## Abstract

A 2D pyramidal metamaterial-based nano-structure is proposed as a wavelength-selective Thermophotovoltaic (TPV) emitter. Rigorous coupled-wave analysis complemented with normal field method is used to predict the emittance as well as the electromagnetic field and Poynting vector distributions. The proposed emitter is shown to be wavelength-selective, polarization-insensitive, and direction-insensitive in emittance. The mechanisms supporting the emittance close to 1.0 in the wavelength range of 0.3-2.0 μm are elucidated by the distribution of electromagnetic field and Poynting vectors in the proposed structure. Finally, thermal stability and radiant heat-to-electricity TPV efficiency for a realistic InGaAsSb TPV system are discussed.

© 2015 Optical Society of America

## 1. Introduction

A basic thermophotovoltaic (TPV) system consists of two key components: an emitter and TPV cells, and it converts thermal radiation directly in to electrical power with the aid of photovoltaic (PV) effect [1], The TPV systems have been considered a promising way of recycling waste heat, harvesting solar energy, and space applications [2–4]. The main challenges for TPV systems are low conversion efficiency and power generation. To enhance the power generation, near-field thermal radiation has been proposed by bringing the emitter and TPV cells in close proximity [2, 5–7]. However, it is still difficult to realize the nanoscale TPV systems in the near future. On the other hand, selective TPV emitter is preferred to improve the conversion efficiency, which has an emittance as high as possible in a certain spectral range that matches with a specific TPV cell, and as low as possible in the remaining spectral range. That is because only photons with energy higher than the bandgap (*E _{g}*) of the semiconductor material used in the TPV cell can produce electron-hole pairs and thereby generate electrical power, whereas those below the bandgap with insufficient energy to produce electro-hole pairs just increase thermal energy of the TPV cells [8]. Obviously, a wavelength-selective TPV emitter is desirable for maximizing power generation and conversion efficiency of TPV systems.

Natural materials usually have broadband thermal radiation and have a much weaker emittance compared with the blackbody, rendering them inefficient for TPV systems. Fortunately, periodic micro/nanostructures of wide profile diversity are able to tailor thermal emittance of an emitter by excitation of unconventional physical mechanisms [2]. There have been a number of micro/nanostructures proposed in literature, and they can be classified into 1D gratings [9–14], 2D gratings [4, 15–18], and 3D photonic crystals [19–21]. It is however more advantageous to design a TPV emitter such that its emittance is high at wavelengths *λ* < *λ _{c}* while maintaining low at

*λ*>

*λ*over all polar angles and polarizations, where

_{c}*λ*means the cutoff wavelength. In this respect, Sergeant et al. [10] designed 1D V-groove gratings such that high emittance is obtained at

_{c}*λ*<

*λ*nearly over all polar angles, however, it is polarization and azimuthal angle dependent. The 2D pyramid structures proposed by Rephaeli et al. [15] works very well over most polar angles and all polarizations, but the cutoff wavelength of this structure only reaches 1.5 μm and is difficult to be increased by modifying the geometric parameters. As a result, there still is plenty space for improvement to reach a more efficient TPV emitter. To further improve the TPV system efficiency, a cold-side selective filter [22, 23] is often used in conjunction with selective emitters.

_{c}In this paper, the cutoff wavelength *λ _{c}* = 2.0 μm is considered, and a 2D pyramidal metamaterial-based structure is used to design the matched wavelength-selective, angle and polarization insensitive TPV emitter. This kind of structure was proposed few years ago [24], and was later used for different purposes [25–27]. However, to the authors’ knowledge this kind of structure has never been used to design TPV emitters.

## 2. Model development and numerical method

#### 2.1 Geometry

Figure 1(a) shows the schematic drawing of a 2D pyramidal metamaterial-based structure placed on the top of a substrate. The space along the *z*-axis is generally divided into three regions as depicted in Fig. 1(b): the free space (region I), the pyramidal structure (region II), and the substrate (region III). The pyramidal structure in region II is composed of alternating metallic layer (red layer) and dielectric layer (yellow layer). From top to bottom the multilayer structure starts with metallic layer and ends with dielectric layer. The pyramidal geometric profile is specified by top width *w _{t}*, bottom width

*w*, period

_{b}*Λ*, metallic layer thickness

*t*, dielectric layer thickness

_{m}*t*, and the total number

_{d}*n*of layers. The structure is assumed to be symmetric along

*x*- and

*y*-direction. The kind of nanostructures as shown in Figs. 1(a) and 1(b) can be fabricated by electron-beam lithography [28, 29]. Usually, the TPV emitter works at temperature between 1000 and 1500 K so that materials, constructing the TPV emitter, should be carefully chosen. In the present emitter, tantalum (Ta) is selected as the substrate and the metallic films in region II, and silicon dioxide (SiO

_{2}), which has refractive index

*n*≈1.45 [30], is used as the dielectric films.

#### 2.2 Numerical method

The radiative properties of TPV emitter, as shown in Fig. 1, are numerically calculated by the rigorous coupled-wave analysis (RCWA) [31–33]. According to Kirchhoff’s law of thermal radiation, the directional emittance can be obtained by calculating the directional absorbance for incident light as depicted in Fig. 2. The incident light is assumed to be linearly polarized and can be characterized by a polar angle *θ*, azimuthal angle *ϕ* and polarization angle *ψ*. Here *ϕ* is the angle between the *x*-*z* plane and the plane of incidence, which is defined as the plane containing the incidence and the *z*-axis, while *ψ* is the angle between the electric field **E** and the incidence plane. Any incident waves with polarization angle *ψ* can be decomposed into the transverse electric (TE) waves with *ψ* = 90° and the transverse magnetic (TM) waves with *ψ* = 0°. According to the configurations above, the final closed linear equations in the matrix form can be constructed by following the steps instructed by Moharam et al. [31, 33] and Li [32], and the detailed mathematical formulations for the 2D RCWA will not be shown here for simplicity except for some important issues.

For the multilayer periodic structure of Fig. 1, the pyramid parts are divided into a large number of thin planar grating slabs parallel to the *x*-y plane, and then the 2D RCWA can be completed by satisfying the boundary conditions at each interfaces: the tangential components of the electric and magnetic fields at the interface between two sublayers are continuous.

The Fourier coefficients of the relative dielectric functions in each sublayer *i* in the region II are classified into four categories below depending on if either *m* or *n* is zero:

*i*th sub-grating along

*x*- and

*y*-direction, respectively, and two integers in the subscript,

*m*and

*n*, denote the diffraction orders in the

*x*- and

*y*-direction, respectively.

The accuracy of RCWA is directly related to the total number of diffraction orders *L* and the convergence of the Fourier series representations of the layer dielectric functions [32]. However, the time and space requirements of the eigenvalue problem make calculations involving large *L* prohibitively expensive [34]. As a result, much work has focused on improving the convergence of the 2D RCWA with respect to *L* such that fewer Fourier components may be used [32, 35–37]. After numerical experiments, it was found that introducing a conditional parameter α [35] or reformulating the Toeplitz matrix of the relative dielectric function [32] does not work for the proposed structure. However, the normal vector method [36, 37] works much better.

For the present structure, the normal vectors as shown in Fig. 3 are used to improve the convergence of 2D RCWA, which can be expressed as follows after normalization:

In Eqs. (2a) and (2b), the parameters *c _{x}* and

*c*equal Λ

_{y}*/Λ*

_{x}*and Λ*

_{y}*/Λ*

_{y}*, respectively. For complex pattern, the normal vectors at the position (*

_{x}*x*,

*y*) can be obtained by following the procedure proposed by Antos et al. [37].

Knowing the analytic expression, the normal fields can be transformed into Fourier space easily. By introducing the normal fields, the third and fourth lines of Eq. (57) in [31] are reformed by:

#### 2.3 Development of structure parameters

Numerical simulation provides an effective and economical way to confirm the geometry parameters as shown in Fig. 1(b) to obtain an efficient TPV emitter. However, it takes a lot of time to plot a complete emittance spectrum with sufficient data points by a 2D RCWA program such that a thorough investigation on all spectra from various parameters is unaffordable. Thus, we choose to reduce the dimension of the structure, i.e., 1D structure similar to Figs. 1(a) and 1(b) is first studied to develop the geometry parameters. This simplification can save the calculation time significantly, but still provides many important information. After a number of numerical experiments, the geometry parameters are confirmed as follows: Λ = 560 nm, *w _{b}* = 460 nm,

*w*= 70 nm,

_{t}*t*= 15 nm,

_{m}*t*= 35 nm,

_{d}*n*= 32. Figure 4 shows the emittance

*ε*of 1D pyramidal structure as a function of

*λ*and polar angle

*θ*for TM waves at

*ϕ*= 0. It can be seen that the proposed metamaterial-based pyramidal structure exhibits ultrahigh emittance in the spectral range 0.1-2.0 um even through the polar angle is beyond 70°. Base on the 1D metamaterial-based pyramidal structure mentioned above, a 2D similar structure can be easily constructed by setting the same geometric values along

*x*- and

*y*-directions.

## 3. Result and discussion

#### 3.1 Determining the number of diffraction orders

The numerical accuracy of RCWA is directly dependent on the number of the diffraction orders (or the number of Fourier terms). Marking *M* and *N* as the highest diffraction orders in the *x*- and *y*-direction, respectively, the total number of diffraction orders *L* used in the calculation is (2*M* + 1) × (2*N* + 1), since –*M* ≤ *m* ≤ *M* and –*N* ≤ *n* ≤ *N* are all considered. Due to the geometric symmetry of the structure, *M* and *N* are assigned the same values. It is obvious that the accuracy will be improved with more orders used. However, both the memory required and the processing time consumed in the calculation increase dramatically with the increasing of diffraction orders. Thus, checking the convergence with different number of diffraction orders is necessary to save computational time and resource.

When conducting the convergence test, it was found that the emittance in the wave range of *λ* < 2.0 μm converges very fast with *M* in contrast with the range of *λ* > 2.0 μm as shown in Fig. 5. Thus, emittance at *λ* = 3.0 μm is chosen to validate the convergence for higher diffraction orders. Results in Fig. 5 are for *ψ* = 90°/0°, *ϕ* = 0° and *θ* = 0°. Spectra for *ψ* = 90° and 0° are identical with *ϕ* = 0° and *θ* = 0° because of the symmetry of the structure in the *x* and *y* directions. It can be seen that the emittance changes very little for *λ* = 3.0 μm when *M* is larger than 24. Thereby, *M* = 24 will be employed hereafter to obtain accurate radiative properties efficiently.

#### 3.2 Normal emittance spectral

Figure 6 plots the emittance spectra of several different configurations for comparison with *ψ* = 90°/0°, *θ* = 0° and *ϕ* = 0°. Both black and red lines represent the emittance characters of the proposed structure, while blue and pink lines represent those of Zhao, et al. [17] and Yeng, et al. [18], respectively. The emittance of flat Ta at room temperature is given by a green line [38]. 1D/2D pyramid refers to the structure depicted in Figs. 1(a) and 1(b) of 1D/2D configuration. It should be noted that, to validate our program, the results denoted by the blue line is recalculated by our program with *M* = 15, which is nearly the same with that from Zhao, et al. [17] (*M* = 35 in [17]). As shown in Fig. 6, the emittance of the proposed structure is close to 1.0 in the whole spectral range of 0.3-2.0 μm, which is superior to that of Zhao’s [17] or Yeng’s [18] structure, and shows dramatically decrease in the spectral range of λ > 2.0 μm.

#### 3.3 The effect of polar angle and azimuthal angle

Figures 7(a) and 7(b) show the emittance spectra from the posed structure at different polar angles with *ϕ* = 0° for TE waves and TM waves, respectively. Three polar angles of 0°, 30° and 60° are examined. Although the emittance vary with *ψ* at oblique direction, only the TE waves and TM waves are needed to be considered. That is because any waves with polarized angle *ψ* can be seen as the superposition of TE waves (*ψ* = 90°) and TM waves (*ψ* = 0°). The emittance spectra at *θ* = 30° is almost the same as that at *θ* = 0°, either for TE waves or for TM waves. When the polar angle is increased from 30° to 60°, the spectral emittance is generally suppressed for both TE waves and TM waves except a small rebound in the range of 2.1-3.1 μm for TE waves. The smallest emittance in the range of 0.3-2.0 μm is 0.95 for TM waves and 0.8 for TE waves even at *θ* = 60°.

Figure 8 is a polar plot of the emittance with respect to *θ* at *ϕ* = 0°, which is also valid for *ϕ* = 0°, 180° or 270°, due to the symmetry of the structure. The left half and right half of the plot show the emittance for TE waves and TM waves, respectively. In the desired wavelength range, the emittance is close to 1.0 at most of the polar angles for both TE waves and TM waves. The largest polar angle with emittance no smaller than 0.9 is 60° for TE waves and 75° for TM waves, respectively in the desired wavelength range. For *λ* = 3.0 μm, the maximum emittance is about 0.4 for TE waves and 0.3 for TM waves, respectively.

In Fig. 9 the left and right sides are for TE waves and TM waves, respectively. Only the emittance for 0° ≤ *ϕ* ≤ 90° is shown, due to its eightfold symmetry with respect to *ϕ*. It is shown that the emittance for both TE waves and TM waves is insensitive to azimuthal angle *ϕ*. Note that the emittance varies with *ψ* at oblique direction, thus there is a discontinuity between TE waves and TM waves at *ϕ* = 0°.

#### 3.4 The excitation of slowlight modes

To understand the mechanisms that support the ultrahigh emittance in wavelength range 0.3-2.0 μm, we study the distributions of normalized fields |H* _{y}*/H

*| in the actual inhomogeneous pyramidal structure illuminated by four different wavelengths with*

_{inc}*ψ*= 0°,

*θ*= 0° and

*ϕ*= 0°; see the contour maps in Figs. 10(a)-10(d) for

*λ*= 0.8 μm, 1.2 μm, 1.8 μm and 3.0 μm, respectively. It is identified that lights of different wavelengths accumulate at different parts of the pyramid along

*z*-direction, and the location where energy is trapped will moves from the top part to the bottom part of the pyramid as the wavelengths increase. Plots of Poynting vectors (S = Re (

**E**×

**H***)/2) (arrow maps) can indicate how the lights propagate in the pyramid. It is shown that energy flows whirl into the pyramid and forms two vortexes for wavelengths shorter than 2.0 μm. In addition, the vortexes locate exactly at the places where the magnetic fields concentrate, which is known as slowlight modes [24, 39, 40]. However, no vortex is formed for wavelength 3.0 μm due to the disturbance of the metal substrate. Based on Fig. 10, the metamaterial-based pyramid can be seen as a group of waveguides of different fixed core widths, which is able to excite lots of solwlight modes to work out the ultrahigh emittance in the broadband. On the other hand, great enhancement of the magnetic fields in the pyramidal ridge indicates the excitation of the magnetic polaritons and the magnetic resonance in the dielectric layers [17] (the largest |H

*| always lies in the SiO*

_{y}_{2}layer).

In view of Figs. 10(a)-10(d), it is predicted that TPV cells with different bandgaps can be matched by the proposed 2D structure via tuning the top width *w _{t}* and bottom width

*w*of the pyramid together with the period

_{b}*Λ*. Figure 11 gives an example of tuning the spectral emittance by changing the parameter

*w*with the other geometric parameters the same as before. It is shown that the cutoff wavelength varies from 2.0 μm to 3.0 μm by increasing the bottom width

_{b}*w*of the pyramid from 0.46 μm to 0.56μm.

_{b}## 4. Thermal stability

Considering the high operating temperature (>1000 K) of TPV emitters, thermal stability of nanostructures is much important for realistic applications. It has been shown that there is a risk of structure degradation at elevated temperatures as a result of surface diffusion, surface reactions, recrystallization, grain growth and migration, and boundary diffusion [41–44]. The effect of evaporation and redeposition can be ignored for refractory metals such as Tantalum due to their low vapor pressures. Recently, Rinnerbauer et al. [44] has experimentally demonstrated that a thin surface protective coating of hafnium oxide (HfO_{2}) which acts as a thermal barrier coating and diffusion inhibitor can effectively suppress surface diffusion and surface reactions. To address the problems of recrystallization and grain growth and migration at high temperatures, Ta substrates were suggested to be annealed at high temperature [44]. In the light of Rinnerbauer’s experiment results, the proposed structure as depicted in Figs. 1(a) and 1(b) is improved by removing the first metal layer and expanding the last dielectric layer to cover the substrate in order to minimize the exposed Ta surface. Besides, the SiO_{2} is replaced by HfO_{2} which has refractive index *n* ≈1.90 [45]. To protect the emitters from oxidation, operation under vacuum or protective atmosphere is suggested. The structure of pyramidal metamaterial-based emitter with thermal stability is shown in Fig. 12(a), and its emittance for TM waves with *ϕ* = 0° and *θ* = 0°, 30° and 60° is given by Fig. 12(b). Here, the optimized geometry parameters are Λ = 500 nm, *w _{b}* = 400 nm,

*w*= 70 nm,

_{t}*t*= 20 nm,

_{m}*t*= 40 nm,

_{d}*n*= 30. It can be seen that the proposed structure after thermal stability improvement still exhibits excellent selective properties.

## 5. Performance of TPV with selective emitter

It is noticed that the room temperature optical properties of Ta were used in the previous sections for evaluation and comparison with different emitter strategies, since both Zhao [17] and Yeng [18] used the optical properties at room temperature. In this section, we will illustrate the benefit of the proposed selective emitter for a realistic InGaAsSb TPV application for emitter temperature *T* = 1250 K, and the optical properties of Ta at elevated temperature [23] is used. The model proposed by Yeng et al. [23] is used to obtain detailed performance predictions of TPV systems. The radiant heat-to-electricity TPV efficiency ${\eta}_{\text{TPV}}$ is given by:

The key physical properties of InGaAsSb cells needed in the performance predictions of TPV systems are taken from literature [46]. The hemispherical emittance of the optimized 2D pyramidal metamaterial-based emitter as discussed in section 4 (Λ = 600 nm, *w _{b}* = 500 nm,

*w*= 70 nm,

_{t}*t*= 20 nm,

_{m}*t*= 40 nm,

_{d}*n*= 30) at elevated temperature is given by Fig. 13. It is shown that the TPV emitter at elevated temperature does not exhibit selective properties, which will definitely lower the TPV system efficiency, but the emittance at short wavelengths is still very high. Fortunately, a cold side tandem filter [22] can be used to improve the TPV system efficiency. The reflectance of Rugate tandem filter for 0.53 eV [23] and the external quantum efficiency (EQE) of InGaAsSb [46] are also given in Fig. 13, which will be used to predict the system efficiency.

Results for greybody (*ε* = 0.9), optimized HfO_{2}-filled ARC 2D TaPhC [18], and the proposed 2D pyramidal metamaterial-based emitter at *T* = 1250 K and view factor *F* = 0.99 with/without a cold-side Rugate tandem filter are listed in Table 1 for comparison. Our calculated ${\eta}_{\text{TPV}}$ and ${J}_{\text{elec}}$ for greybody (*ε* = 0.9) with a cold-side filter are 23.42% and 0.723 W/cm^{2}, respectively, which are coincident with that from literature [18]. Thus, our prediction process is reliable. It is obvious as shown in Table 1 that a cold-side filter is highly recommendatory in view of great improvement in ${\eta}_{\text{TPV}}$. In TPV systems without a cold-side filter, the ${\eta}_{\text{TPV}}$ for the proposed 2D pyramidal metamaterial-based emitter is smaller than optimized HfO_{2}-filled ARC 2D TaPhC emitter and greater than grebody, but the greatest ${J}_{\text{elec}}$ is obtained by the proposed emitter. In TPV systems with a cold-side filter, the proposed 2D pyramidal metamaterial-based emitter enables up to 13% improvement in ${J}_{\text{elec}}$ compared to the HfO_{2}-filed ARC 2D TaPhc emitter and 5% improvement compared to the greybody (*ε* = 0.9), even though the improvement in ${\eta}_{\text{TPV}}$ is small. Therefore, the optimized 2D pyramidal metamaterial-based emitter is the best choice for InGaAsSB TPV systems with a cold-side filter. The reason behind the improvement in ${J}_{\text{elec}}$ is the ultrahigh hemispherical emittance for the proposed emitter, which is greater than 0.95 for *λ* < 2.0 μm, and the TPV cell will receive more useful radiant energy to produce electric power.

## 6. Conclusions

This work successfully proposed a 2D pyramidal metamaterial-based nano-structure for TPV emitters and numerically demonstrated their attractive wavelength-selective radiative properties. The 2D RCWA complemented with normal vector method was used to calculate the emittance of the structure accurately and efficiently. For the proposed TPV emitter, its normal emittance is close to 1.0 in the wavelength range of 0.3-2.0 μm and dramatically decreases at wavelengths beyond 2.0 μm. By calculating the emittance with various polarization, azimuthal angle, polar angle, and wavelengths, the proposed TPV emitter was proved to be direction-insensitive and polarization-insensitive in emittance. The distributions of |H* _{y}*/H

*| and Poynting vectors in the actual inhomogeneous pyramidal structure precisely demonstrate the excitation of slowlight modes at short wavelengths with ultrahigh emittance. In addition, the geometric parameters can be tuned to match TPV cells with different bandgaps. Considering thermal stability for elevated operation temperatures, the proposed TPV emitter is improved from structure and material aspects. For a realistic InGaAsSb TPV system with a cold-side filter, greater electric power density${J}_{\text{elec}}$ is obtained by the proposed TPV emitter compared to the greybody with*

_{inc}*ε*= 0.9.

## Acknowledgements

This work is sponsored by the National Natural Science Foundation of China (NSFC) (51136004).

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