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Abstract
We investigate theoretically the photonic spin Hall effect (PSHE) in bilayer borophene metasurfaces. Based on the combined effect of the Fabry-Perot resonance of the bilayer system and the resonant interaction of individual meta-atoms in borophene metasurface which lead to the topological transition, it is found that there exist giant PSHE shifts of the transmitted beams which can be flexibly regulated by adjusting the twist angle of metasurface bilayers, incident angle, spacer refractive index and spacer thickness. Near the topological transition of borophene metasurface the magnitude of PHSE shifts in bilayer borophene metasurfaces is generally on the order of tens of wavelengths and even on the order of hundreds of wavelengths near the epsilon-near-zero (ENZ) regions. The manipulation frequency range of the large PSHE shifts can reach hundreds of terahertz or even picohertz through adjusting the ribbon width of borophene metasurface or the electron density for borophene. It is found that in bilayer borophene metasurfaces there exist the ultrahigh sensitivity of the PSHE shifts to spacer refractive index, which can be applied to design the refractive index sensors with high performance.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
When a physical beam with finite thickness is transmitted or reflected on an interface, the PSHE is a phenomenon in which right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) components of the light beam split perpendicular to the incident plane [1,2]. The PSHE is an optical analogy of electronic spin Hall effect, and the inherent physical mechanism for PSHE is the spin orbit coupling of light which illustrates the mutual influence of the spin (polarization) and the trajectory of the beam [2]. The PSHE has recently received growing attention and has been extensively investigated in different physical regions, including metasurface [3], wave-vector-varying Pancharatnam-Berry phase system [4], metal-dielectric interface [5], polymer films [6], hypercrystal [7], spin-controlled nanophotonic devices [8] and hyperbolic metamaterial (HMM) [9–12]. It has been reported that the experimental workings for the PSHE shifts such as the experimental observations of the enhanced PSHE caused by the anisotropy of HMM [11,12].
Two-dimensional (2D) materials have been widely studied in recent years [13]. As a new family of nano-materials with unique optical properties, 2D materials are expected to be used to realize various new nanophotonic devices such as biosensors, optical modulators and photodetectors [14–18]. As a new member of 2D materials, borophene or monolayer boron sheet is predicted to be metallic with high electron density [19], in contrast with the semiconducting nature for the bulk boron. Borophene as 2D metal is complementary to those of the 2D materials having access to 2D semiconductors as MoS2, 2D semi-metal as graphene and insulators as hBN. Similar with black phosphorus (BP), borophene exhibits in-plane anisotropic response since along the two crystal axes there exist the different in-plane effective electron masses [20,21]. For graphene and BP, the working frequencies of strong light-matter interactions limit within the terahertz or mid-infrared frequency range due to their low carrier densities (${\sim} {10^{17}}{m^{ - 2}}$) [22–24]. High carrier densities (${\sim} {10^{19}}{m^{ - 2}}$) enable borophene to support strong light-matter interactions in the visible and near-infrared region [25–28].
Optical metasurfaces, the 2D analog of metamaterials, recently have received significant attention. Due to both metasurfaces and 2D materials can be described as one 2D current characterized by one 2D conductivity tensor, metasurfaces can be fabricated by 2D materials [29]. The PSHE has been widely investigated on the corresponding structure containing 2D materials [30–38]. It should be pointed out that how to improve the enhancement for PSHE in the corresponding structure containing 2D materials is still a challenging task. Here we study theoretically the PHSE shifts in bilayer borophene metasurfaces. Based on the combined effect of the Fabry-Perot resonance of the bilayer system and the resonant interaction of individual meta-atoms in borophene metasurface, it is found that there exist giant PSHE shifts of the transmitted beams which can be flexibly manipulated by adjusting the twist angle of metasurface bilayers, incident angle, spacer thickness and spacer refractive index. Near the topological transition of borophene metasurface, it can be found that the magnitude of PHSE shifts in bilayer borophene metasurfaces is generally on the order of tens of wavelengths and even on the order of hundreds of wavelengths near the ENZ regions. By controlling the ribbon width of borophene metasurface or the electron density for borophene, the manipulation frequency range of the large PSHE shifts can reach hundreds of terahertz or even picohertz. The ultrahigh sensitivity of the PSHE shifts to spacer refractive index can be used to design the refractive index sensors with high performance.
2. Model and formulas
The semi-classical Drude model can be used to describe the anisotropy of surface conductivity and electron effective mass for borophene. The conductivity components for borophene can be expressed as follows [20,39]:
where ${D_j}$ denotes the Drude weight along the X or Y directions, $\tau $ denotes the mean free time of electrons, ns denotes the electron density, mj denotes the effective mass of electron along the x or y direction. Here we assume the $\tau = 65fs,{m_x} = 1.4{m_0},\; and\; {m_y} = 3.4{m_0}$, where m0 denotes the static electron mass. Figure 1 shows the schematic for the PSHE in bilayer borophene metasurfaces composed of two same borophene metasurfaces. Here the material platforms for the metasurface are the borophene nanoribbons, where each borophene strip has width W and neighboring strips are separated by an air gap G. For deeply subwavelength periodicity $L = G + W \ll \lambda $ ($\lambda $ denotes the free-space wavelength), the metasurface response is assumed to be homogenized and the corresponding effective tensor for surface conductivity can be expressed as [40] $\tilde{\sigma } = [{{\sigma_{\alpha \alpha }},0;0,{\sigma_{\beta \beta }}} ]$, where ${\sigma _{\beta \beta }} = {({1/{\sigma_Y} + i/({\omega {\sigma_C}} )} )^{ - 1}}$, ${\sigma _{\alpha \alpha }} = W{\sigma _X}/L$. Here $\omega = 2\pi f$ denotes the radial frequency, ${\varepsilon _0}$ denotes the free-space permittivity and ${\sigma _C} ={-} ({2iL\omega {\varepsilon_0}/\pi } )\ln [{1/\sin ({G\pi /2L} )} ]$ is the effective strip conductivity. Here $\beta $ and $\alpha $ denote the prime characteristic directions which are parallel and perpendicular to periodical directions, respectively. It has been reported that there exist the elliptic topology of the equal-frequency contours of the light dispersion in the monolayer metasurface when both conductivity tensor components have a positive imaginary part (i.e., ${\mathop{\textrm {Im}}\nolimits} ({\sigma _{\alpha \alpha }}) > 0$ and ${\mathop{\textrm {Im}}\nolimits} ({\sigma _{\beta \beta }}) > 0$), and hyperbolic dispersion topologies appear when the imaginary parts of the two conductivity tensor components have different signs (i.e., ${\mathop{\textrm {sgn}}} ({Im[{{\sigma_{\alpha \alpha }}} ]} )\ne {\mathop{\textrm {sgn}}} ({Im[{{\sigma_{\beta \beta }}} ]} )$) [29,41]. The rotation angles ${\varphi _{1,2}}$, between the X direction and the principal direction of each metasurface $\alpha $, can be used to some extent to determine the nature of the bilayer metasurface. When the incidence plane of beam is at a rotation angle ${\varphi _{1,2}}({i = 1,2} )$ to the main axis for metasurface, the rotated conductivity tensor can be described as [41]:A monochromatic Gaussian beam with wavelength $\lambda $ and beam waist ${w_0}$ can be written as ${\tilde{E}_i} = \left( {{w_0}/\sqrt {2\pi } } \right)\exp [{ - ({k_{ix}^2 + k_{iy}^2} )w_0^2/4} ],$ where ${k_{iy}}$ and ${k_{ix}}$ denote the wave vector components in the Y and X directions, respectively. Here it is assumed that the incident beam has a very narrow angular spectrum, $\Delta k \ll k$ (i.e., a sufficiently large waist ${w_0} = 600\lambda $). Based on the spin, the linearly beam can be expressed as $\tilde{E}_i^H = \left( {1/\sqrt 2 } \right)({{{\tilde{E}}_{i - }} + {{\tilde{E}}_{i + }}} )$ and $\tilde{E}_i^V = i(1/\sqrt 2 )({{{\tilde{E}}_{i - }} - {{\tilde{E}}_{i + }}} )$, where H and V present the horizontal and vertical polarization states, respectively. ${\tilde{E}_{i\textrm{ - }}}$ and ${\tilde{E}_{i\textrm{ + }}}$ denote the RCP components and LCP components, respectively. The boundary conditions are used and then the transmitted beams for linearly polarized incident beam can be given by [31,42]
Fig. 1. Schematic illustrating the PSHE of a wave packet transmitted from the bilayer borophene metasurfaces. The bilayer metasurfaces are consisting of two same borophene metasurfaces with a spacer h between them. The refractive index of the background and spacer media are ${n_1}\left( {{n_1} = \sqrt {{\varepsilon_1}} } \right)\; $ and ${n_2}\left( {{n_2} = \sqrt {{\varepsilon_2}} } \right)$, respectively. The incident plane is X-Z plane.
3. Results and analysis
Figure 2(a) shows near one particular frequency band there exists the giant PSHE shift of bilayer borophene metasurfaces, whose oscillation amplitude gradually decreases as the angle of incidence increases. Figure 2(b) shows near the same frequency band there exists the relatively large PSHE shift of monolayer borophene metasurface, whose magnitude is significantly smaller than that in the bilayer structure as shown in Fig. 2(a). For simplify, the physical parameters of the monolayer borophene metasurface are the same as those of the borophene metasurface 1 in Fig. 1. In few-layer anisotropic metasurfaces the imaginary part for ${\sigma _{\alpha \alpha ,\beta \beta }}$ is responsible for the polarizability [41]. From Fig. 2(c), it can be seen that the behavior of $Im({\sigma _{\beta \beta }})$ determine the two main topologies implemented by the borophene metasurface when $Im({{\sigma_{\alpha \alpha }}} )> 0$. For $Im({\sigma _{\beta \beta }}) < 0$, obtaining the hyperbolic topology of the equal-frequency contours of the light dispersion in metasurface; for $Im({\sigma _{\beta \beta }}) > 0$, realizing the elliptic topology in borophene metasurface. The topological transition point is given by the pole of ${\sigma _{\beta \beta }}$, which provides a resonant response similar to that found in the graphene metasurface [29]. Figures. 2(a, c) clearly show that the giant PSHE shift of bilayer borophene metasurfaces is closely related to the topological transition point ($f \approx 230THz$). The |tpp| dip in Fig. 2(d) can be attributed to the existence of the topological transition. Figure 2(d) shows that a nonzero |tsp | (and |tps | = |tsp |) appears, which denotes that the transmitted beam contains both RCP and LCP components since the transmitted plane wave becomes elliptically polarized. Only considering the zero-order Taylor series of the transmission coefficients, then the spatial PSHE shift of the transmitted beam can be expressed as [31,43]$\Delta Y_H^t \approx ({1 + Re ({{t_{ss}}} )/Re ({{t_{pp}}} )} )\cot {\theta _i}/{k_0}$, where the existence of the $Re ({{t_{ss}}} )/Re ({{t_{pp}}} )$ and $\cot {\theta _i}$ can be used to explain why large PSHE shifts exist near the topological transition and at extremely small incident angles, respectively. The reason is that when the incident angle or the value of ${t_{pp}}$ tend to zero, there exists the numerical maximal term in the approximate expression for the PSHE shift $\Delta Y_H^t$.
Fig. 2. PSHE shifts $D = \Delta Y_H^t$ of bilayer borophene metasurfaces (a) and monolayer borophene metasurface (b) changing with the incident angle and frequency. (c) Dependence of the effective conductivity ${\sigma _{\alpha \alpha ,\beta \beta }}$ for monolayer borophene metasurface on frequency. (d) Fresnel transmission coefficients of bilayer borophene metasurfaces as function of frequency. Some parameters are ${\varphi _1} = 0.1({rad} ),\; {\varphi _2} = \pi ({rad} )$, ${\varepsilon _{1,2}} = 1,$ $L = 120nm,\; W = 56nm,h = 1.6\lambda $ and ${n_s} = 5 \times {10^{19}}{m^{ - 2}}.$
From Figs. 2(a, b), it can be inferred that the enhancement mechanism leading to the giant PSHE shifts of bilayer borophene metasurfaces might not originate solely from the topological transition of borophene metasurface, although it must be one important factor. From Fig. 3(a), it can be seen that near the topological transition point ($f \approx 230THz$) there exists the oscillation distribution for the PSHE shifts of bilayer borophene metasurfaces, which can be attributed to the Fabry-Perot resonance in bilayer anisotropic metastructures [41]. As a comparison, in layered nanostructures containing 2D materials there exist the PSHE shifts enhanced by the Fabry-Perot resonance [31,42], but the magnitude of the value for the PSHE shift is on the order of a few of wavelengths. It can be considered that the enhanced mechanism responsible for extremely large PSHE shifts in bilayer borophene metasurfaces lies in the combined effect of two resonance system: the Fabry-Perot resonance of the bilayer system and the resonant interaction of individual meta-atoms in anisotropic metasurface which lead to the topological transition [41]. It is worth pointing out that the combined effect of the two resonance systems on the PSHE shifts might be far greater than the effect of either resonance system alone. In addition, it must be noted that one necessary premise for the combined effect of two resonance system to have an obvious effect on the large PSHE shifts is that the topological transition points of the two metasurfaces in the bilayer system should be approximately realized and matched, which requires the borophene strips of two metasurfaces are both approximately perpendicular to the incident plane according to Eq. (1). Figure 3(b) shows that the PSHE shifts of bilayer borophene metasurfaces near the topological transition can be manipulated by the twist angle $\Delta \varphi = {\varphi _1} - {\varphi _2}$. When the twist angle $\Delta \varphi $ tends to zero or an integer value of $\pi $, the value for the PSHE shift tends to be maximum. It is noting that when the distance $h \ll \lambda $ it is difficult to achieve twist-induced photonic dispersion modifications [44] in the bilayer borophene metasurfaces due to the relatively weak plasmonic properties of borophene compared to graphene. Figure 3(c) shows that the oscillation relationship for ${t_{sp}}$ and ${\theta _i}$ appears in some regions, which might be used to explain the slightly large $\Delta X_H^t\; $ in Fig. 3(d) according to $\Delta X_H^t \approx ({\partial {t_{sp}}/\partial {\theta_i}} )/{k_0}({{t_{pp}} + i{t_{sp}}} )$ [45]. However, near the topological transition of borophene metasurface the value of $\Delta X_{H,V}^t$ and $\Delta Y_V^t$ is much smaller than that of $\Delta Y_H^t$, so for simplify they are not illustrated in the following.
Fig. 3. (a) PSHE shift $D = \Delta Y_H^t$ of bilayer borophene metasurfaces changing with the distance h and frequency when ${\theta _i} = 0.2({rad} )\; {and}\; \Delta \varphi = {\varphi _1} - {\varphi _2} = 0.1({rad} )$. (b) The dependence of PSHE shift $D = \Delta Y_H^t$ on the twist angle and frequency for ${\theta _i} = 0.2({rad} )\; {and}\; h = 1.2\lambda $. (c) Fresnel transmission coefficient $|{{t_{sp}}} |$ as functions ${\theta _i}$ and $\Delta \varphi $ for $h = 1.2\lambda $. (d) PSHE shift $D = \Delta X_H^t$ of bilayer borophene metasurfaces changing with the distance h and $\Delta \varphi $ when ${\theta _i} = 0.2({rad} )$. Other parameters are ${\varphi _1} = 0.1({rad} ),\; {\varepsilon _{1,2}} = 1,$ $L = 120nm,\; W = 56nm\; $ and ${n_s} = 5 \times {10^{19}}{m^{ - 2}}.$
By using standard e-beam lithography the proposed bilayer borophene metasurfaces can be fabricated and the corresponding operation frequency of the metasurfaces can be manipulated by mechanical modulations or electrical doping [46]. Figure 4(a) present imaginary parts of conductivity ${\sigma _{\beta \beta }}$ for the metasurface composed of borophene nanoribbons under different ribbon width. It can be inferred that the frequency of topological transition redshift with increasing ribbon width. The electron density for borophene can be manipulated by external voltage or chemical doping, which is similar to BP and graphene [47]. The imaginary parts of conductivity ${\sigma _{\beta \beta }}$ for the metasurface composed of borophene ribbons under different electron densities are provided in Fig. 4(c), which show that the frequency of topological transition blueshift with increasing electron density. Combining Figs. 4(a,b) and Figs. 4(c,d), it can be found that there exist the giant PSHE shifts in bilayer borophene metasurfaces near the topological transition region. Furthermore, it is worth noting that Figs. 4(b, d) show that the manipulation frequency range of the large PSHE shifts can reach hundreds of terahertz or even picohertz by adjusting the ribbon width of borophene metasurface or the electron density for borophene. Such dynamic tunability of the giant PSHE shifts near the topological transition makes it possible to selectively sense specific frequencies of interest under various physical parameter settings over a ultrawide frequency range.
Fig. 4. (a) Imaginary parts of conductivity ${\sigma _{\beta \beta }}$ for the metasurface composed of borophene ribbons under different ribbon widths for ${n_s} = 5 \times {10^{19}}{m^{ - 2}}$. (b) Dependence of PSHE shift $D = \Delta Y_H^t$ for bilayer borophene metasurfaces on the frequency and the ribbon widths for ${n_s} = 5 \times {10^{19}}{m^{ - 2}}$. (c) Imaginary parts of conductivity ${\sigma _{\beta \beta }}\; $ for the metasurface composed of borophene ribbons under different electron densities for $W = 56nm$. (d) Dependence of PSHE shift $D = \Delta Y_H^t\; $ for bilayer borophene metasurfaces on the frequency and the electron densities for $W = 56nm$. Other parameters are ${\varphi _1} = 0.1({rad} ),\; \; {\varphi _2} = \pi ({rad} )$, ${\varepsilon _{1,2}} = 1,$ $L = 120nm,\; h = 1.6\lambda \; \textrm{and}\; {\theta _i} = 0.2({rad} ).$
Figure 5(a) shows the PSHE shift for bilayer borophene metasurfaces as functions as the frequency and the permittivity ${\varepsilon _2}$ for spacer material. It is obvious that near the frequency for topological transition there exist the giant PSHE shift, which is significantly affected by the value of the permittivity ${\varepsilon _2}$ for spacer material, especially near the ENZ region. ENZ material generally is generally considered as a type of artificial material which exhibits novel light-matter interaction, and can be used to enhance the PSHE shifts [36,48–38]. Figure 5(b) show that when the value of the permittivity ${\varepsilon _2}$ is relatively large, the value of the PSHE shift gradually decreases with the increase of the incident angle. In addition, when the value of the permittivity ${\varepsilon _2}$ is near the ENZ region, extremely large PSHE shift occurs over a relatively wide range of incidence angles. As shown in Fig. 5(c), it can be seen that when the value of the permittivity ${\varepsilon _2}$ is relatively large, there exists the periodic behavior for the PSHE shift on the separation distance h. Near the ENZ region, when the separation distance is small the value of the PSHE shift increases with the increase of the distance, and the value of the PSHE shift tends to be constant when the separation distance is large. Figure 5(d) shows that the twist angle has a modulating effect on the PSHE shift regardless of the value of the permittivity ${\varepsilon _2}$ (even near the ENZ region). The magnitudes of the PSHE shifts occurring near the ENZ regions in Fig. 5 are larger than those in the non-ENZ regions. Although the function expression of ${t_{pp}}$ is relatively complex, through complex calculation it can be inferred that the corresponding expression contains a certain value item ${\varepsilon _2}/si{n^2}{\theta _i}$, which is similar to the case in the thin ENZ layer [48]. When the incident angle is not zero the value of ${t_{pp}}$ tends to zero near the ENZ region, and according to $\Delta Y_H^t \approx ({1 + \textrm{Re} ({{t_{ss}}} )/Re ({{t_{pp}}} )} )\cot {\theta _i}/{k_0}$ it can be inferred that there exist extremely large PSHE shift at this time. In addition, it should be pointed out that the oscillatory distribution of the PSHE shifts with respect to ${\varepsilon _2}$ shown in Figs. 5(a, b, d) can be attributed to the Fabry-Perot resonance in bilayer anisotropic metastructures. Losses are generally present in ENZ materials and are likely to affect the PSHE shifts unless they are low-loss [49]. The PSHE shifts as functions of the image part of the permittivity ${\varepsilon _2}$ of ENZ material and the incident angle are shown in Fig. 6(a), which shows that the PSHE shifts decrease with the $Im({{\varepsilon_2}} )$ gradually within the certain angle of incidence. Figure 6(b) shows that when the loss of ENZ material is relatively low there exist the extremely large PSHE shifts, which approach to asymptotic values with increasing the spacer thickness.
Fig. 5. (a)Dependence of PSHE shift $D = \Delta Y_H^t$ for bilayer borophene metasurfaces on the frequency and the permittivity ${\varepsilon _2}$ for spacer material when ${\theta _i} = 0.1({rad} ),\; h = 1.6\lambda \; and\; \Delta \varphi = 0.1({rad} )$. (b)Dependence of PSHE shift $D = \Delta Y_H^t$ on the incident angle and the permittivity ${\varepsilon _2}$ when $f = 230THz,h = 1.6\lambda \; and\; \Delta \varphi = 0.1({rad} )$. (c)Dependence of PSHE shift $D = \Delta Y_H^t$ on the spacer thickness and the permittivity ${\varepsilon _2}$ when ${\theta _i} = 0.1({rad} ),f = 230THz\; {and}\; \Delta \varphi = 0.1({rad} )$. (d)Dependence of PSHE shift $D = \Delta Y_H^t$ on the twist angle $\Delta \varphi $ and the permittivity ${\varepsilon _2}$ when ${\theta _i} = 0.1({rad} ),f = 230THz\; and\; h = 1.6\mathrm{\lambda }$. Other parameters are ${\varphi _1} = 0.1({rad} ),\; {\varepsilon _1} = 1,$ $L = 120nm,\; W = 56nm$ and ${n_s} = 5 \times {10^{19}}{m^{ - 2}}.$
Fig. 6. (a) The PSHE shift $D = \Delta Y_H^t\; $ as functions of the image part of the permittivity ${\varepsilon _2} = 0.01 + ik$ of ENZ material and the incident angle. (b) Dependence of the PSHE shift $D = \Delta Y_H^t\; $ on the spacer thickness and the image part of the permittivity ${\varepsilon _2} = 0.01 + ik$ of ENZ material. Other parameters are ${\varphi _1} = 0.1({rad} ),\; {\varepsilon _1} = 1,$ $L = 120nm,\; W = 56nm$, ${n_s} = 5 \times {10^{19}}{m^{ - 2}},h = 1.6\lambda ,{\theta _i} = 0.1({rad} ),f = 230THz\; and\; \Delta \varphi = 0.1({rad} )$.
Figure 7(a) shows the PSHE shift of bilayer borophene metasurfaces changing with the refractive index for the spacer material. From Fig. 7(a) it can be seen that the PSHE shifts are very sensitive to the index changing of spacer material in the wide sensing range, especially when the refractive index is relatively low. For example, as the spacer index n2 increases from 1.01 to 1.09, the PSHE shift changes from 29$\lambda ({\sim 38\mu m} )$ to 228$\lambda ({\sim 328\mu m} )$. The PSHE shift changing drastically with tiny change for refractive index provides a tough application basis for refractive index sensing. The refractive index sensitivity can be expressed as the ratio of the change in the PSHE shifts with the change in the refractive index $({{n_2}} )$ for the sensing region [50]: $S = \delta D/\delta {n_2}$. The maximum sensitivities of the PSHE shifts to refractive index in different refractive index sensing regions are shown in Figs. 7(b-d), which illustrate the sensitivity increases as the refractive index of the sensing medium decreases. Specifically, the refractive index sensitivity for the corresponding structure shows a peak of $3.03 \times {10^3}\lambda \; 1/RIU({\sim 3.8 \times {{10}^3}\mu m/RIU} )$. In contrast, the sensitivity of previous terahertz biosensors generally reaches the ${10^4}nm/RIU$ order [51,52]. Due to the ultrahigh sensitivity of the PSHE shifts to refractive index, the corresponding structure in Fig. 1 can be applied to design the refractive index sensors with high performance which are urgently demanded in many scenarios such as biochemical detecting and climate monitoring [53]. From Fig. 5 and Fig. 7, it can be reasonably inferred that the sensing region and sensitivity of the corresponding structures can be flexibly regulated by adjusting the twist angle, incident angle and spacer thickness. According to the corresponding phenomenon in Fig. 4, it is possible to obtain the sensing devices composed of borophene metasurfaces with a ultrawide operation frequency range. It is worth mentioning that the two-dimensional properties of borophene enable us to construct the compact sensing devices containing borophene.
Fig. 7. (a) The PSHE shift $D = \Delta Y_H^t$ of bilayer borophene metasurfaces changing with refractive index ${n_2} = \sqrt {{\varepsilon _2}} \; $. (b-d) The corresponding maximum sensitivity of the PSHE shift to refractive index in different refractive index sensing regions. Some parameters are $W = 56nm,{\varphi _1} = 0.0({rad} ),\; {\varepsilon _1} = 1,\; \; L = 120nm,\; \; {n_s} = 5 \times {10^{19}}{m^{ - 2}},\; \; h = 1.6\lambda ,\; {\theta _i} = 0.01({rad} ),f = 230THz\;\textrm{and}\; \Delta \varphi = 0.0({rad} )$.
4. Conclusions and discussions
In conclusion, we have studied theoretically the PHSE shifts in bilayer borophene metasurfaces. It is found that there exist the giant PSHE shifts of the transmitted beams which can be attributed to the combined effect of the Fabry-Perot resonance of the bilayer system and the resonant interaction of individual meta-atoms in borophene metasurface. The giant PSHE shifts can be flexibly controlled by adjusting the twist angle of metasurface bilayers, incident angle, spacer refractive index and spacer thickness. By adjusting the electron density for borophene or the ribbon width of borophene metasurface, the manipulation frequency range of the large PSHE shifts can reach hundreds of terahertz or even picohertz. Near the topological transition of borophene metasurface, the magnitude of PHSE shifts in bilayer borophene metasurfaces is generally on the order of tens of wavelengths and even on the order of hundreds of wavelengths near the ENZ regions. It is expected that the ultrahigh sensitivity of the PSHE shifts to spacer refractive index can be used to design the refractive index sensors with high performance. The platform of the anisotropic metastructures here is selected as borophene nanoribbons, but it is worth noting that other 2D material nanoribbons with topological transition properties can also be used to construct bilayer metasurfaces with the similar phenomena in this paper, such as graphene, black phosphorus, and other van der Waals materials. In this paper near the topological transition for borophene metasurface the value for the longitudinal component $\Delta X$ of the PHSE shift is much smaller than that for the transverse component $\Delta Y$ of the PHSE shift, and similar phenomena have been reported in monolayer graphene metasurface [30]. It is worth mentioning that if the general Gaussian beam in this paper is replaced by the Laguerre-Gaussian (LG) beam, near the topological transition for metasurface the orbit-angular momentum of LG beam might greatly enhance the longitudinal component $\Delta X$ of the PHSE shift due to the orbit-orbit interaction [30,54]. It should be pointed out that here the premise of using the effective conductivity tensor as shown in Eq. (1) is that the near-field coupling of the bilayer system is weak and that only the local response in the metasurface system is accounted. If the distance between bilayer is ultrathin ($h\; $ tends to zero), due to the strong coupling strength of the bilayer system there might exist anti-crossing effect for the dispersion relation of the metasurface bilayer system [41,44], which requires a correction in the form for the effective conductivity tensor of the system. If the granularity for the metasurface is not fine, it must be considered that the nonlocal form of the effective conductivity tensor arising from the inhomogeneity of one metasurface associated with one meta-atom structure [55].
Funding
Natural Science Foundation of Fujian Province (2018J01421, 2022J011129); National Natural Science Foundation of China (11547254, 61575043).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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