拓樸能帶理論在光子晶體與電路系統之研究

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：61

、訪客IP：18.227.48.102

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)

相關論文

★ 平坦化陣列波導光柵分析和一維光子晶體研究	★ 光子晶體波導與藕合共振波導之研究
★ 光子晶體異常折射之研究	★ 光子晶體傳導帶與介電質柱波導之研究
★ 平面波展開法在光子晶體之應用	★ 偏平面光子晶體能帶之研究
★ 通道選擇濾波器之探討	★ 廣義光子晶體元件之研究與分析
★ 新式光子晶體波導濾波器之研究	★ 廣義非均向性介質的光傳播研究
★ 光子晶體耦合濾波器之研究	★ 聲子晶體傳導帶與週期性彈性柱波導之研究
★ 對稱與非對稱波導光柵之特性研究	★ 雙曲透鏡之研究
★ 電磁波與聲波隱形斗篷之研究	★ 次波長成像近場透鏡之研究

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

本電子論文使用權限為同意立即開放。
已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

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

參考文獻

[1] Klaus von Klitzing, G. Dorda, and M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance”, Phys. Rev. Lett. 45, 494 (1980).
[2] F. Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the Parity Anomaly”, Phys. Rev. Lett. 61, 18 (1988).
[3] C.L. Kane and E.J. Mele, “ Topological Order and the Quantum Spin Hall Effect”, Phys. Rev. Lett. 95, 246802 (2005).
[4] B. Adrei Bernevig and Shou-Cheng Zhang, “Quantum Spin Hall Effect”, Phys. Rev. Lett. 96, 106805 (2006).
[5] Rui Yu, Xiao Liang Qi, Andrei Bernevig, Zhong Fang, and Xi Dai, “Equivalent expression of topological invariant for band insulators using the non-Abelian Berry connection”, Phys. Rev. B 84, 075119 (2011).
[6] F. D. M. Haldane and S. Raghu, “Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry”, Phys. Rev. Lett. 100, 013904 (2008).
[7] Long-Hua Wu and Xiao Hu, “Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material”, Phys. Rev. Lett. 114, 223901 (2015).
[8] Jun Mei, Zeguo Chen and Ying Wu, “Pseudo-time-reversal symmetry and topological edge states in two-dimensional acoustic crystals”, Scientific Reports 6, 32752 (2016).
[9] Yuting Yang, Hua Jiang and Zhi Hong Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals”, Scientific Reports 8, 1588 (2018).
[10] Pai Wang, Ling Lu, and Katia Bertoldi, “Topological Phononic Crystals with One-Way
Elastic Edge Waves”, Phys. Rev. Lett. 115, 104302 (2015).
[11] Toshikaze Kariyado and Yasuhiro Hatsugai, “Manipulation of Dirac Cones in Mechanical
Graphene”, Scientific Reports 5, 18107 (2016).
[12] Ching Hua Lee, Stefan Imhof, Christian Berger, Florian Bayer, Johannes Brehm, Laurens
W. Molenkamp, Tobias Kiessling and Ronny Thomale, “Topolectrical Circuits”, Communications Physics 1, 39 (2018).
[13] Sunkai, “Topological insulators part III: tight-binding models”, Phys620 (2013).
[14] T. Thonhauser and David Vanderbilt, “Insulator/Chern-insulator transition in the Haldane
model”, Phys. Rev. B 74, 235111 (2006).
[15] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K. Geim, “The
Electronic properties of graphene”, Rev. Mod. Phys. 81, 109 (2009).
[16] Xiao-Liang Qi and Shou-Cheng Zhang, “Topological insulators and superconductors”,
Rev. Mod. Phys. 83, 1057 (2011).
[17] Shun-Qing Shen, “Topological Insulator-Dirac Equation in Condensed Matters”, (Springs
Series in Solid-State Sciences 187, 2012).
[18] Ming-Che Chang and Qian Niu, “Berry Phase, Hyperorbits, and the Hofstadter Spectrum”,
Phys. Rev. Lett. 75, 1348 (1995).
[19] János K. Asbóth,László Oroszlány,András Pályi Pályi, “A Short Course on Topological
Insulator”, (Springer, Lecture Note in Physics, 2016).
[20] You Wang, Li-Jun Lang, Ching Hua Lee, Baile Zhang and Y. D. Chong, “Topologically
enhanced harmonic generation in a nonlinear transmission line metamaterial”, Nature Communications 10, 1102 (2019).
[21] Feng Liu and Katsunori Wakabayashi, “Novel Topological Phase with a Zero Berry Curvature”, Phys. Rev. Lett. 118, 076803 (2017).
[22] Shuo Liu, Wenlong Gao, Qian Zhang, Shaojie Ma, Lei Zhang, Changxu Liu, Yuan Jiang
Xiang, Tie Jun Cui, and Shuang Zhang, “Topologically Protected Edge State in Two-Dimensional Su-Schrieffer-Heeger Circuit”, Research Article. 8609875 (2019).
[23] Nirmal J. Ghimire & Igor I. Mazin, “Topology and correlations on the kagome lattice”, Nature Materials 19, 137-138 (2020).
[24] Stephan Wong, Matthias Saba, Ortwin Hess, and Sang Soon Oh, “Gapless unidirectional
Photonic transport using all-dielectric kagome lattice”, Phys. Rev. Research 2, 012011 (2020).
[25] Maria Blanco de Paz, Chiara Devescovi, Geza Giedke, Juan Jose Saenz, Maia G. Vergniory, Barry Bradlyn, Dario Bercioux, and Aitzol Garcia-Etxarri, “Tutorial: Computing Topological Invariants in 2D Photonic Crystals”, ADVANCED QUANTUM TECHNOLOGIES 00117 (2019).
[26] John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, Robert D. Meade “Photonic Crystals: Molding the Flow of Light”, 2nd ed. (Princeton Univ. Press, 2008).
[27] COMSOL Multiphysics 5.3a.
[28] Marie S. Rider, Samuel J. Palmer, Simon R. Pocock, Xiaofei Xiao, Paloma Arroyo Huidobro, and Vincenzo Giannini, “A perspective on topological nanophotonics: Current status and future challenges”, J. Appl. Phys. 125. 120901 (2019).
[29] F. Haldane & S. Raghu, “Analogs of quantum-Hall-effect edge states in photonic crystals”, Phys. Rev. A 78, 033834 (2008).
[30] S.-s. Chern, Ann. Math. 47, 85 (1964).
[31] Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt, “Maximally localized Wannier function: Theory and applications”, Rev. Mod. Phys. 84, 1419 (2012).
[32] Hongming Weng, “Band Topology Theory and Topological Materials Prediction”, (2016)
[33] Hai-Xiao Wang, Guang-Yu Guo and Jian-Hua Jiang, “Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology”, New J. Phys. 21, 093029 (2019).
[34] Erhai Zhao, “Topological circuits of inductors and capacitors”, Annals of Physics. 10, 006 (2018).
[35] Arfken, G. “Mathematical Methods for Physicists”, 3rd ed. (Orlando, FL: Academic Press, 1985).
[36] O. Feely, “Chaos, cnn, memristors and beyond, Chap. Chua’s Lagrangian circuit elements, pp. 36-40.” (World Scientific 2013).
[37] Charles kittel, “Introduction to Solid State Physics”, 8th ed (Wiley, 2004).
[38] Alexey A. Soluyanov and David Vanderbilt, “Computing topological invariants without inversion symmetry”, Phys. Rev. B 83, 235401 (2011).
[39] Xiang Ni, Maxim A Gorlach, Andrea Alu and Alexander B Khanikaev, “Topological edge states in acostic Kagome lattice”, New J. Phys. 19, 055002 (2017).
[40] Wei-Min Deng, Xiao-Dong Chen, Wen-Jie Chen, Fu-Li Zhao, Jian-Wen Dong, “Vortex index indentification and unidirectional propagation in kagome photonic crystals”, Nanophotonics 8, 833-840 (2019).
[41] Xiao-Dong Chen, Fu-Li Zhao, Min Chen, and Jian-Wen Dong, “Valley-contrasting physics in all-dielectric photonic crystals: Orbital angular momentum and topological propagation”, Phys. Rev. B 96, 020202 (2017).
[42] Yuting Yang, Hua Jiang & Zhi Hong Hang, “Topological Valley Transport in Two-dimensional Honeycomb Photonic Crystals”, Scientific Reports 8, 1588 (2018).
[43] Pierre A. Pantaleon and Yang Xian, “Effects of Edge on-Site Potential in a Honeycomb Topological Magnon Insulator”, Journal of the Physical Society of Japan 87, 064005 (2018).
[44] 嚴忠波, “高階拓樸絕緣體與高階拓樸超導體簡介”, Acta Phys. Sin. Vol. 68, No. 22, 226101 (2019).
[45] Bi-Ye Xie, Hong-Fei Wang, Hai-Xiao Wang, Xue-Yi Zhu, Jian-Hua Jiang, Ming-Hui Lu, and Yan-Feng Chen, “Second-order photonic topological insulator with corner states”, Phys. Rev. B 98, 205147 (2018).

指導教授

欒丕綱(Pi-Gang Luan)

審核日期

2020-8-19

推文