The anisotropic plasmons properties of black phosphorus allow for realizing direction-dependent plasmonics devices. Here, we theoretically investigated the hybridization between graphene surface plasmons (GSP) and anisotropic black phosphorus localized surface plasmons (BPLSP) in the strong coupling regime. By dynamically adjusting the Fermi level of graphene, we show that the strong coherent GSP-BPLSP coupling can be achieved in both armchair and zigzag directions, which is attributed to the anisotropic black phosphorus with different in-plane effective electron masses along the two crystal axes. The strong coupling is quantitatively described by calculating the dispersion of the hybrid modes using a coupled oscillator model. Mode splitting energy of 26.5 meV and 19 meV are determined for the GSP-BPLSP hybridization along armchair and zigzag direction, respectively. We also find that the coupling strength can be strongly affected by the distance between graphene sheet and black phosphorus nanoribbons. Our work may provide the building blocks to construct future highly compact anisotropic plasmonics devices based on two-dimensional materials at infrared and terahertz frequencies.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Surface plasmons are collective oscillations of electrons in metals with the unique capability of concentrating electromagnetic energy in deep subwavelength scale, enabling tremendous exciting applications in nanophotonics [1, 2]. Recently, graphene, a two-dimensional (2D) material, has been demonstrated to support localized and propagating surface plasmons with strong field confinement and low propagation loss in the infrared and terahertz region [3–6]. Particularly, such plasmonic responses in graphene can be actively tuned by an external gating voltage due to the controllable Fermi energy, a feature that is not available in traditional metallic surface plasmons. These unprecedented properties make graphene a promising one-atom-thick platform for the realization of novel optical functions and excellent 2D photonic devices, such as light harvesting , slow light , plasmonic waveguide , plasmonic modulators , optical polarization  and so forth. Meanwhile, it has been shown that graphene plasmons can hybridize with other fundamental modes such as graphene intrinsic optical phonons , phonons in polar substrates (e.g., SiC , SiNx , SiO2 and h-BN [16–19]), molecular vibrational modes [20, 21], metallic electromagnetic modes [22, 23] and even graphene plasmon itself [24–26]. These hybridizations give rise to many interesting phenomena, which can be used to exploit new device concepts for light manipulation at nanoscale.
Black phosphorus (BP) [21, 22], another newly emerging van der Waals-bonded 2D material, has attracted considerable attention due to its tunable direct bandgap, high carrier mobility, remarkable electrical and optical properties. Unlike graphene, the phosphorous atoms in monolayer BP covalently bond with three others and form a hexagonal lattice with a puckered honeycomb structure. This special atomic structure gives rise to the highly in-plane anisotropic electrical and optical properties [27–30]. Specifically, the in-plane effective electron masses along the two crystal axes can differ by an order of magnitude. These astonishing features have widely been investigated for potential applications including field effect transistors [31, 32], heterojunction p-n diode , photovoltaic devices  and photodetectors  etc. Quite recently, it is theoretically demonstrated that surface plasmon resonance can be excited in nanostructured BP ribbon arrays [36, 37], square arrays [38, 39] as well as monolayer [40–42] and multilayer [43, 44] BP film. Due to the puckered honeycomb lattice, BP possesses a larger conductivity along armchair direction than that along the zigzag direction, which consequently results in anisotropic plasmonic response along armchair and zigzag direction. These plasmon properties allow for the realization of novel 2D direction-dependent plasmonic devices. Therefore, the hybridization of the anisotropic BP plasmons with other fundamental modes, especially with plasmon modes in the other 2D materials such as graphene, may further open up the broad applications of BP in photonics and optoelectronics at infrared frequencies.
In this paper, we theoretically demonstrate the interaction between the graphene surface plasmons and the anisotropic plasmons of BP nanoribbons in the strong coupling regime. We investigated the hybridization of these two plasmon resonances. A coupled oscillator model is applied to quantitatively evaluate the dispersion of the hybrid modes. We show that the strong coupling with Rabi splitting of 26.5 meV and 19 meV can be achieved for BP nanoribbons along armchair and zigzag directions by dynamically varying the Fermi energy of graphene to tune the two modes into resonance. Additionally, the mode splitting energy of the hybrid modes for BP along armchair and zigzag directions is also compared. Finally, the effect of the distance between graphene sheet and BP nanoribbon on the coupling strength is discussed.
2. Structure and simulation method
The schematic of monolayer BP is sketched in Fig. 1(a), where x- and y-direction are donated as the armchair (AC) and zigzag (ZZ) directions, respectively. The hybrid system under study consists of a continuous graphene sheet and a patterned BP nanoribbon, as shown in Fig. 1(b). The inset shows the cross section of the structure. The continuous graphene sheet is first placed on a dielectric substrate, and a monolayer BP is covered on graphene sandwiched by a dielectric spacer with thickness d. The BP layer is then patterned into nanoribbon with period of P and width of W. When a light wave with electric filed polarized along the periodic direction of the BP nanoribbons (i.e., transverse magnetic wave) irradiates upon the system, BP localized surface plasmon (BPLSP) modes can be excited at the resonant frequencies in the BP nanoribbons. Meanwhile, with the assistance of the BP nanoribbons, propagating surface plasmons in continuous graphene sheet can be excited by the incident light once the in-plane wavevector of a graphene surface plasmonics (GSP) wave matches the wavevector of a diffraction order scattered by BP nanoribbon. If the two modes are brought into resonance, the system reaches the strong coupling regime and two new hybrid modes are formed.
To investigate the hybridization behaviors between the anisotropic BPLSP modes in BP nanoribbons and the GSP modes in continuous graphene sheets, simulations are performed using finite element method employing COMSOL Multiphysics. In the simulations, the periodic boundary condition is imposed in the horizontal direction, while the perfectly matched-layer (PML) is applied in vertical direction to achieve the absorbing boundary conditions at two ends of computational space. Non-uniform mesh is used in the simulation regions, where the mesh size gradually increases outside the graphene and BP layer, and the maximum element size in graphene and BP layer is set as 0.1 nm. A polarized light with input power of 1W in a period and electric field perpendicular to the BP nanoribbon vertically irradiates upon the hybrid structure. The conductivity of graphene is modeled by a Drude-like formula as 
The monolayer BP is also modeled with anisotropic conductivity σjj given by a simple semiclassical Drude model 
For monolayer BP, the parameters determined by fitting the known anisotropic mass  are eVm, = 2 eV, , , where m0 = 9.10938 × 10−31 kg is the standard electron rest mass, and a = 0.223 nm is the scale length of BP. The nondispersive refractive index of the spacer and the substrate is assumed to be n1 = 1.7.
3. Results and discussion
3.1 Hybridization between GSP mode and anisotropic BPLSP mode
We first investigate the anisotropic plasmons behavior of the BP nanoribbons. The geometric parameters of the BP nanoribbons for both x- and y-directions are set as P = 200 nm, W = 100 nm. The absorption spectra of BP nanoribbons along the x- and y-direction are shown in Figs. 2(a) and 2(b), respectively. It can be seen that the resonant frequency of the BP nanoribbons along x-direction locates at 28.6 THz, while that along y-direction positions at 18.2 THz. This is due to the smaller electron mass of BP along x-direction than that along y-direction. Additionally, the absorption rate in x-direction (20%) is higher than that in y-direction (10%), indicating that BP is optically more lossy in x-direction at the resonance frequency. The small peak at 36.4 THz observed in Fig. 2(b) is the second order resonance mode for BP nanoribbons along y-direction. The corresponding localized electric field patterns shown in the insets indicate that the mode profiles of the BP nanoribbons along x- and y-direction exhibit the similar electric field distribution, which are strong and mainly concentrate on the BP nanoribbons. Electromagnetic energy of the modes is dissipated on the BP nanoribbon due to the ohmic loss, corresponding to the absorption peaks.
Next, we investigate the case for the graphene sheet placed beneath the BP nanoribbons. The Fermi energy level Ef of graphene and the distance d between graphene sheet and BP nanoribbons are assumed to be 0.4 eV and 50 nm, respectively. The calculated absorption spectra of the hybrid system with BP along x- and y-direction are displayed in Fig. 2(c) and 2(d). Interestingly, two absorption peaks are observed in the spectra for both cases. For x-direction case, a broad one with absorption of 19.4% at 29.7 THz and a narrow one with absorption of 11.3% at 22.2 THz can be observed. The corresponding electric field distributions at the resonance peaks are displayed in Fig. 2(e) (marked with I and II, respectively). It seems that the broad peak is similar to a BPLSP mode, because both the spectral line shape and the electric field pattern (marked with II) are similar to the BPLSP mode in Fig. 2(a), though their resonant frequencies are not identical due to the introduction of graphene changed the dielectric environment. Additionally, the electric field pattern marked with II reveals a localized field around graphene, akin to a GSP mode. Hence, the mode of the broad peak is a hybrid mode and we call it BPLSP-like hybrid mode. As for the mode of the narrow resonant peak in Fig. 2(c), it is also a hybrid mode since the electric field pattern (marked with I) is quite similar to a GSP mode but shows a clear mixing between a GSP mode and a BPLSP mode. Hence, we call this mode a GSP-like hybrid mode. Likewise, for the y-direction (Fig. 2(d)), a broad one with absorption of 9.8% at 17.2 THz and a narrow one with absorption of 4% at 24.4 THz can be observed. The corresponding electric field patterns in Fig. 2(f) also confirm that the broad peak is a BPLSP-like hybrid mode and the narrow one is a GSP-like hybrid mode. Therefore, it can be concluded that once graphene is added to the system, a BPLSP-like hybrid mode with a broad absorption peak and a GSP-like hybrid mode with a narrow absorption peak can be excited, which arise from the hybridization between the BPLSP mode and the GSP mode.
To gain a deeper understanding of the hybridization process, the resonant frequency of the GSP mode in the hybrid system is calculated by changing the electron doping of BP from 5 × 1013 cm−2 to 5 × 1014 cm−2, so that the BPLSP mode is tuned out of resonance from the GSP mode to avoid their hybridization. It is found that a GSP mode exists at 22.9 THz (i.e., near the resonant frequency of the GSP-like hybrid mode at 22.2 THz for x-direction and 24.4 THz for y-direction) with absorption of 7%. This indicates that the GSP mode couples weakly to the free space wave. Based on the above results, we can represent the hybridization process by the energy diagrams in Figs. 2(g) and 2(h) for x-and y-direction, respectively. For x-direction, the hybridization between the high energy BPLSP mode (with resonant frequency of 28.6 THz) and the low energy GSP mode (22.9 THz) give rise to a BPLSP-like hybrid mode with even higher energy (29.7 THz) and a GSP-like hybrid mode with even lower energy (22.2 THz); while for y-direction, the low energy BPLSP mode (18.2 THz) hybridizes with the high energy GSP mode (22.9 THz) and result in a BPLSP-like hybrid mode with even lower energy (17.2 THz) and a GSP-like hybrid mode with even higher energy (24.4 THz).
We further phenomenologically describe the complex absorption coefficient of the coupled system by fitting a sum of oscillator response functions according to the following expression 
Here, cb,d, θb,d, fb,d, and γb,d represent the amplitudes, phases, resonant frequencies and damping rates of the two modes, respectively. Note that the phase factor exp(iθb,d) is introduced to take into account the interference between the two modes. Based on Eq. (4), we fitted the spectra and compared it with the numerical spectrum in Figs. 2(c) and 2(d). Excellent agreement between the numerical spectrum and analytical spectrum is achieved, indicating that the hybrid modes result from the hybridization between GSP mode and the BPLSP mode.
3.2 Dynamical tuning of the hybridization behavior
To investigate the dynamical hybridization between GSP modes and anisotropic BPLSP modes, we calculated the absorption spectra of the system for BP nanoribbons along x- and y-direction by varying Fermi energy of graphene, and the results are mapped in Figs. 3(a) and 3(b), respectively, with consecutively shifted Fermi energy from 0.05 eV to 1 eV. Apparently, two absorption bands can be observed for both cases, which are formed by the absorption peak of the BPLSP-like and the GSP-like hybrid modes. In particular, an important feature is that the absorption bands do not cross to each other, instead, they are separated by an energy gap. This is a typical Rabi splitting phenomenon that results from the hybridization between the BPLSP mode and the GSP mode.
To quantitatively describe this Rabi splitting phenomenon, the dispersion relations of the uncoupled BPLSP mode, GSP mode and the hybrid modes are studied. For the BP nanoribbons along x- and y-direction, the dispersion relation of the uncoupled BPLSP mode can be approximated by Eq. (2) and f = ω/2π, the resonant frequency fBPLSPj of a BPLSP mode can be represented in the following form:
In presence of the BP nanoribbons, the propagating surface plasmons in continuous graphene sheet can be excited by incident light once the in-plane wavevector of a GSP wave matches the wavevector of a diffraction order scattered by BP nanoribbons . The resonance frequency of a GSP mode can be expressed as48]. The GSP mode and the BPLSP mode are taken as two classical oscillators and their hybridization is evaluated by a coupling strength parameter. Based on this model, the dispersion relation of the hybrid modes can be derived from that of the original uncoupled modes as follows
According to these equations, the dispersion relations of the BPLSP mode (black line), GSP mode (red line) and the two hybrid modes (blue line) are calculated and plotted in Figs. 3(c) and 3(d) for x- and y-direction, respectively. The numerical absorption peaks (blue hollow stars) are also extracted for comparison. For the uncoupled GSP mode and BPLSP mode, the resonant frequency of the GSP mode shifts to the higher frequencies as Fermi energy increases, and crosses with that of the BPLSP mode at Fermi energy of about 0.45 eV and 0.18 eV for x- and y-direction. When they coupled together, two hybrid modes are generated. The frequencies of both modes shift to the higher frequencies as Fermi energy increases. However, these hybrid modes do not cross to each other, but exhibit an anticrossing with = 6.4 THz and = 4.6 THz for x- and y-direction, respectively, which matches well with the numerical results. This result confirms that the Rabi splitting phenomenon indeed originates from the hybridization between the GSP and BPLSP modes. Additionally, the energy gap of the mode splitting for x-direction ( = 26.5 meV) is larger than that for y-direction ( = 19 meV), indicating that the hybridization of GSP mode with the BPLSP mode along x-direction is stronger than that along y-direction. This is attributed to the higher energy (larger resonant frequency) for x-direction (28.6 THz) than that for y-direction (18.2 THz).
Another important feature in Figs. 3(a) and 3(b) is the variations of the absorption bandwidths with varying Fermi energy. For instance, the bandwidth of the left band is very narrow at Fermi energy of 0.1 eV, but gradually broadens as Fermi energy increases and eventually becomes much wider at 0.8 eV. To gain a deeper understanding of the evolution of the bandwidths, we take Fig. 3(a) for example and extract three representative spectra at Fermi energy of 0.2 eV, 0.45 eV and 0.8 eV, as shown in Figs. 4(a), 4(d) and 4(g), respectively. At Ef = 0.2 eV, the absorption spectrum has a broad resonant peak and a narrow peak, corresponding to the BPLSP-like hybrid mode and GSP-like hybrid mode that can be confirmed by the electric field patterns displayed Fig. 4(b). In this case, the interaction process between the GSP mode and the BPLSP mode can be represented by the energy diagram in Fig. 4(c), which is similar to the diagram in Fig. 2(g) discussed previously. Note that the energy (frequencies) of the four modes in the diagram can be quantitatively calculated by the dispersion relations of the BPLSP mode [Eq. (6)], GSP mode [Eq. (7)] and the hybrid modes [Eq. (8)].
As Fermi energy increases to 0.45eV, the GSP-like hybrid mode (the left peak in Fig. 4(d)) gets more features of a BPLSP mode. Meanwhile, the localized electric field in BP nanoribbon (marked with III in Fig. 4(e)) are much stronger, which are the typical features of a BPLSP mode. On the other hand, the BPLSP-like hybrid mode (the right peak in Fig. 4(d)) gains more characteristics of a GSP mode since its resonant peak are narrower and mode pattern exhibit a stronger GSP mode feature (marked with IV in Fig. 4(e)). These results, in fact, imply the two hybrid modes are in intermediate states with “half-GSP mode” and “half-BPLSP mode”. The energy diagram for this case is presented in Fig. 4(f), which reveals that the hybrid modes are equally mixed with the two modes as the frequencies of the GSP and BPLSP mode become identical. As the Fermi energy is further increased to 0.8 eV, a BPLSP-like hybrid mode with a broad peak and a GSP-like hybrid mode with a narrow peak again emerge, as illustrated in Fig. 4(g). The electric filed patterns in Fig. 4(h) show that the resonant peak of the GSP-like hybrid mode moves to the right side with respect to that of the BPLSP-like hybrid mode. The corresponding energy diagram is also plotted in Fig. 4(i), which indicates that as the energy of GSP mode is higher than that of the BPLSP mode, the upper hybrid mode becomes a GSP-like hybrid mode instead of the BPLSP-like hybrid mode.
Based on the above discussions, it is clear that the changes in the absorption bandwidths with varying Fermi energy are actually indicative of the conversion process between the GSP-like and BPLSP-like hybrid modes. Taking the left peaks in Figs. 4(a), 4(d), 4(g) as an example, initially the narrow peak at Fermi energy of 0.2 eV is a GSP-like hybrid mode. As Fermi energy increases, the energy of the GSP mode rises and approaches the BPLSP mode. Consequently, the interaction between the two modes becomes stronger, and more features of the BPLSP mode is added into the GSP-like hybrid mode, resulting in the broadening of its absorption peak. When the energy of the GSP and BPLSP mode are identical, the hybrid mode becomes a “half-GSP and half-BPLSP” mode. Finally, if Fermi energy is further increased so that the energy of the GSP mode is higher than that of the BPLSP mode, the GSP-like hybrid mode converts to a BPLSP-like hybrid mode with a much wider absorption peak.
3.3 Effect of the distance on the hybridization behavior
We finally explore the influence of the distance d between graphene sheet and BP nanoribbons on the hybridization phenomenon. Taken the x-direction for example, the absorption spectra mapping with the Fermi energy ranging from 0.05 eV to 1 eV for d = 100 nm, 60 nm, and 20 nm are displayed in Figs. 5(a), 5(b) and 5(c), respectively. Obviously, the Rabbi splitting of the hybrid modes can be observed. As d decreases from 100 nm to 20 nm, the energy separation between the two absorption bands enlarges, indicating the stronger hybridization between the GSP mode and the BPLSP mode. Note that, the mode splitting observed in the right absorption band in Fig. 5(c) results from the hybridization of the BPLSP mode and the second order GSP mode. Quantitatively, the mode splitting energy for x-direction with varying d is calculated and compared with that for y-direction in Fig. 5(d). It can be seen that the anisotropic GSP-BPLSP coupling is obvious when d is 20 nm, with mode splitting energy of 56.2 meV and 45.5 meV for x- and y-direction, respectively. As d increases from 20 nm to 100 nm, the coupling strength becomes weeker and the mode splitting energy decreases from 56.2 meV to 2.48 meV for x-direction and 45.5 meV to 0.3 meV for y-direction. When d is larger than 100 nm, the GSP mode coupled weekly with BPLSP mode for both x- and y-directions, and the coupling vanishes when d reaches 120nm. Additionally, the mode splitting energy for x-direction is larger than that for y-direction for the same distance d between graphene and BP nanoribbons. These results indicate that the coupling strength can be tuned by changing the distance d and the patterned direction of the BP nanoribbons.
In summary, we theoretically investigate the hybridization of graphene surface plasmons and the anisotropic plasmons in black phosphorus nanoribbons by finite element method. By varying the Fermi level of graphene, we obtain the strong coherent GSP-BPLSP coupling in both armchair (x-) and zigzag (y-) directions due to the anisotropic properties of BP. A coupled oscillator model was applied to give a quantitative description of the coherent GSP-BPLSP coupling. It is found that the hybridization of GSP mode with the BPLSP mode along x- direction is stronger than that along y-direction. This results from the larger plasmon energy of the BP nanoribbon along x-direction than that along y-direction. Additionally, the coupling strength decreases for both x- and y-directions as the distance between graphene sheet and BP nanoribbons increases. These results may pave a pathway toward the directional dependent active plasmonic devices based on 2D materials.
National Natural Science Foundation of China (No. 61675037, 61675139), Project supported by graduate research and innovation foundation of Chongqing, China (CYB17018), Chongqing Research Program of Basic Research and Frontier Technology (cstc2017jcyjBX0048, cstc2017jcyjAX0038), National High Technology Research and Development Program of China (2015AA034801), and Visiting Scholar Foundation of Key Laboratory of Optoelectronic Technology & Systems (Chongqing University), Ministry of Education.
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