Evolution of the edge states and corner states in a multilayer honeycomb valley-Hall topological metamaterial

The valley-Hall effect provides topological protection to a broad class of defects in valley-Hall photonic topological metamaterials. Unveiling precisely how such protection is achieved and its implications in practical implementations is paramount to move from fundamental science to applications. To this end, we investigate a honeycomb valley-Hall topological metamaterial and monitor the evolution of the topological valley-Hall edge states and higher-order corner states under different perturbation $\delta R$. The evolutions of the edge states of the armchair and zigzag interfaces are demonstrated, respectively. By adjusting the geometric parameters and introducing disturbances to break the inversion symmetry, we achieve the edge states with different modes including the conventional crossed edge state and the specific gapped edge state. It is found that the edge states of topological valley kinking will gradually separate with the increase of $\delta R$, and finally a complete gap between the edge states appears. The gap has rarely been reported previously in topological materials fabricated by printed circuit board technology. In addition, the higher-order topological corner states can also be observed in the proposed topological metamaterial. The higher-order topological phase is theoretically characterized by nontrivial bulk polarization and the Wannier centers. Our results show that the corner state localization becomes stronger with the increase of $\delta R$. It is expected that our results will provide a platform for the realization of optical topological insulators.

Figure 1

(a) Schematic illustration of the valley lattice structure consisting of three layers. Metallic patterns are shown (indicated by yellow) on a dielectric F4B substrate (indicated by dark gray) with a thickness of $t=0.508\mathrm{mm}$. (b) Top view of the unit cell. Geometry parameters are shown as $a=16\mathrm{mm}, l=2\mathrm{mm}, l_1=0.3\mathrm{mm}$, and $l_2=7\mathrm{mm}$. The radii of the two kinds of metallic circular patch are $R_1$ and $R_2$, respectively. The top right corner inset shows the Brillouin zone. (c) Band diagram of the structure with $R_1=R_2=1.75\mathrm{mm}$. (d) Eigenfrequencies of the $K/K^{\prime }$ point and TE modes as a function of $\delta R\left(\delta R=R_1-R_2,R_1+R_2=3.5\mathrm{mm}\right)$. $\delta R<0$ and $\delta R>0$ correspond to structure $A$ and structure $B$, respectively. The inversion of frequency indicates a topological phase transition as $\delta R$ crosses zero.

Figure 2

(a) Calculated Berry curvature of structure $B$ along the line $k_x=k_y$. (b) A rhombic Brillouin zone in the calculation of the bulk polarization, which shares the same area with the original hexagonal Brillouin zone. The green dotted line indicates $k_x=k_y$ in the reciprocal space. (c, d) Calculated Berry phase of (c) $\delta R=1.5$ mm and (d) $\delta R=–1.5\mathrm{mm}$. Bulk polarization is obtained by the integration of ${\theta }_{\alpha ,k_{\beta }}$ over $k_y$. Red dashed lines show the exact values of $1/3$ and $-1/3$. The corresponding Wannier centers are marked by the blue dots and green dots, respectively.

Figure 3

Schematic diagram of inversion-symmetry-broken honeycomb lattices with (a) zigzag and (c) armchair domain walls. The black solid line regions indicate the domain walls. Two valley topological structures are marked by $A$ and $B$. (b, d) Corresponding band structures of the (b) zigzag and (d) armchair edge domain walls with different $\delta R$ ($\delta R=0.5$, 1.5, 2.5, and $3.5\mathrm{mm}$). Red and green solid lines show the edge states of $AB$-type and $BA$-type domain walls, respectively. The shaded regions represent the projection of the bulk bands.

Figure 4

(a) Simulated transmission of the zigzag domain walls with different $\delta R$. (b) Top view of the fabricated samples of the $AB$-type and $BA$-type domain wall with $\delta R=3.5\mathrm{mm}$. In order to clearly show the proposed sample, the right panel shows three layers of lattice structure (top layer, middle layer, and bottom layer). (c) Measured and simulated transmission for the $AB$-type and $BA$-type zigzag domain walls.

Figure 5

Simulated electric field distributions of the edge states for the interfaces of (a) $AB$-type zigzag domain wall and (b) $BA$-type zigzag domain wall for $\delta R=3.5\mathrm{mm}$. The excitation source is depicted by the white dots. Simulated electric field distributions of (c) cavity defects and (d) disorder defects.

Figure 6

Schematic views of the zigzag edge triangular corner structures with (a) structure $A$ surrounded by structure $B$ and (b) structure $B$ surrounded by structure $A$. Wannier centers are illustrated by the green and blue dots. (c) and (d) Eigenfrequencies as functions of the parameter $\delta R$ of the structure shown in Figs. 6 and 6. The light pink shaded regions, the light blue shaded regions, and the red solid lines represent the bulk states, the edge states, and the corner states, respectively. (e) and (f) Corner state electric field distributions corresponding to the black dots of Figs. 6 and 6, respectively. The upper left inset illustration is a schematic of the whole structure.

Figure 7

Eigenmodes of the structure with (a) $\delta R=1.5\mathrm{mm}$ and (b) $\delta R=3.5\mathrm{mm}$ of the triangular corner structures in Fig. 6. Corner, edge, and bulk states are represented by red, green, and blue dots, respectively. (c) Electric field distributions of the three degenerate corner states corresponding to $L, M, N$, and $O$ points. (d) Top view of the fabricated triangular corner structures with $\delta R=1.5\mathrm{mm}$ (left panel) and $\delta R=3.5\mathrm{mm}$ (right panel). White dots indicate the location of the excitation source placed near the corner, edge, and bulk. (e, f) Measured normalized transmissions of (e) $\delta R=1.5\mathrm{mm}$ and (f) $\delta R=3.5\mathrm{mm}$. (g) Comparative normalized transmission spectra of the corner states with $\delta R=1.5$ and $3.5\mathrm{mm}$.