Trapped Modes and Steered Dirac Cones in Platonic Crystals

TL;DR

This paper explores how Dirac cone topologies in platonic crystals can be manipulated by changing lattice aspect ratios, affecting wave propagation and trapped modes in thin plates with periodic pinning.

Contribution

It demonstrates the steering of Dirac cones along symmetry lines in the Brillouin zone by adjusting lattice aspect ratios, revealing new control over wave behavior in platonic crystals.

Findings

Dirac cones can be moved along symmetry lines by varying lattice aspect ratios.

The band surfaces tilt as the Dirac cones are steered.

The shape of the Dirac cone neighborhood influences trapped mode propagation.

Abstract

This paper discusses the properties of flexural waves obeying the biharmonic equation, propagating in a thin plate pinned at doubly-periodic sets of points. The emphases are on the properties of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates with a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that the Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.