## Abstract

Graphene nanoribbon (GNR), as a fundamental component to support the surface plasmon waves, are envisioned to play an important role in graphene plasmonics. However, to achieve extremely confinement of the graphene surface plasmons (GSPs) is still a challenging. Here, we propose a scheme to realize the excitation of localized surface plasmons with very strong field enhancement at the resonant frequency. By sinusoidally patterning the boundaries of GNRs, a new type of plasmon mode with field energy concentrated on the shaped grating crest (crest mode) can be efficiently excited, creating a sharp notch on the transmission spectra. Specifically, the enhanced field energies are featured by 3 times of magnitude stronger than that of the unpatterned classical GNRs. Through theoretical analyses and numerical calculations, we confirm that the enhanced fields of the crest modes can be tuned not only by changing the width, period and Fermi energy as traditional ribbons, but also by varying the grating amplitude and period. This new technique of manipulating the light-graphene interaction gives an insight of modulating plasmon resonances on graphene nanostrutures, making the proposed pattern method an attractive candidate for designing optical filters, spatial light modulators, and other active plasmonic devices.

© 2016 Optical Society of America

## 1. Introduction

Surfaces plasmons (SPs) are collective oscillation modes of conduction electrons coupled to the oscillations of an electromagnetic wave between the interfaces of conducting and insulating material. These modes have been identified in various kinds of metallic systems, ranging from smooth and corrugated films [1] down to nanostrips [2] and nanoparticles [3]. This collective spring motions have a fundamental role in the dynamic responses of electron systems, and have found their ability to focus light and enhance the electric field intensity near the structure by several orders of magnitude, which paves the way to some important applications in areas such as sensing [4], waveguides [5], absorbers [6,7] and other optical modulators [8–10]. However, metallic materials, even in noble metals, which are widely considered as the best available plasmonic materials, suffer a lot from their limited tunabilities and large Ohmic losses when demonstrating their applicability to optical processing devices.

More recently, graphene has emerged as an alternative, unique two-dimensional plasmonic material that displays a new platform for supporting plasmons [11]. This light-graphene interactions known as GSPs benefit a lot from the extraordinary properties of the graphene, such as extreme confinement, electrical or chemical tunability, and long propagating distances [12]. These characteristics make graphene the most promising alternative plasmonic material to traditional metals [13]. This is because the electronic states in conduction and valence bands of graphene can be described by a linear dispersion relation. This relation strongly depends on the doping level or equivalently, on the Fermi energy *E _{f}* of graphene, and further influences its optical response [14]. Therefore, graphene plasmonics have triggered a plethora of applications in photonic metamaterials [15], light sources [16], nonlinear optics [17], and other graphene plasmonics [13,18] at infrared or terahertz frequencies. However, these intriguing applications are mainly depends on the specific structures of graphene since a finite size of graphene can break up its zero band gap [19]. Thus by patterning graphene into various micro/nano-structures, such as disks [20], triangles [21], rectangles [22,23], and especially, ribbons [12,15,24–27], the external electromagnetic radiations can couple effectively to localized plasmon excitations. Among these engineering techniques, the most common way is to design GNRs. This is because in the case of finite nanoribbon width, the free electrons will be scattered and reflected at the terminations of the graphene sheet, thus generating the Fabry-Perot-type resonance [28]. These plasmon resonant phenomena have been experimentally identified by the recent researches [29,30]. Unfortunately, the geometrical tunability of these bounded surface plasmons is only strictly restricted to ribbon width, since they are designed with two parallel linear boundaries in the reported works [12,15,24–27]. More recently, gratings that are formed by elastic vibrations of GNRs are proposed to excite two kinds of GSP modes named crest mode and trough mode, of which the field concentrations are within the grating crest and trough, respectively [31], however, they are still kept with linear boundaries.

In this paper, we propose to use periodic GNRs with sinusoidally shaped boundaries to excite a new kind of plasmon modes called crest modes, of which the localized field energies are concentrated on the sinusoidal gratings. It will be proved that for these crest modes, the field energies are tightly confined at the boundary of gratings, with the localized fields 3 times of magnitude stronger than that of the unpatterned ribbons. Besides, these modes can be tuned not only by changing the width, period and Fermi energy of the ribbon, but also by varying the grating amplitude and period. The properties of the excited crest modes are numerically and analytically confirmed, and meanwhile, are compared to that of classical GNRs (that are with two parallel linear boundaries). With the capacity to extremely enhance the field confinement, the proposed pattern method can be considered as an attractive candidate for designing THz optical filters and modulators.

## 2. Patterns and simulations

The designed system is schematically depicted in Fig. 1. A layer of coplanar GNRs, 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. The ribbons are designed with period

*P*and presupposed to have a width of

*W*before the upper and lower boundaries being sinusoidally shaped with functions

*A*is grating amplitude,

*Λ*is grating period,

*φ*

_{0}is the initial phase of the modulation and is assumed to be –

*π*/2. 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 sinusoidally shaped GNRs.

To theoretically study the boundary modulations of the GNRs, the proposed structures are numerically simulated with the finite-difference time-domain method using the Lumerical FDTD solutions. Since that the graphene nanostructure under study is shaped non-linearly with tens of nanometers in width, it is necessary to give an explanation to this issue. On the one hand, the results from both the first-principle calculations and the classical local electromagnetic theory show that the optical responses of doped graphene nanostructures are in good agreement when the sizes of the graphene nanostructures exceed 10 nm [32]. On the other hand, both of the experimental [20] and theoretical [32] studies demonstrate that the classical electromagnetic theory can describe quite well the optical properties of the graphene nanostructures with non-linear boundaries, such as rings and disks. Thus in this work, we use the classical descriptions to study the optical response of the nanostructured graphene, reasonably neglecting its quantum finite-size effects as well as the edge effects. To ensure the correctness of the classical electromagnetic theory, the modulated minimum width of the GNR is 20 nm in this paper [32]. Thus in our numerical experiments, the graphene film is modeled within the random-phase approximation [11,18], the dynamic optical response of graphene can be derived from the linear-response theory (Kubo formalism) in a form consisting of interband and intraband contributions: *σ* = *σ _{inter}* +

*σ*, where

_{intra}*σ*is the optical conductivity of graphene. Both interband and intraband conductivities are closely correlated with the frequency of incident light and the chemical potential of graphene (also Fermi level

*E*). The chemical potential of doped graphene is determined by the carrier concentration

_{f}*n*(

_{g}=*μ/ћν*)

_{F}^{2}

*/π*(where

*μ*= 15000 cm

^{2}/(V∙s) is the measured dc mobility,

*ћ*is the reduced Plank’s constant,

*ν*= 10

_{F}^{6}m/s is the Fermi velocity), which can be tuned by chemical doping or electrical gating [24]. Generally, within the random-phase approximation (RPA) in the local limit (i.e., for zero wave vector), the graphene conductivity can be reduced to [11]

*e*is the elementary charge,

*ω*is the angular frequency of the incident light, and

*τ*is the carrier relaxation time, which satisfies the relationship

*τ = μE*(

_{f}/*ev*

_{F}^{2}), while

*E*=

_{f}*ћν*(

_{F}*n*)

_{g}π^{1/2}. The step function

*θ*(

*ћω*−2

*E*) conveys the condition for the interband electron absorption at low temperatures. The first term of Eq. (2) associated with a Drude model response for intraband processes, while the second one describes interband transitions and becomes important only for high energies. Considering the ultrathin thickness, the permittivity of graphene is modeled as an anisotropic dielectric constant described by a diagonal tensor. Its surface-normal component is set as

_{f}*ε*= 2.5 based on the dielectric constant of graphite, and its in-plane component is ${\epsilon}_{xx}={\epsilon}_{yy}=2.5+i\sigma \left(\omega \right)/\left({\epsilon}_{0}\omega t\right)$ [24], where

_{zz}*ε*

_{0}is the vacuum permittivity and

*t*= 0.34 nm is the measured thickness of graphene [33,34].

Besides, in our simulations, periodic boundary conditions are imposed in the *x* and *y* directions, while in the propagation direction (*z* direction) perfectly matched absorbing boundary conditions are applied respectively at the two ends of computational space. Since the existence of a large dimensional difference among the thickness and the width (*W* ~100 nm), non-uniform mesh sizes are adopted to guarantee the computational precision 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 1.5 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. In sddition, several additional losses that may arise from the surface optical phonons from the interactions between graphene and the relevant polar substrate, the possible plasmon dispersion, the damping due to surface polar phonons in the substrate, as well as the many-body interactions, are also ignored. 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 [35].

## 3. Results and discussion

#### 3.1 Excitation by sinusoidally shaping the two boundaries

As is well-known that at the nanoscale, electronic confinement effects are essential to the properties of graphene. When an infinite graphene film is patterned to a finite dimension such as nanoribbons and nanodisks, the motions of the free carriers are restricted critically by their finite dimensions so that it can support resonant oscillation modes as the bound electrons, which leads a major challenge to the control of these modes at the nanoscale. These plasmonic oscillations in the classically patterned GNRs (those are with two parallel linear boundaries, *A* = 0 in Eq. (1)) can be excited by incident waves that are polarized in the direction of ribbon width. The excited resonant behavior of GSPs will create a sharp notch on the transmission spectra as the incident optical waves couple to the localized graphene plasmonic waves, which has been demonstrated both theoretically and experimentally at the THz spectral range [27,36]. However, due to the symmetrical linear boundaries, the geometrical tunability of the GSPs just rely on the ribbon width.

To theoretically describe the plasmonic behavior, the bound electron oscillations in isolated GNR can be analyzed with a quasistatic description, which has turned out to be a good approximation when ribbon widths are much smaller than photon wavelengths (~100 mm) [15]. Under this approximation, the plasmon resonances of the system are solely determined by the geometry and the dielectric function. Thus the resonant frequency must meet the phase-matching conditions and be in strict accordance with the following equation by considering the effective width of the nanoribbon when the damping is not large [31,37]

*ω*is the angle frequency of plasmon resonant frequency,

_{p}*ε*is the average dielectric constant and can be simply reduced by

_{avg}*ε*= (

_{avg}*ε*

_{1}+

*ε*

_{2})/2 (where

*ε*

_{1},

*ε*

_{2}are the permittivities of two sides of the graphene sheet),

*W*=

_{eff}*W*+ 2

*A*is the effective ribbon width,

*η*is the dimensionless parameter, which uniquely determines the electrodynamic responses of the nanoribbon array. Neglecting losses,

*η*can be concluded to

Note that *ω _{p}* is independent of

*W*,

*A*,

*E*and other related parameters. Maximum resonance intensity of plasmons takes place at a specific

_{f}*η*value, which is in charge of the plasmon resonant frequency for the given values of

*W*,

*A*, and

*E*and can be deduced from simulation results [12]. Thus,

_{f}*ω*is only a function of the effective ribbon width, the actual model used for the conductivity

_{p}*σ*(

*ω*), and the physical parameters of the graphene.

To demonstrate the proposed enhanced properties of GSPs, 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. Figure 2(a) shows the simulated normal-incidence transmission, reflection, and absorption spectra. Two peaks occurring at 25.69 THz and 53.07 THz can be clearly identified. The first one is the fundamental mode (M = 1) and is dominant with transmission reaching 2.4% (or absorption reaching 19.62%). While the transmission of the second mode (M = 2) just reaches 91.66% (or absorption reaching 2.83%), meaning that the enhancement of electromagnetic fields of the mode 2 is much weaker than that of mode 1. But it does not mean that the field enhancement of the other higher modes will be much weaker. According to our simulations, the transmission dip of the mode 3 reaches 77.67% (or absorption reaching 14.44%), while the dip of the mode 4 reaches 91.02% (or absorption reaching 4.33%), which are not shown in Fig. 2(a). Thus for the modes excited on the GNRs shaped by two sinusoidal boundaries, the odd-order modes have a stronger enhancements of electromagnetic fields than the even-order modes. However, such a field distribution rule rely on the mode orders is going into reverse in the situation with bigger grating period (for example, *Λ* > 300 nm), which will be discussed later in *part 3.2*. Figure 2(c) shows the distribution of electric field norms and Fig. 2(d) shows the *z* components distribution of the electric field for the first four modes. It can be seen from Fig. 2(c) that the properties of these modes are characterized by strongly localized and resonantly enhanced fields on the crest of the sinusoidally shaped gratings, which are called the crest modes. The *E _{z}* components shown in Fig. 2(d) clearly indicate that the crest modes 1, 2, 3 and 4 have phase shift of 2

*π*, 4

*π*, 6

*π*, and 8

*π*along the

*E*-field polarized direction (

*y*-direction) in each period, respectively (a reverse of the field sign corresponds to a phase shift of π).

Since that the sinusoidally shaped gratings are responsible for the field enhancement of the excited crest modes, to see how these modes are different from the unshaped one, we plot the transmission properties of the classical GNRs (with line-line boundaries) in Fig. 2(b). When compared with Fig. 2(a), the resonant frequencies of the first two modes show a blue shift, which can be understood from Eq. (3) by considering the fact that the effective ribbon width of the shaped GNRs (*W* + 2*A*) is wider than that of the classical one (*W*). For comparation, the electric field norms and the corresponding *E _{z}* components in a unit cell of GNRs with line-line boundaries for the excited modes 1-4 are also shown in Figs. 2(e) and 2(f), respectively. As shown in Fig. 2(e), the electric field energy of the mode 1 is uniformly distributed on the two linear edges, while the other three higher order modes is with the field concentrated within the ribbon area. The near-field distributions in Fig. 2(e) show that the four modes are dipoles, quadrupoles, sextupoles, and octopoles, respectively [27]. Strong electromagnetic field uniformly distributes at the ribbon edges or areas in all cases, which is entirely different from what the crest modes shows in Fig. 2(c). This difference indicate that the sinusoidally shaped grating-grating boundaries have quite different ability to manipulate the field localization and modulate the excited plasmons. Note that even through the field of the modes shown in Fig. 2(e) are concentrated on the edges (mode 1) or within the ribbon area (modes 2-4), they are a kind of localized modes instead of a waveguide one [38].

#### 3.2 Tunabilities of the excited crest modes

In this part, we will show that compared with the plasmons induced at classical GNRs, the enhanced plasmonic modes excited at sinusoidally shaped GNRs can be tuned not only via ribbon width *W*, ribbon period *P*, Fermi level *E _{f}*, but also through grating amplitude

*A*and period

*Λ*. We mainly focus on the fundamental crest mode to illustrate these tunabilities. Higher order modes resonate at higher frequencies and can also be optimally coupled to external illuminations. As shown in Eq. (3), the resonant frequency of the excited crest modes is a function of

*W*,

*A*, and

*E*, which can be thought of tunable device design parameters. Considering that large-area graphene films can be grown using an optimized liquid precursor chemical vapor deposition method before transferred onto a substrate, and many approaches have been developed to prepare GNRs [19]. For example, by using the electron beam lithography technique, ultranarrow aligned GNR arrays with ribbon width

_{f}*W*down to 5 nm [39], and graphene nanorings with inner diameter 60 nm and outer diameter 100 nm had been experimentally fabricated [20]. Thus the periodic GNRs with width

*W*˃ 20 nm,

*P*= 200 nm while modeled by Eq. (1) can be easily achieved. Electromagnetic simulations are carried out to demonstrate the effect of these parameters on the response of the shaped ribbons.

The simulated normal-incidence transmission spectra with grating amplitude *A* increasing from 0 nm to 40 nm are shown in Fig. 3(a). It can be seen that at the resonant frequency, the transmission reaches a dip of the minimum possible value for all grating amplitudes. This suggests an extremely high confinement of the THz field in the shaped GNRs at plasmon frequency, which makes the proposed pattern method an attractive candidate for designing THz transmission filters or modulators. Figure 3(b) shows the tunability of the resonant frequency and the full width at half maximum (fwhm) as a function of *A*. With *A* increases, the effective ribbon width increase too, thus the plasmon resonance frequencies show a redshift. The figure indicates that the simulation data coincide well with the theoretical curve calculated from Eq. (3). Meanwhile, the fwhm of the corresponding transmission spectra increases with larger modulation depth of grating amplitude, which is because the lager *A* is, the more light around the central frequency *ω _{p}* would couple into plasmons through grating. In addition, the increase of the plasmon line width indicates that in contrast to unpatterned GNRs, the plasmons in sinusoidally shaped GNRs become naturally radiative with large grating amplitude.

Except for grating amplitude, the effective ribbon width can also be tuned by changing the presupposed ribbon width, thus could be used to tune the excited plasmons. To show that, simulations with different ribbon widths are carried out and the corresponding transmission spectra are shown in Fig. 3(c). In these simulations, the grating amplitude is fixed at 40 nm, while the global occupation ratios (*W*/*P* = 0.5) remain unchanged. With increasing ribbon width, the resonant frequencies of the crest mode 1 show a redshift. As the tuning curve shows in Fig. 3(d), the numerical results agree well with Eq. (3). Since that the effective ribbon widths increase with increasing ribbon width, this kind of shift is not difficult to understand from Eq. (3), where the resonance frequency is of the form ${\omega}_{p}\propto \sqrt{1/\left(W+2A\right)}$. At the same time, the line width of the plasmons just change a little, because the occupation ratio as well as the grating amplitude remain the same.

In addition to *A* and *W*, the enhanced crest modes can also be tuned by other geometrical parameters, such as the grating period *Λ* and the ribbon period *P*. Though these two variables are not considered in Eq. (3) by the quasi-static analysis, they are still two non-negligible fundamental design parameters. The simulated normally incident transmission spectra with grating periods (*Λ*) ranging from 50 nm to 600 nm are shown in Fig. 3(e). The corresponding scaling rules of the resonant frequency of the fundamental crest mode with respect to *Λ* are shown in Fig. 3(f). The plot shows that with the increase of *Λ*, the resonant frequency increases firstly before reaching 300 nm and then decreases. Thanks to the sinusoidal gratings, the transmission corresponding to this mode still keeps in a very low level as the grating period increasing, meaning that the excitation of the fundamental crest mode is with high efficiency. While for other higher orders of modes, we found that when *Λ* ≥ 300 nm, the transmission minimum corresponding to mode 2 will be below that of mode 3, this situation is different from the one as described earlier in *part 3.1*. For example, when *Λ* = 600 nm, the transmission dip of mode 2 reaches 33.32%, which is much lower than 63.21% of mode 3. These changes indicate that higher modes will be excited more efficiently at large value of grating period, which could be used to tune the excitation efficiency of higher order modes. However, unlike the above change rules, the spectral widths of the resonances decrease gradually with increasing *Λ*. This is understandable since a bigger value of *Λ* will lead to a smoother boundary, which will further leads to a larger carrier relaxation time and thus a similar fwhm [12]. The last geometrical factor that affects the plasmon resonance is the ribbon period *P*. As with isolated GNRs, the resonant frequency can also be changed by tailoring the channel width between ribbons, as predicted in Fig. 3(g). The scaling of the resonant frequency with respect to *P* is shown in Fig. 3(h). With increasing distance between ribbons, the plasmon resonance frequencies show a blueshift due to the weakened modes coupling strength. Meanwhile, the spectral width of the resonance decreases inversely with *P*, because of the decrease of the occupation ratio [27]. These results demonstrate how the shaped GNR resonances can be designed and tuned to produce strong field enhancement at a chosen resonant frequency.

Except for the tunable properties benefitting from the geometrical structures, the most outstanding property of GSPs is its ultrabroad bandwidth and fast electrical tunability. This optical response strongly depends on the doping concentration of the graphene, which is controlled mostly by a potentially fast approach of electrical backgating [15]. By employing an electrolytic gate, carrier concentration as high as 4 × 10^{18} m^{−2} in graphene sheet was observed, meaning *E _{f}* = 1.17 eV [40]. Using this method, the Fermi energy level of graphene could be experimentally modified from 0.2 eV to 1.2 eV after applying a high bias voltage [41]. Thus in this paper, we reasonably assume that

*E*can be dynamically tuned from 0.4 eV to 1.0 eV. Simulated transmission spectra shown in Figs. 4(a) and 4(b) clearly confirm the broad tuning range with the change of

_{f}*E*. For example, when the Fermi energy of graphene is varied from 0.4 eV to 1.0 eV,

_{f}*ω*is tuned from 16 THz to 26 THz for the crest mode 1. Figure 4(b) shows the tuning curve agrees very well with Eq. (3). For a given ribbon width and the amplitude, the plasmon resonance frequencies are described by a scaling behavior of ${\omega}_{p}\propto \sqrt{{E}_{f}}.$ This universal relation is the characteristic of two-dimensional electron gases [15]. Besides the resonance frequency, the fwhm of the transmission spectra are also tuned. Figure 4(b) shows that the plasmon line width decrease as the Fermi energy increase. This can be easily explained by the fact that at high doping levels, graphene-induced loss in the medium decreases, therefore, the line width of the plasmon resonance is significantly improved.

_{p}#### 3.3 Excitation by sinusoidally shaping only one boundary

In the previous part, we have shown the properties of the enhanced plasmonic modes excited on GNRs with two symmetrically shaped sinusoidal boundaries, from which a new-style manipulation of light-graphene interaction is presented. While in this part, we will study the other scheme of field confinements that are based on GNRs with only one sinusoidally shaped boundary. As shown in Fig. 5, the upper boundary is patterned with Eq. (1), while the lower one remains a linear shape, which results into an asymmetrical boundaries with respect to the field polarization direction. Figure 6(a) shows the simulated normal-incidence transmission, reflection, and absorption spectra of asymmetrically shaped GNRs excited by a normally incident plane wave with electronic field polarized along the *y* direction. Two peaks occurring at 31.02 THz and 51.12 THz can be identified. The first one is the fundamental mode (M = 1) and is dominant with transmission reaching 2.55% (or absorption reaching 23.32%). While the transmission of the second mode (M = 2) reaches 43.05% (or absorption reaching 35.89%). To better understand the excited modes, we show the field distribution of the electric fields norms and the *z* components for the first four modes in Figs. 6(b) and 6(c), respectively. Due to the asymmetric boundaries, the field energy is strongly localized and resonantly enhanced mainly on the crest of the sinusoidally shaped boundary, rather than on the unshaped linear boundary, as shown in Fig. 6(b). The *E _{z}* components shown in Fig. 6(c) clearly indicate that the excited crest modes 1, 2, 3 and 4 have phase shift of 2

*π*, 3

*π*, 4

*π*, and 5

*π*along the

*E*-field polarized direction (

*y*-direction) in each period, respectively, as Fig. 6(c) shows. Compared with the crest modes excited on two sinusoidally shaped symmetrical boundaries (see Fig. 2(d)), due to the asymmetrical boundary shape, these types of crest modes have phase losses of

*π*, 2

*π*, and 3

*π*for higher order modes 2, 3 and 4, respectively.

Since that the above properties of the enhanced crest modes are owing to the sinusoidal gratings, it is in imperative need to reveal how the grating amplitude on one boundary modulates the plasmon resonances. Figure 7(a) shows the simulated normal-incidence transmission spectra with *A* increasing from 0 nm to 40 nm. It can be seen that the transmission reaches a dip of the minimum possible value for all grating amplitudes at the resonant frequency. For the ribbon with one sinusoidally shaped boundary, the effective ribbon width should be expressed as *W _{eff}* =

*W*+

*A*, which results in the resonant frequency:

*A*. The figure indicates that the simulation data coincide well with the theoretical curve calculated from Eq. (5). With

*A*increases, the plasmon resonance frequencies undergo a red-shift from 38.98 THz to 31.03 THz. Note that this shift range is smaller than that of the pattern with two sinusoidal gratings, where the resonance frequencies show a red-shift from 39.52 THz to 25.68 THz at the same limit of variation of

*A*(as shown in Fig. 3(b)). Meanwhile, the fwhm of the corresponding transmission spectra increases from 230 nm to 320 nm, which is also smaller than the range corresponding to the shape with two sinusoidal boundaries, where the fwhm undergo a red-shift from 230 nm to 488 nm. The different variation ranges of the frequencies and fwhm are all because of the different number of sinusoidal boundaries that result in different effective ribbon widths.

Different from the classical GNRs, the patterned GNRs here with one or two sinusoidally shaped boundaries present a new picture of field confinements. To confirm the different abilities of the field energy modulation, the distribution of |*E*|-field variations along the polarization direction (as the dotted line shows in the insert) is presented in Fig. 7(c). It clearly shows that for all orders of the crest modes, the field energies are tightly confined at the boundary of gratings, and the localized fields are enhanced by 3 times of magnitude at the grating edges compared to that of the unpatterned ribbons, which suggests an extremely high confinement of the THz field in the shaped GNRs at plasmon frequency. This special ability of the sinusoidally shaped gratings indicates that the light-graphene interactions can be greatly increased by reasonably designing the boundary types of the graphene nanostructures. Since that graphene nanoribbons are fundamental components to the development of graphene nanoelectronics, and at the nanoscale, electronic enhancement effects and nature of the nanoribbon edge shapes become essential to the properties of graphene, the capacity to extremely enhance the field confinement makes the proposed pattern method an attractive candidate for designing THz transmission filters or modulators.

## 4. Conclusion

In summary, we have put forward a concept and a design of using sinusoidally shaped GNRs to excite the localized crest plasmon modes. These modes are shown to be quite different from conventional modes excited on traditional graphene nanoribbons because their field energies are concentrated on the crest of the sinusoidal gratings rather than uniformly distributed on the two linear edges. The excited crest modes possess extremely field enhancements at the edge of the grating that are 3 times of magnitude stronger than that of the classical GNRs. Through theoretical analyses and numerical simulations, we demonstrate that the excited crest GSPs can be tuned by means of their physical (Fermi energy) and geometrical (grating amplitude, grating period, ribbon width, and ribbon period) properties. This new picture of modulating the light-graphene interaction expands our understanding of plasmon resonances on graphene nanostrutures and makes the proposed pattern method an attractive candidate for designing 2D graphene plasmonic devices working at THz frequencies.

## Funding

National 973 Program of China (2012CB315701); National Natural Science Foundation of China (NSFC) (61505052, 11574079, 61176116, 11074069).

## Acknowledgments

The authors thank Hong-Ju Li and Jian-Ping Liu for technical assistance and discussions.

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