當今，石墨烯的研究已成為材料物理領域中的顯學，其新穎的電子、光學與拓樸特性被視為極具潛力的研究方向。由於石墨烯擁有獨特的二維蜂巢晶格結構與狄拉克錐，其在量子傳輸、奈米電子學以及拓樸材料領域中均展現出關鍵的理論與應用價值。近年來，將石墨烯沿徑向捲曲形成奈米卷軸（graphene nanoscroll）的研究亦日益受到重視。此一結構兼具一維與二維系統的特性，卷曲帶來的幾何扭轉與邊界條件改變，使其電子態展現出不同於平面石墨烯的豐富物理現象。本研究即以數學分析為核心，深入探討石墨烯奈米卷軸背後的能帶結構與拓樸性質。
我們首先從手徵對稱性（chiral symmetry）出發，建立具有對稱約束的理論框架，並透過特殊的數學構造，使系統同時滿足對稱條件。進而引入拓樸纏繞數（topological winding number）作為拓樸量化的核心指標，以此刻劃奈米卷軸中電子能帶的拓樸量。藉由嚴謹的數學定義與推導，我們驗證了石墨烯奈米卷軸於能帶結構中所展現的拓樸特性，並進一步釐清其能帶背後的數學意涵與物理本質，揭示幾何纏繞與對稱性之間的密切關聯。
其次，本研究依據卷軸交界處的幾何特性與邊界條件，將石墨烯奈米卷軸區分為三種類 型，並在緊束縛模型（tight-binding model）的基礎上，提出兩個全新的理論模型：「有效位 能模型」與「腰斬的緊束縛模型」。這兩種模型能有效模擬不同結構下電子波函數的行為，並作為後續能帶解析的重要理論工具。我們針對三種類型之奈米卷軸分別進行能帶分析。之後我們對三種類型奈米卷軸進行能帶分析與數值比對，揭示幾何結構對電子能帶的關鍵影響。
最後，我們探討石墨烯奈米卷軸在外加磁場下的電子行為，嘗試建立最符合物理實際情 況的磁通模型。具體而言，本研究比較了兩種磁通量分佈方案，分析其能帶變化差異。並引入皮爾斯相位因子（Peierls phase）以相位累積去分析並判定合理的磁場耦合形式。結果顯示，目前對加磁場方式的理解仍有待深化，尚無法確定最合適的通量模型。未來的研究將持續在理論推導與數值模擬層面上，探討磁場對拓樸態轉換的影響，期望能為奈米尺度下的量 子控制與拓樸電子學發展奠定更穩固的理論基礎。

In recent years, graphene research has become a central focus in the field of materials physics, as its novel electronic, optical, and topological properties present highly promising directions for exploration. Due to its unique two-dimensional honeycomb lattice and Dirac cone structure, graphene exhibits significant theoretical and practical value in quantum transport, nanoelectronics, and topological materials. In particular, the study of graphene nanoscrolls structures formed by rolling graphene sheets along the radial direction has drawn growing attention. These systems possess characteristics of both one and two dimensional materials. The geometric twisting and modified boundary conditions introduced by the scrolling process give rise to rich physical phenomena distinct from those of flat graphene. This work employs a mathematical framework to investigate the underlying band structure and topological properties of graphene nanoscrolls in depth.
We begin with ”chiral symmetry”, constructing a theoretical framework under symmetry constraints, and employ specific mathematical formulations to ensure the system satisfies these conditions. The“topological winding number”is then introduced as the central quantity for topological characterization, serving to quantify the topology of electronic bands within the nanoscroll. Through rigorous mathematical definitions and derivations, we verify the topological properties manifested in the band structure of graphene nanoscrolls, elucidating the mathematical and physical meanings behind these bands and revealing the close connection between geometric twisting and symmetry.
Next, based on the geometric features and boundary conditions at the scrolled interface, graphene nanoscrolls are categorized into three types. Within the framework of the tightbinding model, we further propose two new theoretical models: the ”Effective Potential Energy Model” and the ”Tight- Binding Model in Waist Chop”. These models effectively simulate the behavior of electronic wave functions under different structural configurations and serve as key theoretical tools for subsequent band analysis. We perform band structure analysis and numerical comparisons for the three nanoscroll types, revealing the critical influence of geometry on the electronic band properties.
Finally, we investigate the electronic properties of graphene nanoscrolls in the presence of an external magnetic field, aiming to develop a flux model that accurately reflects their physical behavior in the real world. Specifically, two magnetic flux distribution schemes are compared, and their corresponding effects on the band structure are analyzed. The Peierls phase factor is introduced to account for phase accumulation and to determine the appropriate magnetic coupling form. The results suggest that the current understanding of flux application remains incomplete, and the most suitable flux model has yet to be determined. Future studies will continue to investigate, through theoretical and numerical approaches, the influence of magnetic fields on topological phase transitions, with the goal of laying a solid theoretical foundation for quantum control and topological electronics at the nanoscale.

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