拓樸絕緣體為一種特殊的材料，其本身為絕緣體，但在結構的表面或介面處又能允許電流傳播。近年來許多學者將拓樸絕緣體透過量子霍爾效應、量子自旋霍爾效應以及量子能谷霍爾效應，將其理論類比至光子以及聲子晶體。藉由施加外部場，打破時間反演對稱；或是藉由改變晶格結構，破壞空間反演對稱，使能帶結構中的狄拉克錐張開並使結構發生拓樸相變，進而在介面處激發邊緣模態。本文則探討以正方形散射柱依正方晶格排列所組成之聲子晶體，首先探討散射柱與填充基材之間的填充比對能帶結構之影響，接著進一步旋轉散射柱破壞晶格的空間對稱以拉開狄拉克錐並將結構依散射柱旋轉方向分為兩種不同的結構，在其介面處將可激發邊緣模態，再透有限元素軟體進行全波模擬驗證其應用至彈性波導之可行性。其後再針對填入不同之填充基材，探討其對邊緣模態之影響。
為了進一步分析材料對邊緣模態之影響，引入具有膨脹性質的負蒲松比材料做為填充基材，發現相比於以一般材料做為填充基材的結構，能帶結構相對複雜且在其他頻段發現類似邊緣模態的模態。而透過全波模擬驗證發現其也同樣具有邊緣模態的性質。

In this thesis, we studied the acoustic analogue of quantum Hall valley effect with the phononic crystals composed of square rods in the square lattice arrangement. The topological transition occurs in the periodic system by breaking the space inversion symmetry which caused by rotating the rods. The topological edge states can be excited at the interface surface between two structures that are topologically different. In addition, the effects of filling ratios and the rotating angles on the band structure. The finite element commercial software COMSOL Multiphysics® is employed to simulate the propagation of the edge states and the transmission of the elastic wave propagation is shown.
Furthermore, we introduced the auxetic materials as the matrix of the phononic crystal and the influences of the negative Poisson’s ratio are observed. We found that there exists an edge-like state in the edge-bulk correspondence. It is an edge state because, from the full wave simulation, we observed that the state is immune to backscattering.

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