論文提要內容：
對於現今二維材料的蓬勃發展，摻雜效應能使其材料有微妙的變化。摻雜效應會破壞原本晶體的週期性，進而能使材料有更多樣的發展和應用。利用超晶胞(supercell，SC)方法解決計算中非週期性系統，是常見的方法:隨著超晶胞越大，相對應的布里淵區(Brillouin Zone，BZ)即隨之變小；比較無摻雜之單元晶胞(primitive cell，PC)與SC計算，SC不僅含較多之電子數，而使得其BZ內的能帶數目增加，而且SC中的能帶，會因摺疊BZ而產生糾纏現象。所以，利用能帶展開理論，可以展開糾纏的能帶；此外，我們也引入混合泛函近似(hybrid functional)以及準粒子GW修正，模擬多電子系統之遮蔽效應。，因而，通過此計算之電子能帶結構，將可直接與角分辨光電子能譜曲線(ARPES)做比較。
本論文中，我們分別計算三種二維系統:石墨烯摻雜氮化硼、二硫化鉬摻雜硒、以及二硒化鎢摻雜鉻與硒空缺。在石墨稀摻雜氮化硼中，討論相同濃度不同的摻雜組成，以及不同濃度摻雜情況下，對於其摻雜形成能(formation energy)及電子能帶結構之影響。另一方面，我們發現二硫化鉬在不同晶格常數條件下，有不同的能隙特性；接著，也探討摻雜硒後，對於其形成能與能隙的影響。最後，探討二硒化鎢摻雜鉻與硒空缺；可以透過軌道投影，分析較局域化的能帶，並研究改變摻雜鉻與硒空缺之距離，所造成的影響。

For 2D crystal structures which original planar periodicity is disturbed through the introduction of external influence of defects, a supercell (SC) scheme is a common practice with longer periodic boundary conditions in the computational exploration of aperiodic systems. However, the expansion of primitive periodic cell (PC) in the SC approximation induces a Brillouin Zone (BZ) shrink through the zone folding and usually gives rise to a massively entangled band structures in a tiny size BZ of large SC structures. To probe the weak influence contributed by the imperfection of crystals, we discuss how to visualize supercell band structures in hybrid density functional theory more effectively by incorporating unfolded spectral weights which can be compared with angle-resolved photoemission spectra (ARPES) of 2D materials, such as graphene, silicene, etc. 
In this thesis, we performed the first-principles unfolding calculations to study the defect effects in the electronic band structures of 2D materials: graphene doped with BN, MoS2 doped with Se, and WSe2 doped with Cr and Se-vacancy (VSe).

目錄
第一章 緒論	1
第二章 理論與計算方法	2
2-1密度泛函理論	2
2-2 Kohn-Sham 理論	3
2-3交換相干能	5
2-4混合泛函近似HSE(Heyd-Scuseria-Ernzerhof)與GW修正	6
2-5單元晶胞與超晶胞布里淵區關係	8
第三章  能帶展開定理	11
3-1. Bloch function 與 Wannier function	11
3-2展開定理	13
第四章計算結果	17
4-1石墨烯摻雜氮化硼	17
4-1-1相同濃度不同形式的摻雜	17
4-1-2 不同濃度的摻雜	19
4-2二硫化鉬摻雜硒	21
4-2-1不同的晶格常數比較	21
4-2-2相同濃度不同形式的摻雜	24
4-3二硒化鎢	26
4-3-1二硒化鎢摻雜鉻	26
4-3-2二硒化鎢摻雜鉻與硒空缺	27
第五章 結論	29
參考文獻	31




圖目錄

圖2-5-1(a)(b)布里淵區示意圖	9
圖2-5-2二維倒空間示意圖：	9
圖3-1-1 Bloch functions示意圖	11
圖3-2-1為展開定理示意圖	13
圖4-1-1 (a)圖為3x3x1超晶胞結構，(b)圖為6x6x1超晶胞結構	18
圖4-1-2 (a)圖為3x3x1超晶胞(分散摻雜)能帶結構，(b)圖為6x6x1超晶胞(團簇摻雜)能帶結構	19
圖4-1-3 6x6x1超晶胞結構(a)摻雜濃度為8.33%(b)摻雜濃度為33.3%(c)摻雜濃度為75%	20
圖4-1-4 6x6x1超晶胞能帶結構(a)摻雜濃度為8.33%(b)摻雜濃度為33.3%(c)摻雜濃度為75%	20
圖4-2-1 MoS2結構示意圖	21
圖4-2-2 單元晶胞的電子能帶結構圖(a)晶格常數為3.15Å (b)晶格常數為3.17Å (c)晶格常數為3.19Å	22
圖4-2-3布里淵區路徑	22
圖4-2-4 (a)3x3x1超晶胞結構圖，(b)6x2x1超晶胞結構	24
圖4-2-5 (a)3x3x1超晶胞能帶結構，(b)6x2x1超晶胞能帶結構	25
圖4-3-1 二硒化鎢結構示意圖	26
圖4-3-2 二硒化鎢摻雜鉻結構示意圖	26
圖4-3-3 能帶結構圖(a)二硒化鎢摻雜鉻(b)鉻之d軌道投影	27
圖4-3-4 二硒化鎢摻雜鉻與硒空缺結構示意圖	27
圖4-3-5 二硒化鎢摻雜鉻與硒空缺能帶結構圖(a)VSe^1(b) VSe^2(c) VSe^3	28














表目錄
表3-2-1超晶胞與單元晶胞符號對照表	14
表4-1-1石墨烯摻雜氮化硼之濃度能隙比較表	20
表4-2-1不同晶格常數與不同修正方法的能隙比較表	22
表4-2-2二硫化钼(MoS_2)摻雜硒(Se)之不同形式摻雜能隙表	25

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