Analog Quantum Valley–Hall and Quantum Spin Hall Plasmons in Graphene Metasurfaces with Low Point Group Symmetry

In recent years, topological physics of classical waves in artificial crystals has become an emerging field of research. While Dirac cones and valley‐related physics are conventionally studied in these systems with C6v and C3v point‐group symmetries, here analog quantum valley Hall and quantum spin Hall plasmons in graphene metasurfaces with lower point‐group symmetries are explored. First, it is shown that a single‐layer graphene sheet with rectangle holes respecting to the C2v point group symmetry can host a mirror (σv) symmetry‐protected Dirac cone along the X–M edge of the Brillouin zone. Then we demonstrate that introducing further circular holes to the graphene sheet can break the mirror symmetry (i.e., reducing C2v symmetry to C1v) and thus gap out the Dirac cone, which allows us to explore the valley and layer‐pseudospin related topological plasmons in these graphene metasurfaces with low point group symmetry. Valley‐locking unidirectional propagations along the domain–wall interface of a single‐layer graphene metasurface and layer‐pseudospin converter in a double‐layer graphene metasurface are explicitly demonstrated for graphene plasmons in the THz range. This work provides a new design principle for exploring Dirac cones, valley, and pseudospin related physics using much lower point‐group symmetries.


Introduction
In recent years, motivated by the developments of topological phases in condensed matter physics, [1,2] topological photonics [3,4] DOI: 10.1002/apxr.202200076 has received a great deal of attention, mainly due to its unique topological properties that enable robust optical manipulation immune to defects and disorder. The field of topological photonics started with a theoretical proposal to simulate the quantum Hall edge states of light along an interface between two magneto-optical photonic crystals with different topological properties, which could potentially act as a unidirectional waveguide. [5,6] Subsequently, Wang et al. observed these unidirectional backscattering-immune topological electromagnetic states experimentally in the microwave regime. [7,8] Topological photonic systems in general could be divided into two classes: those that simulate the quantum Hall effect with broken time-reversal symmetry in magnetooptical photonic crystals and those that preserve the time-reversal symmetry in all-dielectric photonic crystals. The research of topological photonics has led to many interesting applications, such as unidirectional waveguides, [8,9] robust photonic delay lines, [10,11] topological lasers, [12,13] and reconfigurable topological switches. [14,15] The macroscopic properties of classical photonic systems coupled with the flexible and tunable symmetry of photonic crystals make them a good platform for studying topological phenomena [16,17] otherwise difficult to achieve in solid-state electronic materials. Topologically protected edge modes are particularly attractive in overcoming some of the limitations associated with the disorder in photonics.
Currently, graphene is an extremely powerful and promising platform for achieving tight optical field confinement beyond the diffraction limit of light due to its physical properties that can be easily controlled by physical and chemical methods, [18][19][20] which is one of the important requirements for the development of efficient nanoscale on-chip photonic integrated circuits. Based on this property, graphene plasmonic excitations exhibiting tight optical field confinement can be used as energy carriers for photonic systems in the terahertz and mid-infrared frequency ranges. [21][22][23][24][25] Periodically patterned graphene nanostructures in static magnetic fields can host topologically unidirectional edge plasmonic excitations at infrared frequencies due to the effect of time-reversal symmetry breaking. [26,27] The valley www.advancedsciencenews.com www.advphysicsres.com degree of freedom has also been shown to be a controllable degree of freedom and has attracted much scholarly attention in 2D layer structures, [28][29][30][31][32] which can easily migrate into classical systems by breaking mirror or inversion symmetry, [33][34][35] thus providing another efficient method for realizing topological edge transport of classical waves. [36][37][38][39][40] Topological valley transport based on graphene plasmonic excitations has been studied using periodically arranged graphene nanodiscs in a honeycomb lattice with inversion-symmetry breaking. [41,42] Moreover, large-area periodic graphene nanopore arrays [43,44] have been developed where the band structure of graphene plasmons can be easily tailored in a purely geometrical manner compared to the periodically aligned graphene nanodisk crystals. [45] Recently, You et al. have studied valley Hall topological plasmons in a graphene nanohole plasmonic crystal waveguide that can host the topological unidirectional edge plasmonic propagations in the absence of a magnetic field. [46] On the other hand, degenerate Bloch modes induced by crystalline symmetries have also been proposed for the realization of pseudospins for polarized light [47] as well as scalar sound, [48] and most recently, bilayer systems have been employed to study both valley [49] and layer-based pseudospins. [50][51][52] However, to the best of our knowledge, most of the works in the literature up to now on Dirac cones, valley and pseudospin related physics are based on C 6v and C 3v point group symmetries. The possibility of realizing these physics in systems with much lower point group symmetry remains largely unexplored.
In this work, we propose analog quantum valley Hall and quantum spin Hall plasmons in graphene metasurfaces consisting of periodically patterned nanohole arrays with low point group symmetry, which are different from, [53][54][55] where no real graphene materials are considered but only that the systems are in graphene-like crystal lattice, that is, hexagonal lattice. Here, we would like to clarify that all quantum Hall related effects mentioned in this work are not the real quantum effects of electronic systems in condensed matter physics, but rather refer to the analog effects in photonic systems, specifically, the analog quantum valley Hall and analog quantum spin Hall effects in the context of topological photonics. [3,4] To achieve this, we first show that graphene plasmonic crystals whose unit cell contains a rectangular hole with C 2v symmetry etched onto a single-layer graphene sheet can host a mirror symmetry protected Dirac cone along the X-M edge of the Brillouin zone. To break the mirror symmetry that protects the Dirac cone and thus to open a bandgap, a further circular hole could be introduced to the graphene sheet. Note that the C 2v point group contains two mirror symmetries, that is, v (the xz plane) and d (the yz plane), and it is the v symmetry that protects the Dirac cone. To demonstrate this, we introduce two circular holes that break the d symmetry but respect the v symmetry to the unit cell (i.e., the two circular holes are located at the same side of the rectangular hole) and show that a bandgap cannot be opened in this case. To distinguish clearly the difference of the two mirror symmetries and to use the fact that the size of the bandgap opened around the Dirac cone is proportional to the strength of the symmetry breaking, in this work, we mainly focus on the two-circular-hole configurations that break the v mirror symmetry but respect the d (i.e., the two circular holes are located at the different sides of the rectangular hole). Indeed, we find that when a single circular hole with a radius of 30nm is used, the size of the bandgap opened around the Dirac cone is about 0.38 THz, whereas two same circular holes respecting d but breaking the v symmetry lead to a bandgap with the size around 0.74 THz. Taking advantage of this wide topological bandgap, a domain-wall interface separating two graphene plasmonic crystals in a mirror-symmetric manner is constructed, and valley Hall plasmonic edge modes within the topological bandgap are created. Through full-wave simulations, the valley-locking unidirectional propagations of valley Hall plasmons along this well-designed interface are demonstrated.
In addition, analog quantum spin Hall plasmons based on layer-pseudospin in a bilayer graphene metasurface are further investigated. Especially, an interlayer converter that flips the layer pseudospin during unidirectional propagations across a heterojunction with three regions in each layer is explicitly demonstrated, verifying that rich physics could be explored in such graphene plasmonic crystals with low point group symmetry. While Dirac points and related topological effects are mostly studied in systems with high point group symmetries, such as C 3v , C 4v , and C 6v , the lower point group symmetries of C 2v and C 1v studied in this paper greatly extend the regimes where Dirac points can exist and could promote the further investigations of low-symmetry-protected Dirac points and their possible practical application not only in the field of topological photonics, but also in other research fields, such as solid-state systems. Thus, in terms of physics research, our work provides a new route to explore Dirac points, valley and pseudospin related physics by exploiting low-symmetry point groups, which has not been explored to the best of our knowledge. Regarding to practical applications, the different ways to break the symmetry of the systems make it very convenient to design domain-wall interfaces for the unidirectional propagation of light, and moreover, the structures are very simple and easier for fabrications due to the reduced constraints imposed by symmetries, which could be an advantage for the design of more compact photonic devices. The new design principle based on much lower point group symmetries is quite general and could be applied to other classical waves as well as intelligent metasurface devices. [56,57]

The System Design
The main aim of the current work is to demonstrate concepts, such as Dirac cones, valley-Hall and layer-pseudospin edge modes could be realized in graphene plasmonic crystals with much lower point group symmetries than those conventionally used in previous works. Figure 1a shows the schematic of a domain-wall interface (highlighted by the light green area with x-axis marked by the green dashed line) that supports valley-Hall plasmonic modes between two graphene plasmonic crystals D1 and D2, whose unit cells are highlighted by the red and blue squares, respectively. As can be seen, domains D1 and D2 are mirror symmetric to each other with respect to the x-axis and thus their band diagrams are the same, that is, they share a common bandgap in the band diagrams. We will demonstrate below that this arrangement will lead to the emergence of valleylocking plasmonic modes along the designed domain-wall interface. The unit cell and first Brillouin zone of the graphene plasmonic crystals D1 and D2 are shown in Figure 1b,c, respectively. Especially, the unit cell consists of a main rectangular hole and two circular holes distributed at different sides (either top Figure 1. a) Schematic of a domain-wall interface supporting valley-Hall plasmonic modes between two graphene plasmonic crystals D1 and D2 with C 1v point group symmetry in the xy plane. The domain-wall interface is highlighted by the green area, whereas the unit cells of D1 and D2 by the red and blue squares, respectively. b) Unit cell of the graphene plasmonic crystals, which consists of a main rectangle hole (pink) and two circular holes (i.e., either the top two blue ones or the bottom two green ones) in a single-layer graphene sheet (grey region). The lattice constant, radius of the circular holes, width, and height of the rectangle hole are, a, r, W, and H, respectively. c) First Brillouin zone of the graphene plasmonic crystals, where the high symmetry points are labeled as Γ, X, and M, respectively.
or bottom) of the rectangular hole, where the rectangular hole in the graphene sheet is used to create a mirror-symmetry protected Dirac cone, whereas the two circular holes serve as mirrorsymmetry-breaking perturbations to open a bandgap around the Dirac cone.
In this work, we use Comsol Multiphysics 5.6 to perform all simulations, in which the lattice constant, radius of the circular holes, width and height of the rectangular holes are set as a = 500 nm, r = 30 nm, W = 180 nm, and H = 400 nm, respectively. We remark that while the rectangular hole is located at the center (0,0) of the unit cell, the centers of the circular holes are at (±(a+w)/4,±a/4).

Results and Discussions
In this section, we present the main results of our work. In specific, we first study the band diagrams of different graphene plasmonic crystals to illustrate the mechanism for the existence of Dirac point and the symmetry-breaking perturbations needed to gap out the Dirac point. Based on the graphene plasmonic crystals with a topological valley gap, we construct a domainwall interface and demonstrate the existence of valley-Hall plasmonic modes along the designed interface via calculating the projected band diagram and simulating the valley-locking unidirectional propagations. Finally, by combining the valley-Hall plasmonic modes at different domain-wall interfaces into a bilayer graphene plasmonic crystal, layer-pseudospin and interlayer converters that can flip the layer pseudospin during unidirectional propagations are explicitly demonstrated.

Band Diagrams of Different Graphene Plasmonic Crystals
We first demonstrate that Dirac point could be obtained in the band diagrams of graphene plasmonic crystals with low point group symmetry of C 2v . To show this, a graphene plasmonic crystal whose unit cell contains a rectangular hole with C 2v symmetry in a single-layer graphene sheet is considered and its band diagram is presented in Figure 2a. As can be seen from the result, a Dirac point labeled by the red dot exists along the X-M edge of the Brillouin edge, and the intensity and phase distributions of the electric field Ez in the graphene plane for the two degenerate frequencies at the Dirac point (A and B) are given in Figure 2b.
To illustrate the symmetry element of the C 2v point group that protects the Dirac point, we write the coordinate of the Dirac point in momentum space as ( /a,ky). The C 2v point group contains four symmetry operations, E (identity), C 2 (180°rotation), and two mirror symmetries, that is, v (the xz plane) and d (the yz plane). As d maps /a to − /a, which essentially are the same point in the Brillouin zone, the existence of this Dirac point can be attributed to the v mirror symmetry.
To demonstrate that the Dirac point in Figure 2a is indeed v protected, we introduce a further circular hole to the unit cell of Figure 2a and present the band diagram of this new graphene plasmonic crystal in Figure 3a. As the circular hole obviously breaks the v mirror symmetry, the Dirac point is gapped out as expected. The intensity and phase distributions of the electric field Ez for the two band edges of the valley gap A and B are given in Figure 3b, from which one can see that the phase distributions of the A and B modes host vortex centers with opposite winding directions. However, the introduced circular hole also breaks the d mirror symmetry. To exclude the possible protection mechanism based on d mirror symmetry, we introduce two circular holes to the unit cell of Figure 2a, which explicitly break the d mirror symmetry but respect the v mirror symmetry. The corresponding band diagram is presented in Figure 3c, from which one can see that the Dirac point restores when compared to the result in Figure 3a (see also the similarity between the intensity and phase distributions in Figures 2a and 3c). Thus by comparing the band diagrams in Figure 3a,c, which involve different symmetrybreaking holes, that is, the gapping out of the Dirac point in   Figure 3c with two holes that break only the d symmetry, we unambiguously demonstrate that the Dirac point in this system is indeed protected by the v mirror symmetry of the C 2v point group.
To clearly distinguish the v and d mirror symmetry breakings and also to enlarge the valley gap, in the following we will focus on graphene plasmonic crystals where the two circular holes are distributed at different sides (i.e., both left and right) of the rectangular hole, located at either the top (see Figure 4a) or at the bottom (see Figure 4c). Compared to the valley gap with a size of 0.38 THz obtained by a single circular hole, the valley gap with two circular holes has a size of around 0.74 THz, that is, almost doubled due to the fact that the bandgap opened around the Dirac point is roughly proportional to the strength of the symmetry breaking.
While the band diagrams of Figure 4a,c are the same because their unit cells are the same under inversion, their phase distributions show vortex centers with opposite windings for both the A and B band edge modes (see Figure 4b,d), which are consistent with previous results [46] and signal different topological properties of the two unit cells. The effect of the circular hole radius on the two band edge frequencies A and B is further investigated and presented in Figure 4e, from which one can see that with the increase of the circular hole radius, while the frequency of the upper band edge A changes slowly, the frequency of the lower band edge B decreases dramatically, reaching a bandgap of about 3 THz at radius r = 60 nm. We would like to note that while the results in Figures 3 and 4 may look similar, they convey very different messages, which are crucial for illustrating the different aspects of the rich physics involved, that is, while Figure 3 mainly unravels the symmetry protection mechanism of the Dirac points, Figure 4 describes the two different ways to gap out the Dirac points that lead to unit cells and plasmonic crystals with different topological properties, which are indispensable for constructing domain-wall interfaces hosting topological interface modes. In the next section, we will show that a domain-wall inter-face between two graphene plasmonic crystals, whose unit cells are taken from Figures 4a,c, respectively, can host valley-locking plasmonic modes around the corresponding domain-wall interface.

Projected Band Diagram of Valley-Hall Domain-Wall Interface Modes
The two different ways to gap out the Dirac point, that is, choosing the two v -symmetry breaking circular holes either at the top (Figure 4a) or at the bottom (Figure 4c) of the unit cell, and the resulting different topological properties of the valley modes at A and B (see Figure 4b,d), provide a convenient way to construct valley-Hall interface modes at the domain-wall interface between two graphene plasmonic crystals built from the unit cell of Figures 4a,b, respectively. The Hamiltonian describing the valley gap can be written as [52] where f D is the Dirac frequency when the v symmetry is preserved, v D is the Dirac velocity (i.e., slope of the Dirac cone), ⇀ k = ( k x , k y ) is the momentum measured from the Dirac point,̂i are the Pauli matrices acting on the two valley bands, and is the valley gap opening coefficient. We would like to note that there are two different ways to construct the domain-wall interface, as illustrated in Figure 5a, where for the domain-wall interface A (i.e., D1/D2), the two sets of circular holes are adjacent to the interface. In contrast, for domain-wall interface B (i.e., D2/D1), the two sets of circular holes are away from the interface.
The projected band diagrams corresponding to the two different domain-wall interfaces in Figure 5a are presented in Figure 5b, from which one can see the emergence of two different branches of interface modes (marked by the red and blue lines) within the topological bandgap. These two branches of interface modes exhibit the typical feature of the valley-momentum locking effect, that is, around one valley at k x = 0.5 /a, the group velocity of the modes is opposite to that of the modes around the other valley at k x = 1.5 /a. For example, for the red branch modes corresponding to domain-wall interface A, around k x = 0.5 /a, the group velocity of the modes is negative (i.e., propagating to the -x direction) whereas the group velocity of the modes is positive (i.e., propagating to the +x direction) at around k x = 1.5 /a. For the blue branch modes corresponding to domain-wall interface B, the valley-momentum locking effect is opposite to that of domain-wall interface A, that is, the k x = 0.5 /a valley is locked to the positive group velocity, whereas the valley at k x = 0.5 /a is locked to the negative group velocity. This is reasonable as the two domain-wall interfaces are constructed from D1/D2 and D2/D1, respectively, thus their topological properties will also be opposite to each other. Furthermore, as the valley is a local effect in the Brillouin zone, the resulting valley-Hall interface modes are not connected to the bulk states above or below the bandgap (i.e., the grey regions in Figure 5b).
The intensity distributions of the two branches of valley-Hall interface modes at ① and ② are shown in Figure 5c. And as can be seen from the results, the mode associated with domainwall interface A is more confined to the interface than the mode at domain-wall interface B. Furthermore, while the intensity of mode ① is mostly distributed in the graphene regions between the shorter sides of the rectangular holes, the intensity of mode ② is mainly distributed in the graphene regions between the longer sides of the rectangular holes. Consequently, the plasmonic hot spots associated with mode ① are more localized than those of mode ②.

Valley-Locking Unidirectional Propagations of Valley-Hall Interface Modes
To illustrate the valley-locking unidirectional propagation feature of the domain-wall interface modes shown in Figure 5b, we choose the blue branch modes as an example (similar analysis here also applies to the red branch modes). For the blue branch modes associated with the domain-wall interface B, the modes around the valley at k x = 0.5 /a have a positive group velocity and thus propagate to the +x direction. On the contrary, the modes around the valley at k x = 1.5 /a have a negative group velocity and thus propagate to the −x direction. In the full-wave simulations, we use a chiral source that can emit either left-or rightcircularly-polarized THz waves to excite the valley-Hall interface modes, which is achieved by placing six electric dipoles at the corners of a hexagon. The phase difference between adjacent electric dipoles is set to be − /3 or + /3, which will launch either a leftor right-circularly-polarized wave.
As shown in Figure 6a, when a left-circularly-polarized source at 11.37 THz is used (with the location of a source at the center of the circular arrow and 25 nm above the graphene metasurface), the valley-Hall interface mode with a positive group velocity is launched and propagate unidirectionally to the +x direction. Similarly, if we switch the phase difference between the adjacent electric dipoles to be + /3, a right-circularly-polarized source will be realized and as a result, the exited valley-Hall interface mode will propagate to the −x direction as Figure 6b shows. Therefore, through full-wave simulations, we have demonstrated that the domain-wall interface modes created in graphene plasmonic crystals with much lower point group symmetry of C 1v indeed exhibit the topological feature of valley-locking unidirectional propagations.

Projected Band Diagram of a Bilayer Graphene Metasurface with Layer-Pseudospin
Apart from the analog quantum valley-Hall edge states, analog quantum spin-Hall edge states are another important class of topological edge states [47] and recently, simulating pseudospin using layer degree of freedom in bilayer systems has emerged as a new method to achieve pseudospin-polarized propagations and applications. [50][51][52] A simple strategy to construct layer-based pseudospin is to put two different kinds of domain-wall interfaces that support valley-Hall interface modes on top of each other. As the valley modes associated with the two different domain-wall interfaces usually have different symmetry properties and intensity distributions, the mode coupling between the two domain-wall interfaces at intermediate separation in general is weak, leading to layer-polarized modes. The key distinction between the valley-Hall edge modes in a single-layer graphene plasmonic crystal and the pseudospin-Hall edge modes in a doublelayer graphene plasmonic crystal is that their associated degrees of freedom and locations in momentum space are different, that is, while the pair of valley-Hall edge modes that propagate oppositely to each other are associated with K/K' at different locations of the momentum space, those of the pseudospin-Hall edge modes are associated with top/bottom-layer at the same location of the momentum space. Graphene plasmonic crystals are ideal to construct such kind of layer-pseudospin due to their one-atomthick nature of a graphene sheet when compared to their more bulky counterparts [50,52] and to the best of our knowledge, layer pseudospin-based analog quantum spin-Hall plasmonic modes in bilayer graphene metasurfaces have not been explored in the literature.
Here, we would like to demonstrate that the graphene plasmonic crystals with low point group symmetry we proposed could not only be used to create valley-Hall interface modes in a single-layer graphene metasurface as studied above, but could also be used to construct layer pseudospin-based analog quantum spin-Hall plasmonic modes in bilayer graphene metasurface. The Hamiltonian for the spin-Hall physics based on the layer pseudospin could be written asĤ = f D + D ( k xŝ0̂x + k yŝ0̂y ) + s 0̂z + wŝ x̂0 , where w describes the plasmonic near field  coupling between the two graphene layers whereasŝ i are the Pauli matrices acting on the two graphene layers. To create layer pseudospin, we put two graphene metasurfaces supporting two different domain-wall interfaces as shown in Figure 5a on top of each other with a separation of h = 300 nm, whose supercell is shown in Figure 7a. The projected band diagram of this bilayer graphene metasurface is shown in Figure 7b, from which one can see the emergence of a pair of interface modes (the red lines) within the bandgap. These interface modes exhibit similar features to the conventional quantum spin-Hall edge state, that is, around each valley at k x = 0.5 /a and 1.5 /a, there are two layer-polarized modes at each frequency. To demonstrate the layer-polarized nature of the interface modes in Figure 7b, we choose two modes labeled as ① and ② in Figure 7b and plot their intensity distributions of the electric field Ez in Figure 7c. The results show that while the field for mode ① is mostly distributed to the upper layer of the bilayer metasurface, the field for mode ② is mainly distributed in the lower layer of it. The weak coupling between modes ① and ② could also be understood from their field distributions in the upper and lower layers, that is, while the field distributions associated with domain-wall interface A are mostly distributed in the graphene regions between the shorter sides of the rectangular holes, those with domain-wall interface B are mostly distributed in the graphene regions between the longer sides of the rectangular holes, which thus have litter overlap. The layer-polarized nature of these interface modes will be useful for practical applications as will be demonstrated in the next section.

Layer-Polarized Propagation and Layer-Polarization Conversion
To show possible applications of the layer-polarized modes in the bilayer graphene metasurface discussed above, we present a layer-pseudospin converter as proposed in. [50] The schematic design is shown in Figure 8a, where each layer contains six domains as marked by ①,②,③,④,⑤, and ⑥, whose unit cells are also given. The domain-wall interface is between domains ①④, ②⑤ and ③⑥, which is also the wave propagation direction. For an excitation , which demonstrate the layer-polarized features of these modes, that is, for mode ① Ez is mainly distributed in the upper layer, whereas for mode ②, it is mainly distributed in the lower layer.
source indicated by the red star in Figure 8a, which is an electric dipole source located at the middle of the left edge and 25 nm directly above the upper layer of the bilayer metasurface, the excited wave will propagate to the right and along the propagation direction, the structure could be viewed as a three-region (①④, ②⑤and③⑥) heterojunction. In both the left and right regions ①④ and ③⑥, the domain-wall interfaces in the upper and lower layers are different, resulting in layer-polarized modes. In the middle region ②⑤, the domain-wall interfaces in the upper and lower layers are the same, resulting in layer-mixed modes (i.e., the field will distribute equally in the upper and lower layers), which serve as a transition region for the layer-pseudospin converter. Full-wave simulation result of the plasmonic wave propagation in this bilayer metasurface is shown in Figure 8b, where the field distributions in the upper and lower layers are shown in the left and right panels, respectively. From the results, one can see that the wave propagation shows both features of layerpolarized propagations and layer-polarization conversion. In the region ①④, the wave propagation shows layer-polarized feature as the domain-wall interfaces in the upper and lower layers of the bilayer metasurface are different and the wave only propagates in the upper layer. In the middle transition region ②⑤, the wave is equally distributed in the upper and lower layers as the domainwall interfaces in the two layers are the same. And in the right region ③⑥, the wave switches completely to the lower layer as the domain-wall interfaces in the upper and lower layers are reversed as compared to those of region ①④. In general, the layer polarization is preserved if no intermediate transition region (such as ②⑤) exists in the bilayer systems. These full-wave simulation results demonstrate that the graphene plasmonic crystals with much lower point group symmetry, as we proposed in this work, can also be exploited for constructing layer pseudospin-based propagations and applications.

Conclusion
In conclusion, we have proposed a new design principle for exploring Dirac cones, valley and pseudospin related domain-wall interface modes in graphene plasmonic crystals based on a much lower point group summary of C 2v and C 1v . In specific, we first showed that a graphene plasmonic crystal whose unit cell has a rectangular hole in the middle of a graphene sheet respecting the C 2v symmetry could host a Dirac cone along the X-M edge of the Brillouin zone. We next studied the effects of different kinds of symmetry-breaking perturbations introduced to the unit cell, such as circular holes respecting or breaking the v and d mirror symmetry, and demonstrated that this Dirac cone is protected by the v mirror symmetry of the C 2v point group. To distinguish the v and d mirror symmetry breaking explicitly and also to enlarge the opened bandgap around the Dirac cone, we proposed to use C 1v symmetric unit cells containing two circular holes located at different sides of the rectangular hole for constructing two different graphene plasmonic crystals, that is, D1 and D2 with the two circular holes located at the bottom and top of unit cell, respectively. Valley-Hall interface modes at two different kinds of domain wall interfaces, that is, D1/D2 and D2/D1, were successfully created within the topological valley gap and their topological features, such as valley-locking unidirectional propagations, were demonstrated through full-wave simulations. The two different kinds of domain-wall interfaces also allow one to implement layer pseudospin-based interface modes and applications. To demonstrate this, we have calculated the projected band diagram of a bilayer graphene metasurface and showed the layer-polarized nature of the interface modes. Explicit applications of such layer pseudospin-based interface modes, such as layer-polarized propagation and layer-polarization conversion, were also demonstrated through full-wave simulations.
Our work not only has the potential to realize more compact photonic devices based on topological plasmonic waves in graphene metasurfaces, but the new design principle based on much lower point group symmetries, such as C 2v and C 1v , is quite general and could also be applied to other classical waves, such as acoustic wave.