1. Introduction

The subwavelength optical antennas support field enhancement within their structure achieved through precise configuration adjustments [1–7]. Incident plane waves induce oscillating electric fields via internal dipole moments [8–10], leading to magnetic resonances in nonmagnetic high-refractive-index materials [11]. This yields pronounced electric and magnetic multipole resonances (often referred to as Mie resonances) at the subwavelength scale facilitated by a refractive-index mismatch of an antenna and its surrounding medium and results in strong resonances with low radiative losses [12–14].

High refractive index (denoted as $n$) enables strong interaction of the antenna with the light, while low absorption losses (denoted by $k$) foster extended resonance lifetimes. By finely adjusting the phase and magnitude of scattered light, it becomes possible to design antennas characterized by high refractive indices for the precise control of light propagation [3,15,16]. Semiconductor materials with high refractive indices and small absorption losses (i.e., illumination below the bandgap) are ideal candidates for metasurface material platforms. The antenna effective polarizability plays a pivotal role in harnessing distinctive phenomena, such as the directional scattering (Kerker effect) arising from the interference of electric and magnetic dipole resonances [17,18] and bound and quasi-bound states resulting from the collapse of Fano resonance linewidth [19,20]. However, the attainable antenna dimensions are constrained by an upper limit on the refractive index, providing the material’s optical losses are low. This limit is determined by the relationship between the bandgap (denoted as $E_g$) and the refractive index $n$ and often expressed as $n^4 E_g=95$ eV (aka the Moss rule [21]). Although many materials adhere to the Moss rule, certain materials possess refractive indices surpassing this relation. Iron pyrite FeS$_2$ has an energy band gap of 1.03 eV and experimental refractive index of 4.53 at the wavelength of 1687 nm and represents a ‘super-Mossian’ material that surpasses the relation [22,23]. In contrast to other high-refractive-index materials, such as silicon and transition metal dichalcogenides, the resonance characteristics within iron pyrite metasurfaces in the infrared region have not received extensive research attention.

When a light wave scattered by an antenna propagates in the direction of the incident light, this phenomenon is termed forward scattering [24]. Conversely, the term backward scattering is used when the scattered wave propagates in the opposite direction to that of the incident light wave [25]. The Kerker effect occurs when the backward scattering is effectively suppressed [17], leading to the propagation of the scattering wave in the forward direction only and resulting in a high (and virtually infinite) forward-to-backward ratio. This intriguing effect requires a zero difference in the phase of the backward scattered waves and equal magnitudes of the electric dipole and magnetic dipole polarizabilities [26,27]. In turn, near-zero forward scattering occurs at the $\pi$ phase difference of the waves scattered by the electric and magnetic dipoles and can be utilized for similar purposes.

Light scattered by individual antennas combined in a periodic array interferes and gives rise to the so-called lattice (or collective) resonances [28–32]. Rayleigh anomalies are distinct features corresponding to different diffraction orders of scattering waves and are determined by the array period, the refractive index of the surrounding medium, and the angle of light incidence [10,33]. Thus, for example, altering the spacing between elements of a transdimensional metasurface with a periodic antenna array allows for efficient control of lattice resonances [34].

Bound states in the continuum (BICs) are associated with a collapse of resonance linewidths [35]: they exist in the continuum of radiation waves in the system and cannot be excited with far-field sources [36,37]. A symmetry-protected BIC can be observed in a system that possesses certain symmetrical properties, preventing the mode from coupling with external radiation or escaping to the far field and, thus, confining it to the continuum of radiation waves. This type of BIC arises due to specific symmetries inherent in the system, leading to a collapse of the resonance linewidth and the formation of an exceptionally high-quality-factor resonant state. In the particular case of the antenna array, resonance linewidth collapses at normal incidence and results in an infinite quality factor in the symmetry-protected BIC [38,39].

When symmetry is disrupted by an asymmetric parameter within a unit cell, the symmetry-protected BIC transforms into a quasi-BIC [40]. This quasi-BIC exhibits a finite resonance width and a high quality factor and can be realized by breaking an in-plane inversion symmetry of a unit cell. The non-zero material absorption results in BIC being smeared out [19]. Consequently, formulating a design strategy becomes essential to realize BICs while accounting for optical losses in materials with high refractive indices. Metasurfaces with suitable aspect ratios of antennas and square-lattice array parameters can mitigate losses in the proximity to BIC [41]. A square-lattice array in the sub-diffraction regime cancels out this leakage by the vertical (out-of-plane) orientation of magnetic dipoles. These out-of-plane magnetic dipoles originate from the in-plane circular current and charge oscillations in the horizontal and vertical directions, generating electric and magnetic dipoles, respectively. In this case, the coupling occurs between out-of-plane magnetic dipoles and a mode bound to the plane of the array and forms symmetry-protected BIC.

The ability to efficiently control the transmission, reflection, and absorption of light waves is a characteristic of certain engineered nanostructures and metasurfaces, which incorporate antenna arrays [18,24,42]. These metasurfaces enable the novel manipulation of electromagnetic waves. This tunability and associated unique ability in manipulating the light can be harnessed in applications of thermal emission [43,44]. Kirchhoff’s law relates the absorptance of a metasurface to its emissivity: Specifically, the law states that in equilibrium, for a given material and at a particular temperature, the absorptance of the material at a certain wavelength is equal to its emissivity at the same wavelength. Thus, one can tailor metasurface absorptance and enhance its emissivity by utilizing Kirchhoff’s law [43,45]. However, Kirchhoff’s law is an approximation that holds under certain conditions, such as thermal equilibrium and diffuse radiation, and applies to a specific wavelength and temperature.

Unlike systems where emission strongly depends on the coherence of an incident light beam, the emissivity of a thermal metasurface is predominantly determined by the presence of spatially incoherent currents within the material [46]. In other words, the thermal metasurface’s ability to emit heat (i.e., thermal radiation) is not reliant on coherent incident beams of light. Instead, it involves the interaction of spatially incoherent currents within the material itself. The induced currents of spatially extended modes can produce directional emission patterns. Localized and delocalized modes in the array of high-refractive-index scatterers [47] are examples of metasurface modes and strongly affect the directional emission.

Efficient thermal emission from metasurfaces in the mid-infrared region finds application in infrared sensing as many molecules have absorptance lines in this range [48–51]. The spectral range of a mid-infrared thermal emitter can be finely tuned, spanning from narrow- to broadband, through the careful engineering of metasurface parameters. This enables the excitation of either narrow or broad resonances, each offering distinct operational bandwidths [45,52]. The imaginary part of the material’s permittivity is responsible for the light absorption in the metasurface, and one of the efficient approaches to control emission is tuning the absorbance of the antenna material. Adjusting the imaginary part of the material’s permittivity by altering doping concentration provides the ability to control emission spectra.

We investigate the properties of thermal metasurfaces composed of iron pyrite within the mid-infrared spectrum. Our examination of the absorption cross-section in the context of an isolated iron pyrite antenna validates the presence of multipole resonances within the mid-infrared domain. We demonstrate an enhanced forward-to-backward scattering ratio (generalized Kerker effect) and confirm the in-phase and compensating amplitudes of multipole polarizability resonances in such iron pyrite antennas. Additionally, when employing truncated-cone antennas made of iron pyrite arranged periodically, we observe distinct peaks in absorptance. This periodic arrangement induces lattice resonances, with the absorptance peaks corresponding to peaks in emissivity. These outcomes hold practical significance and can be effectively applied in real-world scenarios, such as mid-infrared gas sensing applications. We show the coupling between electric-dipole and magnetic-dipole lattice resonances and their Rabi splitting, which occurs due to the asymmetric shape of the truncated cone. We show enhancements of an additional out-of-plane magnetic dipole resonance due to the asymmetric shape of the truncated cone. For oblique incidence, we show the out-of-plane magnetic dipole resonances excited at wavelengths longer than in-plane electric and magnetic dipole resonances. These out-of-plane resonances result in symmetry-protected BIC at the normal incident. We demonstrate that the transformation of a symmetric shape antenna (disk) into an asymmetric shape (cone) has the ability to shift resonance wavelength. The intermediate asymmetric shape (truncated cone) shows a higher emissivity enhancement in the metasurface.

2. Isolated antenna

We design antennas made of high-refractive-index materials with relatively low absorption and analyze multipole resonances that they support. While most high-refractive-index materials follow the Moss rule, we choose iron pyrite, whose refractive index is 40% higher than the Moss-rule limit. Understanding and controlling specific multipole resonances within the metasurface is of paramount importance. These resonances can be harnessed to achieve desired optical functionalities, including wavelength selectivity and enhanced absorption. Thus, the meticulous design of individual antennas within a thermal metasurface is crucial for tailoring their optical properties to meet specific requirements. Fine-tuning parameters such as geometry and material composition allow us to control the resonant characteristics of these antennas.

The spectral position of electric dipole and magnetic dipole resonances is strongly influenced by the geometric characteristics of the scattering elements. In the case of a spherical antenna, the magnetic dipole resonance is excited at a wavelength longer than the one of electric dipole resonance. In a disk antenna, the electric or magnetic dipole resonances can be excited at longer wavelengths, and these excitations are influenced by the antenna’s aspect ratio (the antenna’s height-to-diameter ratio). By modifying the aspect ratio of a disk antenna while keeping its height constant, we can change the spectral positions of the electric dipole and magnetic dipole resonances.

We analyze antennas of the truncated-cone shape, which have been realized in Ref. [23] and take the experimental data of iron pyrite permittivity from there. The emergence of these truncated cone antennas can be attributed to the inherent imperfections and complexities inherent in the nanofabrication process, particularly during etching. It is the sidewall tilts that give rise to their unique truncated cone morphology, diverging from the initially intended disk shape. We perform numerical simulations using a frequency-domain solver in CST Studio Suite (Fig. 1). We observe electric and magnetic multipole resonances in the absorption cross-section of an isolated truncated-cone antenna in free space. The truncated-cone antenna has the flexibility to control multipole resonances and excite them with different strengths at different wavelengths due to its asymmetric form. The possibility to vary the top diameter of the truncated cone provides an additional degree of freedom. The low aspect ratio of truncated-cone antenna results in electric dipole resonances excited at longer wavelengths. According to the Kerker effect, when multipoles maintain an in-phase (zero phase difference) and equal amplitude relation, backward scattering is suppressed with higher forward scattering. Thus, we observe the generalized Kerker effect as a higher forward-to-backward scattering ratio. We see that this nanostructure facilitates highly directional scattering forward, enhancing the efficiency of nanophotonic devices, optical components, and sensors.

 

Fig. 1. Scattering from isolated truncated-cone antenna made of iron pyrite (FeS$_{2}$): Absorption cross-section and forward-to-backward scattering ratio in free space. The plane wave incident normally to the antenna’s top surface. Inset: Schematic of the antenna and the corresponding dimensions (1.5-$\mu$m bottom diameter, 0.82-$\mu$m top diameter, and 0.38-$\mu$m height). Due to the low aspect ratio of the isolated truncated-cone antenna, the electric dipole resonance is excited at the wavelength $\approx 3.5~\mu$m. Magnetic dipole resonance is excited at the wavelength $\approx 2.95~\mu$m. Quadrupole and higher-order resonances are excited at wavelengths shorter than 2.3 $\mu$m. In-phase and approximately compensating polarizability amplitudes of the antenna’s multipole resonances at the wavelength $\approx 3~\mu$m result in the highest forward-to-backward scattering ratio and generalized Kerker effect.

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3. Antenna array

The interaction between neighboring antennas and their arrangement in the lattice plays a vital role in shaping the overall metasurface response. It is not just the individual antennas that matter; the collective effect of the entire array can significantly enhance resonances. Light scatters from individual antennas within the lattice and propagates along the array, leading to mutual coupling, and this collective coupling among antennas gives rise to the lattice resonances. The characteristics of these lattice resonances depend on various factors, including the geometry of the array, the constitutive parameters of the antennas, the refractive index of the surrounding medium, and the periodicity of the structure. The aspect ratio of a truncated cone antenna can be adjusted to achieve the excitation of electric dipole lattice resonance at longer wavelengths.

In designing a thermal metasurface with truncated cone antennas, iron pyrite is selected as the material due to its high refractive index. To enable the observation of enhanced emissivity in the mid-infrared region, we utilize an experimentally determined imaginary part of permittivity, approximately 0.5, obtained from previous research. When light incident on a low-loss high-refractive-index material, it induces an oscillating electric dipole moment within the antenna, giving rise to a circular displacement current responsible for the antenna’s magnetic response. The periodic arrangement within the lattice results in an effective polarizability altered with respect to the isolated antenna polarizability, influencing the characteristics of lattice resonances.

In Fig. 2, we demonstrate the impact of periodicity on a truncated-cone iron pyrite antenna within a rectangular lattice array. Changing the periodic arrangement of the metasurface allows us to adjust absorptance and emissivity at various wavelengths. Field enhancement in high-refractive-index antenna results in strong lattice resonances in a periodic array. In the case of the truncated-cone antenna, the electric dipole is excited at the wavelength of approximately 3.4 $\mu$m, and the magnetic dipole is excited at the wavelength of approximately 2.9 $\mu$m for the dense array with $P_x = P_y = 1.6~\mu$m. First, we change the periodicity in the $x$-direction, and for $x$-polarized light, this causes a shift of magnetic dipole along the Rayleigh anomaly associated with the first diffraction order (Fig. 2(a)). The resonance transforms from being an isolated-antenna resonance to a multipole-specific lattice resonance. Magnetic-dipole lattice resonance is shifted toward the electric dipole resonance, but rather than overlapping, these resonances couple to each other and exhibit a phenomenon known as Rabi splitting (as shown in Fig. 2(a)). The interaction between the magnetic-dipole lattice resonances and electric dipole resonance results in the exchange of resonance order: after experiencing Rabi splitting, the magnetic-dipole lattice resonance is excited at longer wavelengths than the electric dipole resonance.

 

Fig. 2. Resonances in the periodic array of truncated-cone antennas made of iron pyrite (FeS$_{2}$): (a) Emissivity and (b) Reflection corresponding to the changes in the $x$-periodicity $P_x$ and fixed $y$-periodicity $P_y = 1.6~\mu$m. Dimensions of the truncated-cone antenna are the same as in Fig. 1, and the antennas are placed on top of a substrate with refractive index $n_s =$ 1.41. For the smallest period $P_x = 1.6~\mu$m, electric dipole resonance is excited at a longer wavelength $\approx 3.4~\mu$m, and magnetic dipole resonance is excited at a shorter wavelength $\approx 2.9~\mu$m. Magnetic dipole shifts following the Rayleigh anomaly and transforms to magnetic-dipole lattice resonance because of the dominating lattice properties. For $P_x \approx 2.2~\mu$m, the resonances are expected to overlap, but because of the mode coupling, Rabi splitting occurs (marked by the yellow elliptical circle). (c) Emissivity and (d) Reflection corresponding to the changes of the $y$-periodicity $P_y$ and fixed $x$-periodicity $P_x = 1.6~\mu$m. Magnetic dipole resonance remains almost unchanged due to the fixed $x$-periodicity. Electric dipole resonance is excited at the wavelength $\approx 3.4~\mu$m for $P_y = 1.6~\mu$m and shifts following the Rayleigh anomaly transforming to electric-dipole lattice resonance. The cyan dashed and dotted lines in all panels are the Rayleigh anomalies in the substrate with refractive index $n_s =$ 1.41, corresponding to the fixed periodicity and varying periodicity, respectively. The blue dashed-dotted lines in all panels are the Rayleigh anomalies for the varying periods in free space. The Rayleigh anomaly for the fixed periods in free space is below the spectral range of consideration.

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Modifying the periodic arrangement of the antennas in the metasurface provides us with a versatile means to precisely tune absorptance and emissivity across a wide range of wavelengths. By strategically manipulating the lattice parameters, such as periodicity and geometry, we can tailor the metasurface’s optical response to meet specific design requirements. This level of control over absorptance and emissivity allows for the engineering of thermal properties and optical characteristics in a variety of applications.

The interplay between these resonance spectral positions leads to changes in reflection within the metasurface. In Fig. 2(b), one can see that the reflection from the array exhibits a remarkable feature. Specifically, the reflection is near zero in the spectral region between the electric and magnetic dipole resonances (wavelength range approximately from 2.9 to 3.4 $\mu$m). This appears to contradict the conventional situation when the reflection between the electric and magnetic dipole resonances is high because their scattered waves are out of phase in the direction opposite to the incident wave. However, the excitation of the electric quadrupole resonance at the wavelengths shorter than the excitation of the magnetic dipole explains the decrease in reflection down to near-zero values. In fact, the contribution from the electric quadrupole is so pronounced that it results in the increase of reflection between the magnetic dipole and electric quadrupole resonances (wavelength range approximately from 2.3 to 3.9 $\mu$m) and the decrease in reflection outside this range (wavelength range approximately from 2.9 to 3.4 $\mu$m and wavelength shorter than 2.3 $\mu$m).

Next, we analyze the properties of the lattice resonances while altering the periodicity $P_{y}$ and keeping $P_{x}$ constant. In this case, electric dipole resonance is mainly affected by the lattice changes and transforms from an isolated-antenna electric dipole resonance to a multipole-specific (electric dipole) lattice resonance. In this case, the dipole resonances do not overlap as the electric-dipole lattice resonance moves away from an isolated-antenna magnetic dipole (Fig. 2(c)). The reflection properties for this case of changing periodicity $P_y$ are similar to those we observed before changing periodicity $P_x$ (Fig. 2(d)): there is a dip and near-zero reflection between the dipole resonances because of the substantial contribution of the electric quadrupole at shorter wavelengths.

4. Out-of-Plane Magnetic Dipole Resonances and BIC

One particularly intriguing outcome of such antenna design is the potential for BICs. These states, which occur within the continuous spectrum of radiation, can lead to remarkable properties such as perfect absorption or enhanced emission at specific wavelengths. Harnessing BICs offers exciting possibilities for designing metasurfaces for mid-infrared thermal applications.

The enhancement of resonant effects supported by the metasurface can substantially boost its emissivity, leading to a more efficient thermal emitter. By harnessing these stronger resonances, we can significantly improve the metasurface’s ability to radiate thermal energy effectively. In this Section, we investigate the impact of antenna asymmetry within the periodic array constituting the metasurface. To perform this analysis, we transform a disk antenna into a cone by reducing the top diameter of the antenna (Fig. 3). Thus, the initial symmetry of the disk is broken, and the antenna gradually transforms into a cone. In this analysis of the antenna’s asymmetry, we consider its top surface as the plane of incidence and illuminate it with a TE-polarized plane wave (‘TE’ stands for Transverse Electric). An angle equal to zero ($\theta = 0$) corresponds to the wave incident normal to the antenna’s top surface. The oblique incidence with the TE polarization includes one additional component of the magnetic field. Consequently, because of the excitation with the out-of-plane magnetic component, varying the oblique incidence angle gives rise to an additional out-of-plane magnetic dipole resonance at longer wavelengths. This out-of-plane magnetic dipole resonance lies on the longer wavelength side of the in-plane electric and magnetic dipole resonances. For disk and truncated-cone antennas in the metasurface, the additional out-of-plane magnetic dipole resonance appears at $\approx 4.35~\mu$m and $\approx 3.9~\mu$m, respectively (Fig. 3(a and b)).

 

Fig. 3. Impact of asymmetric antenna’s shape on the metasurface emissivity: (a) disk and (b) truncated-cone antenna arrays. The angles are varied from 0$^{\circ }$ to 30$^{\circ }$ under obliquely incident TE-polarized light. The bottom diameter and height of the disk and truncated cone are the same as in Fig. 2. The top diameter of the disk antenna is 1.5 $\mu$m. The periodicity of the antenna array in the $x$- and $y$-directions is 1.6 $\mu$m. An out-of-plane magnetic dipole resonance is excited at wavelengths longer than in-plane electric and magnetic dipoles. Truncated-cone antenna arrays show stronger emissivity compared to disk antennas. Moreover, symmetry-protected BIC appears at normal incidence in (a) disk and (b) truncated cone (black circles). The magenta dotted line represents 15$^{\circ }$ oblique incidence as a visual comparison guide. (c) Effect on the additional magnetic dipole resonance due to symmetry breaking transformation of disk antenna: For 15$^{\circ }$ oblique incidence, the antenna’s shape is varied by changing top diameter from 1.5 $\mu$m (disk-shaped antenna) to zero (cone-shaped antenna). Truncated cones, as an intermediate step, show strong emissivity in comparison to disk and cone antennas. The emissivity peak shifts to significantly shorter wavelengths during the transformation. The colorbar is the same for all panels.

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Due to its asymmetric shape and oblique incidence angles, the truncated cone exhibits strong resonance. Consequently, our straightforwardly designed thermal metasurface demonstrates that the emissivity of a truncated-cone-antenna metasurface surpasses that of a disk-antenna metasurface. At normal incidence angles, the resonance linewidth collapses for both the disk and truncated-cone antennas. The phenomenon of resonance linewidth narrowing and collapsing, attributed to symmetry-protected BIC, may undergo distortion owing to the small but finite light absorption in iron pyrite material in the mid-infrared spectral range.

Our metasurface effectively mitigates the leakage tendency associated with symmetry-protected BIC, due to the emergence of an additional out-of-plane magnetic dipole resonance. We demonstrate that a gradual transformation from a disk antenna to a cone, achieved by reducing the top diameter to zero, points out the antenna shape that maximizes resonance enhancement (Fig. 3(c)). For 15$^{\circ }$ oblique incidence, we show that additional out-of-plane magnetic dipole resonance at longer wavelength shifts to shorter wavelength due to the transformation from disk to cone antenna. In the context of a cone shape (with a top diameter of zero), we observe a decrease in emissivity owing to the relatively weaker localization of the antenna mode. However, when we shift our focus to the truncated cone configuration, a distinct and favorable outcome emerges – a notable increase in emissivity.

This phenomenon is particularly intriguing because the truncated cone metasurface exploits the advantages of symmetry-protected BIC. These BICs, inherent to the design, serve as a powerful approach for enhancing the metasurface’s response, especially in mid-infrared applications. By harnessing this property, we can significantly improve the metasurface’s performance in various technological and practical cases within the mid-infrared spectral region.

5. Emissivity enhancement

When it comes to enhancing emissivity, a multifaceted array of strategies comes into play. One approach involves modifying materials using different fabrication procedures to alter their thermal radiation characteristics. Alternatively, seeking out materials with inherently superior emissive properties or employing advanced metasurface designs can improve emissivity. Additionally, the use of nanomaterials and tailored surface coatings can further enhance a material’s ability to emit thermal radiation for specific applications.

5.1 Iron pyrite processing

Iron pyrite caught the attention of researchers because of its potential applications in photovoltaics. This interest stems from the fact that iron pyrite is exceptionally good at absorbing light in the visible range, that is $\alpha >10^5$ cm$^{-1}$ [53]. Several deposition and synthesis methods have been developed, for example, sulfurization [54] and atomic layer deposition [55], resulting in the high absorption coefficient property of iron pyrite in the visible wavelength range and in its utilization in various practical domains. For photovoltaic applications in the visible range, iron pyrite can be fabricated by substituting sulfur atoms with oxygen impurities [56]. For fabricating iron pyrite metasurface operating in an infrared region, a two-step sulfurization process of $\alpha -$Fe$_2$O$_3$ thin film has been experimentally demonstrated [22,23]. Through the two-step sulfurization process, low oxygen impurities concentration, low loss ($k \approx$ 0.75), and high refractive index ($n \approx$ 4.5) can be achieved in the near- and mid-infrared regions. The use of different fabrication techniques empowers researchers and engineers to fine-tune the properties of iron pyrite, making it a versatile material for a range of practical applications and providing the possibility to enhance the emissivity of iron pyrite metasurfaces operating in the mid-infrared range.

5.2 High refractive index

In general, optical antennas exhibit distinct Mie resonances when operating within a refractive index range of less than 1.5 to 2. This is attributed to the limited mode confinement as a result of this refractive index regime, as discussed in Ref. [11]. Within the range of moderate refractive indices of 2 to 3, the resonances are considered weak, as they still exhibit relatively weak mode confinement. On the other hand, semiconductors with refractive indices exceeding 3, such as silicon ($n \approx$ 3.5) and III-V compounds, offer substantial support for strong resonances. Furthermore, refractive indices higher than this threshold are deemed even more favorable. A high refractive index, such as $n \approx$ 4.5 of iron pyrite, enables strong mode confinement, leading to significant resonant enhancement, high-quality-factor resonances, and ultimately renders these materials more practical for a diverse range of applications.

The appropriate imaginary part of the refractive index can span a wide spectrum, but it is essential for it to remain relatively low. This characteristic is exemplified by gallium arsenide, as discussed in Ref. [42], where variations in $\varepsilon ^{\prime \prime }$ of up to 0.5 do not cause any drastic changes in spectra and variations of the accumulated phase. Notably, gallium arsenide exhibits $\varepsilon ^{\prime } \approx 12$ within the relevant spectral range. Therefore, it is reasonable to expect that iron pyrite, with $\varepsilon ^{\prime } \approx$ 20, might tolerate a slightly higher transition value for $\varepsilon ^{\prime \prime }$ while maintaining its optical characteristics.

In pursuit of enhancing metasurface design and optimizing its performance, researchers consistently strive to identify materials with the highest possible refractive indices. This proactive approach not only unlocks the potential for even stronger mode confinement but also results in more pronounced resonant enhancement, superior quality factor resonances, and, ultimately, greater versatility in metasurface applications. By seeking materials with ever-higher refractive indices, we can continually push the boundaries of metasurface technology, enabling more efficient and innovative solutions for a wide array of practical challenges.

A recent study has shown that the primary approach to achieving a high refractive index in materials involves minimizing the disparity between the Penn gap, denoted as $E_p$ (i.e., the average transition energy with $E_p= \hslash \omega _0$, where $\omega _0$ represents the resonant frequency), and the absorption (or band) edge, represented as $E_g$ [57]. However, the study’s findings suggest that, for the majority of tetrahedral materials, achieving a resonant enhancement of the refractive index remains an elusive goal. For example, the energy gap between $E_p$ and $E_g$ for silicon is approximately 3.7 eV. This discrepancy highlights that the attainment of a resonant enhancement in the refractive index for materials, as indicated by the study, might be unattainable [23,58]. In the case of iron pyrite, its high refractive index is primarily attributable to the substantial density of states stemming from the d-band, particularly within the $t_{2g}$ band. This distinctive property positions iron pyrite as a highly promising candidate for designing metasurfaces operating in the infrared spectrum.

5.3 Metasurface design

Using the periodic arrangement of antennas, we demonstrate the possibility of controlling the spectral position of emissivity peaks and their substantial enhancement by exciting and shifting the electric and magnetic dipole lattice resonances. We attain a peak emissivity of approximately 0.57 at the spectral position associated with quadrupole resonances and the most dense array with spacing 1.6 $\mu$m, and around 0.52 at the spectral position corresponding to dipole resonances within our proposed thermal metasurface. This metasurface is situated on a substrate with a refractive index of $n_s = 1.41$. However, it is essential to note that when implementing this antenna array on a plain glass substrate without incorporating metal within the structure, an inherent impedance mismatch arises between the metasurface and the surrounding free space.

We can enhance the maximum emissivity by introducing slightly higher losses in the iron pyrite antenna. To boost the absorptance, we should increase the imaginary permittivity (with a value greater than 0.5) of the iron pyrite, thereby increasing the maximum emissivity. It is important to note that in the low-loss regime (where $k <$ 0.5), we can anticipate linear changes in resonance and, consequently, linear changes in emissivity, as discussed in [59]. Considering the prevalence of heat dissipation, we can gather meaningful results by varying the real permittivity $n$ in the range of 20 to 17, within a wavelength span from approximately 2 to 5 $\mu$m. This should be accompanied by an imaginary permittivity $k$ of at least 0.5. Such a dataset appears well-suited for engineering thermal emission in the mid-infrared wavelength range.

Owing to the straightforward design of the rectangular lattice array featuring truncated-cone iron pyrite antennas without reflecting back-plate, our proposed thermal metasurface exhibits non-zero reflectance and transmission. Further optimizing the metasurface for thermal emission may include eliminating reflectance and transmission from the structure to achieve a highly efficient thermal emitter (similar to [43,60,61]). A widely used approach for reducing transmission is to augment the contribution of the metal within the structure while aligning the structure’s impedance with that of the surrounding medium, thereby reducing reflection simultaneously. We envision that the subsequent inclusion of a high-reflective layer underneath the antenna array can minimize the transmission in the metasurface, similar to how it has been done in conventional designs. Photonic nanostructures with resonant cavities [62] and/or gratings [63,64] can be effective in enhancing the metasurface resonances and response in general. Furthermore, thermal metasurfaces with the metal-dielectric-metal (MDM) design have been a common approach to gain unity absorptance and significantly enhance the emission [46,65,66]. Tuning different parameters in such MDM-based metasurfaces can result in the efficient light control and utilization of optical properties, such as emission directivity [46,67] and spectral bandwidth (narrow or broadband) [48].

6. Conclusion

Designing individual antennas within a thermal metasurface is critical for tailoring their optical properties. The collective effect of the entire array can further enhance resonances, and controlling specific multipole resonances is key for achieving desired optical functionalities. We have designed a thermal metasurface of a high-refractive-index material, iron pyrite, and have shown the excitation of multipole lattice resonances in the mid-infrared range. Based on Kirchhoff’s thermal law, controlling absorptance provides an opportunity to improve the metasurface emissivity. We have demonstrated the shift in emissivity spectra by changing the periodicity of the antenna array constituting the metasurface. Variations of the array periodicity shift electric-dipole and magnetic-dipole lattice resonances along the Rayleigh anomaly in the thermal metasurface. We have shown coupling between electric-dipole and magnetic-dipole lattice resonances and associated with their Rabi splitting.

One intriguing outcome is the potential for bound states in the continuum, offering exciting possibilities for applications in sensing, imaging, and energy harvesting. Effects of oblique incidence have been analyzed by introducing the antenna’s asymmetry and transforming its shape from disk to cone. We have shown that due to the oblique incidence of TE-polarized light, additional out-of-plane magnetic dipole resonances are excited in the antennas. We have demonstrated that the emergence of the symmetry-protected bound state in the continuum is a result of the collapse in the resonant linewidth of the out-of-plane magnetic dipole resonance under normal incidence. Our results have shown that the truncated-cone antenna can be used in more efficient thermal emitters compared to those employing disks or cones.