## Abstract

Dirac cones discovered in classical periodic systems such as photonic and phononic crystals have exhibited many interesting properties, particularly conical dispersion at the Brillouin zone center can be related to a zero-refractive-index. Here, we theoretically and numerically explore the conical dispersion in plasmonic crystal of graphene nanodisks arranged in triangular lattice. We show that the plasmonic crystal of Dirac-like cone resulted from three-fold accidental degeneracy can be mapped to a zero-refractive-index medium around the Dirac-like point frequency of 65.5 THz. The isotropic behavior of Dirac-like point formed by a monopole and two dipoles is observed by calculating isofrequency contours. Furthermore, numerical simulations including cloaking, focusing and unidirectional transmission are implemented to demonstrate the zero-index characteristics of the graphene plasmonic crystal.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Dirac cone dispersions discovered in classical periodic systems such as photonic and phononic crystals have attracted considerable attentions due to their intriguing properties including pseudo-diffusion [1], Zitterbewegung effect [2, 3], nontrivial topological characteristics and non-zero Berry phase [4, 5], and so on [6–8]. Notably the conical dispersions at the Brillouin zone (BZ) center that can be related to a zero-refractive-index, which paves a new way to obtain artificial zero-index materials (ZIMs) and realize a variety of phenomenon of interesting [9–14]. Huang *et al*. reported the first photonic crystals of Dirac-like cone dispersion at the center of the BZ, theoretically and experimentally demonstrated the zero-index characteristics and the cloaking effect of the photonic crystals near the Dirac-like point frequency of microwave regime [12]. Since then, a lot of related research works e.g. experimentally realization of the first on-chip integrated zero-index metamaterial in the optical regime [10], polarization independent conical dispersion in photonic hypercrystals [13] and zero-index photonic graphene around the double Dirac point frequency [15] were carried out. Very recently, Dubois *et al*. reported the first experimental realization of an impedance matched acoustic double zero refractive index metamaterial induced by a Dirac-like cone at the BZ center [11].

Graphene, a single layer of carbon atoms periodically arranged in honeycomb lattices, has been widely explored as a promising material for plasmonic devices due to its metal-like properties in the terahertz (THz) and infrared ranges [16, 17]. In particular, graphene-supported surface plasmon polaritons (SPPs) exhibit tight field confinement [18], relatively low propagating loss and flexible tunability [19, 20], which is hopeful for realizing a new generation of reconfigurable plasmonic components and potential applications in highly integrated plasmonic systems. In 2016, Shi *et al*. investigated the band-stop properties of one-dimensional plasmonic crystals based on periodically modified chemical potential of graphene [21]. Recently, Jin et al. proposed a two-dimensional plasmonic platform composed of a periodically patterned monolayer graphene, where topologically protected one-way edge plasmons were realized by applying a static magnetic field i.e. breaking the time-reversal-symmetry [22]. Subsequently, topological edge states were achieved along an interface constructed by spatial inversion symmetry broken graphene plasmonic crystals (GPCs) [23].

In this work, we proposed the plasmonic crystal of graphene nanodiisks arranged in triangular lattice, which was realized by periodically modulating the chemical potential of monolayer graphene. The Dirac-like cone dispersion resulted from a three-fold accidental degeneracy of a monopolar mode and two dipolar modes was theoretically and numerically investigated by utilizing commercial software COMSOL Multiphysics based on finite-element method (FEM). Furthermore, we show that the GPCs of Dirac-like cone at the BZ center exhibit isotropic behavior and their finite-sized counterparts can behave like ZIMs around the Dirac-like point frequency as evidenced by numerical simulations including cloaking effect, focusing and unidirectional transmission.

## 2. Simulation methods and models

The two-dimensional plasmonic crystal we considered in this work is composed of an array of graphene nanodisks arranged in triangular lattice, which can be realized by periodically modulating the chemical potential of monolayer graphene. As schematically illustrated in Fig. 1(a), a graphene monolayer sits on a silica layer on Si substrate. The thickness of the SiO_{2} layer is periodically modified to get the chemical potential of the graphene periodical variation when a backgate bias voltage is applied between the graphene sheet and silicon substrate. Here, graphene regions with silica thicknesses *d _{1}* and

*d*have different chemical potentials under an external gate voltage, the chemical potential ratio is

_{2}*μ*/

_{c1}*μ*= (

_{c2}*d*/

_{2}*d*)

_{1}^{1/2}[21]. The unit cell of the GPC is shown in Fig. 1(b), where the graphene nanodisk with chemical potential of

*μ*is surrounded by the same sheet of graphene with chemical potential of

_{c1}*μ*, which is resulted from the triangular lattice patterned silicon pillars.

_{c2}*a*and

*r*represent the lattice constant and radii of the graphene nanodisks respectively. Figure 1(c) depicts BZ with the irreducible zone of 𝛤-M-K-𝛤.

In our investigation, the dispersion relation for transverse magnetic (TM) polarized SPP modes supported on the graphene sheet surrounded with air and silica can be deduced by solving Maxwells equations with boundary conditions, which is expressed as [24]

*ε*and

_{Air}*ε*are the relative permittivity of air and silica corresponding to super and substrates in our work. And

_{SiO2}*ε*is the vacuum permittivity of free space,

_{0}*k*= 2π/

_{0}*λ*is the wave number in free space and

*λ*is the operating wavelength in vacuum. The complex-valued surface conductivity of graphene

*σ*composed of the interband electron transitions

_{g}*σ*and the intraband electron-photon scattering

_{inter}*σ*, is derived from the Kubo formula [25,26]with

_{intra}*ω*is the angular frequency of the plasmon,

*e*and

*k*are the electron charge and the Boltzmann constant respectively,

_{B}*T*is the temperature,

*ℏ*is the reduced Planck constant,

*μ*is the chemical potential, and

_{c}*τ*denotes the electron momentum relaxation time due to charge carrier scattering. Specifically, the chemical potential of graphene can be effectively tuned via chemical doping or external gate voltage [16, 27]. Recent experimental work implemented by Efetov

*et al*. has demonstrated that the chemical potential of graphene as high as 2 eV can be achieved [28]. In graphene,

*τ*depends on the carrier mobility

*μ*, which is expressed as

*τ*=

*μμ*/(

_{c}*ev*) where

_{f}^{2}*v*= 10

_{f}^{6}m/s is Fermi velocity. In order to ensure the reliability of our calculation results, we moderately set

*μ*= 20000 cm

^{2}V

^{−1}s

^{−1}from the experimental results previously reported by Thongrattanasiri

*et al*[29]. And the maximal chemical potential used in this work was 0.6 eV. In the non-retarded regime where

*β*»

*k*, the Eq. (1) can be simplified to [24]

_{0}*β*is the propagation constant of SPPs on graphene layer. In the mid-infrared region, the dielectric constants of SiO

_{2}and air are assumed to be

*ε*= 3.9 and

_{SiO2}*ε*= 1.0 respectively [25]. And the effective refractive index for the SPP modes on graphene layer can be obtained from

_{Air}*n*=

_{eff}*𝛽*/

*k*, which is inversely proportional to

_{0}*σ*. It is well-known that the plasmonic modes possess tight field confinement in graphene, thus the effect of silicon substrate on the SPP mode will be weakened when the silica thickness increases. Actually, it is worth emphasizing that the influence of the silicon substrate on the dispersion relation of the graphene SPP modes becomes negligible since the thickness of the silica layer is always larger than 100 nm [21, 26].

_{g}## 3. Results and discussions

The band structures of the GPC composed of a triangular lattice of graphene nanodisks with *μ _{c1}* = 0.3 eV,

*μ*= 0.6 eV are displayed in Fig. 2. As can be seen from Fig. 2(a), TM2, TM3 and TM4 bands intersect at the center of BZ forming a triply degenerate state marked with D, at the frequency of 65.5 THz when

_{c2}*r*= 0.2256

*a*. We can further confirmed from the enlarged view of the band structure around D (inset in Fig. 2(a)) that this triply degenerate state is a Dirac-like point resulted from monopole and dipole interactions as evidenced by the electric field profiles shown in Figs. 2(e)-2(g). The conical dispersion formed by two linear bands intersect with a third flat band is the so-called Dirac-like cone [10, 12]. Further, the conical behavior of the bands near the Dirac-like point is displayed in a three-dimensional dispersion surfaces exhibiting the relationship between the Floquet propagation vector and frequency as plotted in Fig. 2(b). Next, we show that the Dirac-like point is a result of accidental degeneracy which means it occurs at a certain set of structural parameters. As shown in Figs. 2(c)-2(d), the triply degenerate point split into a doubly degenerate point of dipolar modes (Figs. 2(e)-2(f)) and a single mode of monopole (Fig. 2(g)) at the BZ center when the radius of the graphene nanodisk deviates from 0.2256

*a*. Therefore, we come to a conclusion that the Dirac-like cone in GPC emerges as a consequence of accidental degeneracy at a particular radius of graphene nanodisk.

At the Dirac-like point, the wavenumber *k* approaches zero, indicating that the effective refractive index of the GPC *n _{eff}* is also zero due to the direct proportional relationship between

*n*and

_{eff}*k*. Therefore, the proposed GPC can be mapped into ZIMs around the Dirac-like point frequency. To further prove the zero-index characteristic of the GPC, we retrieve the effective permittivity and permeability near the Dirac-like point frequency by utilizing field averaging of the Bloch modes [9]. As displayed in Fig. 3(a), simultaneous zero permittivity and permeability were obtained at the Dirac-like point frequency. The isofrequency contours (bands of constant frequency) of TM2 and TM4 bands are displayed in Figs. 3(b) and 3(c). It shows that the contours maintain a nearly circular shape for frequency ranges of 61.53-64.93 THz (TM2 band) and 65.95-68.79 THz (TM4 band), which means the GPC has essentially an isotropic behavior near the Dirac-like point frequency. The isofrequency contours originating from the Dirac-like point frequency, increase in size away from this frequency, and are circular over a broad frequency range. Light inside the ZIMs experiences no spatial phase variation and infinitely phase velocity, properties that can be applied for realizing focusing, wave-front shaping, directional transmission and invisibility cloak and so on [9, 12–14].

To further demonstrate the zero-index characteristics of the GPC, we carry out the numerical simulation of cloaking based on FEM. The schematic diagram is shown in the inset of Fig. 4(a), where the GPC is surrounded by the same sheet of graphene with perfect magnetic conductor (PMC) boundary conditions on the upper and lower walls. We first investigate the cloaking effect for PMC obstacles. As can be seen from Figs. 4(e)-4(g), the SPP waves with the Dirac-like point frequency of 65.5 THz transmit through the GPC without distortion when an object with PMC boundary condition is embeded, and we also found that the geometrical structure of obstacles barely affect this cloaking ability. To explore this cloaking effect in more detail, we replace the PMC obstacle with a dielectric object to analyze the transmission properties. Figure 4(a) presents the transmittance/reflectance spectrums of the GPC with a dielectric obstacle of refractive index from 0 to 20 embedded. One can see that the GPC we proposed exhibits extraordinary abilities for cloaking dielectric obstacles in a large range of refractive index at the Dirac-like point frequency as evidenced by the electric distributions shown in Figs. 4(b)-4(d). The SPP waves propagate through the obstacle almost without scattering or reflection after a long distance of 18.2*a*, only subject to the intrinsic loss of graphene material.

The uniform phase inside the ZIM can be utilized to shape the phase front of the electromagnetic wave. This property is illustrated in Fig. 5. We designed two kinds of wave front manipulation by using the proposed GPC. Figure 5(a) presents the focusing property of a concave lens made of ZIM GPC with the configuration shown in the inset of Fig. 5(c). The SPP wave with Dirac-like point frequency coming from the bottom is incident normally on the GPC. As there is no phase change across the ZIM, the phase keeps the same along the concave surface, resulting in the formation of a focal point in front of the lens. The norm field intensity distribution along the focal point is plotted in Fig. 5(c), where the full width at half maximum (FWHM) of 31.5 nm is obtained. Figure 5(b) shows a plane wave generator, where a point source with Dirac-like point frequency of 65.5 THz is excited in the center of the GPC as marked by the white fork. Due to the near zero refraction along the upper and lower interface, two plane waves are generated with the same phase, which provides a new way to design THz signal generator. Notably, the point source can be replaced by a line source or a closed line source, leading to the same effect.

Another demonstration on the zero-index characteristic of GPC can be given through the numerical simulation presented in Fig. 6(a), in which a plane wave incident from the left-side of the prism made of GPC, and then exits from the right surface of prism as a plane wave with little phase distortion. Part of incident energy is reflected back because of the impedance mismatch. One can see that the refractive angle at the right surface equals to zero. According to Snell’s law, *n _{eff}*sinθ =

*n*sin0, where

_{eff2}*n*is the effective refractive index of the surrounding graphene and the incident angle

_{eff2}*θ*equals 30

^{。}, so the zero effective refractive index

*n*can be easily obtained. When the plane wave obliquely incidents from the right surface, almost all the incident energy reflected back as illustrated in Fig. 6(b). Hence, the simulation results in Fig. 6 revealed that the unidirectional transmission of electromagnetic wave can be easily realized by using the GPCs. Actually, such kind of manipulation of the wave transmission direction is an inherent property of ZIM: total reflection is always expectable for any incident angle except for normal incidence when a plane wave is incident from an ordinary medium on a ZIM [30].

_{eff}## 4. Conclusion

In summary, we designed GPC composed of a triangular lattice of graphene nanodisks, which can be realized by periodically modulating the chemical potential of the graphene monolayer. By using Comsol Multiphysics based on FEM, we calculated the band structure of the GPC, and we show that conical dispersion at the center of BZ can be obtained by appropriately designing the radius of graphene nanodisk. A Dirac-like point with frequency of 65.5 THz formed by triply accidental degeneracy at BZ center was accomplished when *r* = 0.2256*a* with *a* = 40 nm is the lattice constant. Furthermore, numerical simulations including cloaking, focusing and unidirectional transmission were carried out to demonstrate the zero-index characteristic of the GPC around the Dirac-like point frequency.

## Funding

National Natural Science Fund of China (NSFC) (61378058, 11774103); Fujian Province Science Fund for Distinguished Young Scholars (2015J06015); Promotion Program for Young and Middle-Aged Teachers in Science and Technology Research of Huaqiao University (ZQN-YX203); Project for Cultivating Postgraduates’ Innovative Ability in Scientific Research of Huaqiao University (1511301022).

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