1. Introduction

Excitations of resonant modes in a photonic structure depend strongly on its symmetry. For a photonic structure with high symmetry, there may exist dark modes that are difficult to be directly excited by external light [1]. However, these dark modes could be excited by breaking the structural symmetry, leading to many novel optical phenomena [2]. Indeed, the influence of symmetry breaking has been widely discussed in different photonic structures such as metallic nanocavities [3], asymmetric cut-wire pairs [4–6], and metallic nanosphere clusters [7].

It is well known that doped or gated graphene can support propagating plasmons [8–10] with interesting optical properties, leading to potential applications in photonics and optoelectronics [11,12]. If structured, graphene nanostructures such as ribbons and disks can support localized plasmonic modes in mid-infrared range [13–22], showing many interesting properties such as tunability, deep subwavelength, and high confinement of optical near fields [23–26]. On the other side, multi-layer graphene possesses multiple plasmon bands [27–32] while there is only one plasmon band in single-layer graphene [33,34]. As a result, nanostructured multilayer graphene such as ribbons and disks can support multiple localized plasmon resonances, offering more degrees of freedom in designing photonic devices over single-layer graphene [17,35,36]. Due to the symmetry constrains, however, not all such plasmon resonances in multilayer graphene nanostructures can be excited. For example, in a stacked pair of either identical graphene ribbons (two-layer graphene ribbons) [37] or identical graphene nanodisks [38], only the symmetric mode can be excited by externally incident light. In patterned arrays of disks consisting of graphene/insulator stacks which intrinsically can support multiple localized plasmonic resonances, only one resonant peak in extinction spectra is experimentally observed [17] while other modes remain unfortunately dark.

In this paper, the effect of structural asymmetry in multilayer graphene ribbons is numerically studied. By introducing structural asymmetry, dark plasmonic modes can also be directly excited by externally incident light, as confirmed by numerical simulations.

2. Methods

To investigate the localized plasmon resonances in multilayer graphene ribbons, full-wave simulations (CST Microwave Studio) are performed. Graphene is modeled as an ultrathin metallic film with a thickness of t = 1 nm, similar to that used in [20,39], and a frequency-dependent dielectric function $\epsilon (\omega )=1+i\sigma (\omega )/\omega t{\epsilon }_0,$ where ω is the angular frequency and ε₀ is the vacuum permittivity. The optical conductivity of graphene σ(ω), obtained from the random-phase approximation [8,9], is used [23]

In our simulations, multilayer graphene ribbons are periodically arranged in a one-dimensional supercell with a period that is five times larger than the graphene ribbon width unless otherwise specified. This can guarantee that the obtained results are for individual multilayer graphene ribbons since the inter-coupling among multilayer graphene ribbons can be neglected. Triangular meshes are used in the simulations. Light is normally incident with the electric field polarized perpendicular to graphene ribbons.

3. Results and discussions

We first consider simplest multilayer graphene ribbons, two-layer graphene ribbons of equal ribbon width, as schematically shown in the inset of Fig. 1(a). The two constituent graphene ribbons are aligned centrosymmetrically and the transverse cross-section is thus rectangular. The dispersion of plasmons for two-layer graphene (two parallel graphene sheets), calculated by a transfer matrix method [40], is shown in Fig. 1(a). Obviously, there are two plasmon bands. The lower band is an acoustic mode that shows a linear dispersion and the upper band is an optical mode that displays a square-root dispersion in the long-wavelength limit.

 

Fig. 1 (a) Plasmon dispersion (solid lines) for two-layer graphene with a interlayer separation of 100 nm. That of a single-layer graphene (dashed line) is also shown for comparison. Dots represent the resonant frequencies of two-layer graphene ribbons of equal ribbon width for different ribbon widths extracted from the simulated absorption spectra, according to Eq. (2). The inset shows the schematics of an individual two-layer graphene ribbon and the cross-sectioned electric field distributions of the symmetric mode. (b) Absorption spectra of two-layer graphene ribbons of equal width for different interlayer separations: 100 nm (red line), 200 nm (green line), and 500 nm (blue line). The ribbon width is 200 nm. The dashed black line represents the absorption spectrum of a single graphene ribbon with a ribbon width of 200 nm. (c) Absorption spectra for two-layer graphene ribbons of unequal ribbon width at different interlayer separation as schematically shown in the inset. (d) Normalized electric (upper panels) and magnetic (lower panels) field amplitude distributions for the two-layer graphene ribbons of unequal ribbon width at two resonant frequencies. Arrows indicate the current direction within the constituent graphene ribbons.

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Basically, in a single-layer graphene ribbon there is only one fundamental plasmonic resonance. However, in multilayer graphene ribbons the fundamental plasmonic modes in the constituent ribbons will couple with each other, leading to hybridized plasmonic modes. In two-layer graphene ribbons, there are two such hybridized plasmonic modes, one symmetric and the other anti-symmetric. Figure 1(b) shows the absorption spectra of two-layer graphene ribbons of equal ribbon width for different interlayer separations. For a fixed separation, there is only one absorption peak in the displayed frequency range, indicating that only one resonant mode is excited. From the field and current distributions, this resonance originates from the symmetric mode with parallel currents within the two constituent ribbons. Unfortunately, the anti-symmetric mode is dark due to the symmetry constraints. With the increase in the interlayer separation the resonant frequency of the excited plasmonic mode shows a red shift, owing to the decrease in coupling. For large separations, it approaches to that of the constituent single graphene ribbon as expected since the coupling can be neglected.

To explore the relation of the plasmon modes between two-layer graphene and two-layer graphene ribbons, plasmons in two-layer graphene ribbons can be described by an effective wave vector k_eff which satisfies$2k_{\text{eff}}W+2\Phi =2n\pi$ [41], where W is the ribbon width, Φ is the reflection phase at the edge, and n is an integer. In most studies, the reflection phase is taken to be $\Phi =-\pi ,$ leading to the frequently used relation $k_{\text{eff}}=(n+1)\pi /W.$ For the fundamental mode (n = 0), $k_{\text{eff}}=\pi /W.$ Theoretical investigations [41] reveal that the reflection phase has a nontrivial value of $\Phi \approx -3\pi /4$ rather than –π. Thus, the effective wave vector should be

In two-layer graphene ribbons of equal ribbon width, the symmetric mode can be excited. The anti-symmetric mode remains unfortunately dark due to the symmetry constrains. To exit the dark mode, one can break the symmetry of the structure, for example, by making the width of the two constituent graphene ribbons different, as schematically shown in the inset of Fig. 1(c) where the width of one graphene ribbon is W₁ = 100 nm and that of the other is W₂ = 200 nm. The two constituent graphene ribbons are still centrosymmetrically aligned and the transverse cross-section is thus an isosceles trapezoid in shape. Obviously, from Fig. 1(c) there are two resonances in the absorption spectra. The resonance of high frequency corresponds to the symmetric mode while that of low frequency is due to the anti-symmetrical mode. The excitation of the dark mode in the two-layer graphene ribbons of unequal ribbon width is due to the symmetry breaking. With the increase in the interlayer separation, the symmetric mode shows a red shift in frequency while the anti-symmetric mode displays a blue shift. For a large separation, e.g., 500 nm, the frequencies of the two modes system approach those of the two constituent individual graphene ribbons as expected. From the field and current distributions shown in Fig. 1(d), for the symmetric mode the currents in the two constituent graphene ribbons are parallel and can be excited. For the anti-symmetric mode, the currents are anti-parallel but with different amplitudes. Interestingly, there are two contributions for the excitation of the anti-symmetric mode: one is the electrical dipole resonance and the other is the magnetic dipole resonance, which can be clearly seen from the electrical and magnetic field distributions. This interesting property will be discussed later.

The detailed optical properties of two-layer graphene ribbons of unequal ribbon width are discussed in Fig. 2 where the width of one of the constituent graphene ribbon is fixed to be W₁ = 200 nm, while that of the other varies from 80 to 320 nm, corresponding to the width difference $\Delta W=W_1-W_2$ changing from −120 to 120 nm. From the absorption spectra shown in Fig. 2(a), it is obvious that the anti-symmetric mode can be easily excited by normally incident light when the structural asymmetry is introduced. The frequency of the symmetric mode increases with the width difference. The anti-symmetric mode shows a similar behavior. For small width differences, the intensity of the anti-symmetric mode is relatively small. The quality factors of both modes are also calculated, shown in Fig. 2(b). In a wide range of the width difference, the quality factor of the anti-symmetric mode is much higher than that of the symmetric mode. When the width difference is larger than about 76 nm, the quality factor of the symmetric mode exceeds that of the anti-symmetric mode.

 

Fig. 2 (a) Absorption spectra for two-layer graphene ribbons with a different ribbon width different ΔW as labeled at intervals of 20 nm. The interlayer separation is 100 nm. The period of the supercell used is 1 μm. (b) Q factors of the symmetric (red dots) and anti-symmetric (black dots) modes as a function of the ribbon width difference. (c) Normalized amplitudes of the electric (upper panel) and magnetic (lower panel) dipole moment densities in a contour plot as a function of ΔW and frequency.

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To investigate the origin of the resonances in two-layer graphene ribbons of unequal ribbon width, electric and magnetic dipole moments are calculated. Considering that the structure under study is basically a two-dimensional system, the electrical and magnetic dipole moment densities are defined as [42]

It should be noted that both resonances in two-layer graphene ribbons are dominantly the electric dipole resonances in nature although there may exist the magnetic dipole resonances. This is because the far-field irradiance resulting from the magnetic dipole resonance is about two orders of magnitude smaller than that from the electric dipole resonance. Optical resonators with magnetic resonances are essential building blocks for constructing metamaterials [43,44]. Thus, how to eliminate electric dipole resonances in such graphene structures in order to attain a magnetic dipole resonance would be an interesting topic for future study.

As shown in previous studies [27–32], multi-layer graphene display multiple plasmon bands. In general, it can be shown by the transfer matrix method [40] that in N-layer graphene there are totally N plasmon bands with one being the optical mode and the remaining N−1 being the acoustic modes [31]. Figure 3(a) shows the dispersion of plasmons in ten-layer graphene. Obviously, there are totally ten plasmon bands: the uppermost band is the optical mode and the rest are the acoustic modes. If structured into multi-layer graphene ribbons, there would exist multiple localized plasmonic modes. However, not all such modes can be excited. Indeed, there are only three or four noticeable absorption peaks in the displayed frequency range as shown in Fig. 3(b) for ten-layer graphene ribbons of equal ribbon width with the transverse cross-section being a rectangle in shape. The resonant frequencies of the excited modes in the ten-layer graphene ribbons of equal ribbon width obtained from the absorption spectra are also plotted in Fig. 3(a) according to Eq. (2), showing the correspondence of plasmonic modes between multilayer graphene and multilayer graphene ribbons. Obviously, Eq. (2) can render a satisfactory correspondence.

 

Fig. 3 (a) Plasmon dispersion for ten-layer graphene. Symbols represent the resonant frequencies obtained from the calculated absorption spectra according to Eq. (2). (b) Absorption spectra for ten-layer graphene ribbons of equal ribbon width. Red, green, and blue lines are for the ribbon width of 200, 300, and 400 nm, respectively. The interlayer separation in ten-layer graphene and ten-layer graphene ribbons is both 100 nm.

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As in two-layer graphene ribbons, if we break the symmetry of ten-layer graphene ribbons of equal ribbon width, those previously dark modes may be excited. In this study, we break the symmetry of the ten-layer graphene ribbons by changing the transverse cross-section from a rectangle to an isosceles trapezoid. The ribbon widths are now not equal but vary linearly from the top to the bottom constituent graphene ribbon. In Fig. 4, the absorption spectra of such ten-layer graphene ribbons of unequal ribbon width are shown for different Fermi energy and relaxation time. For a large relaxation time, there are three or four prominent absorption peaks for the case of equal ribbon width. This indicates that only three or four modes can be efficiently excited and the remaining modes are difficult to excite. In contrast, ten prominent peaks appear in the absorption spectra for the case of unequal ribbon width, suggesting that all resonant modes are excited. At a small relaxation time, the absorption peaks may disappear due to the broadening of the absorption peaks, leading to a broad absorption band. The half width in the case of unequal ribbon width is larger than 10 THz, more than three times larger than that in the case of equal ribbon width. Interestingly, the frequency range of the absorption band can be tuned by varying the Fermi energy, resulting in a tunable broadband absorption.

 

Fig. 4 Absorption spectra for ten-layer graphene ribbons of equal ribbon width (black lines) and of unequal ribbon width (red lines). The ribbon width of the former is 200 nm while that of the latter varies nearly from 120 nm for the top to 300 nm for the bottom graphene ribbon. The interlayer separation is 200 nm. The period of the supercell used is 1 μm. In (a), the Fermi energy is 0.4 eV. The dotted and solid lines are for the relaxation time of 0.4 and 0.04 ps, respectively. In (b), the Fermi energy is 0.6 eV. The dotted and solid lines are for the relaxation time of 0.6 and 0.06 ps, respectively.

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5. Conclusion

In conclusion, we study numerically the excitations of plasmonic modes in multilayer graphene ribbons. These multilayer graphene ribbons can support hybridized plasmonic modes due to the coupling among the constituent graphene ribbons. However, not all plasmonic modes can be excited by external light due to the symmetry constrains. In two-layer graphene ribbons of equal ribbon width, only the symmetric mode can be excited. With the introduction of unequal ribbon width, the anti-symmetric mode can be efficiently excited due to the symmetry breaking. The ideal of symmetry breaking induced excitations of dark modes is extended to multilayer graphene ribbons. By introducing unequal ribbon widths, all hybridized plasmonic modes can be excited including those dark ones in the case of equal ribbon width. We show numerically that in ten-layer graphene ribbons of unequal ribbon width a broadband absorption with a half width more than 10 THz can be achieved. Moreover, this broadband absorption can be tuned by changing the Fermi energy of graphene.