拓樸絕緣體(Topological Insulator)是一種相當特別的材料，它和一般絕緣體都是無法在材料內部導電，但是在表面上拓樸絕緣體具有能夠導電的表面態，因此能了解表面態的特性。而近年來有文獻顯示[1]在靶材為銻化鉍(Bi2Te3)的拓樸絕緣體時，利用雷射光進行的激發-探測實驗(Pump-probe experiment)，穿射區可量測的到聲子振幅的變化，但在反射區的部分，厚度小於20nm幾乎量測不到聲子振幅的變化，因此我們想要了解若是能控制聲子激發後的傳播方向，是否就能量測的到被雷射光激發出的同調聲子。在本篇論文裡，我們先用簡諧近似及虎克定律推導出一維晶格的模型，再用克羅內克內積(Kronecker product)推出二維晶格的模型，把粒子與粒子間作用關係以線性耦合方程式來描繪，並以矩陣串聯在一起，描述晶格振動(Lattice vibration)的行為，接著算出各個模態的自然共振頻率(Nature resonance mode)；其次，我們研究一維單點激發與單方向傳播聲子的性質，以及考慮在不同頻率下傳遞聲子，找出臨界頻率(Critical frequency)，分析單方向傳播能量最高傾向的頻段，以及當外力激發以餘弦輸入，考慮不同相位下的變化；最後，以脈衝式函數做激發，以及在考慮系統具有損耗下，聲子單方向傳遞能量的性質。

The discovery of the property of topological insulator (TI) opens a new opportunity in condensed matter physics. Because the existence of surface state of the topological insulator material, excitation of surface phonon become possible. This enable us to investigate the propagation property of surface phonon. Recent experimental demonstration reported the inability to observe surface phonon by pump-probe setup of reflection type when the thickness of TI sample of antimony bismuth(Bi2Te3) is less than 20nm[1]. Yet later experimental evidence suggested the existence of phonon generation by pump-probe setup of transmission type[2]. We are motivated by the above experimental observation which suggests a possibility of phonon propagation in one direction.
In this thesis one dimensional lattice model of finite number of harmonic oscillators coupled by Hook's law is proposed to investigate the propagation property of phonon generated at the surface of sample. By property select the excitation frequency, oscillators away from the excitation position experience large amplitude, which demonstrate the unidirectional propagating property. The critical frequency to excite the unidirectional propagating property is identified to be within the first and second nature resonance frequencies. It was found that the critical frequency and bandwidth to excite unidirectional propagation wave is inversely proportion to the number of oscillators, which corresponds to the thickness of a TI sample. By numerical simulation, the unidirectional transport behavior is justified.
Furthermore, we investigate the oscillation and propagation property of finite harmonic oscillator array of two dimension by employing point excitation and line excitation. Point excitation analysis showed even lower critical frequency and bandwidth to have unidirectional propagating wave. Yet result of line excitation reveal similar spectral response to the one-dimensional case, suggesting the reducing in dimension which is applicable to three-dimensional problem using surface excitation.

目錄
中文摘要 i
英文摘要 ii
目錄 iv
圖目錄 vii
表目錄 x
第一章 緒論 1
1-1 前言 1
1-2 研究動機 2
1-3 論文架構 4
第二章 理論基礎與模型架構 5
2-1 晶格振動 5
2-1-1 拓樸絕緣體概念 5
2-1-2 晶體中聲子振盪行為 5
2-2 一維原子鏈模型 6
2-2-1 簡諧近似 6
2-2-2 虎克理論 8
2-3 二維原子鏈模型 11
2-3-1 克羅內克理論 11
2-4 模態自然共振頻率 13
第三章 一維單向傳播特性研究 19
3-1 頻域分析 19
3-1-1 模擬條件與參數 19
3-1-2 模擬結果與分析 19
3-2 單向傳播頻段分析 24
3-2-1 模擬條件與參數 24
3-2-2 模擬結果與分析 24
3-3 時域分析-弦波相位 29
3-3-1 模擬條件與參數 29
3-3-2 模擬結果與分析 29
3-4 時域分析-高斯函數激發 33
3-4-1 模擬條件與參數 33
3-4-2 模擬結果與分析 34
3-5 時域分析-傳播加入損耗 38
3-5-1 模擬條件與參數 38
3-5-2 模擬結果與分析 39
3-6 頻域分析-改變激發位置 43
3-6-1 模擬條件與參數 43
3-6-2 模擬結果與分析 43
第四章 二維單向傳播特性研究 47
4-1 二維晶格內單點激發的結果 47
4-1-1 模擬條件與參數 47
4-1-2 模擬結果與分析 48
4-2 二維晶格內多點激發的結果 52
4-2-1 模擬條件與參數 52
4-2-2 模擬結果與分析 52
第五章 現象與討論 56
第六章 結論與未來展望 58
6-1 結論 58
6-2 未來展望 59
參考文獻 60

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