拓樸絕緣體在近 20 年間受到相當大的矚目，凝聚態物理學由於拓樸學的導入，更開啟了科學史上嶄新且令人振奮的一頁。拓樸絕緣體主在研究電子、電磁波在固體材料中因「量子自旋霍爾效應」產生的拓樸相變，致使本來無法激發導態的絕緣體，在其邊界處產生受拓樸保護且高效率傳播的邊緣模態。
本文利用具有拓樸能帶結構的二維聲子晶體，來實現彈性體中的量子能谷霍爾效應，來達到固體材料中使彈性波在特定邊界傳遞的目的。首先建立聲子晶體的模型，並利用有限元素法計算其能帶結構。透過改變幾何尺寸來調整能帶結構，使色散曲線中的迪拉克錐有可以有不受其他能干擾、較好的控制環境。另外改變模型中的空間對稱性，會出現模態反轉與贋自旋的現象，也會得到因贋自旋方向不同而拓樸不等價的兩種聲子晶體。利用這兩種拓樸不等價的聲子晶體製造介面，並以超晶胞法分析其能帶結構，會在邊體關係圖中找出具有清晰邊緣能帶的頻段。為驗證邊緣模態的現象，利用全波模擬模型觀察單一頻率打入結構中，驗證邊緣模態具有高集中、低散射的波傳行為。
本研究得出，拓樸不等價的聲子晶體相連接時，其介面處皆會出現受拓樸保護的邊緣模態。而該邊緣模態具有抑制後向散射、高集中性與高效率穿透行為，具有應用於設計彈性波波導與集能器等裝置的潛力。

A two-dimensional hexagonal phononic crystal is proposed to exhibit the topologically protected elastic edge wave, which only occurs at the interface between two topologically inequivalent phononic crystals. The lattice is designed to imitate the atomic arrangement inside the graphene, in order to form the clear Dirac cones in the band structure. Then, we break the inversion symmetry of C6v into C3v by adjusting the radii’ scaling ratio α in the lattice. In the process of reducing symmetry, the degenerate Dirac cone at K-point splits thus a band gap opens. The sign of α determines modes of pseudo-spin up or down at Kpoints.
The additional bands appear in the band gap at the interface between two phononic crystals with different pseudo-spin modes. A full-wave simulation is conducted to observe these pass-band modes. These interface modes are proved to be the topologically protected edge waves, which are confined and can only propagate along the interface. The transmission spectrum shows great transmissivity of the excited edge mode, even as the wave encounter sharp corners. Because the topological-insulated phononic crystal has robust wave transmission property, we can use it to design highly-efficient acoustic devices such as wave splitters and energy harvesting devices.

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