1. Introduction

As a new family of nanomaterials, two-dimensional (2D) materials with atomic-scale thickness exhibit advantages of robustness, mechanical flexibility and easy integration over traditional bulk material, and thus provide unprecedented possibilities in developing electronic and optoelectronic devices with desired physical properties. Graphene is the most popular member among 2D materials, and possesses the highest carrier mobility (∼20000 cm²/V·s), but the zero-bandgap nature sets the obstacle in the applications where the high on-off ratio is required [1, 2]. Due to the layer-dependent band gap, black phosphorus (BP) thin film is brought to the spotlight since its exfoliation from its bulk crystal in 2014 [3]. In bulk form, BP has a band gap of 0.3 eV, which is expected to increase as the thickness of BP thin film decreases and finally reaches 2 eV in monolayer form [4–6]. Moreover, the band gap of BP can be efficiently tuned by the electrostatic gating and surface charge transfer method via electron/hole doping, offering additional degrees of freedom to design tunable devices [7–10]. Another distinct feature of BP is the high in-plane anisotropy stemming from the puckered structure of phosphorus atoms, which enables the development of novel polarization dependent devices [11,12]. These attractive properties make BP an excellent material in the infrared. However, as the consequence of the intrinsic atomic thickness, the inherent optical absorption of BP is usually quite low, severely limiting the efficiency of BP-based devices.

Inspired by the advancements in graphene plasmons with field confinement merit and tunable benefit in the spectral range spanning infrared to terahertz range, great efforts have been devoted to discovering and exploring the plasmonic resonances of BP in different structures or systems [13–18]. However, most of the BP-based resonance structures exhibit weak light absorption with low doping concentration and low sensitivity. An alternative way for absorption enhancement is to introduce the multilayer structure of BP into the design [19–21]. Nevertheless, the relatively complicated fabrication techniques are required for the high absorption rate in the multilayered structure. In addition, the theory of coherent absorption was also demonstrated for absorption enhancement of BP, while the additional configurations is necessary for the proper phase modulation of two coherent incident waves [22,23]. Most recently, a total absorption scheme in 2D materials based on critical coupled mode theory has attracted extensive attentions [24–29]. Especially, Qing et al. presented a perfect absorber by critical coupling of a monolayer BP with guided resonances of a photonic crystal slab and demonstrated the sensitivity of the absorption to the slab parameters. Given the fabrication processes of the absorption structure composed of the photonic crystal slab with high refractive index contrast are complicated in practice, the design of a simple BP-based structure with the flexible tunability, high absorption and easy fabrication process is still required.

In this work, we propose a BP-based perfect absorption structure via critical coupling in the mid-infrared, where a monolayer BP is coupled to a 2D periodic polymer structure with low index contrast. This simple structure not only efficiently enhances the light-BP interaction with the optical absorption up to 99.9%, but also demonstrates the desirable capability of tailoring the absorption by controlling the conditions of critical coupling, including electron density of BP, geometrical parameters, and incident angle of light. Furthermore, the BP-based structure exhibits distinct absorption characteristics under TM- and TE-polarized illumination due to the in-plane anisotropy of BP. Finally, the perfect absorption is also realized with the multilayer BP by simply adjusting the geometrical parameters. Thus, this proposed structure shows great potentials for the high-performance BP-based optical and photonic devices in the mid-infrared.

2. Structure and model

The schematic diagram of the proposed absorption structure is illustrated in Fig. 1(a). A monolayer BP is sandwiched between a Polydimethylsiloxane (PDMS) layer with a periodic arrays of circular air holes and a MgF₂ layer. A 200-nm thick gold (Au) layer is deposited on the back of the MgF₂ layer to prevent the transmission of incident light. The side-view of the structure and the geometrical parameters are displayed in Fig. 1(b). The refractive indices of PDMS, air and MgF2 are taken to be 1.37, 1 and 1.34, respectively [30, 31]. The permittivity of Au is given by the Drude model [32]. The equivalent relative permittivity of BP can be derived from the surface conductivity with the thickness d_BP = 1 nm, which is dependent upon incident polarization due to the different electron mass along armchair and zigzag direction. The assumption of using the finite thickness to simulate 2D material will accelerate the numerical simulations, and can resemble the real results since it is still small enough compared with the wavelength of interest [14, 17, 19]. Figs. 1(c) and 1(d) displays the polarization-dependent permittivity of BP, and demonstrates its anisotropic optical behaviors.

 

Fig. 1 (a) Schematic diagram and (b) the side-view of the proposed absorption structure. The inset illustrates the structure of a monolayer BP. The lattice period in both x and y direction is denoted as P. The thickness of Au layer, MgF₂ layer, BP layer and PDMS layer are represented by D, d_m, d_BP and d_p, respectively. The radius of the air holes is R. The effective permittivity of BP along (c) x direction and (d) y direction. The black and red lines denote the real and imaginary parts, respectively.

Download Full Size | PDF

The numerical simulations for the absorption characteristics of the proposed structure are carried out using the finite difference time domain (FDTD) method. The periodic boundary conditions are applied in both x and y direction and the plane waves are normally incident from −z direction while the perfectly matched layer are employed to absorb all the light out of the boundaries along the propagation direction. The non-uniform mesh is adopted, and the minimum mesh size inside the BP layer equals 0.25 nm and gradually increases outside the BP layer to balance storage space and computing time. The absorption of the structure can be calculated from A = 1 − T − R where T and R are transmission and reflection, respectively. Because of the Au layer as the metallic mirror to block the transmission, the absorption can be finally simplified as A = 1 − R.

In the proposed configuration, the concept of critical coupling is employed for perfect absorption via coupling the guided resonance to the lossy BP. The in-plane periodicity in the PDMS layer consisting of square lattices of air holes enables phase-matched coupling between the guided mode and free-space radiation, leading to the guided resonance with electromagnetic field significantly confined within the structure. Hence, the incident light can be coupled with the guided resonance and the absorption would be remarkably enhanced in the vicinity of the guided resonance frequency. The coupled mode theory (CMT) can account for the phenomenon of critical coupling for optical absorption enhancement. The reflection coefficient in the coupled system is described as [24,25]

Under the critical coupling condition, the impedance of the proposed structure is supposed to match with that of the free space, i.e. Z = Z₀ = 1. For the proposed one-port structure, the effective impedance can be written as [33,34]

3. Results and discussions

To investigate the absorption characteristics of the structure, the geometrical parameters are optimized to satisfy the critical coupling conditions, and the parameter values are initially set as follows: the radius of air holes is R =700 nm, the PDMS thickness is d_p= 700 nm, and the MgF₂ layer thickness is d_m=1500 nm. Here we adapt the electron doping of BP n_s=3×10¹³ cm⁻² accessible by surface charge transfer doping method [9, 10]. The absorption spectra of the proposed structure are numerically simulated under the normally incident light, and the simulated absorption spectra are depicted in Figs. 2(a) and 2(b). The absorption spectra for TM and TE polarizations exhibit significant direction dependent properties resulting from the inherent in-plane anisotropic feature of BP. For TM polarization, the absorption spectrum with a perfect absorption up to 99.9% is obtained at the resonance wavelength of 4467.5 nm, while absorption spectrum for the case of TE polarization has the peak with 95.8% at 4476.5 nm. By comparison, the resonant wavelength as well as the resonance position for TE polarization shows a clear redshift, especially, the absorption is as low as 3.2% at 4467.5 nm, i.e., the resonance wavelength of TM polarization.

 

Fig. 2 Simulated and theoretically absorption spectra for (a) TM and (b) TE polarization. The insets are the effective impedances of the corresponding absorption spectra in the vicinity of the resonance wavelengths. (c) Distributions of electric field intensity |E| at the resonance wavelength for TM polarization, including x–y cross-section plane above the BP layer and along x–z and y–z cross-section planes.

Download Full Size | PDF

Here we start from the investigation of absorption spectrum in the case of TM polarization due to the perfect absorption merit. The full width at half maximum (FWHM) is Δλ=3.96 nm, indicating the extremely narrow bandwidth for spectral selective absorption. The quality factor Q which can be defined as Q = λ₀/Δλ reaches the value of 1128.17. Meanwhile, the theoretical absorption spectrum based on CMT is also depicted in Fig. 2(a), which exhibits an excellent agreement with the simulation results. According to CMT, the loss and external leakage of the structure is δ = γ_e = 9.34 × 10¹⁰ Hz. Then the theoretical quality factor Q is calculated as 1128.8 by Q_CMT = Q_δQ_γ/(Q_δ + Q_γ), where intrinsic loss is defined as Q_δ = ω₀/(2δ) and the external leakage defined as Q_γ = ω₀/(2γ_e). The nearly identical values of the theoretical Q_CMT and the simulated Q demonstrate the perfect absorption of the structure is attributed to the critical coupling. In addition, we calculate the effective impedance of the system in the vicinity of resonant wavelength, as shown in the inset of Fig. 2(a). It is clearly observed that the impedance of the structure is Z = 1.00 + i8.03 × 10⁻⁴ at resonance, substantially equaling to that of the free space. Under this condition, the reflection of the structure is maximally suppressed because of the impedance matching with free space and the transmission is blocked by the metallic mirror, giving rise to the perfect absorption. In contrast, the loss and external leakage of the structure in the case for TE polarization are calculated as δ = 5.96 × 10¹⁰ Hz and γ_e = 9.34 × 10¹⁰ Hz, respectively. The leakage rate of the structure keeps unchanged during the polarization switch, but the intrinsic loss decreases because the imaginary part of effective permittivity of BP along y direction is lower than that along x direction. As a result, the system is in the state of over coupling and the impedance is Z = 1.23 + i0.78, leading to the nonperfect absorption characteristics for TE polarization. Fig. 2(c) provides the distributions of electric field intensity |E| at the resonance wavelength for TM polarization. When the structure is excited to the resonant state, the electric field is confined as the guided modes near the BP layer, leading to a remarkable absorption enhancement and the final perfect absorption of the structure.

It is well known that the critical coupling depends on the match of the absorption rate of the structure and the leakage rate of the guided resonance, which gives us the clue to control the absorption characteristics by adjusting the two aspects. In the wavelengths of interest, the intrinsic absorption of the structure mainly contributed from the lossy BP layer. With tunable electron doping advantage, effective permittivity of BP can be dynamically controlled and then the critical coupling conditions as well as the absorption characteristics of the structure can be adjusted. Figs. 3(a) and 3(b) illustrates the variations of the absorption peak and resonance wavelength at different electron doping of BP under TM and TE polarizations, respectively. Obviously, the resonance wavelengths for TM and TE polarization both show a slight blue shift as the electron doping increases, which can be explained by the fact that the real part of the effective permittivity of BP becomes small with larger electron doping. It is also found that the absorption peak for TM polarization gradually increases from 94.8% with n_s=1×10¹³ cm⁻² to the perfect absorption of 99.9% with n_s=3×10¹³ cm⁻², then gradually decreases to 87.1% with n_s=11×10¹³ cm⁻². This can be explained by the match between the intrinsic absorption and the leakage rate of the structure. During the variations, the leakage rate of the structure can be considered as unchanged, γ_e= 9.34×10¹⁰ Hz, since only the electron doping n_s of BP layer is changed here. Meanwhile, the other aspect, i.e. the intrinsic loss of the structure would gradually increases because the imaginary part of effective permittivity of BP becomes larger as the electron doping increases. For more details of the increasing intrinsic loss δ and the steady leakage rate γ_e, the δ (5.6×10¹⁰ Hz) is smaller than the leakage rate for n_s=1×10¹³ cm⁻², then equals to γ_e for n_s=3×10¹³ cm⁻², and becomes larger (2.3×10¹¹ Hz) than gamma for n_s=11×10¹³ cm⁻². It concludes that the system evolves from the states of over coupling to critical coupling and then under coupling as the electron doping increases. The above analysis also applies to TE polarization and the system goes through the states of over coupling to critical coupling for increasing electron doping n_s from 1×10¹³ cm⁻² to 7.4 ×10¹³ cm⁻², and then under coupling as n_s further increases. In addition, the slight variation of absorption peak with above 90% amplitudes in a relatively broad electron doping region of BP reveals the robustness of the absorption structure, while the shift of resonance wavelength indicates the feasibility in a broad absorption spectrum regime.

 

Fig. 3 Simulated absorption spectra at different electron doping n_s of BP for (a) TM and (b) TE polarization.

Download Full Size | PDF

On the other hand, we concentrate on the control of the external leakage rate of the structure when the intrinsic loss rate from BP keeps unchanged with electric doping n_s=3×10¹³ cm⁻². The influence of the geometrical parameters on the absorption of the whole structure is investigated via changing the air hole radius, the thickness of PDMS layer and MgF₂ layer, respectively. Here we focus on the absorption characteristics in the case of TM polarization. Fig. 4(a) illustrates the dependence of the absorption peak and the resonant wavelength on the air hole radius with other parameters are fixed. When the radius of air holes increases from 500 nm to 900 nm, the external leakage rate increases and the structure goes through the states of undercoupling, critical coupling and overcoupling in this radius range. Accordingly, the absorption peak of the structure firstly increases from 85.6% to 99.9% and then decreases to 90.2% during the modulation. At the same time, the wavelength of guided resonance i.e., the absorption peak position shows a blue shift from 4492.5 nm to 4443.9 nm, which is attributed to the reduced effective refractive index of the guided resonance as air hole expands. Thus with a proper engineering of the air hole radius, the critical coupling conditions for absorption characteristics can be flexibly tuned.

 

Fig. 4 Dependence of peak absorption and resonance wavelength on geometrical parameters, (a) the air hole radius, (b)the thickness of PDMS layer and (c) the thickness of MgF₂ layer.

Download Full Size | PDF

The dependence of the absorption characteristics of the structure on the thickness of PDMS layer and MgF₂ layer is considered in Figs. 4(b) and 4(c). When the PDMS layer thickness increases from 500 nm to 900 nm with the radius of air hole is fixed as R=700 nm, the absorption peak of the structure rises up from 92.3% to 99.9% and then goes down to 87.3%. Compared to that, the absorption peak shows a drastically increase and then a sharp fall as the thickness of MgF₂ layer increases from 1300 nm to 1700 nm. This can be concluded that the external leakage rate of the structure is more sensitive to the variation of the MgF₂ layer thickness relative to the PDMS layer thickness. During their variations, the resonance wavelengths in Fig. 4(b) and (c) both exhibit redshift tendency with nearly linear variation, because the effective refractive index of the structure increases with the increasing thicknesses. Moreover, simulation results reveal that nearly perfect absorption up to 99% could be maintained with the air hole radius within the range from 650 to 720 nm, the PDMS layer thickness between 670 nm to 730 nm and the MgF₂ layer thickness between 1480 nm to 1530 nm when other parameters have fixed values. Therefore, the proposed structure not only realizes the high efficiency optical absorption with highly tunability, but also exhibits relatively large fabrication tolerances.

The dependence of the absorption characteristics on the angular dispersions is also investigated. Figs. 5(a) and 5(b) show the absorption as the function of incident angle and wavelength for TM and TE polarization, respectively. When the incident angle varies from 0° to 6°, the absorption peak of the structure for TM polarization keeps nearly invariant and the resonance wavelength shows a slight blue shift. The absorption performance for TM polarization is insensitive to the incident angle within this range, and has potentials in designing integrated optoelectronic devices. For TE polarization, the phenomenon of wavelength splitting is observed and two major absorption peaks appear in the absorption spectra when the incident light is oblique. The two absorption peaks originate from the excitation of the guided resonance. In contrast with the stable absorption in TM polarization, the two absorption peaks exhibit distinct changes in resonance wavelength and absorption peak, showing promising applications in multispectral light detection. The dependence of the absorption characteristics on the polarization angle and wavelength is displayed in Fig. 5(c). When the polarization angle is 0°, the perfect absorption is realized at the resonance wavelength of 4467.5 nm for TM polarization. When the polarization angle varies from 0° to 90°, the absorption peak for TM polarization gradually declines, while another absorption peak arises at 4476.5 nm for TE polarization and continuously increases. When the polarization angles finally comes to 90°, absorption peak for TM polarization completely disappears and that for TE polarization reaches its maximum. During the evolution, the resonance wavelengths of both absorption peaks keep intact, which reveals the absence of coupling effect between the guided modes excited by TM and TE polarizations and guarantees the linear superposition of the absorption at respective resonance wavelength.

 

Fig. 5 Dependence of absorption of the structure on angular dispersions, incident angle for (a)TM and (b)TE polarization under the oblique incidence, and (c) polarization angle under the normal incidence.

Download Full Size | PDF

In real case, there would be inevitable disorders during the mechanical exfoliation and transfer processes of the monolayer BP, and the multilayer BP can be found in the mixture as well. Previous investigations have claimed the layer-dependent band gap of BP, which decreases to ∼0.3 eV in the multilayer form. Considering this situation, the absorption characteristics of the structure with the multilayer BP are finally investigated. Given the fact that the imaginary part of effective permittivity of BP becomes larger with the lower band gap, the intrinsic loss rate of the structure would increase, which accordingly requires the larger external leakage rate to fulfill the critical coupling. Therefore, the perfect absorption of 99.6% at the resonance wavelength of 4420.5 nm is realized by adjusting the air hole radius from R =700 nm to 900 nm when other parameters are fixed. As shown in Figs. 6(a)–6(c), the variations of the peak absorption and the resonance wavelength as functions of geometrical parameters exhibit similar tendency as in the structure with the monolayer BP. In addition, due to the decreasing real part of the effective permittivity of the multilayer BP, all the resonance peaks shift to the relatively shorter wavelengths than those with the monolayer BP, confirming the known layer-dependent wavelength shift[13]. These absorption characteristics for the multilayer BP further demonstrate the robustness of the proposed structure in practice.

 

Fig. 6 (Multilayer BP) Dependence of peak absorption and resonance wavelength on geometrical parameters, (a) the air hole radius, (b)the thickness of PDMS layer and (c) the thickness of MgF₂ layer.

Download Full Size | PDF

4. Conclusions

In conclusions, a BP-based anisotropic perfect absorption structure is proposed and theoretically investigated in the mid-infrared. The perfect absorption up to 99.9% is attributed to the critical coupling of the monolayer BP to the guided resonance in the periodic polymer lattice with low index contrast. The polarization-dependent absorption characteristics of the structure is obtained due to the anisotropic properties of BP. Moreover, the absorption peak and resonance wavelength can be flexibly tuned by adjusting the electron doping of BP, the geometrical parameters of the structure and the incident angles of light. Finally, the perfect absorption is also realized with the multilayer BP by simply adjusting the geometrical parameters. In this work, the concerned mechanism of critical coupling for enhanced absorption is universal and the proposed structure shows advantages of high efficiency absorption, the remarkable anisotropy, flexible tunability, and easy-to-fabricate characteristics. Therefore, our results not only provide a good way to improve light-matter interaction for 2D materials, but also will play a significant role in the design of advanced BP-based devices.