1. Introduction

Graphene, a flat monolayer of tightly packed carbon atoms arranged in a honeycomb crystal structure, has been strengthening its irreplaceable position in material sciences for the past ten years [1]. Despite its short history, this strictly two-dimensional (2D) material exhibits exceptionally high crystal and electronic quality, and has already revealed a cornucopia of new physics and potential applications [2]. Surface plasmons (SPs), the propagating modes of oscillating surface electrons coupled to the oscillations of an electromagnetic wave between the interfaces of insulating and conducting media, is one of the most exciting avenues for research on graphene [3]. Compared to the SPs supported by conventional materials (like metal), graphene surface plasmons (GSPs) present appealing properties such as high tunability, extreme confinement, and low losses [4]. Because of these unique properties, it has been recently proposed as a platform for strong light-matter interaction [5]. However, there is one main challenge in realizing the full promise of GSPs: to develop methods for efficiently exciting GSPs. This is a challenging task due to the strong mismatch between wavevectors in graphene and free space at THz and IR frequencies [6]. To this end, several approaches, including classical schemes like the diffractive grating on substrate [7], the use of nanoribbons [4,8 ], have been proposed. And recently, periodic diffraction gratings generated on a thin graphene sheet itself [6,9–11 ] is proposed to overcome wavevector mismatching. However, these structures either suffer from limited tunabilities, or transform the 2D graphene sheet into 3D workspace. Although, SP modes supported by graphene ribbon constructions have been studied and classified [4,12 ], their resonant frequency is strongly dependent on their original size and material properties. Fast modulation and switching of GSPs is limited to be achieved by the tunability of Fermi energy via electrostatic gating.

In this paper, we propose a concept that exploits periodic diffraction gratings generated on in-plane bended graphene nanoribbons (BGNRs) by a mechanical vibrator from one end, which would enable efficient excitation of propagating GSPs. We will show that these in-plane bended gratings allow the perpendicularly polarized light to effectively couple to two kinds of SP modes. The properties of both modes with the field concentration within the grating crest (crest mode) and trough (trough mode) are discussed, and meanwhile, are compared to that of a static nanoribbon. Specifically, it will be proved that the proposed concept permits fast modulation and switching of GSPs by the methods through both turning of the mechanical vibration and the Fermi energy of the graphene. Theoretical analysis confirms the simulated results very well. This consistency is of great help to build plasmonics devices with high tunabilities such as modulators and multichannel optical filters working in a wide range.

2. Methods and Materials

The system under study is schematically depicted in Fig. 1 . The coplanar graphene nanoribbons are periodically distributed along y-coordinate axis and infinitely extended along the x direction in the x-y plane with period P = 100 nm. These nanoribbons are located on top of a substrate with permittivity ε_d = 2.25 and are simultaneously excited in the y-direction from one end by a mechanical vibrator with amplitude A. The experimentally measured Young’s modulus of free-standing monolayer graphene is E = 1.0 TPa, establishing graphene as the strongest material ever measured [13]. This extraordinary mechanical properties help design graphene mechanical resonators in very wide frequencies. The mechanical vibration of graphene can be actuated by using either electrical or optical modulations (experimentally achieving 180 MHz) [14], or a mechanical vibrator (numerically reaching 46 GHz) [6]. In this paper, we suppose to use a mechanical vibrator to simultaneously actuate the nanoribbons from one end by a concentrated line force. This make it possible to model elastic vibrations in direction of nanoribbon width by the scalar biharmonic equation $\left({\Delta }^2-\rho t{\omega }_b^2/D+\kappa /D\right)W=q,$ where ρ, t, ω_b, W, and q represent the density, the thickness of graphene, the vibration frequency, the displacement field and source of vibrations, respectively [15]. D is the flexural rigidity given by $D=Et/\left[12\left(1-{\upsilon }^2\right)\right],$ where ν is the Poisson coefficient. κ accounts for the stiffness coefficient of the substrate, which is modeled as a Winkler foundation exerting an additional reaction force -κW on the thin plate [6]. By assuming q = 0 and defining a new parameter ${\tilde{\beta }}_b^4={\beta }_b^4-\kappa /D,$ the above biharmonic equation can be recast into a simpler one $\left({\Delta }^2-{\tilde{\beta }}_b^4\right)W=q,$ where is the dispersion relation. The elastic wave generated on the graphene nanoribbons has a wavelength $\Lambda =2\pi /Re\left({\tilde{\beta }}_b\right).$ Here, the parameters are set as ρ = 2300 kg/m³, t = 0.34 nm, κ = 10¹¹ Nm⁻¹, ν = 0.2, thus D = 2.34 × 10⁻¹⁸ Pam³ [6].

 

Fig. 1 Geometry under investigation: a layer of coplanar graphene nanoribbons, which lie in the x-y plane while extend along the x direction, are located on top of a substrate with permittivity ε_d = 2.25 and are simultaneously excited in the y direction from one end by a mechanical vibrator with wave number β_b, forming in-plan bended gratings with period Λ. The period of the nanoribbons is fixed as P = 100 nm unless otherwise specified. The inset shows the top view of one period sinusoidal grating of the in-plane BGNRs in the x-y plane and the direction of electromagnetic excitation. Note: the incident light is polarized in the y-direction.

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The incident y-polarized transverse magnetic (TM) laser light with wave number β ₀ normally strikes the surface of the periodically structured graphene system. With the basic parameters and nomenclature labeled on it, the inset of Fig. 1 shows one period grating of the in-plane BGNRs. For electromagnetic analysis at THz and IR frequencies, this flexural wave (no more than GHz in our consideration) of graphene nanoribbon can be accurately modeled as a static grating and is utilized to couple incident electromagnetic fields to propagating GSPs.

The proposed structures are numerically simulated with the finite-difference time-domain method using the Lumerical FDTD solutions. The periodic boundary condition is imposed in the x and y directions, while in the propagation direction (z direction) a perfectly matched absorbing boundary condition is applied respectively at the two ends of computational space. Since there is a large dimensional difference among the thickness (0.34 nm) and the ribbon dimension (~50 nm), non-uniform mesh sizes are adopted in the simulations. The mesh size inside graphene layer along z axis is set as 0.034 nm, and among x and y axis are 0.8 nm, respectively, and the mesh size gradually increases outside the graphene layer. To make sure the validation of the simulation, 9000 femtosecond of simulation time and the highest mesh accuracy are set in the modal. The graphene film is modeled as a thin layer with a measured thickness t = 0.34 nm [16,17 ]. The permittivity of graphene is modeled as an anisotropic dielectric constant described by a diagonal tensor. Its surface-normal component is set as ε_zz = 2.5 based on the dielectric constant of graphite, and its in-plane component is ${\epsilon }_{xx}={\epsilon }_{yy}=2.5+i\sigma \left(\omega \right)/{\epsilon }_0\omega t$ [7]. σ(ω) is the frequency dependence of surface conductivity, which can be approximately written as $\sigma \left(\omega \right)=i{e}^2{E}_f/\left[\pi {\hslash }^2\left(\omega +i{\tau }^{-1}\right)\right].$ Here ε ₀ is vacuum permittivity, and ω is the angle frequency of incident wave, e is electronic charge, ћ is reduced Planck constant. The Fermi energy E_f is determined by the carrier concentration $E_f=\hslash {\nu }_F{\left(n_g\pi \right)}^{1/2}.$ τ is intrinsic relaxation time and follows the relation $\tau =\mu E_f/e{\nu }_F^2,$ where μ = 15000 cm²/(V∙s) is measured dc mobility [6] and υ_F = c/300 is Fermi velocity.

3. Results and discussion

3.1 Excitation by bended graphene nanoribbons (BGNRs)

It is generally known that when an infinite graphene film is patterned to a finite dimension (like graphene nanoribbon and nanodisk), the motions of the free carriers are restricted so that it can support resonant oscillation modes as the bound electrons. These plasmonic oscillations in patterned graphene structures can be excited by incident waves that are polarized in the direction of finite dimension. The excited resonant behavior of GSPs will create a sharp notch on the normal-incidence transmission spectra as the incident optical waves couple to the graphene plasmonic waves, which has been demonstrated both theoretically and experimentally at the THz spectral range [18,19 ]. Here we analyze the optical response of the plasmonic oscillations in BGNR arrays by using the FDTD method and compare the results to the theoretical quasi-static analysis.

To see how the GSPs is generated by the external field, electromagnetic simulations of the setup shown in Fig. 1 excited by a normally incident plane wave with electronic field polarized along the y direction are carried out. For these simulations, W = 50 nm, E_f = 1.0 eV, A = 10 nm, and Λ = 100 nm (when the flexural wave frequency ω_b/2π is 1.31 GHz). The simulated normal-incidence transmission, reflection, and absorption spectra are shown in Fig. 2(a) . One can indeed see two sharp notches occurring at 47.3 THz and 66.3 THz. To identify how the two peaks form, we plot their electric, magnetic field norms and H_z components at the resonant frequencies in a unit cell in Figs. 2(b)-2(d), 2(h)-2(j), respectively. As for the mode dominated with lower frequency (47.3 THz) and transmission reaching 3.7% (or absorption reaching 25.4%), it can be seen from Fig. 2(b) that the properties of this mode are characterized by the field concentrating on the crest of the BGNR, which is called Crest-Mode 1 (C-M 1). The H_z components shown in Figs. 2(d)-2(g) clearly indicate that C-M 1, 2, 3 and 4 have phase shift of 2π, 4π, 6π, and 8π along the E-field polarized direction in each period, respectively (a reverse of the field sign corresponds to a phase shift of π). While for the other mode dominated by higher frequency (66.3 THz) and transmission reaching 13.2% (or absorption reaching 37.4%), it can be seen from Fig. 2(h) that it is characterized by the field concentrating on the trough of the BGNR, which we call Trough-Mode 1 (T-M 1). The H_z components shown in Figs. 2(j)-2(m) clearly indicate that the fields have phase shift of 2π, 4π, 6π, and 8π for T-M 1, 2, 3 and 4 in each period, respectively.

 

Fig. 2 (a) Transmission, reflection, and absorption spectra of the BGNRs. The inset shows the polarization direction of electromagnetic excitation and the two effective ribbon widths of the C-M and T-M. Electric (b), magnetic field norms (c) and Hz components (d) in a unit cell for the C-M 1 mode. Hz components of the magnetic field for C-M 2 (e), C-M 3 (f) and C-M 4 (g), respectively. Electric (h), magnetic field norms (i) and Hz components (j) in a unit cell for the T-M 1 mode. Hz components of the magnetic field for T-M 2 (k), T-M 3 (l) and T-M 4 (m), respectively. The parameters are set as W = 50 nm, E_f = 1.0 eV, A = 10 nm, and Λ = 100 nm (ω_b/2π = 1.31 GHz).

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Note that the crest and trough modes are not only shown to be separated from each other by a gap in several tens of THz (when A = 10 nm), but also characterized by entirely different field concentration due to the bended gratings (that is with the maxima of field intensity on the grating crest―crest mode and trough―trough mode). This is quite different from the situation of a still ribbon where the normally excited fundamental mode is a edge mode characterized by strong field concentration taking place along the ribbon edges. The electric near-fields plotted along the width direction shown in Figs. 2(b) and 2(h) demonstrate that these two modes are 2D dipoles characterized by the two nodes in the induced density across the ribbon width. Both modes result to be the fundamental electromagnetic modes of the BGNR, possessing also an optimum frequency and the highest optical absorption in the graphene sheet (e.g., coupling to electron-hole pairs and phonons).

In order to better describe the observed resonant behavior, the bound electron oscillation in isolated graphene ribbons can be theoretically understood using the quasistatic description, which is a good approximation for ribbon widths (~1 mm) much smaller than photon wavelengths (~100 mm) [18]. Under this approximation, the plasmon resonances of the system are thus solely determined by the geometry and the dielectric function. Based on the simulated results above, the BGNR is described by the effective ribbon width W_eff, that is for the C-M, W_eff = W + 2A, while for the T-M, W_eff = W - 2A, as shown in the insert of Fig. 2(a). To meet the Maxwell equations and the phase match condition, the resonant frequency must be in strictly accordance with the following equation when the damping is not large [4,7 ]

3.2 Dynamically tuning of the graphene surface plasmons (GSPs)

It is clear from Eq. (1) that ω_p is a function of E_f and W_eff. Additionally, W_eff can be thought of as a function of W and A. Among them, A and E_f can be thought of as dynamically tunable device design parameters: A can be controlled through the elastic vibration of the GNRs by the mechanical vibrator, and E_f can be controlled by applying an external gate voltage to the graphene sheet. Electromagnetic simulations are carried out to demonstrate the effect of these parameters on the response of the vibrating graphene sheet.

The simulated normal-incidence transmission spectra with different grating amplitudes (A) are shown in Fig. 3(a) . The transmission spectrum of the static GNRs (A = 0) is also indicated for comparison. It shows that without the GSPs being excited, the highest optical absorption is occurred at the only one fundamental mode. When the still GNRs are mechanically excited to form the sinusoidal gratings, each original mode will disappear and be replaced by two corresponding new modes, one below the original frequency (C-M) and one above (T-M). One can indeed see sharp notches caused by the two fundamental modes, with transmission coefficient comparable to that of the original peak. Figure 3(a) shows that without the in-plan bended gratings being generated (A = 0), these two new modes cannot be excited. This means that the scheme permits fast off-on switching and dynamically tuning of the GSPs corresponding to these two modes. To better show the dynamically tunability, we plot the scaling rules of plasmon resonance frequencies for the fundamental modes of the C-M and T-M with respect to A in Fig. 3(b). It can be seen that the ω_p of the C-M 1 (T-M 1) increases (decrease) with increasing A. Since W_eff = W ± 2A, this phenomenon is not difficult to understand by Eq. (1). The scaling of the ω_p with respect to A agrees very well with the theoretical data. Besides, according to the simulated results, we find that there is a linear relationship between ω_p and A, the simulated data can be fitted as (unit of A: nanometer)

 

Fig. 3 Mechanically tuning of the GSPs. (a) Transmission spectra with different A for E_f = 1 eV, W = 50 nm and Λ = 100 nm (when the flexural wave frequency ω_b/2π is 1.31 GHz). (b) Plasmon resonance frequency for the fundamental modes of the C-M and T-M vs. A. The dots are simulated results from FDTD, and the line is calculated from Eq. (2). (c) FWHM of the transmission spectra in (b) as a function of A. The blue dot is corresponding to the still ribbon (A = 0) in (b) and (c). (d) Transmission spectra with different Λ for E_f = 1 eV, W = 50 nm and A = 15 nm. (e) Scaling rule of plasmon resonance frequency for the fundamental modes of the C-M and T-M with respect to Λ, the insert shows the FWHM of the transmission spectra in (d) as a function of Λ.

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Note that this linear relationship cannot be derived directly from Eqs. (1) and (2) under the given conditions, because η need to be deduced from the simulated results [4]. Supposing A → 0 in Eq. (3), the resonance frequencies of the two fundamental modes will approach to 57.03 THz and 58.274 THz, respectively. These values, however, are higher than 55.448 THz of the static ribbon (A = 0, blue dot in Fig. 3(b)). This difference can be understood by the fact that the effective ribbon width is not exactly the same as the real ribbon width. Grating corrugations on GNRs lead to a matching condition necessary to generate GSPs for exciting frequencies satisfying the phase matching equation. Nevertheless, the BGNR possesses a slightly broader matching condition than that of the static GNR, because the bended gratings allow incident waves near the matched one to couple into the ribbon. Fig. 3(c) shows the full width at half maximum (FWHM) of the two fundamental modes. It shows the value of FWHM of the spectrum for C-M 1 (T-M 1) increase (decrease) as A increases. Earlier research shows that the value of FWHM increases with decreasing occupation ratio [8]. Known that, the change of FWHM can be explained by the change of the occupation ratio that are determined by the ratio of effective ribbon width (W_eff) to the ribbon period (P).

Another factor that can be controlled through the elastic vibration of the GNRs is the grating period (Λ), which can also be thought of as dynamically tunable device design parameter. To show that, we carry out simulations with different Λ but fixed A, and show the results of transmission spectra in Fig. 3(d). The corresponding scaling rules of the resonant frequency of the two fundamental modes with respect to Λ is shown in Fig. 3(e). These two figures demonstrate the potential of our approach since it is shown that as the periods of the grating Λ ranging from 100 nm to 200 nm (corresponding to vibration frequency ω_b/2π varying from 1.31 GHz to 330 MHz), both the resonance frequencies of the C-M 1 and T-M 1 could be dynamically tuned from 42.5 to 35.7 THz and from 70.2 to 58.7 THz, respectively. At the same time, the FWHM of these two modes will also be tuned as shown in Fig. 3(e).

Except for the tunable properties benefitting from the mechanical properties, the most intriguing property of GSPs is its ultrabroad and fast electrical tunability. This optical response strongly depends on the carrier concentration of the graphene, which is controlled mostly by a potentially fast approach of electrical backgating [18]. Using this method, carrier concentration as high as 4 × 10¹⁸ m⁻² in graphene sheet was observed, meaning E_f = 1.17 eV [20]. Here, we reasonably assume that E_f can be dynamically tuned from 0.4 to 1.2 eV. Simulated spectra shown in Figs. 4(a) and 4(b) clearly confirm the broad tuning range with the change of E_f. For example, when the Fermi energy of graphene is varied from 0.4 eV to 1.2 eV, the ω_p is tuned from 30 THz to 51 THz for the C-M 1 while from 42 THz to 72 THz for the T-M 1, respectively. Fig. 4(c) shows the tuning curves agree very well with Eq. (1). For a given ribbon width and the amplitude, the plasmon resonance frequencies are described by a scaling behaviour of ${\omega }_p\propto \sqrt{E_f}.$ This universal relation is the characteristic of two-dimensional electron gases [18]. Besides the resonance frequency, the FWHM of the transmission dips are also tuned. Fig. 4(d) shows that the width of the plasmon resonances decrease as the Fermi energy increase. This can be easily explained by considering the allowed interband transitions [3]. At a Dirac point all interband transitions are allowed leading to higher loss and therefore wider resonance. However, as the Fermi energy is increased some of the interband transitions are blocked and hence the width of the resonance is lowered.

 

Fig. 4 Electrostatic tuning of the GSPs. (a, b) Transmission spectra with different Fermi energy level in graphene when W = 50 nm, A = 15 nm, and Λ = 100 nm (when the flexural wave frequency ω_b/2π is 1.31 GHz) (a) for C-M 1 and T-M 1 and (b) for C-M 2 and T-M 2. (c) Scaling rule of plasmon resonance frequency for the first two modes of the C-M and T-M vs. E_f. (d) The FWHM of the transmission spectra in (a) as a function of E_f.

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3.3 Geometrical tunability of the graphene surface plasmons (GSPs)

The analyses of GSPs manifested above do not take into account the role performed by ribbon width. In this part, we mainly discuss the tunability of GSPs concerning about this geometric parameter. The resonance frequency of the BGNR in the transmission spectrum is largely determined by the effective ribbon width as ${\omega }_p\propto 1/\sqrt{W_{eff}},$ where W_eff = W ± 2A. To show that, simulations with different ribbon width are carried out and the corresponding transmission spectra are shown in Fig. 5(a) . With increasing ribbon width, both the two resonant frequencies of the C-M 1 and the T-M 1 show a red shift, as the tuning curves show in Fig. 5(b). Since that both the effective ribbon widths of the two basic modes increase with increasing ribbon width, this kind of shift is not difficult to understand. On the one hand, according to Eq. (1), the increase of the W_eff will directly lead to the decrease of ω_b. On the other hand, the increase of the W_eff will also result in increasing of occupation ratio and further strengthen the plasmonic coupling from neighboring ribbons, which plays the same function of red shift [8]. But the same changing tendency of the W_eff has a varying effect on the FWHM of these two modes. As Figs. 3(c) and 5(b) show, the FWHM of T-M 1 increases with increasing W_eff, while it almost remains unchanged for C-M 1. Furthermore, since that W_{eff, Crest} > W_{eff, Trough} keeps always true in all conditions, the values of FWHM of C-M 1 keep bigger than that of T-M 1 (which is best confirmed by Figs. 3(c), 4(d) and 5(b) ). This can be understood by the fact that the wider the effective width is, the more wavevector can couple to the plasmonic waves in graphene.

 

Fig. 5 Geometrically tuning of the GSPs. (a) Transmission spectra with different ribbon width of graphene when E_f = 1 eV, A = 15 nm, Λ = 100 nm and P = 100 nm. (b) Scaling rule of plasmon resonance frequency for the fundamental modes of the C-M and T-M vs. W, the insert shows the FWHM of the transmission spectra in (a) as a function of W.

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4. Conclusion

Our analyses have neglected any effects that may arise because of the possible lattice discreteness and distortion (tensile and shear strains) in a real BGNR. This is a common approximation in such kinds of researches [6,9–11,21 ]. It would be worthwhile to study these effects along with the potential stability of the considered geometry. Given that, it is assumed that the vibration amplitude is not too big (A ⩽ 15 nm). For a practical application, devices with large amplitude A can be achieved by etching the graphene gratings directly on a substrate.

In addition, in our calculations, we have reasonably neglect some damping effects due to the loss of the graphene nanoribbons from the intrinsic optical phonon modes and scattering from the edges. Especially for the higher modes with frequencies (up to 140 THz) within mid-infrared or even near-infrared, several damping pathways, such as the surface optical phonons from the interactions between graphene and the relevant polar substrate, and the possible plasmon dispersion and damping due to surface polar phonons in the substrate, are also ignored [22]. Of course the reasons are many sided: on the one hand, the optical parameters of the graphene used in the FDTD simulations are dominated by the permittivity derived from Drude conductivity. Thus the intrinsic relaxation time of the electrons in the graphene are simply assumed to be constant once given the measured dc mobility and the Fermi level. On the other hand, the dielectric parameter of the substrate used in this paper is supposed to be non-dispersive, which is a commonly used method for simplicity in a theoretical calculation [4,6–8,12,16–18 ]. These mechanisms of the damping pathways have been discussed detailedly with some techniques, including the generalized random phase approximation (RPA) theory, by taking finite plasmon/electron lifetime of the graphene and the dispersive dielectric constant of the substrate into consideration [22].

In conclusion, we propose to use periodic diffractive gratings formed by in-plan BGNRs to excite the highly confined plasmonic waves by the incident light that is perpendicularly polarized to the ribbons. We have shown that two new types of GSP modes can be excited: the crest modes, with the field concentrating on the crest of the BGNR, and the trough modes, with the field concentrating on the trough of the BGNR. Our numerical and theoretical calculations confirm very well and meanwhile, demonstrate the possibility to achieve reconfigurable gratings that could have fast and highly tunable response to light. By varying the elastic vibration, it is indeed feasible to turn off-on the two kinds of modes. Furthermore, the proposed concept permits us not only to excite GSPs but also to tune them dynamically (elastic vibration, Fermi energy) and geometrically (ribbon width). This could be a significant step towards the realization of 2D graphene plasmonics devices with broad operation bandwidths and multichannel modulations, as well as prompt further advances in the emerging field of transformational plasmonics.