Tunneling-induced fractal transmission in Valley Hall waveguides

TL;DR

This paper reveals that in Valley Hall waveguides with large band gaps, backscattering at sharp corners exhibits a fractal pattern due to resonant tunneling, challenging the smooth-envelope approximation assumptions.

Contribution

It demonstrates the emergence of fractal tunneling patterns and enhanced backscattering in Valley Hall waveguides with large band gaps and sharp domain walls.

Findings

Backscattering becomes significant for larger bulk band gaps.

Resonant tunneling induces a fractal pattern in tunneling energies.

Fractal patterns are reflected in the density of states and backscattering rates.

Abstract

The Valley Hall effect provides a popular route to engineer robust waveguides for bosonic excitations such a photons and phonons. The almost complete absence of backscattering in many experiments has its theoretical underpinning in a smooth-envelope approximation that neglects large momentum transfer and is accurate only for small bulk band gaps and/or smooth domain walls. For larger bulk band gaps and hard domain walls backscattering is expected to become significant. Here, we show that in this experimentally relevant regime, the reflection of a wave at a sharp corner becomes highly sensitive on the orientation of the outgoing waveguide relative to the underlying lattice. Enhanced backscattering can be understood as being triggered by resonant tunneling transitions in quasimomentum space. Tracking the resonant tunneling energies as a function of the waveguide orientation reveals a self-repeating fractal pattern that is also imprinted in the density of states and the backscattering rate at a sharp corner.