1. Introduction

Absorbers at optical frequencies are of great interest in various applications including photovoltaics and thermophotovoltaics, photodetectors, color filters, and thermal light sources [1]. Enhanced narrow-band absorption is attainable by exploiting various types of configurations, including Salisbury screens, Fabry–Pérot cavities, guided resonances in the photonic crystals, and localized surface plasmon resonances in the metallic nanoparticles [2]. For some specific applications such as sensing, switching, and imaging, multiband, or broadband performance is required [3,4]. Circuit analog absorber (CAA), metamaterial absorber (MA), and frequency selective surface (FSS) absorbers are three main categories of broadband absorbers [5].

Resonant-based absorbers suffer from limited bandwidth of 10-30% [6]. One way of enhancing the bandwidth of the resonant structures is by combining several closely packed narrowband resonance frequencies [7]. This purpose can be achieved in various ways, including super-unit coplanar structures, a single structure with multiple resonating sections, and stacking metal-insulators [8]. In this regard, four all-dielectric cylinders with different radii are considered in the unit cell of a terahertz microstructure reflector [9]. Moreover, fractal cubic nanowires are proposed at the visible spectrum [10]. Specifically considering the graphene-based structures, 2D circular disks with various radii are considered in the design of a wideband invisible cloak [11]. Furthermore, CVD-grown single-crystal graphene Sierpinski carpet fractals are used to experimentally realize a broadband photodetector [12]. In another design, multiple graphene ribbons are stacked on an optically thick golden substrate [13].

In this research, graphene-covered fractal cylindrical nanoparticles are used as the unit cells of a broadband optical absorber. Previously, we had proposed a tunable optical absorber and a single negative (SNG) metamaterial reflector using assemblies of graphene-coated spherical nanoparticles [14,15] and now we are extending the research to cylindrical particles due to the geometrical flexibility via the length of the cylinders. The degree of freedom provided by the lengths of the cylinders is successfully used in the bandwidth enhancement using the fractal concept.

Another exotic property of the graphene-coated cylinders is the excitation of localized surface plasmons in their top, bottom, and lateral surfaces with different frequencies [16]. An innovative approach for obtaining a multispectral absorption response is to use different resonance modes in a single structure [17]. For instance, in an all-metal plasmonic perfect absorber, excitation of guided modes with different orders is responsible for the dual-band absorption [18]. Moreover, in another broadband structure, fundamental and second-order surface plasmons of graphene-based complementary metasurface are effectively excited at two resonance frequencies [19]. Interestingly, we have attained broadband performance by exciting multiple resonances of various orders in different sections of the unit cell.

To further improve the performance, the fractal geometry is re-arranged as an oligomer. Clusters of nanoparticles can provide much more degrees of freedom in manipulating the optical response [20]. Toroidal dipole resonances, magnetic Fano resonances (FRs) illustrated as hot and cold spots in the nano-gaps, and induced charge transfer plasmons (CTPs) are a few applications of such molecules [21–23]. These assemblies have a great potential to enhance the light trapping of organic solar cells [24,25] and now we will extend their application to the infrared spectrum by using the plasmonic mode engineering in graphene. Importantly, experimental realizations of nanoparticle clusters show a good agreement with the simulation results [26]. It is worth noting that with the current fabrication technology, it is feasible to experimentally realize graphene-coated cylinders with the diameters around tens of nanometers [27,28].

The paper is organized as follows. In section 2, the unit cell of our proposed broadband absorber is introduced. The underlying mechanism of the performance is discussed based on impedance matching technique and the excitation of multiple first and second-order plasmonic resonances. Moreover, the substrate height is introduced as a means for exciting the desired number of Fabry-Perot cavity resonances with high absorption. Later, various aspects of the practical realization of the absorber are clarified. Finally, the concluding remarks are mentioned in section 3.

2. Results and discussions

The structure under analysis is a periodic arrangement of graphene-coated hollow cylinders. In this structure, the fractal nano-cylinders are assembled as an oligomer and an optical reflecting cavity is used as the substrate. The side and top views of the unit cell are illustrated in Fig. 1 and it is assumed that a plane wave with the wavenumber k normally illuminates the structure. The numerical simulations are carried out with the frequency domain solver of CST commercial software package. This solver performs based on the finite integrating technique (FIT). Moreover, unit cell boundary conditions are applied in x-y directions to account for the infinite planar array and Floquet ports are considered along the open z-directions. The absorption (A) of the structure is calculated using the simulated reflectance (R) and transmittance (T) as $A = 1 - T - R$ [7], where $T = {|{{S_{21}}} |^2}$ and $R = {|{{S_{11}}} |^2}$. The scattering parameters S₁₁ and S₂₁ respectively are the simulated reflection and transmission coefficients. Since the particles have resided on top of a metal-backed dielectric layer, the transmitted waves are fully blocked.

 

Fig. 1. The unit cell of the proposed broadband absorber constructed by graphene-coated fractal nano-cylinder oligomer. (a) Side-view of the particles, (b) side view of the substrate, and (c) top-view of the unit cell. The wavenumber of the incident wave is denoted by k.

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2.1 Broadband continuous/discrete spectrum absorption enhancement

To obtain a wide absorption spectrum, the geometrical and optical parameters of our proposed structure in Fig. 1 are optimized. The geometrical parameters are r₁=90 nm, h₁=1200 nm, r₂=36 nm, h₂=400 nm, h_s=1.2 μm, g = 20 nm, and p = 312 nm. Graphene cover is considered as a thin sheet with the temperature T = 300 °K, relaxation time τ = 3.29e − 2 ps, and chemical potential μ_c=0.6 eV. The relative permittivity of the substrate is considered ε=2. Also, as another factor for broadband absorption enhancement, the distance of adjacent unit cells are reduced to 0.2 g for stronger coupling. The absorption, reflection, and transmission spectrum of the optimized structure are shown in Fig. 2.

 

Fig. 2. The absorption, reflection and transmission spectrum of the proposed broadband absorber shown in Fig. 1. The optimized geometrical and optical parameters are r₁=90 nm, h₁=1200 nm, r₂=36 nm, h₂=400 nm, h_s=1.2 µm, g = 20 nm, p = 312 nm, T = 300 °K, τ = 3.29e − 2 ps, and μ_c=0.6 eV.

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The fractional bandwidth of 88.67% (21.11-54.74 THz) for the absorption above 90% is obtained by our proposed structure. In comparison to the broadband tunable metamaterial graphene absorber with − 10 dB bandwidth spanning from 22.02 to 36.61 THz on the same substrate [29], our proposed structure has a wider bandwidth. The ultra-thin electrical length of the former is 0.09λ at the lowest frequency while the latter is 0.169λ at the lowest frequency by neglecting the reflecting ground [29]. From the result of Fig. 2, it is inferred that the absorber has as a fourth-order band-pass response with the plasmonic resonance frequencies of f₁=22.83 THz, f₂=29.80 THz, f₃=40.00 THz, and f₄=50.05 THz. To provide further insight into the mechanism of the absorption, the real and imaginary parts of the retrieved surface impedance Z are illustrated in Fig. 3(a). The normalized surface impedance is extracted via [30,31]

 

Fig. 3. (a) The real $(\Re )$ and imaginary $(\Im )$ parts of the retrieved surface impedance and (b) the phase of S₁₁ parameter in the absorption band to illustrate the impedance matching with the free-space impedance.

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The absorption spectrums of the two sub-structures of the fractal geometry contributing to the bandwidth enhancement are shown in Fig. 4. As expected, the long center cylinder contributes to the low-frequency performance while the oligomer constructed by the short cylinders provides a high-frequency absorption. The mutual interaction of these two structures results in broadband performance, as discussed before.

 

Fig. 4. The absorption spectrums for the two sub-structures of the fractal geometry contributing to the bandwidth enhancement of the proposed absorber in Fig. 1. The blue curve represents the contribution of the center rod with the optimized geometrical and optical parameters as in Fig. 2. The red curve shows the same information for the oligomer in which the center cylinder has an identical size with the surrounding ones.

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Due to the large height difference in the elements of the unit cell, it is expected that the structure performs differently for various incident angles. As Fig. 5 shows, the broadband absorption maintains up to the oblique incidence of around 35°. Moreover, due to fourfold symmetry, the performance is polarization insensitive. Interestingly, for the grazing angle of 90°, the operational mode of the structure is switched to the dual-band reflector with an efficiency of 80%.

 

Fig. 5. Absorptivity of structure in Fig. 1 for different incident angles of the illuminating wave. The optical and geometrical parameters are the same as Fig. 1.

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As a final comment, by considering the designed unit cell on top of the thicker substrates, depending on the height of the substrate, multiple orders of Fabry-Perot cavity resonances are excited and enhanced considerably. The realization of double, triple, and quadruple absorption bands by engineering the substrate height is illustrated in Fig. 6.

 

Fig. 6. Realization of the multi-band absorber by engineering the substrate height. The substrate heights h_s=8.4, 10.8, and 13.2 µm respectively result in the double, triple, and quadruple absorption bands.

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2.2 Revealing the underlying mechanism through surface plasmon engineering

In order to reveal the underlying absorption mechanism through surface plasmon engineering, each finite length graphene-wrapped cylinder can be considered as the combination of two circular disks and one cylindrical shell. Using full electrodynamics calculations, it can be proved that dipolar plasmon resonance of a 2D graphene disk leads to strong near-field enhancement [33]. Plasmonic resonances originate from charge accumulation and can trap the electromagnetic waves, to be dissipated in the ohmic losses of graphene [34]. In the disk assemblies, dipolar hybridization can excite the higher-order modes which are not excitable with a single disk [35]. Similarly, in an infinite graphene-coated cylinder, optical absorption resulting from the plasmonic resonances can be manipulated by the cylinder radius [36]. The necessary condition for plasmonic excitation in graphene wires is the induction of electric currents in the azimuthal direction [37]. For each mode order $m \ne 0$, 2 m nodes could be observed in the electric field, along the cross-section of the wire [38]. In the infinite length graphene-coated cylinders, there is no variation of the electric field along the length of the cylinder. By considering finite-length graphene-based rods, the standing waves are also formed. In the following paragraphs, the above-mentioned resonances will be illustrated. It is interesting to note that although absorber design using metallic nanoparticles is a well-known technique [39], the superiority of our proposed structure is using 2D graphene material which leads to plasmonic resonances of various orders in the different sections of a single structure.

The x, y, and z components of the electric fields in four absorption resonances in Fig. 2 are illustrated in Fig. 7. As the figure shows, the bandwidth is enhanced by exciting the localized surface plasmons of multiple orders in different sections [40,41]. The excitations of localized surface plasmons are inferred via the charge oscillations which are exhibited by the change in the sign of the induced electric fields. The red color shows the positive values and the blue color represents the negative values. In this figure, the second-order plasmons are excited in the center cylinders of Figs. 7(a), (g), and (j). In these figures, the plasmons are also modulated in the longitude direction, which is occurred due to the finite length of the cylinder. In contrast, in the center cylinders of Figs. 7(d)-(e), there are not variations of the electric fields along the lengths of the cylinders which resemble an infinite length cylinder. In the subfigures (c), (f), (i), and (l) the dipolar localized surface plasmon resonances of the top (bottom) disks of the center cylinders are excited. Moreover, in Figs. 7(d), (g), and (j), the quadruple plasmons of the top and bottom disks are excited. The same interpretations can be used for the surrounding cylinders. Localized surface plasmons in all surrounding cylinders are not excited simultaneously and each one contributes diffidently in the bandwidth enhancement. Note that these figures are associated with the first excited mode. By investigating the second mode, these interpretations will correspond to y-directed electric fields.

 

Fig. 7. The electric field distributions of various absorption peaks f_N for N = 1 − 4, in x, y, and z directions to illustrate the contribution of LSPRs of different parts in the broadband absorption.

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From the experimental point of view, in our proposed structure, cylindrical particles with various heights, side dimensions, and aspect ratios are exploited, which are realizable due to the current fabrication technologies [10,42]. For instance, the particles can be thinned down to the desired height with a defocused ion beam [43]. Moreover, graphene can be wrapped around particles with various shapes and sizes due to the van der Waals force [44]. The nano-gaps can be tuned with sacrificial poly-Si filling, which is then removed with highly selective poly-Si wet etchant [45].

To bias the graphene-based periodic structures with the electrostatic biasing scheme, it is essential to electrically connect the elements. In this regard, cross-like elements linked through graphene strips have been proposed [46]. Moreover, complementary structures are used as easier alternatives [47]. The single-gate biasing scheme by employing connecting strips can be exploited for the 3D structures [48]. To prevent the complexities arising from the biasing network, chemical doping is the best biasing technique for our proposed structure. This method does not pose any limitation in the practicability of the structure. On the other hand, since the real and imaginary parts of the graphene surface impedance Z_s, shown in Fig. 8, drastically change with the chemical potential, the absorption rate, and the resonance frequency is not fixed for various chemical potentials. Therefore, the same performance cannot be attained for various chemical potentials. Note that the graphene surface impedance is the inverse of its surface conductivity. The data are extracted from the built-in data of the software.

 

Fig. 8. (a) The real $(\Re )$ and (b) the imaginary $(\Im )$ parts of the graphene surface impedance for various chemical potentials.

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3. Conclusion

To provide multiple plasmonic resonances in a unit cell of the broadband absorber, the fractal concept is applied to the lengths and radii of an oligomer constructed by graphene-coated nano-cylinders. Each graphene-coated cylinder supports two kinds of multipolar LSPRs excited on the top-bottom disks and cylindrical shells. Using our proposed absorber, the fractional bandwidth of 88.67% for the absorption rate above 90% is obtained at the optical regime. The operation of the absorber is based on the wide-band impedance matching technique which is a well-known absorber design. Finally, the proposed structure can perform in the reflecting and multiband modes respectively by using grazing angle illumination and higher-order Fabry-Perot cavity resonances.