石墨烯是近十幾年來最重要的材料之一，它廣為人之的物理特性被許多人看好於不同領域的應用，但在2018年後，石墨烯被發現居然有超導的特性，人們從沒想過將兩層石墨烯堆疊並且旋轉到一些特定的角度，石墨烯居然會出現超導相變而變成超導體。這個超導機制雖然現在還不是完全明瞭，但卻讓我們對於超導的機制有了一些不同的啟發，此外也暗示了凝態物理之後發展的可能方向-旋轉電子學(twistronics)的理論與應用。
本篇論文主要從石墨烯慢慢開始，先討論單層再討論雙層，接著拓展到旋轉雙層甚至多層(三層以上)石墨烯，我觀察到旋轉雙層石墨烯在小於1.5°時將會出現平帶(flat band)，而真正出現超導的角度則為1.16°, 1.12°, 1.08°, 以及1.05°,這些角度有個特殊的名稱-magic angle，而旋轉三層石墨烯或是類似的系統也有發現屬於它們自己的magic angle。
我 將 利 用 第 一 原 理(first principle)和 緊 束 縛 模 型(tight binding model)分析這些系統的能帶，其中一些結果是複現已經發表的論文，還有一些是我在我的模型中加上額外的效應，例如考慮垂直加壓和施加水平應力的情況，我觀察這會對能帶產生特殊的影響，並且將層數拓展也會出現很多有趣的現象，它們在小角度依然會有平帶的產生，也有拓樸，甚至在最近的實驗中也有發現磁性。總之，這樣的系統有許多有趣的物理，而本篇論文將著重在電子結構、能帶的計算。

Graphene is one of the most important materials in recent decades, known for its remarkable physical properties with widespread applications in various fields. However, after 2018, graphene was unexpectedly found to exhibit superconducting properties.
It was previously unimaginable that by stacking two layers of graphene and rotating them to specific angles, graphene would undergo a superconducting phase transition, turning into a superconductor. Although the exact mechanism behind this superconductivity is not yet fully understood, it has provided novel insights into the mechanisms of superconductivity. Moreover, it has hinted at potential directions for future developments in condensed matter physics, particularly in the field of twistronics, both in theory and applications.

This master’s thesis begins with a comprehensive exploration of graphene, first discussing single-layer graphene and then delving into bilayer graphene. It further extends to twisted bilayer graphene and even multilayer systems (tri-layers and above). Notably, I observed that twisted bilayer graphene exhibits a flat band when the twist angle is less than 1.5°. The actual angles leading to superconductivity are 1.16°, 1.12°, 1.08°, and 1.05°, which are referred to as ”magic angles”. Similar ”magic angles” have also been discovered in twisted tri-layer graphene and related systems.

To analyze the electronic band structures of these systems, I employ first-principles calculations and tight-binding models. Some of my results replicate from previously published papers, while others involve additional effects incorporated into my models. These effects include the vertical pressure and horizontal tensile strain, all of which have unique influences on the band structures. Moreover, as the number of layers increases, intriguing phenomena continue to emerge. Even at small twist angles, flat bands persist, and various topological features appear. Recent experiments have also revealed magnetic properties in these systems.

In summary, these systems exhibit a wealth of intriguing physics, with a particular focus on electronic structures and band calculations in this thesis.

Abstract (Chinese) I
Acknowledgements (Chinese) II
Abstract III
Contents V
List of Figures VII
1 Introduction 1
1.1 Two-dimensional (2D) materials . . . . . . . . . . . . . . . . . . . . 1
1.2 Twisted bilayer graphene . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Theory 5
2.1 Tight binding method . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . 7
3 Twisted Bilayer Graphene Structure 17
3.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Corrugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Lattice Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 DFT calculation for corrugation . . . . . . . . . . . . . . . . . . . . 25
4 Band Structure 28
4.1 Tight binding model for TBG . . . . . . . . . . . . . . . . . . . . . 28
4.2 TBG band structure . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Taking corrugation into account . . . . . . . . . . . . . . . . . . . . 31
4.4 Vertical pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Tensile strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 Twisted bilayer SiC . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.7 Twisted trilayer graphene . . . . . . . . . . . . . . . . . . . . . . . 35
5 Conclusion and future works 52
Bibliography 54

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