The Dirac-like Point In Two-dimension Phononic Crystal

Dirac points, where two bands cross over each other linearly, have been studied inclassical systems. A lot of interesting phenomena have been discovered, which imply potentialapplications in diverse fields. In this Master thesis, we have studied the properties ofanisotropic Dirac points and semi-Dirac points in two-dimensional(2D) phononic crystals(PCs).These 2D PCs consisting of a triangular array of square-shaped cylinders embedded inwater host. We utilize the different spatial symmetry between the crystal lattice and thescatters to create the corresponding Dirac points.(1) We found that the linear dispersions at the low-symmetry points in the Brillouin zone(BZ) of the PCs can form a tilted and anisotropic cone, which means not only that the bandslopes can be very different along different k directions, but also that for the same k direction,the band slopes above and below the degenerate frequency can be different. The anisotropicbehaviors in the reciprocal space can be understood in terms of the effective Hamiltonian,which is constructed from the Bloch eigenstates at that degenerate point. In addition, thelinear slopes of the anisotropic Dirac dispersions along any direction can be accuratelypredicted by the k.p method. Based on the peculiar anisotropic band structures, someinteresting acoustic wave transmission phenomena are investigated.(2) Using different side length of the iron cylinders, a new 2D PCs is designed, in whichthe accidental degeneracy of the eigenstates is utilized to realize a semi-Dirac point at the BZcenter. In the vicinity of the semi-Dirac point, the dispersion relation is linear along theGY direction but quadratic along the GX direction. Rotating the iron cylinders around theiraxis by 45 oand slightly tuning the side length of the cylinders, a new semi-Dirac point canbe realized at the BZ center, where the dispersion relation is quadratic along the GY directionbut linear along the GX direction. A k.p perturbation method is used to investigate thispeculiar dispersion relation and study how these semi-Dirac points are formed. Based on theperturbation method, effective Hamiltonians can be constructed around these semi-Diracpoints, so that Berry phase can be calculated and is found to be zero. Furthermore, acousticwave transmission through corresponding PCs is given, and switch-like behavior of thetransmittance is observed along different directions. The work presented here is helpful to theunderstanding of semi-Dirac point in phononic crystals and suggests possible application indiverse fields.