拓樸能帶理論在光子晶體與電路系統之研究

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姓名

陳翔駿(Hsiang-Chun Chen) 查詢紙本館藏

畢業系所

照明與顯示科技研究所

論文名稱

拓樸能帶理論在光子晶體與電路系統之研究
(The Research of the Topological Band Theory Applying to Photonic Crystals and Circuit Systems)

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摘要(中)

本論文參考經典的 Su-Schrieffer-Heeger (SSH) 模型，探討在一維與二維緊束縛模型 (tight binding model) 系統以及光子晶體 (photonic crystal) 中，藉由改變晶胞內的跳躍振幅 (intracell hopping amplitude ) 與相鄰晶胞間的躍遷振幅 (intercell hopping amplitude) 之比例而導致的拓樸相變 (topological phase transition)。此外，我們也分析此拓樸相變中的拓樸不變量 (topological invariant) 與對應的拓樸邊緣態 (topological edge state)。
第一章主要是介紹石墨烯蜂窩晶格 (honeycomb lattice) 的能帶特色，知道可以利用K跟K’有不同方向的假自旋 (pseudospin) 產生量子能谷霍爾效應 (quantum valley Hall effect)。第二章則把一維和二維的 SSH 模型應用在電路系統上，計算出它們的哈密頓量 (Hamiltonian) 以便求出能帶結構 (band structure)。第三章是說明COMSOL Multiphysics 商用模擬軟體如何計算能帶結構，並介紹用來判斷拓樸性質 (topological properties) 的拓樸不變量 (topological invariants)。第四章通過數值計算出電路系統的能帶結構與捲繞數 (winding number) 和用超晶胞法 (supercell method) 模擬出可果美晶格 (Kagome lattice) 光子晶體 (photonic crystals) 的拓樸邊緣態 (topological edge states)。之後，改變SSH 模型的邊界位能，會出現不同於拓樸邊緣態的邊緣態，這是一種稱為 "Tamm mode" 的缺陷態 (defect states)。
在電路系統的 SSH 模型得知，增加兩邊邊界的當地位能 (on-site potential)，會把非平庸拓樸 (nontrivial topology) 變成平庸拓樸 (trivial topology) 的拓樸性質，等於我們只藉著增加邊界的位能，就能使其發生拓樸相變，最後在光子晶體系統中，也能通過改變邊界位能，造成邊緣態的改變。

摘要(英)

In this thesis, we treat the Su-Schrieffer-Heeger (SSH) model as a porotype, and study the topological phase transition occurring in some one-dimensional (1D) and two-dimensional (2D) tight binding models and photonic crystals. The topological phase transition is due to changing the ratio between the intracell and intercell hopping amplitude in the models. We also study the topological invariants in the topological phase transition and analyze the properties of the topologically protected edge states.
In the first chapter we mainly study the characteristics of the band structure of the honeycomb lattice system, and discuss how to use the two different pseudospins at the K and K’ point of the Brillion zone to generate the quantum valley Hall effect. In the second chapter we mimic the 1D and 2D SSH models to define the corresponding circuit systems, formulate their Hamiltonians, and then calculate the band structures. The third chapter explains how to use the commercial simulation software COMSOL Multiphysics to calculate the band structures, and introduce the topological invariants used to analyze the topological properties. In chapter four, we first numerically calculate the band structures and winding numbers of the circuit systems. We then use the supercell method to simulate the topological edge states of the photonic crystal defining by the periodically arranged dielectric rods located on the Kagome lattice. We also study the influence of the boundary potentials in the SSH model. We find that by adding the boundary potentials, new edge state emerges, which is different from the topological edge state. This might be the defect state named " Tamm mode".
In the SSH model of circuit systems, just by adding the on-site potentials on the two boundaries, the topological property of the system turns from nontrivial into trivial. This indicates that we can have a topological phase transition only by increasing the on-site potential at the boundary. Corresponding to this, in the photonic crystal systems, the edge states can also be changed by changing the “on-site potential” at the boundaries.

關鍵字(中)

★ 拓樸絕緣體
★ 拓樸光子學
★ 拓樸電路學
★ 拓樸邊緣態
★ 拓樸相變
★ 札克相位

關鍵字(英)

★ Topological Insulator
★ Topological Photonics
★ Topological Circuits
★ Topological edge state
★ Topological phase transition
★ Zak phase

論文目次

摘要....I
Abstract........II
致謝....III
目錄....IV
圖目錄.. VI
第一章緒論......1
1-1 前言........1
1-2 緊束縛 (tight binding) 模型計算石墨烯 (graphene) 能帶..3
1-3 石墨烯的狄拉克錐與谷..6
1-4 布洛赫定理 (Bloch Theorem) 和貝瑞相 (Berry phase).....9
第二章研究模型與理論.....12
2-1 一維拓樸絕緣體模型—The Su-Schrieffer-Heeger(SSH) Model ...12
2-2 一維SSH電路系統模型...13
2-3 二維 SSH 模型........14
2-4 二維SSH電路系統模型...16
2-5 二維可果美晶格 (Kagome lattice) 光子晶體......18
三、研究方法.....21
3-1 光子頻帶結構 (photonic band structures)......21
3-1-1 以COMSOL計算二維可果美 (kagome) 光子晶體頻帶結
構...23
3-2 拓樸不變量...25
3-2-1 瓦尼爾函數與札克相 (Zak phase) 的關係........27
3-2-2 威爾森迴圈方法計算札克相.....31
四、研究結果與討論........33
4-1 一維拓樸模型的數值分析........33
4-1-1 以傳遞矩陣法求解SSH模型的能帶結構..33
4-1-2 以拉格朗日(Lagrangian)方程計算SSH電路模型..37
4-1-3 電感並聯SSH電路能帶圖與札克相......40
4-2 二維拓樸模型的數值分析........41
4-2-1 二維SSH模型能帶圖與拓樸不變量......41
4-2-2 二維SSH電路模型能帶圖與拓樸不變量..43
4-2-3 二維可果美晶格光子晶體的能帶結構與漩渦指數 (votex
index)..46
4-3 拓樸模型的拓樸邊緣態..52
4-3-1 二維可果美光子晶體的拓樸邊緣態.......52
4-3-2 改變邊界位能的拓樸邊緣態.... 54
4-3-3 改變SSH電路系統的邊界電感的拓樸邊緣態...57
4-3-4 增加左右兩邊的邊界位能，造成邊緣態改變的原因...62
4-3-5 大能隙的可果美光子晶體.......62
五、結論與未來展望........66
5-1 結論........66
5-2 未來展望.....67
參考文獻...68

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指導教授

欒丕綱(Pi-Gang Luan)

審核日期

2020-8-19

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