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Abstract
The polarization independent and non-reciprocal absorption is particularly crucial for the realization of non-reciprocal absorption devices. Herein, we proposed and studied the absorption response of two- and three-layer anisotropic black phosphorus (BP) metamaterials by using the finite-difference time-domain (FDTD) simulation and radiation oscillator theory (ROT) analysis. It is shown that, due to unequal surface plasmon resonant modes excited in zigzag (ZZ) and armchair (AC) directions of the anisotropic BP layer, tunable polarization independent and dependent absorption can be achieved for the proposed multi-layer anisotropic BP metamaterials with AC-AC, AC-ZZ, ZZ-AC, AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ configurations. Especially, the polarization independent absorption also can be realized for odd-layer BP nanostructures. Unlike previous reports, polarization independence only can be achieved in the even-layer BP nanostructure. Moreover, tunable non-reciprocal absorption with the extremely large non-reciprocal degree (NRD) is also found in the case of AC-ZZ and ZZ-AC configurations and AC-ZZ-φ and ZZ-AC-φ configurations. These results may open up the possibility of realizing tunable polarization independent and non-reciprocal plasmonic devices based on 2D materials.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Benefit from the advantages of high charge carrier mobility and atom-scale thickness, two-dimensional (2D) materials have attracted enormous attention in the past years [1,2]. 2D materials show strong light-matter interactions owing to the strong field confinement and low loss [3,4]. Especially, surface plasmons (SPs) have been also found in many 2D materials [5–7]. Based on the atomic arrangement and lattice symmetry, SPs are distinguishable for different kinds of 2D materials [6,8]. The one is the isotropic 2D materials with the isotropic optical conductivity tensor in plane [9,10], such as graphene. In recent years, SPs based on graphene have been reported [11–14]. The other is the anisotropic 2D materials with anisotropic optical conductivity tensor in plane, such as black phosphorus (BP) and Rhenium disulfide [8,15,16]. Compared with the isotropic 2D material, the anisotropic 2D material has one more degree of freedom in-plane, so their properties are more plentiful, such as the linear dichroism and anisotropic plasmons [8,17,18].
BP is a typical anisotropic 2D material, which not only has pure planar anisotropy characteristics, but also shows outstanding photonic and electronic properties [18,19]. Similar to graphene and novel metal materials, BP also can support the propagation of SPs [20–22]. But the atomically thin layer for BP leads to the strong quantum confinement effect [23] and large surface-to-volume ratio [24], which are unmatched by bulk materials [25,26]. In addition, unlike the graphene, the bandgap of BP is layer-dependent, with a single-layer bandgap of 2.0 eV and a multi-layer bandgap of 0.3 eV [27,28]. Thus, the advantages of BP are also attractive for the future device implementation, such as plasmonic sensors [29,30,31], perfect absorbers [15,32], optical storages [33] as well as polarization selectors [8,34]. The previous studies about anisotropic BP were focused on the strong anisotropy related applications. BP with a specific orientation of even-numbered layers can realize polarization-independent optical responses [8]. However, how to realize the polarization independent plasmonic behaviors in odd-numbered BP layers is still a challenge.
In this paper, we propose and analyze six multi-layer anisotropic BP metamaterials to realize the tunable polarization independent and non-reciprocal absorption. Firstly, the absorption and reflection spectra of AC-AC (Both BP1 and BP2 layers are AC directions in the x-axis), AC-ZZ (BP1 is AC direction in the x-axis, and BP2 is ZZ direction in the x-axis), and ZZ-AC (BP1 is ZZ direction in the x-axis, and BP2 is AC direction in the x-axis) configurations are investigated through the FDTD simulation and ROT analysis. Secondly, dependences of the polarization angle, coupling distance, and carrier density on the absorption and non-reciprocal degree in the AC-AC, AC-ZZ, and ZZ-AC configurations are clarified in detail. At last, the odd-numbered layer anisotropic BP metamaterials (AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ configurations) are proposed to realize the polarization independent absorption.
2. Structure and theory
Figure 1(a) shows the schematic diagrams of the AC-AC, AC-ZZ, and ZZ-AC configurations for two-layer anisotropic BP metamaterials. The top and front views of the proposed two-layer BP metamaterials are shown in Figs. 1(b) and (c), respectively. Figure 1(d) depicts the schematic diagram of the AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ configurations for the three-layer anisotropic BP metamaterials, where φ represents the counterclockwise rotation angle of AC direction for the BP3. The top and front views of the proposed three-layer BP metamaterials are plotted in Figs. 1 (e) and (f), respectively. The bottom metal is Au, and the substrate is chosen to be SiO2. The structural parameters are set as follows: h1=600 nm is the thickness of the substrate, h2=20 nm is the thickness of the Au layer, w=100 nm is the width of the BP layer, the thickness of BP layer is set as 1 nm in our paper, d is the coupling distance of the two adjacent BP layers, θ is the polarization angle of input plane wave, and px=200 nm and py=200 nm are the periods in x and y directions, respectively. The spectra of the anisotropic BP metamaterials are simulated by FDTD methods. In the simulations, the effective area is divided into uniform Yee cells with Δx=Δy=Δz=0.5 nm. The perfectly matched layer is set for the z direction, and the periodic boundary condition is chosen for x and y directions. The permittivities of Au and SiO2 are taken from [35]. Here, a semi-classical Drude model is introduced for demonstrating the surface conductivity of monolayer BP and expressed as [36,37]
Fig. 1. (a) Schematic diagram of AC-AC, AC-ZZ, and ZZ-AC configurations for two-layer anisotropic BP metamaterials and their (b) top view and (c) front view. (d) Schematic diagram of AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ configurations for three-layer anisotropic BP metamaterials and their (e) top view and (f) front view. (g) Anisotropic conductivity of BP in AC and ZZ directions as the carrier density n=1.2×1014 cm−2.
To describe the optical responses of the two-layer anisotropic BP metamaterials in Fig. 1(a), a ROT in frequency domain is developed as [8,41]
3. Results and discussions
3.1 Absorption of two-layer anisotropic BP metamaterials
The reflection and absorption spectra and their generating mechanism are studied in the AC-AC, AC-ZZ, and ZZ-AC anisotropic BP metamaterials, as depicted in Fig. 2. The red solid line in Fig. 2(a) shows an extremely strong absorption at the wavelength of 7.75 μm for the AC-AC configuration in the case of θ=0°. Electric field distributions show that SPs are strongly excited on the BP1 forming the obvious absorption peak, as shown in the inset of Fig. 2(a). The weak electric field for BP2 is caused by the infinite of BP2 along the polarization direction when θ=0°. Thus, compared with BP1, BP2 can be regarded as a quasi-dark mode in the case of θ=0°. Therefore, the parameters of ROT analysis in the case of θ=0° for the AC-AC configuration can be simplified as: ω2x=0, ω1y=ω2y=0, γ1x=γ2x, γ1y=γ2y, m1x=m2x, m1y=m2y, Q2x≈0, and Q1y=Q2y=0. Thus, χeff can be reduced as χeff≈Q1×2m1xcosθ/ε0B1x. From Fig. 2(a), we can find that the FDTD simulation results are well agreement with the ROT results. In Fig. 2(b), we can see that two absorption peeks appear near the wavelengths of 6.2 μm and 11.9 μm for the AC-AC configuration when θ=90°, respectively. The electric field in the inset of Fig. 2(b) shows that the two absorption peaks are caused by the first- and second-order SPs excitation on BP2. Here, BP1 can be seen as quasi-dark mode. Therefore, the parameters for ROT analysis can be expressed as: ω1x=ω2x=0, ω1y=0, γ1x=γ2x, γ1y=γ2y, m1x=m2x, m1y=m2y, Q1x=Q2x=0, Q1y≈0, and thus χeff≈Q2y2m2ysinθ/ε0B2y. Compared with Fig. 2(a), a similar absorption peak also appears, as shown in Fig. 2(c). That is because the SPs along the AC direction on BP1 are strongly excited, and SPs in the ZZ direction on the BP2 can hardly be excited by the incident light for the AC-ZZ configuration when θ=0°, as shown in the inset in Fig. 2(c). At this time, the parameters of ROT analysis can be fitted as: ω1x=ω2x=0, ω1y=0, γ1x=γ2y, γ1y=γ2x, m1x=m2y, m1y=m2x, Q2x≈0, Q1y=Q2y=0, and thus χeff≈Q1×2m1xcosθ/ε0B1x. At last, the similar absorption and reflection spectra appear for the ZZ-AC configurations in the case of θ=0°, as shown in Fig. 2(d), compared with AC-AC configuration when θ=90°. That is because the SPs in the ZZ direction on BP1 are excited, and SPs in the ZZ direction on BP2 can not be excited. Thus, ω2x=0, ω1y=ω2y=0, γ1x=γ2y, γ1y=γ2x, m1x=m2y, m1y=m2x, Q2x≈0, Q1y=Q2y=0 and χeff≈Q1×2m1xcosθ/ε0B1x. From Figs. 2(c) and 2(d), obvious non-reciprocal spectra, caused by the anisotropic surface conductivity for the BP layer, can be observed.
Fig. 2. Reflection and absorption spectra for the AC-AC configuration with d=200 nm, h1=600 nm, h2=20 nm, and w=100 nm when θ=0° (a) and θ=90° (b), Reflection and absorption spectra for the AC-ZZ (c) and ZZ-AC (d) configurations with d=200 nm h1=600 nm, h2=20 nm, and w=100 nm when θ=0°. The inset show the electric field distributions on BP1 and BP2. Here, the black and red solid lines are the FDTD results, and the black and red circles are the ROT results.
Then, the dependence of polarization angle θ of input plane wave on the absorption of AC-AC, AC-ZZ, and ZZ-AC configurations are studied. The obvious periodic change for absorption spectra can be seen in the AC-AC configuration when θ increases from 0° to 360°, and the period is equal to 180°, as depicted in Fig. 3(a). The absorption peak 1 (AP1) caused by SPs in the AC direction can reach a maximum absorption value of 99.99% when θ=0° and 180°. However, the absorption peak 2 (AP2) caused by SPs in the ZZ direction can reach a maximum absorption of 47.40% when θ=90° and 270°. Figures 3(b) and 3(c) show the absorption spectra as a function of polarization angle θ for the AC-ZZ and ZZ-AC configurations, respectively. Interestingly, both AP1 for AC-ZZ configuration and AP2 for ZZ-AC configuration can realize the polarization independent absorption, which is different from the reported polarization independence absorption for metamaterial caused by circular symmetric components [43,44]. However, for anisotropic materials, structural symmetry and electromagnetic parameter symmetry are the necessary conditions to realize polarization independence properties.
Fig. 3. Dependence of θ on the absorption of (a) AC-AC, (b) AC-ZZ, and (c) ZZ-AC configurations with d=200 nm, h1=600 nm, h2=20 nm, and w=100 nm.
These results are beneficial for designing polarization independent optical devices. Then, we can see that AP1 for the AC-AC configuration decreases sharply, and AP2 for the AC-AC configuration increases as θ increases from 0° to 90°, as shown in Fig. 4(a). In addition, the AP2 for ZZ-AC configuration decreases very slowly, but the AP1 for the AC-ZZ configuration continues to stay above 99%. The almost perfect absorption is of great significance for achieving high-performance absorption devices. At last, we study non-reciprocal absorption properties of AC-ZZ and ZZ-AC configurations when θ=0° and 90°, as shown in Fig. 4(b). Here, we define the non-reciprocal degree NRD=|(AAC-ZZ-AZZ-AC)/(AAC-ZZ+AZZ-AC)| to describe the non-reciprocal absorption properties [45]. From Fig. 4(b), we can see that NRD can reach the maximum of 0.91 and 0.84 at the wavelength of AP1 and AP2, respectively. Especially, the reciprocal absorption also can be achieved at the wavelength of 9.89 μm and 10.12 μm when θ=0° and 90°, which is caused by the different excitation order of resonance mode for AC and ZZ directions. Thus, the two-layer anisotropic BP metamaterials can realize tunable non-reciprocal absorption properties.
Fig. 4. (a) Absorption ratio at AP1 and AP2 for the AC-AC configuration, AP1 for the AC-ZZ configuration, and AP2 for the ZZ-AC configuration with d=200 nm, h1=600 nm, h2=20 nm, and w=100 nm. (b) NRD of AC-ZZ (ZZ-AC) configuration when θ=0° (90°) with d=200 nm, h1=600 nm, h2=20 nm, and w=100 nm.
In the above study, we set a long coupling distance d for simplification of calculation in ROT analysis. In fact, the effect of coupling distance on the absorption can not be ignored in the case of strong coupling. Here, we discuss the absorption spectra versus the coupling distance d for AC-AC, AC-ZZ, and ZZ-AC configurations of two-layer anisotropic BP metamaterials. Figure 5(a) shows the absorption when d=25 nm, 50 nm, 100 nm, and 200 nm. The same absorption spectra can be seen when d=100 nm and 200 nm, which is caused by the weak coupling between BP1 and BP2. The absorption curve splits into two peaks when d=50 nm, and the two splitting absorption peaks move far away from each other when d decreases as shown in Fig. 5(a). The absorption shows a slight blue-shift, and the other SP mode on BP2 is also excited when d decreases for the AC-ZZ configuration, as depicted in Fig. 5(b). However, the absorption peak shows a red-shift for the ZZ-AC configuration as d decreases. Comparing with Figs. 5(a)–(c), we can see that the influence of the coupling distance on the AC-AC configuration is greater than those of AC-ZZ and ZZ-AC configurations as a result of intense SPs along the AC direction of the BP layer. At last, the result about NRD versus d is depicted in Fig. 5(d), we can see that the blue-shift and red-shift appear for the non-reciprocal peaks of the AP1 and AP2 when d decreases, respectively. In addition, NRD at AP1 and AP2 also slightly decreases with the decrease of d.
Fig. 5. Dependence of d on the absorption for AC-AC (a), AC-ZZ (b), and ZZ-AC (c) configurations with θ=0°, h1=600 nm, h2=20 nm, and w=100 nm. (d) NRD of AC-ZZ and ZZ-AC configurations with θ=0°, h1=600 nm, h2=20 nm, and w=100 nm when d=25 nm, 50 nm, 100 nm, and 200 nm, respectively.
Here, we investigate the effect of carrier density n on the absorption for the proposed AC-AC, AC-ZZ, and ZZ-AC configurations of two-layer anisotropic BP metamaterials. Obvious blue-shift absorption spectra can be observed for AC-AC, AC-ZZ, and ZZ-AC configurations when the carrier density n increases from 0.8×1014 cm−2 to 1.2×1014 cm−2, as shown in Figs. 6(a)–(c). Except for the blue-shift, the absorption ratios at AP1 and AP2 almost keep a constant. The NRD as a function of carrier density n is also depicted in Fig. 6(d). We can see that non-reciprocal peaks show blue-shifts as n increases. Especially, the position of NRD=0 can also be effectively tuned by the carrier density n in the two-layer anisotropic BP metamaterials.
Fig. 6. Dependence of carrier density n on the absorption in AC-AC (a), AC-ZZ (b), and ZZ-AC (c) configurations with d=200 nm, θ=0°, h1=600 nm, h2=20 nm, and w=100 nm. (d) NRD of AC-ZZ and ZZ-AC configurations with d=200 nm, θ=0°, h1=600 nm, h2=20 nm, and w=100 nm when n=0.8×1014 cm−2, 0.9×1014 cm−2, 1.0×1014 cm−2, and 1.1×1014 cm−2, respectively.
3.2 Absorption of three-layer anisotropic BP metamaterials
Here, we further discuss the dependence of the rotation angle φ on the absorption spectra for the AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ three-layer anisotropic BP metamaterials when θ=0°. Figures 7(a) and 7(b) show the same absorption of AC-AC-φ and AC-ZZ-φ configurations, which are caused by the same excitation of SPs on BP3 in the weak coupling case among the three BP layers. For the ZZ-AC-φ configuration, new strong absorption peaks appear at the wavelength of 8 μm when the rotation angle φ are around 45°, 135°, 225°, and 315°. Then, the NRD based on the rotation angle φ is also investigated for the AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ configurations, as shown in Fig. 7(d). We can see that the non-reciprocal peak of AP1 first decreases and then increases, but the non-reciprocal peak of the AP2 decreases as the rotation angle φ increases from 0° to 90°. Interestingly, the position of NRD=0 always keep a constant when φ increases.
Fig. 7. Dependence of rotation angle φ on the absorption of (a) AC-AC-φ, (b) AC-ZZ-φ, and (c) ZZ-AC-φ configurations with d=200 nm, θ=0°, h1=600 nm, h2=20 nm, and w=100 nm. (d) NRD of AC-ZZ-φ and ZZ-AC-φ configurations with d=200 nm, θ=0°, h1=600 nm, h2=20 nm, and w=100 nm. when φ=0°, 30°, 45°, 60°, and 90°, respectively.
At last, to realize the polarization independent absorption for odd-layer BP metamaterials, we study the dependence of the polarization angle θ on the absorption of the AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ configurations with d=200 nm as the rotation angle φ=0°, 45°, and 90°, respectively. When φ=0°, the absorption spectra of the AC-AC-φ configuration are the same as those of the AC-AC configuration, as shown in Figs. 3(a) and 8(a). For φ=45°, an extra SPs mode named as AP3 is excited on BP3 forming another absorption peak at the wavelength of 12.3 μm, as depicted in Fig. 8(b). In addition, compared with the case of φ=0°, the maximum of absorption for the AP1 rotates 15° along the counterclockwise, as shown in Figs. 8(a) and 8(b). Figure 8(c) shows the polarization independent absorption of the AP2 for the AC-AC-φ configuration when φ=90°, which is in contrast to the inability to achieve the polarization independent absorption of the AC-AC configuration. Here, we also introduce three ROT to fit the numerical results marked by squares, as shown in Figs. 8(a)–(c). The parameters of ROT analysis for the AC-AC-φ configuration can be expressed as: γ1x=γ2x, γ1y=γ2y, γ3x=γ1xcos(φ)+γ1ysin(φ), γ3y=γ1xsin(φ)+γ1ycos(φ), m1x=m2x, m1y=m2y, m3x≈|m1xcos(φ)+m1ysin(φ)|, and m3y≈|m1xsin(φ)+m1ycos(φ)|. According to Eq. (8), we fit the FDTD results with the ROT analytical results. Then, we study the absorption spectra versus the polarization angle θ as φ=0°, 45°, and 90° for the AC-ZZ-φ configuration. The polarization independent perfect absorption for the AP1 can be observed when φ=0°, 45°, and 90° for the AC-ZZ-φ configuration, as plotted in Figs. 8(d)–(f). Interestingly, the absorption for the AP3 also exhibits the polarization independent response when φ=45°, as shown in Fig. 8(e). Interestingly, the absorption spectra rotate 90° for φ=0° and 90°, as shown in Figs. 8(d) and 8(f). For the AC-ZZ-φ configuration, the parameters can be expressed as: γ1x=γ2y, γ1y=γ2x, γ3x=γ1xcos(φ)+γ1ysin(φ), γ3y=γ1xsin(φ)+γ1ycos(φ), m1x=m2y, m1y=m2x, m3x≈|m1xcos(φ)+m1ysin(φ)|, and m3y≈|m1xsin(φ)+m1ycos(φ)|. We can see that the simulation results are in good agreement with the ROT results, as shown in Figs. 8(d) and 8(f). At last, the effect of the polarization angle θ on the absorption of the ZZ-AC-φ configuration as the rotation angle φ=0°, 45°, and 90° are discussed in Figs. 8(g) and 8(i). Owing to the same SPs on the BP1 and BP2, we can see that the polarization independent absorption for the AP2 can be achieved for the ZZ-AC-φ configuration when φ=0°, as shown in Fig. 8(g). For φ=45°, the AP2 still keeps the polarization independent absorption, but another two polarization dependent absorption peaks occur at the wavelength of 8.0 μm and 12.3 μm. They are caused by the SPs excited on the BP1 and BP3. In the case of φ=90°, the absorption for the AP2 shows a slight change when the polarization angle θ increases from 0° to 360°. The square marking shows the ROT analysis results, which are in consistent with the numerical results, as shown in Figs. 8(g)–(i). From the results in Fig. 8, we can see that the polarization independent absorption can also be realized for odd-layer BP metamaterials.
Fig. 8. Dependence of polarization angle θ on the absorption of AC-AC-φ configurations with d=200 nm, h1=600 nm, h2=20 nm, and w=100 nm when (a) φ=0°, (b) 45°, and (c) 90°. Dependence of polarization angle θ on the absorption of AC-ZZ-φ configurations with d=200 nm, h1=600 nm, h2=20 nm, and w=100 nm when (d) φ=0°, (e) 45°, and (f) 90°. Dependence of polarization angle θ on the absorption of ZZ-AC-φ configurations with d=200 nm, h1=600 nm, h2=20 nm, and w=100 nm when (g) φ=0°, (h) 45°, and (i) 90°.
4. Summary
In summary, we have investigated the absorption properties of two- and three-layer anisotropic BP metamaterials through FDTD simulation and ROT analysis. We can see that the SPs can be excited on the surface of BP layer forming the obvious absorption, which can be well described by the proposed ROT theoretical analysis. Then, the dependence of polarization angle θ on the absorption is discussed for the AC-AC, AC-ZZ, ZZ-AC, two-layer anisotropic BP metamaterials. We can find that the polarization independent absorption can be realized in the case of AC-ZZ and ZZ-AC configurations. Especially, the perfect absorption can be realized in the case of AC-ZZ configurations. We can also find that the AP1 for AC-ZZ and AP2 for ZZ-AC stones insensitive to polarization angles. In addition, the obvious non-reciprocal absorption also can be observed at the wavelength of 9.89 μm and 10.12 μm when θ=0° and 90° AC-ZZ and ZZ-AC configurations, respectively. Then, the absorption and NRD for the AC-AC, AC-ZZ, ZZ-AC can be effectively tuned by the coupling distance and carrier density of the BP layer. We can see that the absorption increases as the coupling distance d increases. Moreover, the absorption responses for the AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ three-layer anisotropic BP metamaterials are studied in detail. The tunable absorption spectra can not only be realized by tuning the rotation angle φ, but also show the polarization independent absorption at different wavelengths in the AC-AC-φ, AC-ZZ-φ, and ZZ-AC-φ configurations. These results prove that the polarization independent absorption can not only be achieved in the even-layer BP, but also in the odd-layer configurations. The results may play important role in designing tunable polarization independent and non-reciprocal plasmonic devices in anisotropic 2D materials.
Funding
National Key Research and Development Program of China (2017YFA0303800); National Natural Science Foundation of China (11774290, 11974283, 61705186, 62065017); China Postdoctoral Science Foundation (2019M653722); Fundamental Research Funds for the Central Universities (310201911FZ049); Natural Science Basic Research Program of Shaanxi Province (2020JM-130).
Disclosures
We declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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