拓樸絕緣體為一種特殊材料，內部不導電，但在結構表面或界面處可允許能量通過，並且透過量子霍爾效應可類比至聲子晶體。藉由改變結構對稱性或施加外部場、打破空間或時間反演對稱，使能帶結構圖中的雙狄拉克點被拉開並產生拓樸相變，創造出兩種拓樸不等價的材料，便可利用其界面處產生的邊緣模態控制波的傳遞方向。本文設計一基材中間嵌入空氣散射柱組成單位晶胞，接著改變其結構的對稱性，同時引入量子能谷霍爾效應，研究聲波的波傳現象。首先，建立結構模型，利用有限元素法軟體COMSOL Multiphysics® 計算出其能帶結構，並設計材料性質與結構參數找出雙狄拉克點，並討論改變結構參數對於雙狄拉克錐的影響。接著改變散射柱旋轉的角度拉開雙狄拉克點並討論能帶反轉現象。利用超晶胞法分析兩種拓樸不等價聲子晶體所組成的界面，並於邊體關係圖尋找能量集中於界面處之邊緣模態。最後，於結構中激發於邊緣模態頻率範圍之聲波，並使用直線與Z字型的彎角路徑搭配缺陷、亂序，藉由比較幾種波傳路徑，驗證邊緣模態對於抑制後向散射及缺陷免疫的穩健性，並使用穿透頻譜圖確認其高傳輸效率之性質。同時也結合量子自旋霍爾效應，使用不同的偽自旋波源搭配四通道之結構，透過控制波源的偽自旋方向來決定波傳方向。

In the present study, we design a topological insulator with quantum valley Hall effect, which is proposed to exhibit the topologically protected edge mode in acoustic wave. The unit cell of the topologically phononic crystal is composed of six rectangular scattering pillars. We verify that the topological phase transition occurs around the double Dirac cone and present the topological phase diagram as a function of the rotation angle of the meta-atoms. Once the coupling between adjacent metamolecules is sufficiently strong, the mode inversion of topological states emerges. We design straight and Z-shaped paths and add two kinds of defects and disorders to prove its strong wave propagation property. All simulation results in this thesis are obtained by the finite element analysis software COMSOL Multiphysics®. It is also shown that robust pseudospin-dependent one-way transmission is immune to defects in the topological phononic crystals, which can be applied to acoustic wave transmissions and communications. Our approach in acoustic systems provides a strategy to explore topological states in two-dimensional systems. The robustness of the edge mode for limiting the backscattering and defect immunity is verified.

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