Tailoring anisotropic absorption in borophene-based structure via critical coupling

The research of two-dimensional (2D) materials with atomic-scale thicknesses and unique optical properties has become a frontier in photonics and electronics. Borophene, a newly reported 2D material provides a novel building block for nanoscale materials and devices. We present a simple borophene-based absorption structure to boost the light-borophene interaction via critical coupling in the visible wavelengths. The proposed structure consists a borophene monolayer deposited on a photonic crystal slab backed with a metallic mirror. The numerical simulation and theoretical analysis show that the light absorption of the structure can be remarkably enhanced as high as 99.80$\%$ via critical coupling mechanism with guided resonance, and the polarization-dependent absorption behaviors are demonstrated due to the strong anisotropy of borophene. We also examine the tunability of the absorption behaviors by adjusting carrier density and lifetime of borophene, air hole radius in the slab, the incident angles, and polarization angles. The proposed absorption structure provides novel access to the flexible and effective manipulation of light-borophene interactions in the visible, and shows a good prospect for the future borophene-based electronic and photonic devices.


I. INTRODUCTION
Recent decades have witnessed the rapidly growing interests in two-dimensional (2D) materials, a new family of nano-materials with peculiar band structures and unique optical properties 1 . The development of 2D materials has enabled effective manipulation of incident light over a wide wavelength scale with the promise to achieve next-generation nanophotonic devices such as optical modulators, field effect transistors, photodetectors, and biosensors [2][3][4][5][6] . To boost the light-matter interactions limited by their monolayer nature, one intuitive solution is to excite the plasmonic response of the 2D materials, for instance, graphene plasmons in the terahertz and infrared and the localized anisotropic surface plasmon resonance of nanostructured black phosphorous in the mid-infrared [7][8][9][10][11][12][13] . Alternatively, the interactions can also be enhanced by integrating 2D materials with the resonant structures such as plasmonic nanoantennas, Fabry-Perot cavity, hyperbolic metamaterials, et al. [14][15][16][17][18][19] . Among them, the critical coupling mechanism becomes an excellent candidate for absorption enhancement of 2D materials due to the simple design, the remarkable field confinement, and the flexible tunability merits. This method has been theoretically and experimentally employed in the whole 2D material family, providing a good way to improve the light-matter interaction of 2D materials and holding great potential for the development of novel functional devices with superior performance [20][21][22][23][24][25][26][27][28][29][30][31][32][33] .
Very recently, borophene, the two-dimensional boron polymorphs, has become a new member of the wonderful 2D material family and provides novel building blocks for nanoscale materials and devices. In contrast with the semiconducting nature of the bulk boron, borophene sheet is predicted to be metallic with highly electron density, which has been demonstrated by the atomic-scale characterization of the synthesized sheets on silver surfaces [34][35][36][37][38] . Such 2D metal characteristic is complementary to those of the existing 2D materials only accessible to 2D semi-metal as graphene or 2D semiconductor as MoS 2 .
Moreover, similar with black phosphorus, borophene shows highly anisotropic electronic properties and leads to interesting physical phenomena 39,40 . The high electron density and strong anisotropy of borophene has motivated the further investigation of manipulating light-matter interaction. Inspired by the plasmon modes in other existing 2D materials, the patterned borophene is demonstrated to exhibit anisotropic plasmonic behavior in visible wavelengths 41 . However, the nanostructured borophene in the isolated fashion is usually involved with complicated fabrication techniques. For practical applications, it is highly desirable to exploit the borophene monolayer form for the interaction enhancement.
Following this idea, we propose a borophene-based absorption structure via critical coupling in the visible range, where borophene monolayer is covered on top of a photonic crystal slab backed by a metal mirror. We show that by critical coupling with guided resonance, the light-borophene interaction is highly enhanced with the total absorption up to 99.80%.
The absorption behaviors can be tailored by tuning the carrier density and carrier lifetime of borophene, the air hole radius of the slab, and the incident light angles. Specially, owing to the strong anisotropy of borophene, the structure exhibits distinct absorption spectra under TM and TE polarizations. The proposed structure presents a promising way to enhance the interaction of light in borophene, which may promote the development of the novel borophene-based nanodevices.

II. STRUCTURE DESIGN AND NUMERICAL MODEL
On the periodic table, boron is the neighbor of carbon and they have the same short covalent radius and the flexibility to adopt sp 2 hybridization, which is favorable to form the low-dimensional borophene. As boron has one less electron than carbon, the monolayer structure needs to be stabilized by balancing out the two-center bonding in the hexagonal regions and three-center bonding in the triangular 42,43 . Depending on the connectivity of the boron, various polymorphs of borophene have been theoretically predicted and confirmed by experimental synthesis, such as the lowest-energy monolayer structure by Mannix et al., the β 12 and χ 3 polymorphs by Feng et al. on Ag(111) surface using molecular beam epitaxy 36-38 . Since our main aim is to explore the total absorption of borophene via critical coupling as other 2D materials, the borophene monolayer with high density of electrons and strong anisotropy is adopted as the building block in this work, where its schematic could be found in Fig. 1(a) and (b). In the visible wavelengths, the surface conductivity of the borophene monolayer can be modeled using a simple semiclassical Drude model as 41 where j is the direction concerned that is taken to be x or y in this study, ω is the incident light frequency, τ is the mean free time of electron ranging from 10 fs to 65 fs, e and n s represent electron charge and free carrier density, respectively, D j is the Drude weight, m j is the effective electron mass along different directions, and m x = 1.4m 0 , m y = 5.2m 0 with m 0 is the rest mass of electron. The effective permittivity of the borophene monolayer can be derived from the surface conductivity along each direction as where ε r = 11 is the relative permittivity of boron, ε 0 is the permittivity of free space, and d B  In the proposed one-port coupling configuration, borophene monolayer is deposited on top of a photonic crystal slab backed with a metallic mirror, as illustrated in Fig. 1 (a) and (b). In order to achieve critical coupling absorption in the visible, the lossless photonic crystal slab made from Si 3 N 4 with periodic air holes is adopted as the resonator, where the guided modes in the slab couple to the external radiation and leads to the leaky guided resonances. The placement of the borophene monolayer on the top introduces the lossy thin film, while it shows little impact on the field distribution of resonance inside the slab.
The metallic mirror is employed to reflect light and suppress the transmission. We perform numerical simulations using the finite-difference time-domain (FDTD) method within the wavelengths between 530-610 nm, while the refractive index of n = 2 is used for the Si 3 N 4 slab and the permittivity of the silver mirror in this wavelengths is given by Drude model 44 .
The plane waves are normally incident from -z direction, and the mesh inside the borophene monolayer is 0.15 nm that is fine enough to resemble the real results. Then the absorption of the proposed structure is simplified as A = 1 − R because of the blocked transmission from the metallic mirror.

III. RESULTS AND DISCUSSIONS
In the initial simulations, the borophene monolayer is set with the free carrier density the intrinsic loss rate δ in the system, the reflection coefficient can be described as 20 where y and u represent output and input wave amplitudes, ω 0 is the resonance frequency.
The absorption is calculated as From the equations, the critical coupling condition can be achieved when the external leakage and intrinsic loss rates are the same as δ = γ at the resonant frequency of ω 0 . That is to say, the reflection coefficient of the system will vanish and all the incident light will be totally absorbed under this condition. At the same time, the effective impedance of the whole structure is supposed to equal with that of the free space, i.e. Z = Z 0 = 1. The effective impedance of the one-port configuration is given by 45,46 where T 11 , T 12 , T 21 , and T 22 are the elements of the transfer matrix of the structure calculated from the scattering matrix elements . The two roots of Z correspond to the two paths of light propagation with plus sign denoting the positive direction.
Based on CMT, the theoretical absorption spectra are provided and depicted as the dashed lines in Fig. 2(a)  Given the fact that the perfect absorption is achieved by manipulating the critical coupling condition, the absorption behaviors can be actively tailored by adjusting the intrinsic loss and external leakage rates of the proposed structure. In the wavelength of interest, the borophene monolayer plays the essential role of lossy thin film and contributes to the intrinsic loss in the coupled system. This motivates us to investigate the influence of the borophene properties on the absroption behaviors under different polarizations. In Fig. 3, the dependences of the absorption peak and the resonance wavelength of spectra on the free carrier density n s and the carrier lifetime τ in borophene are depicted. For TM polarization in Fig. 3(a), the absorption peak shows an increases from 94.88% to 99.80% as the carrier density n s reaches 4.3×10 15 cm −2 , and then the system keeps in critical coupling state with maximum absorption depsite of the further increase of n s . In Fig. 3(b), the absorption peak for TE polarization is unchanged firstly and then shows a maximum of 97.11% at n s = 6×10 15 cm −2 . The resonance wavelengths of both polarizations show linear blue shift owing to the smaller real part of the effective permittivity of borophene as ns increases, implying the feasibilty within a broad wavelength range. In Fig. 3(c) and (d), as the carrier lifetime τ varies within the emperical range from 10 fs to 65 fs, the absorption peak increases at first and keeps at the maximum of 99.80% for TM polarizaiton, while it decreases slightly from 96.30% to 92.05% for TE polarization. It is also observed that the variations of τ show no impact on the resonance wavelength, which can be explained by its little influence on the real part of the borophene permittivity. Moreover, all the absorption peaks in Fig. 3 show large amplitudes with above 90%, revealing the robustness of the proposed absorption structure. In addition to the modulation of the instinct loss resulting from borophene properties, the absorption behaviors can also be controlled by the variations of the external leakage rate.
In the guided resonance system supported by the photonic crystal slab, the external leakage rate γ is mainly controlled by the ratio between the air hole radius and lattice period, i.e.
R/P 20 . Here we investigate the variations of absorption peak and resonance wavelength with respect to the air hole radius while other parameters are fixed, as shown in Fig. 4(a) and (b). Obviously, the absorption behaviors show more sensitive dependence on this geometrical parameter than that on borophene properties. In detail, for TM polarizaiton, the absorption peak exhibits significant increases from 21.83% to 99.80% as radius varies from 50 nm to 70 nm and then shows decline tendency. As radius increases, the external leakage rate also increases while the instinct loss rate keeps almost unchanged. As a result, the coupled system evolves from the state of under coupling, through critical coupling and to over coupling.
Due to the reduced effective refractive index of the guided resonance with the larger radius, the resonance wavelength of the absorption spectra displays an obvious blue shift from 572.83 nm to 556.61 nm. The influence of air hole radius on the absorption behaviors for TM polarization also applies to that of TE polarizations. Thus, the leakage rate could be engineered by adjusting geometrical parameters, leading to the tunable absorption in borophene-based structure.

IV. CONCLUSIONS
In conculsion, we propose and theoretically demonstrate the enhanced light-matter interaction in the borophene monolayer via critical coupling in the visible wavelengths. In the proposed simple compact absorption structure consisting of borophene covered on a photonic crystal slab backed with metallic mirror, the numerical results show that the absorption can be remarkably enhanced up to 99.80% at normal incidence. The physical regime of the total absorption is the field confinement through the principle of critical coupling with guided resonance. The proposed structure exhibits polarization-dependent absorption spectra along the x and y directions owing to the anisotropic nature of borophene. We also examine various parameters such as borophene properties including carrier density and lifetime, geometrical parameters including air hole radius in the slab, the incident angles, and polarization angles to analyze the tunablity of the absorption behaviors. Though it is still a challenge in experimental realization to isolate the borophene monolayer from the siliver substrates, our proposed structures opens a new path to improve the light-borophene interaction in the visible for future borophene-based electronic and photonic devices.