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研究生： 趙博文
Chiu, Pok-Man
論文名稱： 拓樸相的產生及其新穎的傳輸性質
Emergence of Topological Phases and Their Novel Transport Properties
指導教授： 李定國
Lee, Ting-Kuo
口試委員: 仲崇厚
Chung, Chung-Hou
牟中瑜
Mou, Chung-Yu
鄭弘泰
Jeng, Horng-Tay
張明哲
Chang, Ming-Che
學位類別： 博士
Doctor
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2019
畢業學年度： 107
語文別： 英文
論文頁數： 83
中文關鍵詞： 空間群對稱性 、拓樸半金屬 、電導率 、自旋-軌道耦合 、非傳統外爾半金屬 、表面態
外文關鍵詞： nonsymmorphic symmetry, topological semimetals, conductivity, spin-orbit coupling, unconventional Weyl semimetals, surface states
相關次數： 點閱：2  下載：0 
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    • 在凝聚態物理領域中拓樸與對稱是兩個重要的概念。它們共同作用決定了大部分的新穎傳輸現象，例如: 量子霍爾效應，量子反常霍爾效應，量子自旋霍爾效應以及它們的非量子化的對應效應。本論文提出一個方法可以產生不同種類的拓樸相，同時探討它們的奇異的傳輸性質。在引言一章，我們給出一個新的貝理相與同倫群之間的關係式。本論文的第一部分修改由「楊」與「肯」提出的具有空間群對稱性的正方形晶格模型。該模型可視為在空間對稱群晶格中最簡單的具有自旋-軌道耦合的二維狄拉克半金屬模型。在磁場的作用下，會產生二維外爾半金屬與陳絕緣體相。接著我們探討上述三種拓樸相的傳輸性質。拓樸半金屬有眾多的相，它們的費米面可以是點狀，線狀和閉合圈。在論文的的第二部分，透過垂直疊合二維的空間對稱群晶格，疊合過程中對稱性將會保持或破壞，我們構造出四類最簡單的三維模型。結果，從我們的簡單模型可以產生出四類不同的拓樸相，例如：狄拉克線半金屬，外爾線半金屬，非傳統外爾半金屬及弱拓樸絕緣體。意外地，無需對稱性保護的外爾圈半金屬也同時產生，其中鼓膜狀的表面態出現於圈之間。最後，我們探討它們的奇異傳輸性質。

    Topology and symmetry are the two important concepts in the field of condensed matter physics. They working together determine most of novel transport phenomena, e.g. quantum Hall effect, quantum anomalous Hall effect and quantum spin Hall effect, and their non-quantized counterpart. In this dissertation, we propose a method to generate various topological phases. In the meantime, we explore their novel transport properties. In the introduction chapter, we give a new relation between Berry phase and homotopy groups (a topological invariant in algebraic topology). In the first part of this dissertation, we adopt one of the nonsymmorphic square lattice models proposed by Young and Kane. This model can be viewed as a minimal model for two-dimensional (2D) Dirac semimetals with SOC in nonsymmorphic crystals. When applying exchange field, 2D Weyl semimetal and Chern insulator phases can be generated. Subsequently, we explore the transport properties of this three topological phases. Topological semimetals have a variety of phases, whose Fermi surfaces can be nodal points, nodal lines and nodal loops. In the second part of this dissertation, we construct four classes of 3D minimal models via vertically stacking a 2D nonsymmorphic lattice with and without breaking crystalline symmetries. As a result, four distinct topological phases can be generated in our minimal models, such as Dirac nodal line semimetals, Weyl nodal line semimetals, unconventional Weyl semimetals with topological charge $C=2$, and weak topological insulators. Unexpectedly, Weyl nodal loops are generated without mirror symmetry protection, where nontrivial "drumhead" surface states emerge within the loops. Lastly, we explore their novel transport properties.

    List of figures xi List of tables xvii 1 Introduction 1 1.1 Crystalline and global symmetries . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Topology: Ten symmetry classes of single-particle Hamiltonians and topological invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Topological materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Some experimental results of electronic transport in topological materials . 7 2 2D Dirac semimetal with SOC and its exchange field induced topological phases 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Tight-binding model of 2D Dirac semimetal with SOC . . . . . . . . . . . 12 2.3 Exchange field induced topological phases: 2D Weyl semimetals and Chern insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.1 2D Weyl semimetals . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.2 Chern insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Transport properties of 2D Dirac semimetals and their exchange field induced topological phases 21 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 AC longitudinal conductivity in 2D Dirac semimetals and Weyl semimetals 23 3.2.1 Theory of AC longitudinal conductivity in the clean limit: Kubo- Greenwood formula . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 2D Dirac semimetal . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.3 2D Weyl semimetal . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 DC longitudinal conductivity in 2D Dirac semimetals and Weyl semimetals 28 3.3.1 Theory of DC longitudinal conductivity . . . . . . . . . . . . . . . 28 3.3.2 2D Dirac semimetal . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.3 2D Weyl semimetal . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 AC and DC transverse conductivity in Chern insulators . . . . . . . . . . . 34 4 Four classes of 3D minimal models via vertically stacking a 2D nonsymmorphic lattice 37 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 3D minimal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.1 Dirac nodal line semimetals . . . . . . . . . . . . . . . . . . . . . 39 4.2.2 Weyl nodal line semimetals . . . . . . . . . . . . . . . . . . . . . 40 4.2.3 Weyl semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.4 Weak topological insulators . . . . . . . . . . . . . . . . . . . . . 42 4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Transport properties of the four classes of 3D minimal models 47 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Theory of AC and DC conductivity in clean limit: Kubo-Greenwood formula 48 5.3 AC and DC longitudinal conductivity in the four classes of 3D minimal models 51 5.3.1 Dirac nodal line semimetals . . . . . . . . . . . . . . . . . . . . . 51 5.3.2 Weyl nodal line semimetals . . . . . . . . . . . . . . . . . . . . . 53 5.3.3 Unconventional Weyl semimetals . . . . . . . . . . . . . . . . . . 53 5.3.4 Normal semimetals and weak topological insulators . . . . . . . . . 55 6 Conclusion 57 References 59 Appendix A Tight-binding Hamiltonian of 2D Dirac semimetal 65 A.1 Tight-binding approximation . . . . . . . . . . . . . . . . . . . . . . . . . 65 A.2 Derivation of each term in the Hamiltonian . . . . . . . . . . . . . . . . . 67 A.2.1 Derivation of the nearest-neighbor and next-nearest-neighbor hopping term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A.2.2 Derivation of the next-nearest-neighbor Rashba SOC and in-plane SOC 67 Appendix B Nonsymmorphic space group and its representation 69 Appendix C Berry-Zak phase, Chern number and numerical Wilson loop 73 Appendix D Linear response theory and Kubo-Greenwood formula for conductivity 77

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