1. Introduction

With more than 80% of power consumption currently coming from fast-depleting fossil fuels, there is a global challenge due to the ever-increasing power demand, which is predicted to be doubled and tripled by mid- and late-21st century [1]. Abundant and clean solar energy provides a viable solution to alleviate the energy crisis and reduce the pollution of greenhouse gases. Solar absorbers, which convert solar radiation to heat, are vital components affecting the performance of solar thermal, solar thermoelectric and solar thermophotovoltaic systems. An ideal solar absorber should have unity absorptance within a wide spectral range from UV to near-infrared, where most of solar energy is distributed. On the other hand, zero emittance in the mid-infrared is highly desirable for solar absorbers to minimize the heat loss from self-emission. Therefore, perfect broadband solar absorption and low mid-infrared emission are key characters for a high-performance solar absorber. In addition, independence on the direction and polarization state is also crucial for a perfect solar absorber to maximize the absorption of solar energy.

Recently, electromagnetic metamaterial absorbers have attracted great attention since selective absorption or emission can be achieved by exciting plasmonic resonances at particular wavelengths inside the structures [2, 3]. A great number of metamaterials are made of micro/nanostructures with subwavelength metallic patterns on a metal film separated by a dielectric spacer. Strong electromagnetic coupling could occur between the metallic pattern and the film at selected wavelengths due to electric and magnetic responses of the metamaterial. Perfect metamaterial absorbers made of electric ring resonators coupled to metal wires were proposed by Landy et al. [4], and exhibited selective absorption in the terahertz region. By replacing the metal wires with a continuous film, Tao et al. [5] improved the design to achieve wide-angle absorption for both transverse electric (TE) and magnetic (TM) polarized waves. Different pattern designs such as chiral metamaterial [6], fishnet structure [7], and cut-wire array [8] were proposed to achieve omnidirectional and polarization-independent absorption in the THz regime. By shrinking the sizes of the metamaterial absorbers, the near-perfect selective absorption can be obtained in the infrared and visible region. Liu et al. [9] experimentally demonstrated absorption of 97% at the wavelength of 6 μm in a subwavelength perfect absorber made of a film-coupled crossbar structure. A plasmonic absorber made of a layer of gold patch array with the width less than 200 nm on a thin Al₂O₃ layer and a gold film showed an absorption peak of 88% at the wavelength of 1.58 μm [10]. By depositing a two dimensional (2D) Ag grating with a period of 300 nm on a 60-nm SiO₂ and a Ag film, Aydin et al. [11] demonstrated an ultra-thin plasmonic absorber in the visible spectrum. Strong visible light absorption has also been achieved by film-coupled colloidal nanoantennas [12], circular plasmonic resonators [13], and nanoparticles [14–16], by exciting magnetic resonance inside the metamaterial absorbers. It is worth noting that, selective absorption can also be used for controlling thermal emission, indicated by Kirchhoff’s law [17], and selective thermal emitters made of film-coupled micro/nanostructures [18–23] have been studied. These structures with 2D symmetric patterns are proved to exhibit strong wavelength selectivity, angular independence, and polarization-independent behaviors.

Since the absorption behaviors such as the resonance wavelengths in the plasmonic metamaterial absorbers strongly depend on the shape and geometric sizes of the top patterns [20, 24, 25], dual-, multi-, and broad-band absorption can be obtained by employing the multisize effect. Cui et al. [26] experimentally demonstrated a broadband absorber made of 1D metal strips with four different widths. Four resonances are excited at nearby wavelengths in the infrared such that the resonance peaks are coupled to form a broader absorption band from 9 μm to 11 μm. Wu and Shvets [27] theoretically showed a similar design with three different metallic strip widths to achieve broadband absorption in the near-infrared region. Based on the same concept, 2D film-coupled multi-sized disk arrays [28] and multi-sized patch arrays [29] were also shown as broadband plasmonic absorbers with polarization independence by exciting multiple resonances in the infrared. Dual- and broad-band absorption can also be achieved with asymmetric [30], double [31], or multi-sized [18] cross-bar structures. Moreover, by designing stacked metal-dielectric structures, plasmonic resonances can be excited inside distinct dielectric spacers at different wavelengths to achieve dual- [32] or multi-band [33] absorption, and an ultra-broadband light absorption can be achieved in the infrared region by combining the multisize and multilayer effect [34]. By using 2D trapezoid arrays, Aydin et al. [11] demonstrated the broadband polarization-independent light absorption in the entire visible spectrum with an average absorptance of 0.71 from the measurement. Furthermore, Wang and Zhang [21] proposed a high-performance selective TPV emitter made of 1D film-coupled tungsten grating structure. The spectral emittance is more than 0.8 in the wavelength region from 0.6 μm to 2 μm with quasi-diffuse behavior due to the excitations of surface plasmon polariton (SPP) and magnetic polariton (MP) as well as the high intrinsic loss of tungsten, indicating the potential to be an efficient infrared broadband absorber. However, it is still a great challenge to design wide-angle, polarization-independent, selective solar absorbers with unity absorptance from UV to near-infrared and zero emittance in the mid-infrared region.

In this work, we numerically design metamaterial structures made of 2D tungsten gratings on a thin SiO₂ spacer and an opaque tungsten film as perfect selective solar absorbers. The radiative properties of these metamaterial solar absorbers are investigated in a broad region from UV to mid-infrared based on the finite-difference time-domain method. The effects of geometric parameters, such as the tungsten patch width, grating period, grating thickness, and the spacer thickness, on the spectral absorptance of singled-sized metamaterial absorbers are studied. The physical mechanisms responsible for the enhanced selective absorption in the visible and near-infrared region are discussed, and the behaviors of magnetic polaritons are elucidated with the electromagnetic field distribution and an inductor-capacitor (LC) model. The absorptance of metamaterial absorbers with two different patch sizes for the top tungsten grating is also studied, and the performance of optimized metamaterial structures is evaluated as selective solar absorbers at normal direction. The effects of oblique incidence and polarization states on the spectral absorption are further analyzed. Finally, the absorptance of multi-sized metamaterial absorbers with three or four different patch sizes is explored.

2. Metamaterial structures and numerical methods

Figure 1 depicts the structures of the proposed metamaterial solar absorbers made of 2D periodic tungsten gratings on a thin SiO₂ spacer and a tungsten thin film. A unit cell of the metamaterial structure with single-sized tungsten patches is shown in Fig. 1(a). The geometric parameters include grating period Λ, tungsten patch width w, grating height (or patch thickness) h, and SiO₂ spacer thickness t. The tungsten thin film could be a couple of hundred nanometers thick as long as it is opaque. The whole structure could be deposited on silicon, quartz or other substrates. The 2D periodic gratings have the same geometric parameters (i.e., Λ and w) in the x and y directions, since the geometric symmetry is crucial to realize the polarization independence at normal direction. A wavevector K_inc represents the electromagnetic wave with a free-space wavelength λ incident onto the metamaterial structure at a polar angle (or incidence angle) θ, polarization angle ψ, and azimuthal angle ϕ. The polar angle θ denotes the angle between K_inc and the surface normal of the structure (i.e., z direction). The angle ψ between electric field vector E and the plane of incidence, defined by K_inc and the structure surface normal, is the polarization angle. ψ = 0° indicates the transverse magnetic (TM) polarized wave while ψ = 90° gives the transverse electric (TE) polarized wave. Azimuthal angle ϕ is the angle between the x axis and the plane of incidence, and is taken as ϕ = 0° here for simplicity. Otherwise, conical diffraction has to be considered due to the non-zero wavevector components in both x and y directions for the incident wave. Figure 1(b) shows the schematic of a unit cell for the metamaterial absorber with double-sized tungsten patches of different widths w₁ and w₂. The patches with the same width are arranged diagonally such that the structure behaves exactly the same at normal incidence for either TE or TM waves. Each patch is centered in its quadrant, and the period ${\Lambda }^{\prime }$ of the double-size metamaterials is twice that of single-sized ones, i.e, ${\Lambda }^{\prime }=2\Lambda$. Following [21, 23], a set of geometric parameters: Λ = 600 nm, w = 300 nm, h = t = 60 nm is considered here as the base values for the present study.

 

Fig. 1 (a) Schematic of proposed single-sized metamaterial solar absorbers made of 2D periodic tungsten gratings with period Λ, patch width w and grating height h, on a thin SiO₂ spacer with thickness t and an opaque tungsten thin film. The electromagnetic wave is incident at a polar angle θ, polarization angle ψ, and azimuthal angle ϕ. The structure is assumed to be geometrically symmetric in the x and y directions, and ϕ is taken as 0° for simplicity. (b) Schematic of double-sized metamaterial solar absorbers with tungsten patches of different widths w₁ and w₂, and period ${\Lambda }^{\prime }=2\Lambda$ Tungsten patches with the same size are arranged diagonally and each patch is centered in its quadrant.

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The finite-difference time-domain (FDTD) method (Lumerical Solutions, Inc.) is used for calculating the radiative properties of the proposed metamaterial absorbers in a broad spectral region from UV to mid-infrared (i.e., 0.3 μm to 20 μm) by numerically solving the Maxwell equations. The optical constants of tungsten and SiO₂ are both taken from Palik [35]. A broadband linearly polarized plane wave source simulating the incident electromagnetic waves is placed one micron away from the structure. Periodic boundary condition is applied at normal incidence in the x and y directions of the simulation domain, while Bloch boundary condition is used for oblique incidence to account for the phase difference in the periodic structures. Perfectly matched layers with reflection coefficients less than 10⁻⁶ are placed at the boundaries along the z direction. Non-uniform meshes with minimum mesh size of 5 nm are used, and the relative difference in absorptance is within 0.3% compared with that obtained by a minimum mesh size of 3.3 nm. A frequency-domain field and power monitor is placed above the plane wave source to collect the reflected waves, from which the spectral-directional reflectance ${R^{\prime }}_{\lambda }$ at different polarization states can be obtained. The transmittance is also checked from a field and power monitor underneath the metamaterial structures, to verify the opaqueness of the structures. The spectral-directional absorptance can be calculated from ${{\alpha }^{\prime }}_{\lambda }=1-{R^{\prime }}_{\lambda }$ since the metamaterial structures are opaque, and the spectral-directional emittance is obtained from ${{\epsilon }^{\prime }}_{\lambda }={{\alpha }^{\prime }}_{\lambda }$ according to the Kirchhoff’s law. A spectral resolution of 5 nm is used for the numerical simulations and is sufficient to resolve the spectra of the radiative properties of studied metamaterial absorbers.

3. Spectral absorptance of single-sized metamaterial structures at normal incidence

3.1 Geometric effects and optimization

We first investigated the geometric effects on the spectral absorptance of single-sized metamaterial solar absorber at normal incidence, aiming to elucidate the physical mechanisms responsible for the enhanced absorption and to optimize geometric parameters to achieve higher absorption in the visible and near-infrared region. The effects of patch width w, grating period Λ, grating height h, and spacer thickness t are considered starting from a set of base geometric parameters of Λ = 600 nm, w = 300 nm, and h = t = 60 nm. Other parameters are fixed at the base values when a specified geometric parameter varies during the numerical simulations. The polarization angle ψ is set to be 0° with the oscillating magnetic field H along the y direction, and the absorptance at TM incidences is obtained. The radiative properties for TM and TE-polarized waves would be the same at normal incidence due to the geometric symmetry. Figures 2(a)-2(d) show how the normal absorptance changes with patch width w, grating period Λ, grating height h, and spacer thickness t, respectively, in the spectral region from 0.3 μm to 4 μm, where most of solar energy is confined.

 

Fig. 2 Spectral absorptance of the single-sized metamaterial solar absorber at normal incidence as a function of (a) tungsten patch width w, (b) grating period Λ, (c) grating height h, and (d) spacer thickness t. The base values of the geometric parameters are Λ = 600 nm, w = 300 nm, and h = t = 60 nm. The broadband high absorption in the spectral region from 0.3 μm to 2 μm is due to several physical mechanisms including SPP, CMP, intrinsic loss of tungsten, and MP.

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When the patch width w changes from 200 nm to 400 nm, as shown in Fig. 2(a), the normal absorptance is enhanced significantly due to the enhancement of an absorption peak around λ = 2 μm. Note that, the peak magnitude increases up to 1 at the width of w = 350 nm, and the peak wavelength shifts to longer wavelengths with larger width values. In the meantime, another peak with absorptance more than 0.95 exists around 0.7 μm, and shows much less dependence on the patch width, although the peak shifts slightly to longer wavelengths with larger width. Similar emittance peaks have been found previously in the similar tungsten metamaterial TPV emitters made of 1D [21] and 2D gratings [23] with w = 300 nm, and are attributed to the excitations of MP and SPP modes at longer and shorter wavelengths, respectively. In fact, the absorption peak around 0.7 μm is the hybrid of a SPP mode and a coupled magnetic polariton (CMP) mode, to be explained later with the help of electromagnetic field distribution. The minor absorption peaks around λ = 0.4 μm and 1.4 μm between the two major ones are due to the intrinsic loss associated with the interband transitions in tungsten [35]. It can be clearly observed from this figure that the absorptance (or emittance) drops sharply at wavelengths beyond the MP resonance.

Figure 2(b) shows the effect of grating period Λ on the normal spectral absorptance of the single-sized metamaterial solar absorber when it varies from 400 nm to 800 nm. It can be observed that the MP peaks around λ = 1.8 μm remain almost un-shifted, except for Λ = 400 nm, in which case the coupling across the small gap between neighboring patches has some effects on MP peaks. On the other hand, the SPP peaks shift to longer wavelengths with increased grating period. Surface plasmon polaritons or SPPs are the coupling of the collective oscillation of surface charges at the interface to the external electromagnetic waves at specific wavelengths. The excitation of SPPs between two nonmagnetic materials is determined by the dispersion relation $k_{\text{spp}}=(\omega \text{/}{c}_0)\sqrt{{\epsilon }_1{\epsilon }_2/({\epsilon }_1+{\epsilon }_2)}$, where ε₁ and ε₂ are the dielectric functions of materials at each side of interface respectively, and their real parts should have opposite signs to excite SPPs [17]. The SPP resonance wavelengths strongly depend on the grating period Λ and incidence angle θ. Zhao et al. [23] have provided a detailed discussion on the behavior of SPPs at normal and oblique incidence for the 2D TPV emitter with the similar structure. Therefore, the SPP behavior will not be elaborated here again. In addition, it’s interesting to notice that as grating period increases, the CMP peaks also shift to longer wavelengths.

Figure 2(c) presents the effect of grating height h on the normal absorptance of the single-sized metamaterial solar absorber. When h varies from 60 nm to 200 nm, the absorptance spectrum from 0.6 μm to 1.8 μm is enhanced with larger h values. The MP peak around λ = 1.8 μm depends little on the grating height, and shifts slightly toward shorter wavelength with increasing h. The absorptance at the MP peak could be close to 1 with h = 90 nm, and the absorptance drops abruptly beyond the MP peak, resulting in absorptance values of 0.5 around λ = 2 μm and below 0.05 at λ = 4 μm. On the short wavelength side, the sharp SPP peak at λ = 0.6 μm does not change with the grating height. However, another peak, associated with the CMP mode, starts to separate from the SPP mode around λ = 0.6 μm, and shifts to longer wavelengths with thicker tungsten patches or larger h values. As a result, a broad absorption band from 0.6 μm to 1.8 μm with ${{\alpha }^{\prime }}_{\lambda }>0.9$ is achieved with h = 150 nm.

As shown in Fig. 2(d), spacer thickness t yields a similar effect as the grating height on the normal absorptance of the single-sized metamaterial solar absorber. When the spacer thickness changes from 40 nm to 150 nm, the MP peak shifts to shorter wavelength, while the peak amplitude first increases to a maximum close to 1 with t = 80 nm and then drops with further thicker spacers. The SPP peak locations do not change with spacer thickness but the amplitudes change with different t values. The CMP peak separates from the SPP peak around λ = 0.6 μm, and shifts slowly towards the longer wavelength with increasing t. As a result, the absorptance in the spectral region between 0.6 μm to 1.8 μm is greatly enhanced, with the minimum value of spectral absorptance increases from 0.6 at t = 40 nm to 0.92 at t = 120 nm. However, the absorptance starts to decrease with further thicker spacers.

Besides the MP, CMP and SPP resonance modes, which enhance the absorptance around several particular wavelengths, another important factor for the broadband high absorption is the high intrinsic loss of tungsten used here. Tungsten has several interband transitions around the wavelengths of 0.4 μm, 0.6 μm and 1.4 μm. Metals with relatively low losses such as Ag and Au have been commonly considered for constructing plasmonic metamaterials for potential sensing, imaging and cloaking applications. However, for solar thermal applications, high intrinsic loss is actually beneficial to enhance the absorption of solar radiation across a wide spectral range, therefore tungsten is chosen here. The most important factor to achieve almost perfect absorption in a broad spectral band from visible to the near-infrared region with the designed metamaterial absorber is the coupling effect between different resonance modes and interband absorption of tungsten. One would expect that the absorptance will be reduced if the resonance modes are far apart and could not effectively couple with each other.

Clearly, the absorptance of the singled-sized metamaterial solar absorber strongly depends on the geometric parameters. Several absorption peaks, associated with the excitations of MP, CMP and SPP modes as well as the intrinsic loss of tungsten, can be clearly seen, and the coupling between these modes results in a broad and enhanced absorption band in the visible and near-infrared region. The peak wavelengths of the MP and CMP modes also show strong dependence on the patch width w, grating period Λ, grating height h, and spacer thickness t, which could be potentially employed to further broaden the absorption peak. The absorption could be also maximized by optimizing the geometric parameters. In order to further understand the physical mechanisms for the absorption enhancement, the behaviors of MP and CMP modes are elucidated below.

3.2 Physical mechanisms of magnetic polariton (MP) and coupled magnetic polariton (CMP)

Figure 3(a) presents the electromagnetic field distribution when the magnetic polariton (MP) is excited at λ = 1.75 μm inside the single-sized metamaterial solar absorber with geometric parameters of Λ = 600 nm, w = 300 nm, h = 120 nm, and t = 60 nm, calculated from FDTD at normal incidence. The electromagnetic field in two unit cells is shown at the x-z cross section in the middle of the tungsten patch. The contour represents the strength of magnetic field normalized to the incidence, i.e., log₁₀|H/H₀|², suggesting the local H field enhancement or suppression. The arrows are the electric field vectors, indicating the direction and strength of induced electric current. Clearly, there is a strong confinement of electromagnetic energy inside the SiO₂ spacer between the top tungsten patches and bottom tungsten film. The strongest field enhancement occurs at the center of the spacer with 1.5 orders of magnitude higher than the incident H field. At the same time, the electric field vectors indicate an induced current loop around the anti-node of the magnetic field. This field pattern is exactly the characteristics of excitation of magnetic resonance, which has been discussed in detail in similar grating structures from previous studies [20, 21]. The basic mechanism is that, the free charges at the tungsten surfaces resonate with incident electromagnetic waves and induce oscillating electric current, which results in resonant magnetic field according to Lenz’s law.

 

Fig. 3 Contour plots of electromagnetic field distribution in the x-z cross-sectional view at the middle of the tungsten patches, when (a) MP is excited at λ = 1.75 μm and (b) CMP is excited at λ = 0.78 μm for the single-sized metamaterial absorber with Λ = 600 nm, w = 300 nm, h = 120 nm, and t = 60 nm. Two unit cells are shown and different layers are delineated. The contour represents the logarithmic values of magnitude square of normalized magnetic field to the incidence, while the arrows indicate the electric field vectors.

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Figure 3(b) shows the electromagnetic field distribution when the CMP is excited at λ = 0.78 μm in the single-sized metamaterial absorber. Besides the strong magnetic field enhancement and an induced current loop which can be seen inside the spacer between the upper tungsten patches and the bottom tungsten film, which is similar to the behavior of MP, the electric field vectors form another current loop inside the gap between the neighboring tungsten patches, along with strong magnetic field enhancement mainly inside the spacer layer. The localized field could be one order of magnitude stronger than the incident field. The field distribution indicates that two magnetic polaritons are excited in one unit cell, one is between upper tungsten patches and the film and the other is between neighboring patches. Therefore, coupled magnetic polariton (CMP) is named for this resonance mode. In fact, similar CMP mode has been seen in double-layer grating structures [37].

The previous studies on the MPs in the 1D grating structures employed an analytical capacitor-inductor (LC) model based on the charge distribution when the MP is excited, and the predicted magnetic resonance conditions match reasonably well with numerical simulations, confirming the excitation of MPs and resulted resonant absorption or emission [20, 21]. Note that, the magnetic resonance condition from the LC model is independent on the structure length along the direction of magnetic field, i.e, the y direction for TM waves. Therefore, even though 2D grating structures are considered here, the same LC model can be used to confirm the MP mode since the patch length along the y direction has no effect. The interaction between the upper tungsten patches and the bottom tungsten thin film can be represented by a parallel-plate capacitor with $C_{\text{m}}=0.22{\epsilon }_{{\text{SiO}}_{\text{2}}}{\epsilon }_{0}w\text{/}t$ per unit length, and a parallel-plate inductor with $L_{\text{m}}=0.5{\mu }_{0}wt$ per unit length. The interaction between the neighboring tungsten patches can be modeled as a gap capacitor $C_{\text{g}}={\epsilon }_{0}h/(\Lambda -w)$ per unit length. Therefore, the impedance for the LC circuit is:

The magnetic resonance wavelengths are calculated based on Eq. (1) and plotted as a function of different geometric parameters in Figs. 4(a)4(d). The predicted MP resonance wavelengths agree well with the FDTD simulation on the effects of patch width w, grating period Λ and spacer thickness t. The dependence of MP wavelength on the width can be understood by the fact that, larger width will result in larger values for capacitance C_m and inductance L_m and L_k, and thus increasing MP resonance wavelengths. Similarly, thicker spacers will lead to smaller L_kC_m values, while the other term L_mC_m is independent on the spacer thickness t in the LC model. Therefore, the MP wavelength decreases with larger t values. The LC model indicates that resonance wavelength slightly decreases as grating period increases, which matches with the FDTD simulation quite well. On the other hand, the LC model indicates that increasing grating height will slightly increase the MP wavelength, however, the FDTD simulation suggests that the MP wavelength decreases slightly with increasing h. Note that, the LC model is based on several approximations and could not consider the coupling effect between MP and other modes, which may account for the discrepancy on the effect of grating height between the FDTD simulation and LC model.

 

Fig. 4 The resonance wavelengths of MP and CMP modes are summarized in (a−d) with varied geometric parameters. The solid curves are predicted geometry-dependent resonance wavelengths from the analytical LC model for the MP mode.

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4. Performance of optimized single- and double-sized metamaterials as solar absorbers at normal direction

Figure 5(a) shows the absorptance of two single-sized metamaterial absorbers with the same geometric parameters of Λ = 600 nm, h = 150 nm and t = 60 nm but different patch widths w₁ = 250 nm and w₂ = 300 nm, respectively. The grating height h is optimized from the geometric study such that the single-sized metamaterial could have close-to-unity absorptance in the visible and near-infrared region. The single-sized metamaterial absorber with larger patch width w₂ = 300 nm has a broader band of absorption but a little bit lower absorptance in the near-infrared, than the one with smaller patch width of w₁ = 250 nm.

 

Fig. 5 (a) Spectral normal absorptance in the spectral region from 0.4 μm to 4 μm for a double-sized metamaterial solar absorber with tungsten patch widths of w₁ = 250 nm and w₂ = 300 nm, in comparison with that of single-sized metamaterial solar absorbs with w₁ or w₂. Other geometric parameters are the same: Λ = 600 nm, h = 150 nm, and t = 60 nm. The inset depicts the arrangement of the tungsten patches for the double-sized absorbers. (b) Spectral normal emittance of single-sized and double-sized metamaterial solar absorbers in the longer wavelength region from 4 μm to 20 μm.

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Here we further consider a double-sized metamaterial consisting of patches with two different widths of w₁ = 250 nm and w₂ = 300 nm. Since the MP resonance wavelengths highly depend on the patch width, MPs could be excited at two different wavelengths determined by different patch sizes. Coupling of these two MP peaks could potentially result in a broader absorption band. The calculated absorptance for the double-sized metamaterial is plotted in Fig. 5(a). By comparison, the double-sized metamaterial has a broader absorption band than the single-sized one with w₁ = 250 nm, and higher absorptance than the single-sized one with w₂ = 300 nm. The minimum absorptance of the double-sized metamaterial is higher than 0.95 in a wide spectral range from 0.6 μm to 1.8 μm. As a selective solar absorber, the low emittance in the longer wavelengths is very crucial to minimize the thermal energy loss from the re-emission of the absorber itself. Figure 5(b) shows the spectral emittance of the single-sized metamaterials and the double-sized one at normal direction. Clearly, the emittance for the metamaterial solar absorbers is below 0.04 from 4 μm to 20 μm in wavelength. The small peak emittance from 8 μm to 12 μm is due to the phonon absorption of SiO₂.

To better illustrate the double size effect, the electromagnetic field distributions inside the double-sized metamaterial absorbers are plotted at the MP resonance wavelengths for the single-sized metamaterials with w₁ = 250 nm and w₂ = 300 nm, respectively. As shown in Fig. 6(a) at λ₁ = 1.6 μm, MP with local field enhancement can be excited under both patches of different sizes. The excitation of magnetic resonance can be also seen at λ₂ = 1.8 μm in Fig. 6(b), while the field localization is much stronger under the larger patch since this wavelength matches well the MP resonance condition for patch with width of w₂. Since the MP wavelengths for w₁ and w₂ are very close, the MPs can be seen under both patches at both resonance wavelengths, indicating strong coupling effect. As a result, the absorption is further enhanced in a broader spectral range inside the double-sized metamaterial absorber.

 

Fig. 6 Electromagnetic field distributions inside the double-sized metamaterial solar absorber at (a) λ₁ = 1.6 μm and (b) λ₂ = 1.8 μm, which are MP resonance wavelengths of the single-sized metamaterial absorbers with w₁ = 250 nm or w₂ = 300 nm, respectively. The MPs could occur inside the double-sized metamaterial absorbers at both resonance wavelengths under the tungsten patches with different widths of w₁ = 250 nm and w₂ = 300 nm.

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To quantitatively evaluate the performance of proposed metamaterial structures as solar absorbers, the total solar absorptance (or the fraction of absorbed solar energy) at the normal incidence is calculated by

While the total absorptance represents the performance to collect solar energy, the total emittance should also be considered as a measurement of thermal energy loss from the thermal emission of the absorber itself, which can be calculated at normal direction by

By neglecting convection and conduction heat loss and assuming 1 sun condition, the total photon-to-heat conversion efficiency of the solar absorbers can be calculated by:

5. Diffuse-like and polarization-independent behaviors of metamaterial solar absorbers

The directional behavior of the solar absorbers is important for the solar energy absorption at oblique angles. In addition, polarization independence is also critical for a perfect solar absorber to maximize the solar energy absorption since solar radiation is randomly polarized. Similar single-sized metamaterials have been demonstrated to exhibit quasi-diffuse-like [21] and polarization-independent behaviors on the emission spectrum for the TPV applications [23]. Here, we focus on the effects of direction and polarization on the performance of double-sized metamaterial solar absorbers.= Figure 7 plots the spectral absorptance of the double-sized metamaterial absorber as a function of polar angle θ at several representative wavelengths of λ = 0.6 μm, 1.2 μm and 1.8 μm for TE (i.e., ψ = 90°) and TM (i.e., ψ = 0°) polarized waves, respectively. At λ = 1.8 μm where the MPs are excited under the tungsten patches, the absorptance is 0.982 at normal incidence, and decreases slightly to 0.975 for TE waves and 0.964 for TM waves at θ = 30°. Even at θ = 60°, the absorptance could be as high as 0.848 for TE waves, and 0.854 for TM waves. The directional-insensitivity at this wavelength is attributed to the directional independence of MPs, which has been discussed previously [20, 21, 23]. The absorptance at λ = 1.2 μm, which is mainly associated with the interband absorption of tungsten, is 0.949 at normal direction. Although the absorptance tends to decrease slightly when the incidence angles increases, the absorptance at θ = 30° is 0.894 for TE waves and 0.941 for TM waves. When the incidence angle changes to 60°, the absorptance is still as high as 0.837 for TM waves, but drops to 0.698 for TE waves. At λ = 0.6 μm there is an absorption peak of 0.98 at normal direction due to the SPP. Since SPP resonance wavelength has strong dependence on the direction, the absorptance then slightly drops but maintains around 0.88 at a broad angular range from 5° up to 60° or so for both polarizations. It can be clearly seen that, the absorptance of the double-sized metamaterial absorber is insensitive to the incidence angle, and high absorptance exists over a large range of incidence angles for both polarizations.

 

Fig. 7 Spectral absorptance of the double-sized metamaterial solar absorber as a function of polar angle θ at several representative wavelengths of λ = 0.6 μm, 1.2 μm and 1.8 μm for (a) TE and (b) TM polarized waves.

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The effect of polarization angle ψ on the spectral absorptance of the double-sized metamaterial solar absorber is also studied at the normal incidence, shown as the contour plot in Fig. 8.High absorptance region represented by bright colors can be clearly seen in the short wavelength region from 0.3 μm to 2 μm or so. At a given wavelength, the absorptance does not show any variations with different polarization angles, which changes from 0° to 90°, suggesting the polarization independence of the metamaterial solar absorbers. This is can be understood by the identical behavior between TE (i.e., ψ = 90°) and TM (i.e., ψ = 0°) waves at normal incidence due to the geometric 4-fold symmetry of the double-sized metamaterial structure. For any given polarization with ψ between 0° and 90°, the incident electric field E can be always decomposed into TE and TM polarized waves, resulting in the polarization independence. Therefore, it is crucial to maintain the 4-fold symmetry for designed metamaterial absorbers to achieve polarization independence.

 

Fig. 8 Contour plot of the spectral absorptance of the double-sized metamaterial solar absorber as a function of wavelength λ and polarization angle ψ at normal incidence (θ = 0°). The metamaterial absorber shows polarization independence.

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6. Spectral absorptance of multi-sized metamaterial solar absorbers

We have demonstrated that, by using two tungsten patches at different sizes, the absorptance of the metamaterial absorber could be further enhanced in a broader spectral region compared to the single-sized ones. Is that possible to further broaden the band with enhanced absorptance by multiple-sized patches? Figure 9 shows the spectral absorptance at normal incidence for multiple-sized metamaterials with three or four different patch sizes, in comparison to that of double-sized metamaterial absorber. The patch width values are w₁ = 250 nm, w₂ = 300 nm, and w₃ = 350 nm for the 3 by 3 patch array, and w₁ = 250 nm, w₂ = 300 nm, w₃ = 350 nm, and w₄ = 400 nm for the 4 by 4 patch array. The patches are arranged in a 3 by 3 or 4 by 4 array to be diagonally symmetric. Clearly, with additional larger patch sizes, the high-absorptance band can be further broadened to longer wavelength compared to that of the double-sized metamaterial. This is because the MP resonance wavelength increases with strip width, and additional MPs can be excited at the longer wavelengths with larger patches. However, the absorptance values starts to decrease. This can be explained by the fact that, with more patch sizes, the filling fraction of each patch size becomes less, which leads to less confined solar energy when MP is excited under each patch. As a result, there exists a trade-off between broadening absorption band and achieving high absorption values when more patch sizes are used to design the metamaterial absorbers.

 

Fig. 9 Comparison on the spectral normal absorptance between the double-sized metamaterial absorber and multi-sized ones with three or four different tungsten patch widths. The insets depict how to arrange the different patches such that they are diagonally symmetric. The patch width values are: w₁ = 250 nm, w₂ = 300 nm, w₃ = 350 nm, and w₄ = 400 nm.

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7. Concluding remarks

In the present work, we have numerically designed selective solar absorbers made of metamaterial nanostructures consisting of periodic tungsten square patches on a SiO₂ thin film and a tungsten thin film. High absorptance in the visible and near-infrared region and low emittance in the mid-infrared can be achieved at normal incidence from the single-sized metamaterial absorbers. The physical mechanisms responsible for the high absorption include the excitations of SPP, MP and CMP modes as well as the intrinsic bandgap absorption of tungsten have been elucidated in detail along with the geometric effects on the absorptance spectra. The absorptance can be further enhanced to be close-to-unity for single-sized metamaterial solar absorbers with optimized geometric parameters such as grating height or spacer thickness, and in a broader spectral region with double-sized metamaterial absorbers. The spectral absorptance of the designed double-sized metamaterial absorber is higher than 0.95 in the wavelength region from 0.6 μm to 1.8 μm, while the spectral emittance is lower than 0.04 from 4 μm to 20 μm in the mid-infrared. The total solar absorptance of the metamaterial absorbers could be more than 88% at normal incidence, while the total normal emittance is around 3% at the absorber temperature of 100°C, resulting in the excellent performance as selective solar absorbers with the total photon-to-heat efficiency around 86% under one sun condition. In addition, the effects of incidence angle and polarization angle have been studied and the results show the direction-insensitive and polarization-independent behaviors of the designed metamaterial solar absorbers. The multi-size effect on the absorptance of the metamaterial absorbers is also investigated, and a trade-off between high absorptance and broad absorption band with multiple patch sizes is identified. The design of perfect metamaterial solar absorbers here would be beneficial to enhance the performance of solar energy harvesting and conversion systems.