本文中，我們利用第一原理計算來研究研究石墨烯相關系統以及第四族系統的幾何與電子性質，範圍包括一維邊緣綴飾石墨帶、二維的皺褶石墨烯在不同的曲率、周期、皺褶方向。不同的邊緣綴飾原子、曲度、及鋸齒狀的邊緣結構導致了石墨帶的三種幾何結構，這些結構源起於各種不同的邊界-邊界交互作用。皺褶石墨烯態密度上低能的峰與掃描穿隧能譜的實驗做比較，可以用來區分兩種不同型態的皺褶石墨烯。對應於石墨烯，矽烯在不同的氫化濃度和氫原子分部之下可以達成能隙的調控與產生不同數量的局域電子，這也提供了一個方法來指認氫濃度。另一方面，我們利用自我開發的廣義化緊束模型計算石墨烯在外加磁場之下與各種不同堆疊的石墨塊材的熱電子性質。溫度與磁場的競合效應決定了石墨烯的磁比熱，其臨界溫度與臨界磁場展現了一個簡單的線性關係。在塊材石墨中，層與層間原子的交互作用決定了電子比熱的溫度相依性。AA、AB、與ABC堆疊的石墨塊材分別展現了線性的溫度相關在高溫、中溫、與低溫的範圍。我們的計算結果也與實驗上的研究做詳細的比對，此外也得到一些嶄新的結果尚待實驗進一步驗證。

In this dissertation, we use first principle calculations to study the geometric and electronic properties in group IV graphene-related systems, ranging from one-dimension edge-decorated graphene nanoribbons to two-dimensional graphene ripples with different corrugated directions, periods, and curvatures. Various decorating atoms, different curvature angles, and the zigzag edge structure have resulted in three types of geometric structures, depending on the edge-edge interactions. The existence of low-lying peaks of densities of states can be used to distinguish specific types of graphene ripples obtained in STS experiments. As a counterpart of graphene, hydrogenated silicenes with different hydrogen configurations and concentrations lead to tunable band gaps and strong localized states at the Fermi level, providing a method to identify the hydrogen concentrations. A self-developed generalized tight-binding model is used to study the electronic thermal properties of the nanostructures from two-dimensional graphene to three-dimensional graphites with different stacking configurations. The temperature and the magnetic field compete with each to determine the specific heat of graphene, revealing a simple linear relation between critical temperature and critical magnetic field. In bulk systems, the interlayer atomic interactions dominate the temperature-dependent specific heat, exhibiting linear T dependences in low-, middle-, and high-temperature for Bernal, rhombohedral, and simple hexagonal graphites, respectively. We have paid attention to compare the calculations with the experiments and have obtained some novel features not yet verified experimentally.

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