Quantum valley Hall effect-based topological boundaries for frequency-dependent and -independent mode energy profiles

Topological artificial crystals can exhibit one-way wave-propagation along the boundary with the wave being localized perpendicular to the boundary. The control of localization of such topological wave propagation is of great importance for enhancing coupling or avoiding unwanted coupling among neighboring boundaries toward topological integrated circuits. However, the effect of the geometry of topological boundaries on localization properties is not yet fully clear. Here, we experimentally and numerically demonstrate valley-topological transport on representative valley-topological boundaries with micro-electro-mechanical systems. We show that the zigzag and bridge boundaries, which have highly efficient wave transport, exhibit frequency independent and dependent wave localization, respectively. A simple analytic model is presented to capture the different behaviors of the two boundaries observed in the experiments. Our results provide opportunities to engineer frequency responses in topological circuits including frequency selective couplers through proper selection of boundary geometries. The authors numerically and experimentally investigate the transport properties of a quantum valley-Hall effect in a micro electromechanical system. The zigzag and bridge boundaries, which have highly efficient wave transport, exhibit frequency independent and dependent wave localization, respectively.