第一原理密度泛函理論，是材料領域研究學者一個強大的理論模擬基礎。基於該理論發展出的演算法，僅需要事先知道材料的晶格結構以及原子種類，完全不需要其他實驗數據，就能夠預測出該材料種種電子的量子行為。現今，在第一原理模擬以及實驗的雙方合作下，科學家們對電子在材料內拓樸行為的認識有著飛速的成長。
本論文利用第一原理密度泛函理論，從原子級的尺度探討Mn2Bi2Te5的電子能帶，並進一步預測其拓樸性質，以及拓樸相變。該材料在一個單位晶胞內具有兩個帶磁矩的錳原子(Mn)，並且晶胞內的Bi與Te原子共同提供非平凡的拓樸性質。目前Mn2Bi2Te5的基態具有何種磁性結構還有待後續實驗進一步驗證。但先前，科學家在DFT (density functional theory) 計算下 [1]預測了當Mn2Bi2Te5的兩個Mn原子為鐵磁性偶合，並且其磁矩平行該材料的堆疊方向的話，材料的最低能量態會是外爾半金屬(Weyl semimetal)。我接著進一步利用改變晶格長度來模擬材料在受到應變的情況下，拓樸性質會有何改變。計算結果發現當晶格長度變長(模擬受到tensile strain)，Mn2Bi2Te5的拓樸性質會消失，從外爾半金屬，相變為不帶有拓樸性質的一般鐵磁性絕緣體。但當晶格長度收縮時(模擬受到compressive strain)，Mn2Bi2Te5會發生拓樸相變進入軸子絕緣體的拓樸相。
一般的絕緣體若具有特定的對稱性，會使得軸子偶合項 (axion coupling term) θ 被量子化至0 或 π。而當 θ=π 時，且時間返演對稱被破壞的情況下，該材料被稱作軸子絕緣體。軸子絕緣體因為並不具有時間返演對稱並且擁有量子化的 θ=π，該材料表面存在半整數的量子霍爾電導。而該電導的存在將使得材料內部的電場與磁場偶合在一起，即為，當對該材料外加一個電場時，將會導致材料內部產生一個平行於電場的磁矩。且反之也是成立，當外加一磁場時，將導致平行方向電偶極矩的產生。這種特殊的電、磁場響應被稱作為磁電效應(magnetoelectric effect)。軸子絕緣體的磁電效應其實原自於其表面上的半整數霍爾電導。軸子絕緣體由於電子在半整數電導的傳導下，其傳導方向與電子的自旋會互相偶合，並且又具有特殊的磁電響應，使得軸子絕緣體在將來的自旋電子元件上有著不小的應用潛力。因此發現合適的軸子絕緣體候選材料，是一個重要的課題。

The advent of the density functional theory offers a efficient way to predict the electronic structure of the material with only the knowledge of structure of unit cell and the type of the atoms within such unit cell. Therefore, it is a very powerful tool to simulate the material before the experiment to cost down the experiment cost, or explaining the experimental result after one found the weird things in the experiment. To date, the cooperation of DFT simulation and the material experiment have made a large progress in the knowledge of topological material, which possess many of the exotic quantum phenomenon that have the application potential of spintronic device in the future.
In this thesis, I exploit the DFT package VASP [2] to simulate and predict a novel topological material called axion insulator. The material Mn2Bi2Te5, possesses non-zero magnetic moment on the two Mn atoms, and the atoms Bi and Te together offer the non-trivial topological property. The topological property is strongly corelated with the symmetry of material, therefore, the magnetic configuration can affect the topology of material largely. The true magnetic configuration of Mn2Bi2Te5 have not been defined yet, it still needs more experiment to identify. But in the ref [1], by the DFT calculation, they predicted that when the magnetic moment of two Mn atoms are ferromagnetic coupled and are parallel to the stacking direction, the Mn2Bi2Te5 is in the Weyl semimetal phase. I further simulate the Mn2Bi2Te5 under the strain with several magnitude. From the results, I found that Mn2Bi2Te5 is topologically trivial insulator under the tensile strain, but is axion insulator under the compressive strain. The topological phase transition tuned by applying strain and the new axion insulator candidate is the main topic of this thesis.

目錄
摘要 2
Abstract 3
致謝 4
目錄 5
1. Introduction of calculation techniques 7
1.1 Hohenberg -Kohn theorem 7
1.2 Kohn-Sham equation 9
1.3 Local density approximation(LDA) 11
2. Calculation methods for topological properties 13
2.1 Wannier function & Wannier Charge Center(WCC) 13
2.1.1 Tight-Binding basis 13
2.1.2 Wannier90 14
2.1.3 Wannier Charge Center(WCC) 15
2.2 Chern number 16
2.2.1 Berry connection & Berry curvature 16
2.2.2 Chern number in term of Berry curvature 18
2.2.3 Relation of Chern number and WCC 19
2.2.4 Layer resolved Chern number 21
2.3 Z4 symmetry indicator 34
2.4 Iterative Green`s function for calculating surface density of state 42
3. Introduction of basic topological material 46
3.1 Integer quantum Hall effect(IQHE) 46
3.1.1 Some experimental results 46
3.1.2 The extistence of edge state 47
3.1.3 Connecting to the topology 50
3.2 Quantum anomalous Hall effect (QAHE) 54
3.3 Axion insulator 55
3.3.1 Some knowledge about 3D Strong topological insulator 55
3.3.2 Axion insulator related properties 57
4. Results and Discussion 60
4.1 Introduction 60
4.2 Weyl semimetal phase 62
4.3 The other phases controlled by strain 67
4.3.1 The Z4 symmetry indicator 71
4.3.2 The topologically trivial (-4% strain) case 73
4.3.3 The topologically non-trivial (+2% strain) case 75
4.3.4 The distribution of Chern number in the real space 77
5. Summary 81
6. Reference 82

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