## Abstract

A two-dimensional graphene plasmonic crystal composed of periodically arranged graphene nanodisks is proposed. We show that the band topology effect due to inversion symmetry broken in the proposed plasmonic crystals is obtained by tuning the chemical potential of graphene nanodisks. Utilizing this kind of plasmonic crystal, we constructed N-shaped channels and realized topologically edged transmission within the band gap. Furthermore, topologically protected exterior boundary propagation, which is immune to backscattering, was also achieved by modifying the chemical potential of graphene nanodisks. The proposed graphene plasmonic crystals with ultracompact size are subject only to intrinsic material loss, which may find potential applications in the fields of topological plasmonics and high density nanophotonic integrated systems.

© 2017 Optical Society of America

## 1. Introduction

Photonic crystals (PhCs) have been widely studied due to their unprecedented ability for wave manipulation and applications in optical communication and sensing [1–5]. More recently, the exploration of topology on band structures has revolutionized our understanding of PhCs. A typical example is topological PhCs that have an extraordinary ability for molding the flow of light [6–11]. By breaking time-reversal symmetry or designing sophisticated metamaterial structures, one can open a complete photonic band gap in the Dirac cone dispersion to achieve nontrivial topological bands which are also known as topologically protected photonic states [12–17]. Such topologically protected photonic states endowed with topological robustness, are immune to structural imperfections and backscattering. Notably, a new kind of topological PhC purely composed of conventional dielectric material was derived simply by deforming a honeycomb lattice of cylinders, where photonic topological bands and pseudo-time-reversal symmetry were investigated [13]. Furthermore, a topological metacrystal structure comprised of an array of metal rods attached to two parallel copper plates was successfully utilized to control and transmit waves along any desired path without back-reflection at microwave frequency [6]. However, conventional dielectric materials based PhC structures, usually in micrometer scale, are challenging to realize further reductions on the size of devices due to the diffraction limit of light. Therefore, new techniques to effectively manipulate light on the nanometer length scale are indispensable for the miniaturization and development of efficient on-chip integration of optical components.

Surface plasmon polaritons (SPPs), the electromagnetic waves propagating along an interface between a metal and a dielectric, are regarded as a promising physical mechanism to overcome the optical diffraction limit and to advance the miniaturization of devices [18–20]. Recently, graphene has been introduced to the field of plasmonics due to its unique electrical and optical properties including tight field confinement, versatile tunability and relatively low ohmic loss, which offers a new way to manipulate and concentrate light on the nanoscale [21–25]. Particularly, unlike conventional plasmonic materials, graphene supported SPPs exhibit highly field confinement and low-loss at terahertz (THz) to mid-infrared frequency range, which makes graphene a promising alternative to the all-integrated plasmonic devices at THz frequencies. Fei *et al*. systematically investigated the edge plasmon modes inside graphene nanoribbons [26]. Very recently, Jin *et al*. designed periodically patterned monolayer graphene, where topological one-way edge states were realized by introducing a static magnetic field to break time-reversal-symmetry [27].

In this work, we propose a two-dimensional graphene plasmonic crystal consisting of an array of graphene nanodisks in the same sheet of graphene with a honeycomb lattice, which can be used to realize band topology effect in THz to mid-infrared frequency. Through analysis of dispersion relation and modal properties near the Dirac point, it is found that a complete band gap from 47.3 to 50.5 THz can be obtained by tuning the chemical potential of graphene nanodisks, i.e. breaking the inversion symmetry of plasmonic crystals. Further, we construct N-shape channels by using the inversion symmetry broken graphene plasmonic crystals, which supports internal topological edge transmission within the band gap. And the topologically protected exterior boundary transmission is also simulated numerically.

## 2. Simulation methods and models

As shown in Fig. 2, the graphene plasmonic crystals considered in this work are constructed by a monolayer graphene, where two periodically arranged graphene nanodisks are surrounded by the same sheet of graphene with different chemical potential. The radii of the graphene nanodisks are *r*, and *a* is the lattice constant. The domain view on the lower right of Fig. 1(b) plots the Brillouin zone (BZ) and *k* path for the honeycomb unit cell.

In our investigation, the dispersion relation for transverse magnetic (TM) modes supported on the monolayer graphene can be obtained by solving Maxwells equations with boundary conditions, which is expressed as [20]:

*ε*= 1 and

_{Air}*ε*= 3.9 are the dielectric constants of air and silica corresponding to super and substrates in our work. And

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

_{0}*k*=

_{0}*ω*/

*c*is the wave number in free space. In the non-retarded regime where

*β*»

*k*, the Eq. (1) can be simplified to [20]

_{0}*β*is the propagation constant of SPPs on graphene layer. And the effective index for the SPPs on graphene layer can be obtained from

*n*= 𝛽 /

_{eff}*k*, which is inversely proportional to

_{0}*σ*. The surface conductivity of graphene

_{g}*σ*composed of the interband electron transitions

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

_{inter}*σ*, is obtained from the Kubo formula [28]: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*is the electron momentum relaxation time. Specifically, the chemical potential of graphene can be effectively tuned via chemical doping or external gate voltage [29, 30]. Recent experimental work implemented by Efetov D K

*et al*. has demonstrated that the chemical potential of graphene as high as 2 eV can be achieved [31]. And Low T

*et al*. have reported that a relaxation time as high as 3 ps was experimentally obtained [32]. In order to ensure the reliability of our numerical study, we moderately set

*τ*= 1 ps, and the maximal chemical potential used in this work was 0.6 eV.

## 3. Results and discussions

We explore the band topology of graphene plasmonic crystals constructed by periodically patterned graphene nanodisks. The unit cell consists of two graphene nanodisks surrounded by the same sheet of graphene with different chemical potentials as shown in Fig. 1(b). Figure 2(a) displays the band structure of graphene plasmonic crystal with *μ _{c2}* = 0.6 eV,

*μ*=

_{c1}*μ*= 0.3 eV, where there is no band gap obviously. Two bands linearly intersect at a two-fold degenerate Dirac point marked with

_{c3}*D*, at frequency of 49.65 THz at the BZ corner

*K*point. We also can see from the enlarged view (inset of Fig. 2(a)) of the band structures near the

*K*point, which exhibits the conical dispersions clearly. The Dirac cone dispersion protected by the combination of the spatial inversion symmetry and time-reversal symmetry will be split when one of symmetry is broken. Notably, an overwhelming advantage of such plasmonic crystal can be tuned without changing geometrical structure. To open a gap between the two bands, we modify the chemical potentials of the two nanodisks, i.e. breaking the inversion symmetry. As shown in Fig. 2(b), one can see that a complete band gap from 47.3 THz to 50.5 THz emerged when the chemical potentials of the two nanodisks differs from each other (

*μ*= 0.6 eV,

_{c2}*μ*= 0.33 and

_{c1}*μ*= 0.27 eV). From the group theory point of view, the group symmetry of

_{c3}*K*(or

*K`*) point is reduced from C

_{3v}to C

_{3}when the inversion symmetry is broken. And the degenerate irreducible representation

*E*will transform into two non-degenerate irreducible representations

*and*

^{1}E*. The insets in Fig. 2 illustrate the evolution of eigen-field*

^{2}E*E*distributions at

_{z}*K*point, where the

*E*concentrates in the nanodisk of lower (higher) chemical potential in band 1 (band 2).

_{z}To further investigate the band topology of graphene plasmonic crystals, we constructed a zigzag interface formed by two inversion symmetry broken plasmonic crystal structures with *μ _{c2}* = 0.6 eV,

*μ*= 0.33 and

_{c1}*μ*= 0.27 eV (shown on the top of Fig. 3(a)). A plane wave with the Dirac frequency of 49.65 THz is incident from the bottom indicated with red arrow. From the electric field intensity distribution shown in Fig. 3(a), one can see that the SPP waves are well-confined within the zigzag interfaces. Breaking the inversion symmetry of the graphene plasmonic crystal leads to the opening of the Dirac point forming a complete band gap where there exist topologically protected edge modes. To verify that, we calculated the dispersion relations of the edge modes (Fig. 3(b)) with a super-cell of finite period (

_{c3}*N*= 19) indicated by the green dashed lines (shown on the top of Fig. 3(a)). The shaded regions represent the projected band diagrams. The inset plots the representative eign-field intensity of the edge modes. One of the most distinguishing features of topological edge states is their robustness against perturbations. Such kind of topologically protected edge modes highly confined within the zigzag interfaces can transmit through sharp corners without scattering into bulk modes. Figure 3(c)-3(e) display the electric field intensity distributions of a plane wave with frequency of 48, 49 and 50 THz propagating through the N-shape channels, where we can see the SPP waves transmit along the N-shape channels constructed by the zigzag interfaces. The SPP waves vanish after a long traveling distance due to the intrinsic loss of graphene material. It should be noted that such kind of topological mode is different from the defective state constructed by introducing defects into the crystals. Unlike the edge states that propagate in the topological band gap and are protected against scattering, the defective states suffer strong backscattering though allowing propagation in the band gap.

Endowed with band topology effect, many interesting phenomena can be realized through zigzag interfaces constructed by the inversion symmetry broken graphene plasmonic crystal. Figure 4(a) illustrates schematic structures of multidirectional emission and localization mode. A point source with frequency of 49.65 THz located at the center of hexagon. One can see from the electric field intensity distribution shown in Fig. 4(b) that highly confined SPP waves propagate along the six zigzag interfaces formed by the six bulk inversion symmetry broken graphene plasmonic crystals without scattering into bulk crystals. Also, we constructed localization mode by connecting three zigzag interfaces shown in the right side of Fig. 4(a). A point source with frequency of 49.65 THz, located at one of the vertexes of triangle (indicated with red lines), is used to excite the localization mode. The corresponding electric field intensity distribution is shown in Fig. 4(c), where the SPP waves are confined within the triangular zigzag interfaces.

All the above discussed topological states are propagating along the internal edges between two inversion symmetry broken bulk graphene plasmonic crystals, we can also construct external boundary supporting edge modes. As shown on the top of Fig. 5(a), by modifying the chemical potential of the outmost array of graphene nanodisks, one can achieve topologically protected edge modes propagating along the exterior boundary. The electric field intensity distribution in Fig. 5(a) exhibits the edge states excited by a nearby point source (indicated with red fork) with Dirac frequency of 49.65 THz, where the SPP waves are confined within the zigzag edges and propagating towards both sides. Such kind of edge modes protected by band topology effect can even transmit through two edges with a sharp bend. As illustrated on the top of Fig. 5(b), a point source with Dirac frequency located at the vertex of the triangle (indicated with red fork) is utilized to excite the edge modes. As can be seen from electric field intensity distribution, the edge states simultaneously propagate along the two zigzag boundaries without scattering into “free space”.

## 4. Summary

In summary, we designed graphene plasmonic crystals constructed by periodically arranged graphene nanodisks surrounded by the same sheet of graphene with different chemical potentials. The band structure of the graphene plasmonic crystals was calculated. A complete band gap was obtained by breaking the inversion symmetry of the plasmonic crystals, which exhibits band topology effect. Further, we simulated the topologically protected edge states along N-shape channel and multi-channels in different directions constructed by the bulk inversion symmetry broken graphene plasmonic crystals. And we also constructed graphene plasmonic crystals supporting exterior boundary transmission. The simulation results reveal that such kind of topologically protected edge states propagating with no scattering into bulk modes are only subject to intrinsic loss of graphene material.

## Funding

Natural Science Fund of China (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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