我們在本論文中探討應用單狄拉克粒子模型來了解石墨單層膜 (graphene，簡稱石墨單膜) 相關系統 (包括了石墨雙膜等等)。首先我們以考慮最鄰近躍移的最簡單緊密束縛法推導出石墨單膜相關系統的低能量哈密頓矩陣。之後，我們仔細討論可運用超對稱量子力學處理的課題，包括石墨單膜相關系統在有外加磁場的情況下其低能量的能階與能態，及理解系統中所觀測到的反常量子霍爾效應。最後，利用超對稱量子力學中的形狀不變位能方法，我們討論狄拉克費米子受具長程引力庫倫位能的單一帶電雜質，所產生的束縛態問題。以及其在石墨單膜系統中的應用。 In this thesis, we discussed the application of the single-(Dirac) particle model to understand some physical behaviors of the graphene related systems (including also bilayer graphene, etc.) At first, we derive the low-energy Hamiltonians by the minimal tight-binding model that only includes hopping between nearest neighbor carbon atoms. We then discuss in detail the topics that can be treated by supersymmetry quantum mechanics (SUSY QM), including the low-energy eigenvalues and eigenfunctions of the graphene related systems in the magnetic field and the application to understand the anomalous quantum Hall effects observed in these systems. Finally, by making use of shape invariant potentials approach of SUSY QM, we discuss the bound-state problem of the Dirac particle scattered by a single impurity with the long-range attractive Coulomb potential and its application to the graphene system.
