Type-Ⅱ Dirac Points And Bilayer Couplings In Square Phononic Crystals

Phononic crystals,form by periodically arranged acoustic materials,have the similar band structure as electronic crystal,and thus can be used to control sound or vibration.Its geometric scale enables us to have precisely control on the geometric structures and the couplings between different units,so that various kinds of band structures for phononic crystals can be artificially designed.With the advantage,together with the topological theory in condensed matter physics,researchers have discovered and realized a series of acoustic topological states in phononic crystals.Phononic crystals with type-I Dirac points have been widely studied and the type-II ones have also recently been realized for acoustic waves in air.Compared with the airborne phononic crystals,elastic wave phononic crystals show better performances in anti-interference and thus have lower propagation loss.But the designs of elastic wave phononic crystals are more difficult in applying the tight-binding model.In this paper,we design two-dimensional phononic crystals in square lattice with type-II Dirac points.Type-II Dirac points,induced by the mirror symmetry of the phononic crystals,can slightly move along the boundary of the Brillouin zone,when slightly adjusting structural parameters.By tuning structures to break the mirror symmetry,the degeneracies of the type-II Dirac points can be lifted,leading to a band inversion.The phononic crystal before and after the band inversion belong to different topological valley phases,whose direct consequence is the existence of topologically protected gapless interface states between two distinct topological phases.Moreover,inspired by the similar stress field distributions in the interface and the free boundary,we find the boundaries of a single phononic crystal phase can similarly host the gapless boundary states.We further introduced twisted angle into a bilayer phononic crystal in square lattice.A special flat bands structure was found.Quasiperiodic or periodic lattice can be achieved with different twist angle.For any quasiperiodic structure caused by an arbitrary twist angle,we can approximate it with a periodic one with required degree of accuracy by using a Pythagorean triple.In this way we can studied the band structure of quasiperiodic phononic crystal and achieved acoustic local modes.The above works have expanded the investigations of valley related phononic crystals.The observed phenomena about acoustic transports enrich the methods of controlling acoustic waves and elastic waves,and,at the same time,pave the ways for the design of new acoustic topological devices.