emitter}*η*, where subscripts

_{Real}/η_{Ideal}*Real*and

*Ideal*refer to TPV systems with thermal emitter of consideration and an idealized thermal emitter at the same temperature. So,

*β*doesn’t elaborate anything about the efficiency of the TPV system as a whole, it is a good measure to compare the effectiveness of different thermal emitters for a given PV cell and emitter temperature. Note that if

_{emitter}*β*> 1, it implies that the TPV system is efficient but only a part of the spectrum of the EQE is being utilized and the output power is less than what an ideal emitter would produce. Therefore, the goal is to get

_{emitter}*β*close to 1 but not exceed 1. It is assumed that the emitter has a much higher temperature than the PV cell, which has a negligible reflection coefficient. While calculating effectiveness indices for different thermal emitters and a given PV cell, it is assumed that PV cell temeprature and its EQE do not vary.

_{emitter}We calculate the effectiveness of the thermal emitter for GaSb PV cell and at emitter temperature 1500 K for the designs presented in Fig. 3(b). Our proposed design has an effectiveness index *β _{emitter}* = 0.72 while its value for the 1-D design and the one considered in [13] is 0.78 and 0.65 respectively. It is important to note that despite having lower emissivity, the 1-D design has higher effectiveness index because it has lower emissivity beyond 2

*μ*m, thermal wavelength at 1500 K being 1.93

*μ*m. Structure discussed in [13] shows lower effectiveness index as it has higher emissivity below the bandgap. Effectiveness index of the aforementioned designs at 1300 K is

*β*= 0.6, 0.68 and 0.58 respectively. As the thermal wavelength shifts to 2.3

_{emitter}*μ*m, losses increase which clearly demonstrates the role of temperature and the dominant thermal wavelength governing the efficiency.

Thus, we have presented for the first time a thermal emitter which consists of Miemetamaterial based thin film. Dielectric properties of SiO_{2} thin films are influenced by the inclusions of W nanoparticles. This manifests into the spectral shaping of thermal radiation from the surface of the emitter. The best possible emission spectrum is realized by adjusting the thickness of the film, radius and volume fraction of nanoparticles. The emission spectrum of the design matches well with the EQE of GaSb and InGaAs based PV cells that indicates high conversion efficiency. Emission spectrum of the present design is on par with previously proposed design based on 2-D grating structures. Very low emission for *λ* > 2 *μ*m minimizes the leakage of long wavelength photons below the TPV band-gap. Materials chosen for the design (SiO_{2} and W) are suitable from fabrication perspective and can withstand high operating temperatures of the TPV system. Moreover, fabrication of the proposed nanostructure is relatively simple as compared to surface gratings making it cost effective. Nanoparticle radius, volume fraction and film thickness offer a good tunability for tailoring of emission spectrum. We also explored a new parameter named as effectiveness index that can be used as a good indicator to compare efficacy of thermal emitter for a given PV cell and operating temperature. This study gives valuable insights into design opportunities for selective thermal emitters that can be applied for TPV systems.

## Acknowledgments

This work is partially funded by the Rhode Island STAC Research Grant and the Start-up Grant through the College of Engineering at the University of Rhode Island.

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