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Tailoring of topological states via boundary selection

Chen WeiQiu's team realized the tailoring of topological states via boundary selection
(a, b) Schematics and parameters of the sonic crystal. (c, d) The band structures of the sonic crystal with r=0 (c) and r=0.1/-0.1 (d), respectively. Credit: ©Science China Press
The wave in a topological state can propagate only along the structural boundary or interface. In the case of the valley Hall effect, the topological acoustic waves can propagate in both the directions. The selection of different boundaries along with appropriately designed interfaces provides the acoustic waves in the band gap range with abilities of one-way propagation, dual-channel propagation, immunity from back-scattering at sharp corners, and/or transition between propagation at interfaces and boundaries.

In the paper, a C6v-symmetric structure is thoroughly studied in which the valley Hall effect plays an important role when the Dirac cone at the corner point of the Brillouin region is opened, accompanied by a new band gap, by changing the symmetry of the structure from C6v into C3v. Further, the researchers analyzed the acoustic wave propagation in two types of topological states in the range of this new band gap. The first type appears at the top or bottom boundary of the single structure, whereas the second type is confined to the interface between the two structures exhibiting different topological properties.

As depicted in Figures 1(a) and 1(b), the sonic crystal is composed of a triangular-lattice array. The side lengths of two adjacent regular triangles connected by rectangular waveguides are d1 and d2, respectively and d1/d2=(1-r)/(1+r). Figures 1(c) and 1(d) provide the dispersion relations of the sonic crystal along the typical boundaries of the Brillouin zone when r=0 and r=0.1/-0.1, respectively. In Figure 1(c), double degeneracy occurs, which is located at the K-point due to the C6v symmetry of the structure. In constrast to Figure 1(c), a band gap appears in Figure 1(d) by breaking the C6v symmetry into the C3v symmetry. Therefore, an acoustic topological insulator based on the mechanism of the valley Hall effect is constructed. Both the two types of topological states in the range of this new band gap can propagate in both the directions—i.e., dual-channel propagation.

In Figure 2(a), the three boundaries are all selected to be Type 8 (please refer to the paper for the classification of the boundaries), and the researchers apply a point source at the middle point of the top boundary. Then the acoustic wave propagates in both the directions along the boundaries as shown in the figure. Although the edge state has the characteristic of dual-channel propagation, one-way propagation can be achieved by tailoring the boundaries as shown in Figure2(b). Further, the output position can be adjusted arbitrarily by changing the types of the boundaries as shown in Figures 2(c) and 2(d). In Figure 3, combining the edge and interface states together, the acoustic wave propagation path can be programmed in various ways.

Chen WeiQiu's team realized the tailoring of topological states via boundary selection
Simulations of acoustic wave propagation in a triangular structure (r=0.1) with different boundaries as indicated in (a)-(d). Credit: ©Science China Press
Chen WeiQiu's team realized the tailoring of topological states via boundary selection
Simulation of acoustic wave propagation along several designed paths. Credit: ©Science China Press
In Figures 3(c-e), the researchers only change the selections of the two parts of the top boundary to control the propagation behavior of acoustic waves. The acoustic wave first propagates from bulk to the boundary of the composite structure, and then propagates on the top boundary to the right, to the left or in both the directions, according to the boundary types in Figures 3(c-e). The structure in Figure 3(f) is the same as that in Figure 3(e), but the point source is set at the left side of the top boundary. The acoustic wave first propagates along the top boundary of the structure with r=-0.1. When the wave meets the interface between r=0.1 and r=-0.1, there yields a bifurcation in the wave propagation. In the superposed composite structure, the boundary types only affect the existence of the edge state and have no influence on the interface state.

By combining the edge state and the interface state, the acoustic wave propagation path in the band gap can be tailored in more flexible, diverse, and intriguing ways. It provides a new concept for the design of tunable acoustic devices.

More information:
Jiao Wang et al, Tailoring edge and interface states in topological metastructures exhibiting the acoustic valley Hall effect, Science China Physics, Mechanics & Astronomy (2019). DOI: 10.1007/s11433-019-9601-6

Provided by Science China Press