1. Introduction

Two-dimensional materials (2DMs) have attracted extensive attention as plasmonic materials due to their remarkable electrical and optical performance over the past decade [1]. Among these 2DMs, graphene is the most popular one since its isolation by the Manchester group in 2004 [2]. Graphene is a single-layered carbon atoms arranged in a honey comb lattice. In the terahertz and infrared regimes, graphene can support surface plasmons [3–7], which have been widely utilized to design novel optoelectronic devices, such as sensors [8], field effect transistors [9], polarizers [10], modulators [11], photodetectors [12], and absorbers [13].

Although graphene has the highest carrier mobility among 2DMs, its zero or near-zero band gap limits its applications that require high on-off ratio and strong light-matter interaction [14]. As an alternative 2DM, black phosphorus (BP) has also attracted enormous attentions because of its unique optoelectronic properties, such as thickness-dependent tunable band gap [15], high carrier mobility and carrier density [16]. Besides, the monolayer BP shows in-plane anisotropic properties due to its hexagonal lattice with a puckered structure formed by phosphorus atoms [17,18]. Recently, various structures and methods have been investigated to strengthen the light-BP interaction in the BP-based metamaterials in the terahertz and infrared range [19–29]. In [19], plasmonic excitations were realized in BP, and the plasmon frequency was related to the carrier concentration and effective couplings between conduction band and valence band. Liu et al. designed and theoretically demonstrated localized surface plasmon in monolayer BP using periodic nanoribbon and nanopatch [20]. Qing et al. proposed and investigated a metamaterial perfect absorber consisting of a BP monolayer, a photonic crystal, and a perfect electric conductor (PEC) to enhance light absorption at terahertz frequencies [21]. Han et al. exploited the behaviors of anisotropic plasmons inside BP in different systems including individual nanoribbon, vertically offset paired ones, and nanoribbon/sheet hybrid system [22]. Fang et al. numerically proposed a periodic bowtie structure based on BP, in which the localized surface plasmons can be excited in the BP nanoantennas at terahertz regime [23]. Wang et al. proposed an infrared absorber composing of multilayered BP sandwiched between dielectric layers, in which the incident infrared light can be efficiently dissipated by increasing the number of BP layers [24]. Afterwards, Wang et al. further designed a dual-band absorber consisting of orthogonally stacked BP nanoribbons, which absorbs incident infrared light at two frequency bands with polarization-independent property [25].

However, most of the above BP-based structures have the disadvantages of weak absorption rate with low doping concentration, or require relatively complicated fabrication technique. Moreover, the resonance band cannot be tuned effectively and flexibly. This is mainly because the plasmon resonances in individual doped and patterned BP are relatively weak, which limits its anisotropic potentials. Therefore, Nong et al. investigated the hybridization of surface plasmons between graphene and BP in the strong coupling regime [30]. Hong et al. combined graphene and BP plasmonic properties to realize polarization-dependent perfect absorption [31]. Both strong and anisotropic plasmonic absorption are numerically demonstrated in the graphene-BP hybridization structures. Nevertheless, absorption bandwidths of the mentioned structures are often narrow due to the single resonance generated from graphene and BP.

In this paper, we propose an anisotropic infrared plasmonic broadband absorber based on graphene-BP multilayers. Utilizing the advantages of graphene and BP plasmons, the absorber shows both strong and anisotropic plasmon resonance that are not available in either individual graphene or BP layer. By stacking multilayer graphene-BP pairs, the absorption band can be broadened with absorption rate larger than 90%. Furthermore, absorption spectrum can be tuned by geometric parameters and doping levels. The physical mechanism is also revealed by electric field distributions. To illustrate the significance and novelty of the proposed structure, main characteristics of state-of-the-art absorbers based on 2DMs are listed in Table 1 for comparisons.

Table 1. Comparisons between plasmonic absorbers at infrared frequencies.

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2. Models and method

Here, we propose and investigate a three-layer graphene-BP-sandwiched absorber (GBPSA) based on sub-wavelength plasmonic configuration as shown in Fig. 1. The proposed GBPSA consists of stacked graphene-BP bilayers supported by Al₂O₃ layers with a relative permittivity of ~3.2 [32]. The monolayer graphene and BP are separated by a hexagonal boron nitride (hBN) layer, which is used as an insulating spacer to maintain higher carrier mobility of graphene and BP and prevent carrier transport between graphene and BP. The optical parameter of lossless hBN in simulations is obtained from the literature [33]. The thickness t₂ is much smaller than the wavelength in the entire range of interest, thus the top dielectric layer has a negligible influence on the absorption spectrum. However, it prevents the top graphene-BP bilayer from harmness induced by environment, which enable GBPSA to work in different circumstances with high stability for a long period. The finite element method (FEM)-based frequency domain solver of the commercial software COMSOL is used to investigate the broadband absorption properties of GBPSA. Floquet periodic boundary conditions are applied in the x- and y-axis directions. The incident infrared light is imposed from the top surface of structure. PEC boundary condition is applied in the bottom surface. Triangular meshes of user-defined size are used for graphene and BP layers due to their localized enhanced electromagnetic fields. Tetrahedral meshes are applied for the remaining domains of the structure. The absorption rate of the absorbers can be expressed as A = 1−T–R, where transmission T = |S₂₁|², reflectance R = |S₁₁|², and S represents the scattering parameter, respectively. Owing to the PEC used in the substrate, the transmission T is equal to zero in the infrared frequency range. As a result, absorption rate expression can be simplified as A = 1−R. In this work, we assume graphene and BP as 2D conductive surfaces with zero thickness, which requires much less discrete meshes surrounding graphene and BP [34].

 

Fig. 1 Schematic of the proposed three-layer graphene-BP-sandwiched absorber (GBPSA) in (a) perspective view and (b) cross-section view. t₁ is the thickness of hBN layer between graphene and BP. t₂ and d are the thickness of the upper three Al₂O₃ layers and bottom Al₂O₃ layer, respectively. w₁, w₂ and w₃ are the widths of graphene-BP nanoparticles in different layers (Three layers can be ranked in arbitrary order). p is the periodicity of the periodic GBPSA structure.

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Kubo formulas are commonly used to describe the surface conductivity of graphene σ(ω) as below [35]:

For BP model, its surface conductivity can be calculated by employing a simple semi-classical Drude model as [19]

For single layer BP, we have $\gamma =\frac{4a}{\pi }$eVm, $\Delta$ = 2 eV, ${\eta }_c=\frac{{\hslash }^2}{0.4m_0}$,${\nu }_c=\frac{{\hslash }^2}{1.4m_0}$. The electron doping is chosen as $n_s$ = 2$\times$10¹³ cm⁻², while the relaxation rate is chosen as $\eta$ = 10 meV. $a$ denotes the scale length of the BP and $\frac{\pi }{a}$ denotes the width of the Brillouin Zone.

3. Results and discussions

In order to elucidate the anisotropic absorption properties of the proposed GBPSA, light absorption rate of single-layer GBPSA with different nanoparticle widths is plotted in Fig. 2(a). Both transverse electric (TE) and transverse magnetic (TM) polarized infrared incidence could excite the localized surface plasmons in the finite length of graphene-BP nanoparticles [20]. However, due to different effective mass along different crystal directions in BP, the optical loss under TM and TE incidence shows obvious differences. When the incident electric field is along the armchair direction of BP crystal (TE), the absorption peaks are red-shifted from 16.21 μm to 19.91 μm as w increases from 105 nm to 145 nm, with near-unity absorption. On the other hand, the resonant wavelength is red-shifted from 14.01 μm to 17.31μm under TM incidence, whose electric filed is along the zigzag direction of BP. The redshift is mainly attributed to the increase of the effective length for carrier vibration inside graphene-BP nanoparticles. The perfect absorption occurs when the impedance matching condition between the GBPSA and free space is satisfied. The width-tunable anisotropic absorption capacity of GBPSA could be utilized to design reflective polarizers or polarized light filters working in different frequency regions.

 

Fig. 2 Absorption rate of (a) single-layer and (b) three-layer GBPSA. (c) Optical losses inside different materials of GBPSA under TE incidence. The parameters are w₁ = 135 nm, w₂ = 115 nm, w₃ = 125 nm, p = 250 nm, t₁ = 5 nm, t₂ = 100 nm and d = 1.65 μm, under normal incidence. The width of single layer graphene-BP is 115 nm.

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By stacking graphene-BP nanoparticles of different widths layer by layer, the bandwidth can be significantly improved as shown in Fig. 2(b). For TE polarization, three-layer GBPSA has broadband absorption with a 90% absorption bandwidth of 2.01 μm, from 17.11 μm to 19.12 μm. The center wavelength λ_c can be obtained by λ_c = (λ _- + λ ₊)/2 = 18.12 μm, where λ _– and λ ₊ represent the low and upper wavelength edges of 90% absorption, respectively. The fractional bandwidth, the ratio of the absolute bandwidth to the center wavelength, is about 11.1%. For TM polarization, the corresponding bandwidth is 2.91 μm, from 13.71 μm to 16.62 μm. The center wavelength λ_c is 15.17 μm and the fractional bandwidth is 19.2%.

For comparison, Fig. 2(b) also plots the absorption rate in a three-layer BP-sandwiched structure without graphene. The plasmonic resonance of BP with low doping is relatively weak for both TE (2.3%) and TM (41.3%) polarization. The polarization-dependent capability is attributed to the anisotropic plasmonic dispersion of BP. Meanwhile, the absorption rate of three-layer graphene-sandwiched structure without BP is also shown in Fig. 2(b). Three strong resonances in graphene are observed with absorption rates of 84.2%, 92.1% and 86.3%, respectively. The optical response is insensitive to the incident polarization due to the in-plane isotropic surface conductivity of graphene. Accordingly, the advantages of graphene and BP are combined in the GBPSA configuration, which reveals both strong and anisotropic plasmon resonances.

As can be observed in Fig. 2(c), both the losses in BP and graphene contribute to the total absorption of GBPSA. Due to the strong plasmon resonance in graphene, the incident infrared light could be highly concentrated around graphene layers. BP layers are placed in the vicinity of graphene, so the confined and enhanced electromagnetic fields penetrate BP and dissipate in the lossy material. Therefore, the enhanced absorption inside BP is significant considering the weak plasmon resonance in individual BP as shown in Fig. 2(b). However, the loss in graphene is a little stronger than that in BP, which can be mainly attributed to the relatively larger surface conductivity of graphene than BP in the frequency range.

In order to reveal the physical mechanism of anisotropic broadband absorption in GBPSA, the electric field distributions of three-layer GBPSA are plotted in Fig. 3. For TE polarization, at the resonance wavelength of 18.8 μm as shown in Figs. 3(a) and 3(d), the incident wave could excite carriers to oscillate along the finite particle side (y-axis) and induce an independent electric dipole response both in graphene and BP layers. The dipole electric responses inside graphene layer and BP layer have an opposite sign, which will generate antiparallel surface current on the graphene and BP plane. Therefore, the incident electromagnetic fields are focused around the edges of the 1st graphene-BP layer and induce the effects of near field enhancement and energy concentration. The energy assumption inside lossy material such as graphene and BP can be calculated by the following formula:

 

Fig. 3 Electric field distributions of three-layer GBPSA, (a)-(c) are the cross-section views and (d)-(f) are the top views, where w₁ = 135 nm, w₂ = 115 nm, w₃ = 125 nm, p = 250 nm, t₁ = 5 nm, t₂ = 100 nm and d = 1.65 μm.

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On the contrary, for λ = 15.8 μm as shown in Fig. 3(e), few enhanced near field appears in graphene-BP nanoparticles since this wavelength is far away from all the plasmonic resonance wavelengths of these three graphene-BP particles in the TE absorption spectrum. However, for TM polarization, λ = 15.8 μm is exactly right the resonance wavelength for the 1st graphene-BP particle. Hence, the incident TM wave could induce the carriers in graphene and BP to vibrate in the x-axis direction as shown in Fig. 3(f). The graphene and BP layers are positioned in the vicinity of each other, the evanescent near field scattered by graphene and BP nanoparticle is considerably intense compared to the exciting field, which leads to the strong coupling of scattered field from each graphene-BP nanoparticle. Consequently, the superposition of the inverse electromagnetic fields induced by electric dipole that is excited by the incident light makes contributions to the suppressed reflectance. Thus, perfect absorption occurs at the resonance wavelength of 15.8 μm for TM polarization as demonstrated in Fig. 2(b).

According to Figs. 2 and 3, therefore, it is predicted that an anisotropic ultra-broadband absorption spectrum can be further realized by increasing the number of stacked layers in the proposed GBPSA and changing the dimensions of each graphene-BP particle.

By altering the geometric dimensions of GBPSA configurations, the anisotropic broadband absorption spectra can be tuned as demonstrated in Fig. 4. The resonance properties of Fabry-Perot resonator formed by the graphene-BP layers and PEC substrate are highly sensitive to the thickness variation of the dielectric spacer layers, which facilitate the tuning of absorption band by slightly changing the spacer thickness, as shown in Figs. 4(a) and 4(b). When the thickness d<1.6 μm for TE polarization in Fig. 4(a), the middle absorption peak is relatively weak with an absorption rate below 90%. As d increases from 1.6 μm to 2.2 μm, the middle absorption peak is strong enough and the broad bandwidth decreases because the left and right peaks become closer to the middle peak. When d>2.2 μm, the broadband is divided into two separated bands. For TM polarization in Fig. 4(b), as d increases from 1 μm to 2.2 μm, the bandwidth increases at first and then decreases. There is an optimal thickness of the dielectric layer, at which the absorption bandwidth reaches the maximum. When d>2.2 μm, the bandwidth is divided into two narrow bands as well as the TE polarization. This is attributed to the increase of the effective thickness of these multiple Fabry-Perot resonators, which are strongly coupled between each graphene-BP layer and the gold substrate. The phenomenon can be utilized for designing both dual-band and broadband anisotropic absorbers for novel optoelectronic devices.

 

Fig. 4 Absorption spectra as a function of geometric parameters: (a) and (b) for dielectric thickness d, (c) and (d) for insulator thickness t₁, (e) and (f) for periodic spacing p, under TE and TM incidences, respectively. Other parameters are as in Fig. 2(b).

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The coupling strength between graphene and BP layers depends on the thickness of the insulator layer t₁. Therefore, Figs. 4(c) and 4(d) plot the absorption spectra versus t₁. In Fig. 4(c), the absorption band tends to be slightly red-shifted when t₁ increases from 2 nm to 13 nm, and then keeps almost unchanged when t₁ still rises up to 40 nm. The bandwidth remains almost constant with the variation of t₁. In Fig. 4(d), the absorption band is obviously red-shifted as t₁ increases from 2 nm to 42 nm, while the bandwidth gradually increases at first until it reaches the maximum and then decreases afterwards. This originates from the fact that for TM polarization, the plasmonic resonance in BP layer is stronger than TE polarization as shown in Fig. 2(b). As a consequence, the coupling between graphene and BP layer for TM polarization is more sensitive to the thickness of the insulator than TE polarization.

As a critical parameter of GBPSA configuration, periodic spacing p affects the anisotropic broadband performance, thus the absorption spectra under different values of p are illustrated in Figs. 4(e) and 4(f). When the periodic spacing is less than 280 nm, the coupling between adjacent graphene-BP nanoparticles is relatively strong, so the bandwidth above 90% absorption rate is broad for both of TE and TM polarizations. As p increases from 280 nm to 1000 nm, the absorption efficiency of GBPSA has a dramatic drop, while the broadband is divided into two narrow and weak absorption bands. The absorption efficiency is related to the filling factor of graphene-BP nanoparticles, which can be defined as F = w/p. F becomes smaller as p increases, then the intensity of the electric field concentration is significantly weaken inside graphene-BP nanoparticles, so the absorption efficiency remarkably drops.

It is an effective way to tune the absorption spectrum via w, d, t, p and other dimensions in the design of a GBPSA with specific requirements. Besides, according to graphene and BP model formulas in Eqs. (1)-(4) and Eqs. (5)-(8), the surface conductivities of graphene and BP are determined by μ_c and n_s, respectively, which represent the doping level of corresponding material. Therefore, the performance of GBPSA can be manipulated by changing μ_c and n_s as shown in Fig. 5. From a practical point of view, we choose μ_c between 0.4 eV and 0.8 eV which are verified by experiments in previous works [36]. A maximum theoretical carrier density was demonstrated to be n_s = 2.6 × 10¹⁴ cm⁻² [16,37], so we use a moderate value between 10¹³ cm⁻² and 10¹⁴ cm⁻². In Figs. 5(a) and 5(b), as μ_c increases from 0.4 eV to 0.8 eV, the absorption spectra for both TE and TM polarizations are blue-shifted obviously, with center wavelength changing from 20.1 μm to 14.8 μm for TE polarization, and from 16.6 μm to 12.7 μm for TM polarization. The bandwidth above 90% absorption gradually decreases from 2.8 μm to 0.8 μm for TE polarization, and from 3.9 μm to 2.3 μm for TM polarization. This is in accordance with the previous work [38], which can be explained by the higher conductivity of graphene with increasing μ_c,, resulting in electromagnetic energy loss and absorption. On the other hand, for monolayer BP, the plasmonic resonance wavelength λ_p can be calculated as ${\lambda }_p\propto \sqrt{w/n_s}$, where w is the length of nanoparticles [19]. Therefore, while w is fixed, as n_s increases from 10¹³ cm⁻² to 10¹⁴ cm⁻², the center wavelength is blue-shifted from 18.7 μm to 15.2 μm for TE incidence as shown in Fig. 5(c). For TM incidence, the bandwidth tends to be narrower and narrower as n_s increases as demonstrated in Fig. 5(d). The explanation is similar with that of graphene above. Generally, both real and imaginary parts of BP surface conductivity become higher with the increase of n_s [24,25], which means more optical loss and absorption in BP.

 

Fig. 5 Absorption spectra as a function of doping level: (a) and (b) for different μ_c of graphene, (c) and (d) for different n_s of BP, under TE and TM incidences, respectively, where w₁ = 135 nm, w₂ = 115 nm, w₃ = 125 nm, p = 250 nm, t₁ = 5 nm, t₂ = 100 nm and d = 1.65 μm.

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In the discussions above, anisotropic broadband absorption properties of GBPSA were investigated under normal incidence. However, the tolerance of wide incident angles is necessary for broadband infrared absorbers in the application of novel photonic devices. Hence, we further investigate the optical response of GBPSA under oblique infrared incidences. In Fig. 6(a), the dependence of absorption spectra on the incident angles θ for TE polarization is demonstrated. It is observed that GBPSA has stable absorption bandwidth with θ up to 64°, while the middle absorption peak decreases gradually. When θ>70°, the broadband absorption is divided into two absorption peaks because the middle absorption peak drops sharply. For TM polarization shown in Fig. 6(b), as θ increases from 0° to 62°, the absorption bandwidth will be wider slightly while the broadband absorption rate remains larger than 80%. The peak absorption rate maintains larger than 70% with a sufficient bandwidth of 4 μm while the incident angle increases up to 74°. Besides, for both TE and TM incident infrared lights, the center wavelengths of the absorption band keep almost unchanged with the increase of θ. The excellent angular stability is attributed to the tightly confined surface plasmon resonances in graphene-BP nanoparticles, which is not sensitive to the incident angle.

 

Fig. 6 (a) and (b) are absorption spectra as a function of incident angles θ for TE polarization and TM polarization respectively. (c) is absorption spectra for various polarization angles β under normal incidence. The parameters are w₁ = 135 nm, w₂ = 115 nm, w₃ = 125 nm, p = 250 nm, t₁ = 5 nm, t₂ = 100 nm and d = 1.65 μm.

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To further investigate the polarization dependence of GBPSA, we plot the absorption spectra under incidence of different polarization angles β. The polarization angle of TE incidence is assumed to be 0°. As can be seen from Fig. 6(c), when β = 0°, the spectrum is the same as the TE polarization in Fig. 2(b). As β increases to 90°, the incident light turns out to be a TM incidence. Thus, the absorption spectrum is the same as the TM polarization in Fig. 2(b). When the polarization angle is between 0° and 90°, the incident electric field has both x and y- components. Therefore, the hybridization mode will induce the electrons in BP to vibrate in both armchair and zigzag directions. As a consequence, localized plasmon resonances in armchair and zigzag directions can be excited simultaneously with reduced light intensity at corresponding x and y-directions. Interestingly, when the polarization angle is between 30° to 60°, three absorption peaks can be obviously observed in the absorption spectra. This is mainly because the resonance modes in three graphene-BP particles are not closely positioned together.

4. Conclusions

In conclusion, we have theoretically proposed an anisotropic broadband infrared absorber based on multilayer graphene-BP nanoparticles in a sandwiched metamaterial configuration. Combining the advantages of graphene and BP localized surface plasmons, the GBPSA shows highly anisotropic plasmonic responses that are available in neither individual graphene nor BP sheets. By stacking slightly different widths of graphene-BP nanoparticles in a sandwiched configuration, the absorption bandwidth can be increased due to the different resonant modes closely positioned together. The absorption spectra can be tuned either by the geometric parameters of GBPSA or by the doping levels of graphene and BP. Moreover, GBPSA can tolerate a wide range of incident angles for both TE and TM polarizations. Our approach can be used in the design of novel infrared sensors and tunable reflective polarizer in symmetrical structure.