本研究由第一原理出發，以局域化之Wannier函數為基底，描述強束縛激子的波函數。接著利用雙粒子動能核心的構想，透過求解 Bethe-Salpeter Equation (BSE)，得到激子之關聯函數與響應函數，進而可與非彈性光散射實驗比較，以了解強束縛激子的動態傳播特性。我們主要模擬二維單層片狀結構的碳化矽與氮化硼，其強束縛激子的動態行為。此外，由本研究之結果發現，當維度增加時，結構由單層片狀結構，變為雙層片狀結構時，而原本強束縛的激子將不再受限於只能在平面上傳播。

In this thesis ,we perform first-principles calculations to explore the properties strongly bound Frenkel-like excitons. According to the local character ,the wavefunction of Frenkel-like exciton can be represented by the Wannier function in our study. Furthermore ,electronic correlation function and response function can be obtained by solving the Bethe-Salpeter equation with the help of developing two-particle kinetic kernel(T). Our result can be compared with the measurement of inelastic X-ray scattering. Practically ,we focus on the dynamical properties of strongly bound exciton in two-dimensional SiC sheet and BN sheet. Meanwhile ,our results show that exciton could be propagated along the direction perpendicular to the sheet plane by reducing the inter-sheet distance.

目錄
第一章　低維度材料激子性質.................................1
1.1 目的與研究動機.........................................1
1.2 強束縛激子特...........................................2
1.3 低維度材料特性.........................................6
1.4 實驗方法：非彈性散射實驗...............................8
第二章　理論背景..........................................10
2.1 密度泛函理論(DFT).....................................10
 2.1.1 Hohenberg-Kohn 定理................................11
 2.1.2 Kohn-Sham方程......................................14
2.2 Wannier函數...........................................16
2.3 線性密度響應理論......................................20
 2.3.1 時間相關微擾理論...................................20
 2.3.2 密度響應函數.......................................21
 2.3.3 Lehmann representation.............................24
 2.3.4 電子電洞關聯函數...................................28
2.4 Bethe-Salpeter方程(BSE)...............................34
第三章　計算方法與流程....................................39
 3.1 求解Wannier 函數.....................................39
  3.3.1 超級原子近似......................................39
  3.1.2 最大局域化法......................................42
 3.2 格林函數以Wannier 函數為基底.........................44
  3.2.1 格林函數以SAWFs為基底............................44 
  3.2.2 格林函數以MLWFs為基底.............................45
 3.3 雙粒子動能核心(T)....................................45
 3.4 動量空間下的關聯函數(L)..............................46
 3.5 線性密度響應函數.....................................51
 3.6 計算流程.............................................53
第四章　二維平面材料的激子特性............................54
 4.1 基態電子結構.........................................54
 4.2 以Wannier函數分析能帶結構............................56
 4.3 激子動態特性.........................................58
 4.4 平面間距對動態激子性質的影響.........................61
第五章　結論..............................................65

圖表目錄
圖1.1 本質吸收示意圖.......................................2
圖1.2 激子形成示意圖.......................................4
圖1.3 (a)Wannier激子示意圖, (b)Frenkel激子示意圖...........5
圖1.4 非彈性X光散射實驗設置圖..............................9
圖1.5 鋰化氟沿著動量空間中(H,0,0)方向的IXS實驗之實驗數據..10
圖2.1 Bethe-Salpeter equation以費曼圖表示.................38
圖3.1 (a)鋰化氟能帶結構 ,(b)鋰化氟的第一布里淵區及能帶結構使
      用的高對稱點 ,(c)價帶超級原子Wannier函數 ,(d)導帶超級
      原子Wannier函數.....................................41
圖3.2 (a)鋰化氟能帶結構 ,(b)鋰化氟單元晶胞 ,(b)鋰原子的s軌
      道Wannier函數 ,(c)氟原子的p軌道Wannier函............43
圖3.3 電洞波函數乘上電子波函數,得到激子波函數.............43
圖3.4 求解 的Bethe-Salpeter方程費曼示意圖.................46
圖3.5 雙粒子核心動能(T) ,紅線來自SAWFs基底 ,黑線為來自
      MLWFs基底...........................................47
圖3.6 求解 的Bethe-Salpeter方程費曼示意圖.................47
圖3.7 L之誤差修正示意圖...................................49
圖3.8 關聯函數(L) (a)來自SAWFs基底 ,(b)來自MLWFs基底......50
圖3.9 (a) 實空間激子Wannier函數 ,(b)動量空間下,激子波函數的
      絕對值平方圖........................................52
圖3.10 (a)鋰化氟IXS實驗數據 ,(b)激子虛部密度響應函數之計算
       模擬數據...........................................53
圖4.1 (a)二維單層片狀結構碳化矽之能帶結構 ,(b)二維單層片狀結
      構氮化硼之能帶結構..................................55
圖4.2 (a)第一布里淵區及能帶結構中使用的高對稱點 , (b)二維單
      層片狀結構碳化矽單元晶胞, (c) 二維單層片狀結構氮化硼單
      元晶胞..............................................55
圖4.3 (a)二維單層片狀碳化矽能帶結構 ,(b)二維單層片狀氮化硼能
      帶結構 ,紅色線為最高佔據分子軌道 ,綠色線為最低未佔據分
      子軌道..............................................57
圖4.4 (a)矽原子在最低未佔據分子軌道的pz軌道對稱Wannier函數, 
      (b)碳原子在最高佔據分子軌道的pz軌道對稱Wannier函數, 
      (c)硼原子在最低未佔據分子軌道的pz軌道對稱Wannier函數, 
      (d)氮原子在最高佔據分子軌道的pz軌道對稱Wannier函數 (紅
      色為正,藍色為負)....................................57
圖4.5 激子Wannier函數 (紅色為正,藍色為負).................58
圖4.6 二維單層片狀結構碳化矽近鄰示意圖....................58
圖4.7 激子由on-site晶格點 ,運動到第一近鄰之示意圖.........59
圖4.8 二維單層片狀結構碳化矽理論計算關聯函數(a)沿(100)方向 
      ,(b)沿(010)方向 , (c)沿(001)方向....................60
圖4.9 二維單層片狀結構氮化硼理論計算關聯函數(a)沿(100)方向 
      ,(b)沿(010)方向 ,(c)沿(001)方向.....................61
圖4.10 (a)二維單層片狀碳化矽能帶結構圖 ,(b)雙層片狀碳化矽能
       帶結構圖 ,兩層片狀結構的間距為4  ,單元晶胞如(c)所示62

圖4.11 雙層片狀結構碳化矽理論計算關聯函數(a)沿(100)方向 ,(b)
       沿(010)方向 ,(c)沿(001)方向........................63
圖4.12 三種可能的碳化矽雙層片狀結構之總能量...............64

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