## Abstract

Two-dimensional (2D) black phosphorus (BP) with direct band gap, bridges the characteristics of graphene with a zero or near-zero band gap and transition metal dichalcogenides with a wide band gap. In the infrared (IR) regime, 2D BP materials can attenuate electromagnetic energy due to losses derived from its surface conductivity. This paper proposes an IR absorber based on 2D BP metamaterials. It consists of multi-layer BP-based nano-ribbon pairs, each formed by two orthogonally stacked nano-ribbons. The multi-layer BP metamaterials and bottom gold mirror together form a Fabry-Perot resonator that could completely inhibit light transmission to create strong absorption through the BP metamaterials. Unlike previously reported BP metamaterial absorbers, this new structure can operate at two frequency bands with absorption > 90% in each owning to the first-order and second-order Fabry-Perot resonant frequencies. It is also polarization independent due to the fourfold rotational structural symmetry. To our best knowledge, this is the first report on using BP metamaterials in an absorber that operates independent of polarization and in dual bands.

© 2017 Optical Society of America

## 1. Introduction

Two-dimensional (2D) materials with atomic-scale thicknesses, such as graphene, transition metal dichalcogenides (TMDs) and black phosphorus (BP), have shown outstanding potentials in many fields such as photonics, optoelectronics, imaging, and telecommunications [1–8]. The applications include sensors [1], surface plasmon polariton (SPP) waveguides [2, 3], phase shifters [4], transistors [5, 6], and absorbers [7, 8]. Among these materials, graphene has the highest carrier mobility, but its zero or near-zero band gap limits its applications that require high on-off ratio and strong light-matter interaction [9]. Although some treatments such as silicon doping have been adopted to open a band gap in graphene [10], they introduce other limitations [11]. By comparison, TMDs such as MoTe_{2}, MoSe_{2}, and MoS_{2} offer noticeable band gaps, resulting in extraordinary on/off ratios (e.g. >10^{8}) [5]. However, the moderate carrier mobilities of TMDs are also a limitation in their applications [6]. Reported methods to increase the carrier mobility in TMDs have involved extremely difficult preparation processes [9].

As an alternative 2D semiconductor, BP has been exfoliated from bulk-BP by mechanical or chemical method [9, 12–14]. Recently, BP has been extensively investigated in many areas, owing to its unique optoelectronic properties, such as high carrier mobility (up to 50,000 cm^{2} V^{−1} s^{−1} in bulk at 30 K) [15], thickness-dependent direct band gap (from ~0.3 eV in bulk to ~2 eV in monolayer) [16–19], and a maximum theoretical carrier density of *n*_{s} = 2.6 × 10^{14} cm^{−2} [20, 21]. However, the potential of BP in absorbers has been underutilized. BP-based saturable absorbers have been developed in Q-switching and mode-locking to achieve pulse emission of lasers. These studies demonstrated the BP fabrication method by exfoliation [9, 12, 14], and its application as a saturable absorber [22–24]. Nevertheless, the patterning of BP materials as metamaterials has rarely been examined. Lately, an infrared (IR) absorber has been reported, which is composed of 2D BP metamaterials sandwiched between dielectric layers and mounted on a full reflective gold mirror. Thus, incident IR light can be efficiently dissipated [25]. This BP metamaterial absorber, however, has the disadvantages of polarization dependence and single frequency band. Nevertheless, it opens a door for further research.

In this work, a BP-based 2D metamaterial is proposed as an absorber that has dual frequency bands and polarization independence. The absorber is mounted on an optically thick gold mirror, which inhibits all transmission, and forms a Fabry-Perot resonator to enhance the light-matter interaction. The trapped electromagnetic energy is dissipated through the loss in BP material. The simulation results show that the proposed absorber can operate at two frequency bands with absorption greater than 90% in the IR regime. Owing to the fourfold rotational symmetry of the structure, the absorber is polarization-independent. Moreover, the mechanisms of the dual frequency bands and polarization independence are illuminated by using Fabry-Perot resonance and field distribution, respectively.

## 2. Electrical model of a 2D BP layer

The atoms in a 2D BP layer are arranged to form a puckered hexagonal honeycomb structure with ridges due to sp^{3} hybridization, which leads to strong in-plane anisotropic electrical and optical properties [25–28]. The permittivity tensor of monolayer BP takes the form [25]

*ε*

_{1},

*ε*

_{2}, and

*ε*

_{3}are the effective permittivities along the

*x*,

*y*, and

*z*axes, respectively. Mathematically,

*ε*(

_{i}*i*= 1, 2, 3) can be derived as [25–28]where

*ω*is the incident light frequency,

*ε*

_{0}is the vacuum permittivity,

*d*is the thickness of the BP layer, and

*j*= √−1.

*ε*

_{i}_{,}

*= 5.76 is the relative dielectric constant of BP [28], and σ*

_{r}*is the in-plane conductivity of BP (σ*

_{i}*$\cong 0$). From Eq. (2), it can be deduced that the in-plane anisotropy of BP is mainly caused by σ*

_{z}*. Only the real part of σ*

_{i}*causes electromagnetic loss [25]. σ*

_{i}*can be approximated by the Drude model as [25–28]Here,*

_{i}*ħ*is the reduced Planck constant,

*η*(eV) describes the relaxation rate, and the Drude weight

*D*is given aswhere

_{i}*e*is the electron charge,

*n*is the carrier density, and

_{s}*m*denotes the in-plane effective electron masses near the

_{i}*Γ*point within the Hamiltonian model, which is stated as

*α*of monolayer BP is 2.23 Å and π/α is the width of Brillouin zone [26], the parameters in Eq. (5) can be set as

*γ =*4

*α/π η*

_{c}=

*ħ*

^{2}/(0.4

*m*

_{0}),

*v*=

_{c}*ħ*

^{2}/(0.7

*m*

_{0}), and the band gap Δ = 2 eV by assuming a standard electron rest mass

*m*

_{0}= 9.10938 × 10

^{−31}kg and a fixed

*η*= 10 meV. Figure 1 shows

*σ*(

_{i}*i*= 1, 2) in a broad spectrum for different

*n*for monolayer BP. It can be found that

_{s}*σ*is approximately proportional to

_{i}*n*,

_{s}*σ*

_{1}is greater than

*σ*

_{2}for all

*n*, and rates of real parts are greater than that of imaginary parts. We should notice that only the real part of

_{s}*σ*results in electromagnetic loss.

_{i}## 3. Theoretic model and research method

As shown in Fig. 2(a), the upper part of the proposed absorber cell consists of *N* layers (*N* = 5) of BP-based nano-ribbon pairs embedded in a dielectric layer. In the middle of the cell there is a layer of the same dielectric material, with a fully reflective gold mirror at the bottom. Here, each nano-ribbon pair (magenta and navy) consists of two orthogonally stacked nano-ribbons, and the adjacent pairs are separated by the dielectric (cyan). It is assumed that each BP nano-ribbon has a width of *w*, infinite length, and identical effective permittivity. The periodicity *p* of the absorber lattice in Fig. 2(b) is 0.5 μm. There is also a top dielectric layer (cyan, thickness: *t*_{1}) in Fig. 2(c) to prevent environmental damage to the BP material [29, 30]. It should be noticed that, as shown in Figs. 2(b) and 2(c), the ridges of the BP nano-ribbons along *x*-direction and *y*-direction are perpendicular to *y*-axis and *x*-axis, respectively. In this work, we fixed the refractive indices (*n*) of all dielectrics to 1.7, and the total thickness (*t*) of the proposed absorber to 8.5 μm. Therefore, the thickness *t*_{0} of the dielectric in the middle of the absorber is 6.0 μm when *t*_{1} is 0.5 μm and *N* = 5, as shown in Fig. 2(d). Due to the strong in-plane anisotropic electrical and optical properties of BP, the responses from BP-based absorbers without special treatment are usually polarization dependent. In our design, the orthogonally stacked BP nano-ribbon pairs are the key to achieve polarization independence.

The absorber was simulated using CST Microwave Studio, where the thickness of BP nano-ribbon can be set as 1 nm [28] and were meshed using fine grids. Periodic boundary conditions were used in the *x* and *y* directions. The incident plane wave points downwards normal to the top surface of the absorber, with the magnetic and electric fields *H _{y}* or

*E*perpendicular to the

_{y}*x*-

*z*plane (i.e., transverse-magnetic (TM) or transverse-electric (TE) polarizations, respectively). The wavelength (

*λ*)-dependent absorption

*A*(

*λ*) is expressed as 1 −

*R*(

*λ*) −

*T*(

*λ*), where

*R*(

*λ*) = |

*S*

_{11}(

*λ*)|

^{2}and

*T*(

*λ*) = |

*S*

_{21}(

*λ*)|

^{2}are the spectral reflection and transmission, respectively. Owing to the gold mirror used in this platform, the light transmission is inhibited completely. Therefore, the absorption can be simplified as

*A*(

*λ*) = 1 −

*R*(

*λ*).

## 4. Results and discussion

In the first simulation for *N* = 5, *n _{s}* was set to be 1.0 × 10

^{14}cm

^{−2},

*t*

_{1}was fixed to 0.5 μm, and BP is monolayer. The wavelength-dependent absorption spectra for

*w*= 0.5, 0.38, 0.34, 0.26, 0.22, 0.16, and 0.14 μm are simulated and shown in Fig. 3(a). The two main resonant peaks are labeled by λ

_{1}and λ

_{2}. For the first resonance λ

_{1}, the peak absorption intensity monotonically decreases with decreasing

*w*due to the larger distance between the nano-ribbons and weaker inter-ribbon coupling, and it is greater than 90% while

*w*≥ 0.22 μm. Moreover, this absorption peak exhibits a redshift with decreasing

*w*. For the second resonance λ

_{2}, the absorption peak intensity first increases and then decreases by decreasing

*w*due to the over coupling and under coupling [31] between the nano-ribbons with oversized and undersized

*w*. An optimal

*w*range of 0.22 ~0.26 μm was found in this work, which both

*A*(λ

_{1}) and

*A*(λ

_{2}) are greater than 90%, allowing the device to function as a dual-frequency IR absorber. The spectra for

*w*= 0.22 μm are colored red in Fig. 3(a). When

*w*= 0.5 μm, the structure is no longer a

*metamaterial*because all orthogonal BP nano-ribbons now merge into a 2D slab extending infinitely in the

*x-y*plane.

The thickness of dielectric (cyan) *t*_{1} also has effects on the absorption performance. For *w* = 0.22 μm with the same *N* and *n _{s}* as above, the wavelength-dependent absorption spectra for

*t*

_{1}= 0.1, 0.2, 0.5, 0.7, and 0.9 μm are shown in Fig. 3(b). It can be noticed that there are little change in peak position for both λ

_{1}and λ

_{2}because the total thickness

*t*is almost constant (there are significant change for λ

_{1}). For the first resonance λ

_{1}, the absorption peak increases with decreasing

*t*

_{1}. The absorption peak for the second resonance λ

_{2}firstly increases and then decreases. The optimal

*t*

_{1}range of 0.2 ~0.5 μm was found in which both

*A*(λ

_{1}) and

*A*(λ

_{2}) are greater than 90%.

The value of carrier density *n*_{s} directly determines the surface conductivity of BP, which results in electromagnetic energy loss and absorption. In general, surface conductivity becomes higher with increasing *n*_{s} [25]. When *N* = 5, *t*_{1} = 0.5 μm, *w* = 0.22 μm and BP is monolayer, the simulated wavelength-dependent absorption spectra are shown in Fig. 3(c) for *n*_{s} = 1.0 × 10^{12}, 1.0 × 10^{13}, 5.0 × 10^{13}, 7.0 × 10^{13}, and 1.0 × 10^{14} cm^{−2}. At *n*_{s} = 1.0 × 10^{12} cm^{−2}, despite the poor absorption, we can notice that λ_{1} exhibits a redshift with decreasing *n*_{s}, while the position of λ_{2} is essentially invariant. It can also be seen that the absorption performance is enhanced with increasing *n*_{s}, and the proposed absorber with *n*_{s} = 1.0 × 10^{14} cm^{−2} can achieve dual-band absorption (see the red line). While an even higher absorption performance is possible with *n*_{s} > 1.0 × 10^{14} cm^{−2}, other factors such as optical phonons or electron-electron scattering might become dominant.

In addition, Δ is thickness-dependent. From bulk to monolayer BP, Δ monotonically increases from ~0.3 eV to ~2.0 eV. For trilayer and bilayer BP, their band gaps are ~1.07 eV and ~1.3 eV, respectively [18]. When *N* = 5, *t*_{1} = 0.5 μm and *w* = 0.22 μm, the simulated absorption spectra for BP material with trilayer, bilayer and monolayer thickare shown in Fig. 3(d). It can be found that the absorption peak intensity for λ_{1} monotonically increases with decreasing Δ. Moreover, despite absorption peak intensity for λ_{2} is very small when *n*_{s} = 1.0 × 10^{13} cm^{−2}, it can be discovered that the absorption peak positions for both λ_{1} and λ_{2} show blueshift with decreasing Δ.

After fixing *n _{s}* = 1.0 × 10

^{14}cm

^{−2}, Δ = 2.0 eV,

*t*

_{1}= 0.5 μm and

*w*= 0.22 μm, now we investigate the relationship between the absorption performance and the number of BP-metamaterial layers (

*N*). The simulated absorption spectra for

*N*= 1–5 are shown in Fig. 3(e). It is obvious that the peak position of λ

_{1}exhibits a redshift with decreasing

*N*, while that of λ

_{2}is almost constant. It can be seen that both absorption peaks become stronger with increasing

*N*, which is caused by the electromagnetic energy loss in the BP arising from the real part of σ

*. In other words, the BP metamaterials attenuate the electromagnetic fields, as shown in Fig. 4 for*

_{i}*N*= 5. Therefore, we conclude that better absorption performance can be achieved by increasing

*N*.

In this design, the gold mirror does not allow the transmission of incident IR light, and it forms Fabry-Perot resonance with the BP metamaterials. Therefore, the reflected energy is dissipated through electromagnetic losses in the 2D BP and leads to strong absorption. In the Fabry-Perot resonance condition [32], the simulated resonant wavelength satisfies λ* _{k}* = 4

*nH*/(2

*k*− 1), where

*k*is the resonant order,

*n*is the refractive indices, and

*H*is the thickness of resonator. λ

_{1}and λ

_{2}are respectively the first-order and second-order resonance wavelengths, while other weaker resonances (e.g.

*k*= 3, 4, and 5) were also observed as shown in Fig. 3. Due to the multilayer structure of the absorber, there is no single fixed

*H*for all resonant wavelengths. For different

*k*, values of the variable equivalent thickness

*H*' (

*H*' should less than or approximately equal to

*t*) are given in Table 1. Now the reason for the redshift of the peak position at λ

_{1}(see Figs. 3(a) and 3(c)) is apparent:

*H*' increases with decreasing

*w*and

*n*

_{s}, due to the weaker inter-ribbon coupling. Analogously, the reason for the almost constant position at λ

_{2}in Figs. 3(b), 3(c) and 3(d) is that the equivalent thickness

*H*' keep almost invariable.

Finally, we examine the absorptions for both TM and TE polarizations shown in Fig. 3. The excellent agreement between these two polarizations indicates that the polarization-independent performance is achieved. This is due to the presence of the fourfold rotational symmetry about the *z*-axis, not only the geometric shape in the unit structure but also the atomic arrangement in the BP material [33, 34] as shown in Fig. 2. These symmetries lead to the symmetrical field distributions induced by TM and TE polarizations, as shown in Fig. 4.

## 5. Conclusions

In summary, we have theoretically proposed an IR absorber operating in dual-frequency bands with polarization independence, based on a metamaterial of orthogonal BP nano-ribbons arranged with fourfold rotational symmetry. The dual frequency bands were achieved by optimizing the nano-ribbon width *w* and carrier density *n*_{s}. Two absorption peaks were found at the first- and second-order Fabry-Perot resonant wavelengths. Simulation results demonstrated that increasing the number of BP absorber layers can significantly affect the intensities of both absorption peaks. Moreover, the polarization-independent property is attributed to the fourfold rotational symmetry in this BP metamaterial. The proposed BP-based absorber can be used as key components in the applications of biosensing, imaging, and communications systems.

## Funding

National Natural Science Foundation of China (NSFC) (Nos. 61661012, 61461016, 61361005, and 61561013); Natural Science Foundation of Guangxi (2017JJB160028); Program for Innovation Research Team of Guilin University of Electronic Technology; Dean Project of Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing.

## Acknowledgments

Y. J. gives special acknowledgment to Prof. Yuanbo Zhang from Fudan University for his help.

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