博碩士論文 103226051 詳細資訊




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姓名 蔡雅雯(Ya-Wen Tsai)  查詢紙本館藏   畢業系所 光電科學與工程學系
論文名稱 彈性波系統的拓樸相位與拓樸邊緣態之研究
(The Research of the Topological Phase and Topological Edge State in the Elastic system)
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摘要(中) 本論文主要討論低維彈性系統的拓樸相變 (topological phase transition) 與拓樸邊緣態 (topological edge states) 之研究。第一部份主要透過連續調變一維週期繩波系統的單位繩段長度以及繩線密度使繩波系統產生拓樸相變,再藉由數值方法來計算系統的札克相位 (Zak phase) 來探討拓樸相並透過系統波阻抗的邊界條件來預測界面態的存在,最後實際建構一條由兩種不同拓樸相所構成的繩波週期系統來驗證界面態的存在。第二部份是利用COMSOL Multiphysics模擬軟體中的弱形式方程 (weak form) 模組來進行拓樸純剪波邊緣態的研究。此處的二維週期彈性波系統是由釔鐵石榴石 (Yttrium iron garnet, 簡稱 YIG) 以三角晶格方式排列於鎢 (Tungsten, 化學符號為 W) 背景中所構成的聲子晶體。藉由純剪波的頻帶結構圖找到一個狄拉克簡併點 (Dirac point) 後,再透過加入與純剪波振動方向平行的外加磁場來破壞系統的時間反演對稱性 (time-reversal symmetry),使頻帶結構中的的狄拉克點處開啟一個拓樸非平庸的頻隙,最後透過超晶胞方法 (supercell method) 驗證拓樸純剪波邊緣態的存在,也討論一些未來可進一步探討的方向。
摘要(英) In this thesis, we discuss the topological phase transition and topological pure shear modes in low-dimension elastic systems. First part, we investigate the topological phase transition by varying the length and the linear densities of the segment strings in one unit string. We can distinguish the topological phase of the periodic string system by calculating the Zak phases and predict the existence of the interface state by the boundary condition of the interface impedances in the string system. At the last, we build a string system which consists of two different topological phases of periodic strings at the interface to verify the existence of the interface state numerically. Second part, we use the weak form module in the COMSOL Multiphysics software to show that the topological pure shear modes can exist in a two dimensional triangle lattice phononic crystal which is composed of yttrium iron garnet (YIG) rods embedded in the tungsten (W) background. In the absence of external magnetic field, the shear wave band structure shows the existence of Dirac cones that are well-known in graphene. As a uniform magnetic field is applied to the system, due to the time-reversal symmetry breaking arising from the magnetoelastic interactions in YIG, the Dirac points are spitted to topologically non-trivial band gaps. The nontrivial topological nature is verified by the numerical calculation of Chern numbers for each band, and simulation of the propagation of the edge states. A possible design for realization of such a system is discussed.
關鍵字(中) ★ 拓樸絕緣體
★ 拓樸邊緣態
★ 拓樸界面態
★ 拓樸相變
★ 繩波
★ 彈性波
★ 聲子晶體
★ 貝里相位
★ 札克相位
★ 陳數
關鍵字(英) ★ Topological Insulator
★ Topological edge state
★ Topological interface state
★ Topological phase transition
★ String wave
★ Elastic wave
★ Phononic crystal
★ Berry phase
★ Zak phase
★ Chern number
論文目次 摘要 I
Abstract II
致謝 III
目次 IV
圖目錄 V
表目錄 VI
第一章 緒論 1
1-1拓樸絕緣體的崛起 1
1-2貝里相 (Berry phase) 與札克相 (Zak phase) 3
1-3 頻帶理論與拓樸絕緣體 4
1-4 本文架構 6
第二章 研究模型 7
2-1 拓樸絕緣體模型 7
2-2一維拓樸絕緣體模型─彈性繩波版本 7
2-2-1週期彈性繩波系統中的拓樸相變 8
2-2-2 札克相 (Zak phase) 10
2-2-3 界面態 (Interface State) 11
2-3 二維拓樸絕緣體模型─彈性波版本 12
2-3-1 彈性波系統中的拓樸純剪波邊緣態 13
2-3-2 貝里相 (Berry phase) 與陳數 (Chern number) 15
2-3-3 邊緣態 (Edge state) 16
第三章 研究方法 17
3-1一維拓樸模型研究方法 17
3-1-1 傳遞矩陣法 17
3-2 二維拓樸模型研究方法 25
3-2-1弱形式方程的有限元素法 25
3-3 拓樸相位的數值分析 30
3-3-1 札克相 (Zak phase) 與貝里相 (Berry phase) 30
3-2-2陳數(Chern number) 30
第四章 研究結果與討論 31
4-1一維彈性繩波拓樸特性研究結果與討論 31
4-1-1 頻帶結構 31
4-1-2 拓樸界面態 36
4-2二維彈性波拓樸特性研究結果與討論 38
4-2-1 頻帶結構 38
4-2-2拓樸邊緣態 (Edge state)與陳數 (Chern number) 40
第五章 結論與未來展望 45
5-1 結論 45
5-2 未來展望 47
參考文獻 48
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指導教授 欒丕綱(Pi-Gang Luan) 審核日期 2018-1-25
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