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Abstract
We study the plasmonic properties of face-to-face phosphorene pairs, including their optical constraints and optical gradient forces. The symmetric and anti-symmetric plasmonic modes occur due to the strong anisotropic dispersion of phosphorene. Compared with the anti-symmetric mode, the symmetric mode has a stronger optical constraint and much larger gradient force. Especially, the optical constraint of the symmetric mode can even reach as high as 96% when the two phosphorene layers are along the armchair and zigzag direction respectively. We also propose a scheme of an ultra-small phase shifter using phosphorene-based photonic devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When light travels in two parallel waveguides, the attractive or repulsive optical gradient force can be generated, depending on the relative phase difference of corresponding guided modes [1,2]. The optical force provides a new manner for manipulation of the light in photonics, whereas a strong force favors efficient manipulation [3–14]. Enhancement of the optical force have been shown by employing slow light in photonic crystals and the cavity resonance of a high-Q (quality factor) optical resonator [15]. Metamaterials and surface plasmon polaritons (SPPs) can also substantially enhance the optical force as the result of the strong optical energy constraint of metallic surface plasmons (MSPs) [16]. Moreover, optical force interfered by a Bessel beam [17] and Fano resonant [18–21] has been reported.
It is discovered that the graphene can support the formation and propagation of SPPs in infrared and terahertz ranges [22–33], which have great innovations in recent years.. Furthermore, as a newly found 2D material, single-layer black phosphorus (BP), the so-called phosphorene, has attracted intense research interest owing to its natural semi-conductor properties [34]. An enormous amount of studies are still carried out to obtain a deeper understanding of this material [35–37]. In 2015, Xu et al made an in-depth study of the thermal conductivity of phosphorene [38], which caused tremendous repercussions throughout the world. Phosphorene is promising for optoelectronics and nanophotonics because it has a thickness-dependent direct electronic bandgap [39–44]. In phosphorene, the covalent bonds of the phosphorus atoms make a hexagonal lattice of a folded cellular structure with other three atoms [45]. This special atomic structure causes extremely anisotropic electronic dispersion and direction-dependent conductivity [46–48]. Compared to graphene, phosphorene may be more suitable for series of applications, such as nano-resonators [49], flexible electronics [50], charge trap memories [51], Terahertz photodetection [52], and amplitude modulation (AM) demodulators [50]. In 2017, Lu et al. studied the optical gradient force between face-to-face infinite-width phosphorene pairs [48] and took full consideration of the strong orientation dependence in BP's optical properties. However, optical gradient forces in the side-by-side configuration haven't been considered.
In this paper, we study the plasmonic properties of face-to-face finite-width phosphorene nanoribbon pairs, including their optical constraints and optical gradient forces. We set the two phosphorene layers in three different relative orientations, i.e., the armchair-armchair, zigzag-zigzag and armchair-zigzag configurations, and the symmetric and anti-symmetric modes are found in all three configurations. Results show that the symmetric modes exhibit larger optical constraints and gradient forces compared to the anti-symmetric modes in BP pairs. More interestingly, we find that the zigzag-zigzag and armchair-zigzag configurations have the largest optical gradient force and optical constraint ratio, respectively. In addition, we add the discussion of side-by-side configuration, which can also support the symmetric and anti-symmetric modes. Our results may be of benefit to the implementation of the nanoscale optical manipulation and devices based on BP. Finally, we propose an ultra-small phase shifter as an example application of BP device.
2. Method and analysis
The conductivities of phosphorene can be described as
Here, Dj = πe2n/mj represents the Drude weight, where j can be x, y, z denoting the three directions respectively. ω is the angular frequency of light. ς is the electronic relaxation rate of phosphorene. n is the electronic doping. The electronic masses in the zigzag and armchair directions of phosphorene layers are respectively given by [53,54]
where ςc and υc are relevant to the effective masses, γ is the effective coupling of band edges, and Δ describes the energy band gap. By fitting the mass of the anisotropy, the above parameters of the phosphorene layer are set as γ = 4a/π eVm, Δ = 2 eV, ςc = ћ2/0.4m0, υc = ћ2/1.4m0, n = 1013/cm2, and ς = 10 meV [46]. Here, a (≈0.223 nm) represents the scale length of phosphorene (hence π/a stands for the width of Brillouin Zone), m0 describes the electronic mass, and ћ is the reduced Planck's constant. The equivalent relative permittivities are given by [53]where εr = 5.76, t is the thickness of the BP layer. From above equations, we can learn the relative permittivities are dependent on conductivities and the direction of light propagation. Simulations are performed using finite element method employing COMSOL Multiphysics.The field expressions of the transverse magnetic (TM) waves can be given by the following equations [55]
Here, d is the distance between the BP pair. Ψ(y) = cosh(χy)/cosh(χd/2) and sinh(χy)/sinh(χd/2) for the symmetric and anti-symmetric mode, respectively. Ψ'(y) = sinh(χy)/cosh(χd/2) and cosh(χy)/sinh(χd/2) for the symmetric and anti-symmetric mode, respectively. θ is the phase shift of the field sandwiched by the BP pairs due to the coupling behavior. β is the propagation constant of the modes. The parameter ky is the wavevector of light in the structure along y-axis direction, and χ is the field decay rate in the space, which are governed by
where k0 is the wavevector of incident light in vacuum. Because of the continuity conditions of tangential component Ez at the interfaces (i.e., y = ± d/2 and y = ± (d/2 + t)), the characteristic equations of basic plasmonic modes in the BP pairs are determined byThrough integrating the Maxwell Stress Tensor (MST) around a discretional surface containing the phosphorene pairs, and the gradient force fn can be calculated [1]. By defining [56]
where δ is the Kronecker delta function, the normalized gradient force along y direction is defined aswhere S is the surface of the volume containing the phosphorene pairs and ny is the unit vector along the y direction. In accordance with the energy conservation law in two parallel BP sheets, the force can also be described as3. Results and discussion
Figure 1(a) shows the molecular structure of the phosphorene. As can be seen from Fig. 1(b), the phosphorene has different structures along the armchair (z-axis) and the zigzag (x-axis) directions. Figure 1(c) shows the schematic of two separated BP layers, in a face-to-face configuration with a gap d between the pair. For simplicity, the surrounding material is considered as vacuum. The width and the thickness of the phosphorene are respectively defined as w and t. The semi-classical Drude model can effectively describe the optical properties of the phosphorene film [47, 57].
Fig. 1 (a) The molecular structure of three-layer phosphorene; (b) Top view of BP. The x and z axes are along the zigzag and armchair directions, respectively; (c) The first diagrammatic sketch of the structure employed.
The relative permittivities of phosphorene are plotted in Fig. 2. It is worth noting that the difference between the relative permittivities in the x- and z-axis directions is caused by the directional dependence of the effective electron mass. Similar to the graphene, in the infrared region, the real parts of the relative permittivities of BP are negative [58].
Fig. 2 (a) Real and (b) imaginary parts of the relative permittivities of BP in the z-axis (armchair) and x-axis (zigzag) directions in the infrared regions.
Here we consider the wavelength λ0 = 20 μm. Figure 3(a) shows the electric field y-components of the symmetric mode and of the anti-symmetric mode with d = 50 nm for the armchair direction. From Fig. 3(b), we can see the effective refractive index of the symmetric mode is larger than that of the anti-symmetric mode. Meanwhile, as the gap d decreases, the inclination of the former becomes larger, which indicates a larger optical field gradient.
Fig. 3 (a,b) The electric field y-component with d = 50 nm and effective refractive index of the symmetric mode and the anti-symmetric mode for the Armchair configuration, respectively. (c,d) Optical constraint ratio(ξ) of the two modes in the BP pair with various d. The inset shows the intensity of normalized Pz with d = 50 nm. (e) The gradient force of the two modes with different d.
Besides, we calculated the optical constraint ratio in the BP pair, which is defined as the ratio of the integrated Poynting vector Pz in the regions between and outside the phosphorene layers,
The distributions of Pz in the BP structures can be obtained by using the FEM simulations, and the optical constraint ratio ξ can be obtained by integrating Pz vertical to the phosphorene layers.
Figures 3(c) and 3(d) respectively show the optical constraint ratio of the two modes. We can see that the optical constraint ratio of the symmetric mode is rather high. The optical constraint ratio can even exceed over 90% when d < 10 nm. When d = 200 nm, the optical constraint ratio is still over 60%. In contrast, the optical constraint ratio of the anti-symmetric mode is quite low—just less than 30%. Just as the inset of Fig. 3(c), for the symmetric mode, most light power can be concentrated between the phosphorene pair, while the majority of the light power spreads outside of the BP pair for the anti-symmetric mode as the inset of Fig. 3(d) shows. The strong constraint of light in the BP pairs makes the contribution to the formation of giant force [59]. Figure 3(e) shows the gradient force fn as functions of the gap d for the two modes. In comparison with the symmetric mode, the gradient force fn under the anti-symmetric mode is orders or magnitudes smaller at a small gap distance.
Figure 4(a) shows the effective refractive index as functions of the gap d for different width w. We can see that the width of phosphorene pair can hardly influence the effective refractive index, therefore we choose a constant w = 600 nm in this study. Similarly, we can obtain that the thickness of the phosphorene pair can scarcely affect the effective refractive index from Fig. 4(b). That means the width and the thickness of the phosphorene pair nearly have no effect on the gradient force.
Fig. 4 (a,b) The effective refractive index of several different widths and thickness, respectively.
Figure 5(a) shows the electric field y-components of the symmetric mode and of the anti-symmetric mode with d = 50 nm for the zigzag direction. From Fig. 5(b), we can see the effective refractive index of the symmetric mode is larger. Meanwhile, with the gap d decreasing, the slope is larger, which indicates a larger optical field gradient. Figures 5(c) and 5(d) respectively show the optical constraint ratio of the two modes. We can find the optical constraint ratio of the symmetric mode under the zigzag direction is a little less than that under the armchair direction. Conversely, the optical constraint ratio of the anti-symmetric mode under the zigzag direction is much larger than that under the armchair direction. Just as the inset of Fig. 5(c) shows, for the symmetric mode, most light power can be concentrated between the BP pair, while the majority of the light power spreads outside the phosphorene pair for the anti-symmetric mode as the inset of Fig. 5(d) shows. Figure 5(e) shows the gradient force fn as functions of the gap d for the two modes. Similar to that of armchair-armchair configurations, in comparison with the symmetric mode, the gradient force fn under the anti-symmetric mode can be a few orders of magnitudes weaker for a small gap d.
Fig. 5 (a,b) The electric field y-component with d = 50 nm and effective refractive index of the symmetric mode and the anti-symmetric mode for the Zigzag configuration, respectively. (c,d) Optical constraint ratio(ξ) of the two modes in the BP pair with various d. The inset shows the intensity of normalized Pz with d = 50 nm. (e) The gradient force of the two modes with different d.
Then we set the two phosphorene nanoribbons in different orientations—one is the armchair while the other is the zigzag. The electric field y-components of the symmetric mode and of the anti-symmetric mode with d = 50 nm are shown in Fig. 6(a). Simultaneously, the slope of the effective refractive index of the symmetric mode is much larger compared with that of the anti-symmetric mode with the gap d increasing, which means a larger optical field gradient. Figures 6(c) and 6(d) show the optical constraint ratios of the two modes respectively. It is shown that the optical constraint ratios of the two modes are less than that of the armchair-armchair and the zigzag- zigzag configurations. It can be clearly seen from the insets of Figs. 6(c) and 6(d) that most light power can be concentrated between the phosphorene pair for the symmetric mode. It is worth to note that the optical constrain ratio in Fig. 6(d) is negative when the gap is small. This is because the Pz between the phosphorene layers is small but negative, while the Pz outside the BP layers is large and positive. On the contrary for the anti-symmetric mode, the majority of the light power spreads outside the BP pair. Figure 6(e) shows the optical forces of two modes similar to that of the previous two configurations.
Fig. 6 (a,b) The electric field y-component with d = 50 nm and effective refractive index of the symmetric mode and the anti-symmetric mode for the Armchair + Zigzag configuration, respectively. (c,d) Optical constraint ratio(ξ) of the two modes in the BP pair with various d. The inset shows the intensity of normalized Pz with d = 50 nm. (e) The gradient force of the two modes with different d.
It is noted that the side-by-side configuration of two phosphorene pair as Fig. 7(a) shows can also support the symmetric and anti-symmetric modes, but the gradient forces between them are much smaller than those in the face-to-face configuration. The blue area represents the phosphorene nanoribbons, and in order to simplify the simulation, we still regard the surrounding media as vacuum.
Fig. 7 (a) The first diagrammatic sketch of the structure employed. (b) The electric field x-component of the symmetric mode and the anti-symmetric mode for the phosphorene nanoribbons arranged in three ways with d = 20 nm, respectively. (c,d) The effective refractive index and the gradient force of the above modes with various d.
The width of the phosphorene is defined as w, and the thickness is t. The gap d is the distance between the edges of the separated phosphorene pair. Again, we put the two phosphorene nanoribbons in three different orientations, i.e., the armchair-armchair, zigzag-zigzag and armchair-zigzag configurations. The electric field x-components of the symmetric modes and the anti-symmetric modes with d = 20 nm for the three situations are shown in Fig. 7(b). It can be seen that light powers are mostly concentrated between the two nanoribbons for the symmetric modes. By contrast, for the anti-symmetric modes, the light powers spread out on both sides of gap between the two nanoribbons. Figure 7(c) shows the effective refractive indices of the three situations. Figure 7(d) shows the optical forces of the three situations. In Fig. 7(c), we can see the effective refractive index of the symmetric mode is larger than that of the anti-symmetric mode in the same configuration. With the gap d decreasing, slopes of the effective refractive index of the symmetric modes are larger, indicating larger optical field gradient. However, for the effective refractive index of the anti-symmetric modes, changes of slopes are quite small, which indicates a small optical field gradient. As a result, in comparison with the symmetric modes, the gradient forces fn under the anti-symmetric modes are orders or magnitudes smaller at a small gap distance, which is shown in Fig. 7(d).
4. The optomechanical applications
In virtue of the strong gradient force between the phosphorene layers, we propose a realization of ultra-small phase shifter working in mid-infrared region. The 3D schematic illustration is shown in Fig. 8(a). It is worth noting that cleanroom fabrication of this BP-based structure can be similar to that of the graphene-based photonics with additional measures to avoid undesired oxidation and doping. In this structure, we adopt the silicon waveguide to help the propagation of light and maintain the optical gradient force between the BP nanoribbon and sheet with low material loss. In our analysis, we set the width and the height of the free-standing silicon waveguide as 100 nm and 50 nm, respectively. The gap d0 between two phosphorene sheets is 32 nm. In consideration of the propagation loss, the length of the Si waveguide is set as 3.5 µm. The propagation length (1/[2|Im(β)|]) is shown in Fig. 8(b), and the inset shows the electric field x-component at wavelength λ0 = 20 μm calculated by COMSOL. By comparing the propagation length and the size of the device, light can transmit through the structure completely. Both the phosphorene sheet and nanoribbon are set along the zigzag direction. The deflection of the double-clamped beam is shown in Fig. 8(c). The separation of the phosphorene sheet and nanoribbon becomes narrower owing to the change of deflection, and causes the change of the effective refractive index as Fig. 8(d) shows. At low laser power levels, the magnitude of the deflection is almost linear with respect to the incident laser power. We can observe that the deflection is quite small, while it brings large variation for the effective refractive index.
Fig. 8 (a) The 3D schematic illustration of the optomechanical phase shifter. (b) The propagation length of the mode given by the inset. The inset shows the electric field x-component at wavelength λ0 = 20 μm. (c,d) Deflection and the effective refractive index along the free-standing waveguide under the circumstance of different incident power.
Finally, we can obtain the phase shift resulting from different incident power with the following formula:
where Δφ = k0∙S and S is the region sandwiched by two curves with propagation constants β1(z) and β2(z).5. Conclusion
In this paper, we analyzed the plasmonic properties of phosphorene pairs, including their optical constraints and gradient forces. The strong anisotropic dispersion of phosphorene gives rise to the symmetric and anti-symmetric plasmonic modes. We found the symmetric modes have larger optical constraint and optical gradient force than the anti-symmetric modes. It is worth noting that the optical constraint of the symmetric mode can even reach as high as 96% when we set the two phosphorene layers along different orientations—one is the armchair and the other is zigzag. Overall, shrinking the gap between a phosphorene pair can effectively enhance the gradient force. Meanwhile, the parallel phosphorene pair can provide a larger optical gradient force than the silicon waveguides. Finally, we also proposed an ultra-small phase shifter as an example application. Our results may be helpful for optical manipulation at nanoscale and development of phosphorene-based photonic devices.
Funding
National Science Foundation (NSF) (CMMI-1405078, CMMI-1554189, CMMI-1634832), the National Natural Science Foundation of China (Grant Nos. 11504139, 11504140, 11811530052), the China Postdoctoral Science Foundation (2017M611693, 2018T110440), and the Key Laboratory Open Fund of Institute of Semiconductors of CAS (Grant No. KLSMS-1604).
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