We find that a stacked pair of graphene ribbon arrays with a lateral displacement can excite plasmon waveguide mode in the gap between ribbons, as well as surface plasmon mode on graphene ribbon surface. When the resonance wavelengthes of plasmon waveguide mode and surface plasmon mode are close to each other, there is a strong electromagnetic interaction between the two modes, and then they contribute together to transmission dip. The plasmon waveguide mode resonance can be manipulated by the lateral displacement and longitudinal interval between arrays due to their influence on the manner and strength of electromagnetic coupling between two arrays. The findings expand our understanding of electromagnetic resonances in graphene-ribbon array structure and may affect further engineering of nanoplasmonic devices and metamaterials.
© 2014 Optical Society of America
Surface plasmons (SPs), which are collective oscillations of electron density at metal-dielectric interface , have attracted tremendous interest due to their capability to concentrate light to subwavelength scale  and their potential applications . Noble metals such as silver and gold have traditionally been the available materials of choice to support SPs. If a metal film is corrugated with a periodic array of nanoholes or nanoslits, SP resonance modes can been achieved with the help of reciprocal lattice vectors provided by the periodic structure [4–6]. In recent years, graphene, a single layer of carbon atoms arranged in honeycomb lattice, emerges as an alternative, unique plasmonic material that displays a wide range of extraordinary properties [7–9]. Graphene SP resonances in periodic patterned structures (e.g., graphene micro-ribbon , micro-disk , and antidot/hole arrays [12, 13], and dielectric grating covered by a graphene sheet [14–16]) have been predicted theoretically and demonstrated experimentally. Compared to SP resonance in conventional plasmonic materials, graphene SP resonance has an appealing property: tunability via electrostatic gating or chemical doping [17, 18].
Besides its tunability, an additional handle to control SP excitation is provided by the dielectric environment  or relative arrangement of the interacting graphene ribbons . In such complexes, SP interaction among neighboring graphene structures becomes more significant and important, its underlying electromagnetic coupling has been addressed to fully exploit SP resonances [21, 22]. It has been revealed that the plasmon interaction and hybridization in pairs of neighboring aligned ribbons dramatically modify mode profiles and spectral responses . We recall that the symmetry breaking in a metallic metamaterial has a great influence on its optical response [24–26]. Motivated by it, in this paper we explore the effect of a lateral displacement in a stacked pair of graphene ribbon arrays on its near-field coupling and resonance spectrum. We find that, in the structure of two graphene ribbon arrays with a lateral displacement, besides the SP resonance on graphene surface, there also exists a localized plasmon resonance (i.e., plasmon waveguide mode resonance) in the gap between two ribbons. The plasmon waveguide mode resonance can be tuned with a small change in lateral displacement or longitudinal interval between arrays. Our study gives an insight into the transmission resonances and the charge oscillation picture of near-field coupling in graphene-ribbon array structure, which has broad and significant impacts on graphene plasmonics and potential applications in plasmonics devices and metamaterials.
2. Model and numerical method
Figure 1 depicts the schematic cross-section of two identical periodic arrays of graphene ribbon. A plane wave of TM polarization (its magnetic field is perpendicular to the x–z plane) impinges normally on the bottom of the double-layer structure. The geometry is defined by five parameters: t for thickness of graphene sheet, w for width of graphene ribbon, p for period of ribbon array, G for longitudinal interval and L for lateral displacement along the x axis. For the sake of simplicity, we assume that the system is suspended in air. In whole paper, t, w and p are taken as 0.5, 160 and 200 nm, respectively. The graphene sheet may be modeled as an anisotropic dielectric constant expressed by a diagonal tensor . Its surface-normal component is set as εzz = 2.5 based on the dielectric constant of graphite, and its in-plane component is expressed by εxx(yy) = 2.5 + iσ(ω)/(ε0ωt), where σ(ω) is the frequency dependence of surface conductivity, ε0 the permittivity of vacuum, ω the angle frequency of incident wave, and t the thickness of graphene sheet. The surface conductivity σ(ω) is calculated with the Kubo formula, which considers both contributions from intraband electron-photon scattering and direct interband electron transitions . The Fermi energy level Ef and carrier relaxation time τ of graphene are taken as 0.6 ev and 1.2 ps, respectively.
We employ finite-difference time-domain (FDTD) method by Lumerical FDTD solutions to simulate the interaction between graphene system and incident plane wave. The periodic boundary condition is imposed in the x direction, while in the propagation direction a perfectly matched absorbing boundary condition is applied at the two ends of computational space. Considering the computer memory, we use non-uniform mesh in the FDTD simulation regions. The mesh sizes inside graphene layer along the x and z axes are set as 5 and 0.05 nm, respectively, and the mesh size gradually increases outside the graphene layer.
3. Results and discussion
First, we study the effect of displacement L on the transmission spectrum of a stacked pair of graphene ribbon arrays. Figure 2 displays the transmission spectra at different displacements L, with G = 10 nm. The transmission spectrum of a corresponding single graphene ribbon array is also shown for direct quantitative comparison. In the transmission spectrum of single graphene ribbon array, there are two transmission dips located at 9.215 and 4.091 μm, respectively. The two dips are originated from the resonances of fundamental and second-order SP modes, respectively . When two graphene ribbon arrays are close to each other, the fundamental SP resonance modes of two arrays interact and hybridize, and then new plasmon eigenmodes (i.e., symmetric and asymmetric modes) are formed . The transmission dip at 7.249 μm (for L = 0) corresponds to the symmetric plasmon mode. Since the net electric dipole moment of the asymmetric plasmon mode is zero in the quasistatic limit, its corresponding transmission dip does not appear in the spectrum. As one ribbon array laterally moves L = 10 nm along the x axis, the transmission dip associated with the symmetric plasmon eigenmode has only a slight redshift about 20 nm. Surprisingly, on its right side there exists a new transmission dip. In this paper, we focus on the two dips, and they will be referred to as SW dip (7.268 μm) and LW dip (8.573 μm) for simplicity’s sake. When L increases, the SW and LW dips are blueshifted and redshifted, respectively. They move about 3.11 and 7.54 μm, respectively, as L changes from 10 to 100 nm.
In order to better visualize resonance picture, we calculate the spatial distributions of the extremum of electric field Ez for transmission dips. Figures 3(a)–3(e) correspond to the dips at 7.249 (L = 0 nm), 7.268 (L = 10 nm), 6.935 (L = 30 nm), 5.860 (L = 60 nm) and 4.153 (L = 100 nm) μm in Fig. 2, respectively. From the electric field distributions in Fig. 3, we can deduce the surface oscillating charge density on graphene surface (σ = ε0n0 · ΔE, n0 is the unit vector normal to graphene surface, E is the electric field perpendicular to graphene surface, and ΔE is the difference of E on graphene-air boundary) and the polarity of surface charge. For a single graphene ribbon array illuminated by a TM wave, a fundamental SP mode is resonantly excited in a graphene ribbon with dipole orientation along the ribbon width, Ez-field has a 2π phase shift in each period. When two graphene ribbon arrays are stacked with a small gap, the plasmonic interaction between two arrays leads to a hybridized symmetric SP mode, where the dipole orientation of the upper array is the same as that of the lower array (Fig. 3(a)). The longitudinal electromagnetic interaction between two arrays mainly concentrates at the two ends of ribbon, it causes an enhancement of energy of bare SP mode. Thus the resonance wavelength of the hybridized SP mode is shorter than that of the fundamental SP mode in single array. As one graphene ribbon array is laterally displaced along the x-axis (i.e., L ≠ 0), the electromagnetic coupling gets complex. In the case of L = 10 nm (Fig. 3(b)), besides the oscillating charges located at the two ends of each graphene ribbon, there are also a few charges on the inner surface of the middle of graphene ribbon and the inner side of the right (left) end of the upper (lower) graphene ribbon (which is just situated above (below) the right (left) end of the lower (upper) graphene ribbon). In the overlapping regions of two arrays along the z-direction, the polarities of the charges in the upper ribbon are oppositive to those in the lower ribbon. The appearance of charges in the overlapping regions enhances the attraction between two graphene arrays. Therefore, the transmission dip resulted from the hybridized symmetric SP mode has a redshift as L changes from 0 to 10 nm. When L increases from 10 to 30 nm (Fig. 3(c)), the charges at the right (left) end of the upper (lower) ribbon decrease, and the induced charges in the inner side of the right (left) end of the upper (lower) ribbon move along the negative (positive) direction of x-axis due to the reduction of the overlapped area between two graphene ribbons.
We consider that more attention should be payed to the standing-wave pattern in air gap between the upper and lower ribbons (Figs. 3(b)–3(e)). It is found that, for the case of L ≠ 0, besides confined plasmonic waves on graphene ribbon surface (which have been discussed in [10, 12, 14, 15]), there is also a localized plasmon mode (i.e., plasmon waveguide mode) in air gap between two ribbons. The localized plasmon resonance is similar to the Fabry-Pérot resonance of guide mode in metal slit . If there isn’t a lateral displacement (i.e., L = 0, Fig. 3(a)), graphene slits of two arrays provide a direct path for energy flowing, plasmon evanescent waves would not propagate along the x direction, and a waveguide mode can not be created in the gap between the upper and lower ribbon. At L ≠ 0, the direct path for energy flowing is broken, the exit of every graphene slit in the lower array may be regarded as an evanescent-field source. The emitted evanescent waves from those sources propagate along the x direction and interfere with each other. Thus a standing-wave is formed in the air gap. The condition of waveguide resonance is β(w − L) = nπ, where β stands for the in-plane wavevector of evanescent field, n is an integral number. β is a function of interval G between two arrays and wavelength λ of incident light, and it increases with decreasing λ or G . For fixed n and G, a larger L requires a larger β according to the above resonance condition. Thereby the increase of L results in a decrease of the wavelength λ of waveguide resonance, which is consistent with the shift of the SW dip in Fig. 2. Strictly speaking, the SW dip is originated from the interaction between plasmon-waveguide resonance mode and hybridized fundamental SP resonance mode. For a small G, the plasmon-waveguide resonance is prominent and important due to the strong electromagnetic coupling between ribbons. It is well known that the SP resonance wavelength is mainly determined by the periodicity of graphene ribbon array . Therefore, it is reasonable that the shift of SW dip is mainly attributed to the change of resonance wavelength of waveguide mode. As the wavelength of SW dip is far away from that of SP resonance mode, the field pattern exhibits a waveguide mode character (Figs. 3(d) and 3(e)).
Figures 4(a)–4(d) show the electric field distributions of the LW dips at 8.573 (L = 10 nm), 9.917 (L = 30 nm), 14.196 (L = 60 nm) and 16.122 (L = 100 nm) μm in Fig. 2, respectively. We observe that the Ez-field and charge distributions of the LW dip at 8.573 (L = 10 nm, Fig. 4(a)) are different from those of the SW dip at 7.268 μm (L = 10 nm, Fig. 3(b)). No electric dipole is resonantly excited in each ribbon. There is a few charges of the same sign at the two ends of each ribbon, and more charges are distributed at the right (left) end of the upper (lower) ribbon. Such charge distribution enhances the attraction between two arrays in comparison with the case of the SW dip at 7.268 μm. Consequently, the wavelength of the LW dip is longer than that of the SW dip. With the increase of L, the positive (negative) charges at the left (right) end of the upper (lower) ribbon gradually disappear, then a few negative (positive) charges start to accumulate near this end (Fig. 4(d)). The near-field patterns in Fig. 4 suggest that the LW dip stems from the resonance of waveguide mode in air gap instead of the excitation of SP mode on graphene surface. We note that increasing L causes a decrease of antinode number of SP standing-wave. According to the resonance condition, the increase of L and the decrease of antinode number require a larger and smaller β, respectively. From the increment of L and the decrement of antinode number in Fig. 4, one may infer that antinode number is a main factor influencing β. A smaller β corresponds to larger wavelength, so the LW dip is redshited with increasing L.
Naturally, one also wonders the influence of interval between two arrays on electromagnetic resonances. Figures 5(a) and 5(b) show the transmission spectra at different values of G with L = 10 and 60 nm, respectively. For the case of L = 10 nm, both SW and LW dips exhibit a blueshift, as G increases from 10 to 15 nm. Once G exceeds about 15 nm, the moving direction of the LW dip changes, i.e., the LW dip shifts to low frequency region. Meanwhile, the SW and LW dips gradually become narrow and broad, respectively. As G increases to 100 nm, the SW and LW dips approach to the second-order and fundamental SP resonance dips of single array, respectively. With respect to the case of L = 60 nm (Fig. 5(b)), as G increases from 10 to 100 nm, both SW and LW dips have a blueshift, and they move about 1.768 and 4.947 μm, respectively. Similarly, at G = 100 nm the SW and LW dips are close to the second-order and fundamental SP resonance dips of single array, respectively.
Figures 6 and 7 display the side-view Ez-field profiles of the SW and LW dips in Fig. 5(a), respectively. Figures 6(a)–6(e) correspond to the SW dips at 7.268 (G = 10 nm), 6.953 (G = 15 nm), 6.504 (G = 20 nm), 5.209 (G = 60 nm) and 4.071 (G = 100 nm) μm, respectively. The electric field pattern of the SW dip at 7.268 μm shows that both SP mode on graphene ribbon surface and waveguide mode in air gap are excited effectively, and the two modes are coupled to each other (Fig. 6(a)). With the increase of G, the charges situated at the right (left) end of graphene ribbon of the upper (lower) array rapidly decrease. The disappearance of charges and the increasing G cause an increase and decrease of longitudinal attraction between the upper and lower ribbon arrays, respectively. Since the decrement of attraction due to the increase of air gap exceeds the increment of attraction resulted from charge change, the SW dip moves toward short wavelength region. With the further increase of G, the induced charges located in the middle part of ribbon gradually decrease, the longitudinal attractive force between two array gets weak, so the SW dip continues blueshifting. At G = 60 nm, some negative (positive) charges start to accumulate the right (left) end of graphene ribbon in the upper (lower) array. At the same time, the charges in the outer boundary zone of the middle of each graphene ribbon increase. When G increases to 100 nm, the charges in the middle of ribbon disappear, and there are different types of charges at the left and right sides of graphene’s middle. Meanwhile, the polarity of the charge at the right (left) end of graphene ribbon in the upper (lower) array changes. At this time, the Ez-field has a 4π phase shift in each period, the second-order SP mode is excited resonantly. The near-field patterns in Fig. 6 tell us that, as G increases, the waveguide resonance is less obvious and the SP mode becomes dominant. The propagation constant β of SP evanescent wave is sensitive to G and decreases with increasing G, and it increases with decreasing wavelength λ. Thus as G increases, the free space resonance wavelength must decrease according to the afore-mentioned resonance equation if L is fixed. The decrease of waveguide resonance wavelength is accompanied with a blueshift of the SW dip. When G is larger than the decay length of SP evanescent wave, the contribution of waveguide mode resonance to the SW dip becomes weak, the wavelength of the SW dip is mainly determined by the SP mode on graphene surface.
Figures 7(a)–7(e) depict the Ez-field of the LW dips at 8.573 (G = 10 nm), 7.947 (G = 15 nm), 7.971 (G = 20 nm), 8.739 (G = 60 nm) and 9.059 (G = 100 nm) μm in Fig. 5(a), respectively. As G increases from 10 to 15 nm, the charges at the left (right) end of graphene ribbon in the upper (lower) array disappear. Meanwhile, the electric field in the middle of air gap between two ribbons becomes weak, correspondingly the charges located in the middle of ribbon decrease. In this situation, the attraction between two graphene arrays decreases with the increase of G, thus resulting in a blueshift of LW dip. If G keeps increasing, the charges in the middle of ribbon continue decreasing, and a few negative (positive) charges start to accumulate the left (right) end of ribbon in the upper (lower) array. Eventually, the distributions of Ez-field and charge are characterized by a fundamental SP mode. We note that the antinode number of SP standing-wave decreases as G changes from 10 to 15 nm. Combined with the waveguide resonance condition and the relation between β and G/λ, it isn’t difficulty to understand the redshift of the LW dip. For a larger G, the waveguide resonance disappears due to the weak optical coupling between the upper and lower ribbons. With respect to the SW and LW dips in Fig. 5(b), their behaviors can be well explained by a similar way.
The electromagnetic resonances in our considered structure can also be modulated by changing the ribbon width w and the array periodicity p. More specifically, the resonance wavelengths of plasmon waveguide mode and surface plasmon mode increase with the increases of w and p. The change of plasmon waveguide resonance wavelength can be explained by the waveguide resonance condition. The redshift of surface plasmon resonance mode can be analysed by the results in . In addition, the width and depth of transmission dip are largely determined by the optical loss in graphene. From the expressions of graphene surface conductivity  and dielectric tensor , we know that the values of graphene surface conductivity σ(ω) and dielectric constant εxx(yy) depend on the carrier relaxation time of graphene (τ). Numerical results (not shown here) indicate that a lower τ is associated with a larger imaginary (real) part of εxx(yy) (σ(ω)). It means that the lower τ corresponds to higher loss and lower intrinsic quality factor for plasmon resonance. As τ decreases, the transmission dip becomes broad and shallow [10, 15]. In this paper, we do not discuss substrate and superstrate effects. If our proposed structure is in a symmetric dielectric environment, the transmission dips exhibit a redshift in comparison with the case of air environment, and similar phenomena mentioned above can also be observed. In the process of practical application, we could start with CVD-grown graphene transferred onto a dielectric substrate (such as SiO2) . A ribbon array may be fabricated in the graphene sheet using standard optical lithography followed by oxygen plasma etching . Then a dielectric layer is deposited on top of the graphene ribbon array by atomic layer deposition, and the second graphene ribbon array is formed on top of the dielectric layer in the same way as above. Finally, the sample is covered by a dielectric layer using atomic layer deposition.
In summary, we have investigated the electromagnetic resonances in a stacked pair of graphene ribbon arrays with a lateral displacement. It is found that, besides SP mode resonance on graphene ribbon surface, there also exists a coupled-plasmon waveguide mode resonance in the gap between graphene ribbons due to the interference of SP evanescent waves in gap. If the difference between the resonance wavelengths of waveguide mode and SP mode is small, the two modes interact with each other and contribute together to transmission dip. The waveguide mode resonance depends largely on the values of displacement and interval between ribbons due to their impact on the manner and strength of electromagnetic coupling between two arrays, so the transmission dips can be adjusted by changing the displacement and interval between two arrays. Our study gave a physical mechanism for electromagnetic resonances in two graphene ribbon arrays and provided a further insight for developing graphene-based plasmonic devices and metamaterials.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11174372, 11074069 and 11264021), the State Key Program for Basic Research of China (Grant No. 2013CB632705), the Youth Foundation of Hunan Provincial Education Department, China (Grant Nos. 10B118 and 11B134) and the Scientific Project of Jiangxi Education Departments of China (Grant No. GJJ13732).
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