Topological Properties Of Phononic Crystals

The study of the topological properties of classical waves in materials and metamaterials is an active area of research,inspired by quantum spin Hall effect and quantum valley Hall effect in electronic system.Recent years,topologically protected wave propagation in quantum systems has quickly extended to other classical systems of physics.The topological properties of these systems can be exploited to achieve unique and exciting functionalities,such as unidirectional transmission,backscattering suppression and defects of immune.We study the quantum spin Hall effect in sound waves and the quantum valley Hall effect in Lamb waves.The specific research is as follows:1.In chapter 2,we design a two-dimensional acoustic crystal（AC）to obtain topologically protected edge states for sound waves.The AC is composed of a triangular array of a complex unit cell consisting of two identical triangle-shaped steel rods arranged in air.The steel rods are placed on the vertices of the hexagonal unit cell so that the whole lattice possesses the₆symmetry.We show that by simply rotating all triangular rods around their respective centers by 180 degree,a topological phase transition can be achieved,and more importantly,such a transition is accomplished with no need of changing the fill ratios or changing the positions of the rods.By utilizing the spatial symmetry of the p and d states in the AC,we can construct the pseudo-time reversal symmetry and find pseudospin states in the interface between topologically trivial and nontrivial ACs,which are very similar to the real spin states of quantum spin Hall Effect in electronic systems.We also develop an effective Hamiltonian for the associated bands to characterize the topological properties of the AC around the Brillouin zone center with the help of the k?p perturbation method.We calculate the spin Chern numbers of the ACs,and reveal the inherent link between the band inversion and the topological phase transition.With full-wave simulations,we demonstrate the one-way propagation of sound waves along the interface between topologically distinct ACs,and demonstrate the robustness of the edge states against different types of defects including bends,cavity and disorder.Our design provides a new way to realize acoustic spin Hall Effect.Potential applications and acoustic devices based on our design are expected,where people can manipulate and transport sound waves in a more efficient way.2.In chapter 3,we design a phononic crystal thin plate（PCP）,the PCP is composed of a triangular array of hexagonal silicon pillars embedded in a rubber plate.The silicon pillars can rotate around its center,when the rotation angle is zero,the point group of point K in the first Brillouin zone is C_3v,and the energy band has a Dirac-cone of A₀ mode at point K and also has third band in the vicinity of this Dirac-cone.When the rotation angle isπ/6,inversion symmetry is broken and the symmetry of point K is reduced to C₃,and the Dirac-cone will be opened to form a band gap,valley pseudospin degree of freedom is formed around K and K’valleys for the A₀ Lamb mode,which is decoupled from the S₀ and SH₀ modes in the low frequency regime.Then we develop an effective Hamiltonian to describe the dispersion relation of A₀ mode and calculate spin Chern number,the Chern number of upper and lower band is±1/2,the Chern number of middle band is 0.Then we can obtain topologically valley-protected edge states.Chiral edge states are explicitly demonstrated,which are immune to defects and exhibit unidirectional transport behaviors when inter-valley scattering is weak.Quantum valley Hall effect is thus simulated in a simple way in the context of Lamb waves.