Valley-dependent topologically protected elastic waves using continuous graphene membranes on patterned substrates

We present a novel structure for topologically protected propagation of mechanical waves in a continuous, elastic membrane using an analog of the quantum valley Hall effect. Our system involves a thin, continuous graphene monolayer lying on a pre-patterned substrate, and as such, it can be employed across multiple length scales ranging from the nano to macroscales. This enables it to support topologically-protected waves at frequencies that can be tuned from the kHz to GHz range by either selective pre-tensioning of the overlaying membrane, or by increasing the lattice parameter of the underlying substrate. We show through numerical simulations that this continuous system is robust against imperfections, is immune to backscattering losses, and supports topologically-protected wave propagation along all available paths and angles. We demonstrate the ability to support topologically-protected interface modes using monolayer graphene, which does not intrinsically support topologically non-trivial elastic waves.