方法發展：對於固體的光學性質計算、其躍遷矩陣，必須用動量算符表示，而因為使用 Pseudopotential 的緣故動量算符也就必須做相關修正。修正動量算符的方法不只一種，常見的有 commutator 修正法與核修補方法兩種。在我們研究群使用的程式中前者已經存在，也給出了合理結果，而我們想要進一步把後者也作進去同一程式中，以便比較兩者的優劣異同

應用計算：介電係數在現今半導體及光電材料裡是一個重要的物理量，很多新型元件需要針對材料的介電係數來加以調控設計，以便達到所需要的效果，材料中會影響到其介電常數之相關因素因此成為重要的研究議題。我們選擇了幾種 high k、low k 材料來進行第一原理計算，並建立新的分析方法，讓我們可以在三度空間中呈現出影響介電常數貢獻的部份來自何處，所選擇得材料包括了一系列不同結晶相的二氧化矽、常見的氧化物晶體、以及近年來受矚目的 HfO2。另外，固態材料受外在靜電場影響的物性變異，隨著奈米科技的發展材料尺寸及元件都大幅縮小，在相當之電位差之下，產生的電場強度就極為巨大，因而其引發的物性變易也就明顯重要。在這一部分的研究中，我們計算分子光電特性之一的超極化率並與實驗比較，也探討它是如何受到外場強大靜電場而有所改變。

關鍵詞：密度泛函理論；贗勢；倍頻係數；核修補；密度泛函微擾理論；介電常數；超極化率；EFISH

Methodology :

To calculate the optical properties of solids, one has to use the expectation values of momentum between bands instead of that of position operator for the evaluation of transition matrix elements. Therefore, when pseudopotential is used, the expectation value needs to be corrected because for r < rc the wavefunction is very different from the true wavefunctions. There are more than one methods to realise such correction, two commonly seen are the so called commutator correction, which to suppress error at small-r pseudo core region, choose to use the position operator and express it in terms of momentum operator plus a commutator term between r and nonlocal (pseudo) potential; another is to subtract the effect of pseudo wavefunction to momentum matrix elements and add back the matrix elements calculated by using (approximate)
atomic true wavefunction Before this work, the commutator term correction has already been implemented into the computational code used in our group, the main methodology development of my thesis is to implement the second method into the same computer code, so that their results can be compared.
Applications :

Dielectric constant is an important physical quantity of semiconductor and opto-electronic industry nowadays, many new devices relay on tuning dielectric constants of its constituent materials in order to achieved the desired properties. Therefore the factors that will affect dielectric properties of a material become an important subject of research. In this work we have used first principles methods to study several chosen high k and low k materials, we have also developed new analysis scheme which allow us to visualised spatial distribution of dielectric contribution in real-space. We have chosen several SiO2 crystals, some commonly seen oxide crystals, and HfO2 which received broad attention recently. On the other hand, we are interested in how physical properties change under the influence of static electric field. Extremely large electric field is possible today, thanks to the development of nanotechnologies which reduce the dimension of devices and cause a regular potential step to generate very large electric field, it is therefore important to study such electric field induced variation of physical properties. In the last part of this work, not only did we calculate hyperpolarisability of molecule and compare with experimental results, but also we investigated how this quantity is changed under large external static electric field.

第一章 簡介 
1.0前言                                                 1
1.1密度泛函理論與平面波贗勢方法簡介                     1
1.2 CASTEP標準光學性質計算                              2
1.3介電常數的應用與計算                                 5
1.4非線性光學計算--分子的超極化率                       9
參考文獻                                               10

第二章 第一原理膺勢方法在處理光學性質上所需之修正方法的比較與探討
2.1動機與背景                                          11 
2.2公式推導                  			   11
2.3.1 NewCASTEP架構簡介                                15
2.3.2為了進行移植所做的追蹤工作（舊 CASTEP 部分）      18
2.4 core-repair term修正前後的光學性質比較             26
2.5結論	                                             28
參考文獻                                               29

第三章 氧化物的 high k，low k 分析
3.1動機與背景                                          30
3.2計算與分析方法                                      30
3.3 SiO2 系列計算結果                                  42
3.4 HfO2 計算結果  　                                  54
3.5其他氧化物的計算     	                       57
3.6結論	                              	             60
參考文獻                                               61

第四章 外加電場下分子材料之光電性質研究
4.1動機與背景  			                  62
4.2方法介紹						   62
4.3硝基苯胺（pNA）外加電場計算      		   63
4.4結論          		                             69
參考文獻                                               70

附錄A 核修補公式推導    		                      71
附錄B 氧化物的 Born effect charges 與 Population analysis 資料    79
附錄C 參考文獻的分子β實驗值掃圖                       87
圖目錄
圖2.3-1 Functional Modules  16
圖2.3-2 Fundamental Modules  17
圖2.3-3 Utility Modules  17
圖2.3-4 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第131行程式碼 18
圖2.3-5 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第200行程式碼 18
圖2.3-6 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第284行程式碼 19
圖2.3-7 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第1003行程式碼 20
圖2.3-8 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第1030行程式碼 20
圖2.3-9 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第1101行程式碼 21
圖2.3-10 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第1118行程式碼 22 
圖2.3-11 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第1129行程式碼 23
圖2.3-12 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第1141行程式碼 23
圖2.3-13 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第1228行程式碼 24
圖2.3-14 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式cst_inp_geom.F第3325行程式碼 24
圖2.3-15 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第367行程式碼 25
圖2.3-16 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第825行程式碼 25
圖2.3-17 核修補版本CASTEP “castepexe_v4.2_csyang_0601”的副程式makematel.F第865行程式碼 25
圖2.4-2 SiO2_quartz 的介電函數 27
圖2.4-3 SiO2_quartz 的介電函數 27
圖3.2-1 SiO2_cristobalite_high 晶體結構立體對圖 30
圖3.2-2 SiO2_cristobalite_low 晶體結構立體對圖 31 
圖3.2-3 SiO2_quartz_beta 晶體結構立體對圖 31
圖3.2-4 SiO2_quartz 晶體結構立體對圖 31
圖3.2-5 SiO2_stishovite 晶體結構立體對圖 32 
圖3.2-6 SiO2_quartz Energy cutoff test  32
圖3.2-7 SiO2_quartz 加1倍空 bands 的吸收光譜 34
圖3.2-8 SiO2_quartz 加2倍空 bands 的吸收光譜 34
圖3.2-9 SiO2_quartz 加3倍空 bands 的吸收光譜 34
圖3.2-10 HfO2 結構立體對圖 35
圖3.2-11 HfO2 Energy cutoff test  36
圖3.2-12 HfO2 加2倍空 bands 的吸收光譜 37
圖3.2-13 HfO2 加3倍空 bands 的吸收光譜 37
圖3.2-14 HfO2 加4倍空 bands 的吸收光譜 37
圖3.2-15 Al2O3  38
圖3.2-16 HfSiO4  39
圖3.2-17 La2O3  39
圖3.2-18 SrTiO3  39
圖3.2-19 Ta2O5 (beta phase)  40
圖3.2-20 TiO2_rutile  40
圖3.2-21 Y2O3  40
圖3.2-22 ZrO2_monoclinic  41
圖3.3-1 SiO2_quartz valence band 介電密度分佈 43
圖3.3-2 SiO2_quartz conduction band 介電密度分佈 43
圖3.3-3 quartz beta valence band 介電密度分佈 44
圖3.3-4 quartz beta conduction band 介電密度分佈 44
圖3.3-5 SiO2 cristobalite_high valence band介電密度分佈 45
圖3.3-6 SiO2 cristobalite_high conduction band介電密度分佈 45
圖3.3-7 SiO2 cristobalite_low valence band介電密度分佈 46
圖3.3-8 SiO2 cristobalite_low conduction band介電密度分佈 46
圖3.3-9 SiO2 stishovite valence band 介電密度分佈 47
圖3.3-10 SiO2 stishovite conduction band介電密度分佈 47
圖3.3-11 SiO2_quartz shg_density virtual electron occupied  48
圖3.3-12 SiO2_quartz shg_density virtual electron unoccupied  48
圖3.3-13 SiO2_quartz shg_density virtual hole occupied  49
圖3.3-14 SiO2_quartz shg_density virtual hole unoccupied  49
圖3.3-15 SiO2_quartz 吸收光譜 50
圖3.3-16 npts310  valence band 吸收密度分佈 50
圖3.3-17 npts310  conduction band 吸收密度分佈 50
圖3.3-18 npts696  valence band 吸收密度分佈 51
圖3.3-19 npts696  conduction band 吸收密度分佈 51
圖3.3-20 npts1356  valence band 吸收密度分佈 52
圖3.3-21 npts1356  conduction band 吸收密度分佈 52
圖3.4-1 HfO2 valence band 介電密度 54
圖3.4-2 HfO2 conduction band 介電密度 54
圖3.4-3 HfO2 電子貢獻部分與總貢獻部分的介電常數值 55
圖3.4-4明顯可見 HfO2 的比 SiO2 複雜，並且在低波數（低能量）HfO2 有比 SiO2 更低的 peaks 存在（在250 cm-1附近） 55
圖3.4-5 HfO2 Born Effective charge  56
圖3.4-6 HfO2 population analysis  56
圖3.5-1將表3.5-2 改以圖的方式呈現，以便觀察其趨勢 58
圖4.3-1 pNA 分子立體對圖 63
圖4.3-2 pNA Energy cutoff 測試結果 63
圖4.3-3不同電場的能帶解析比較 65
圖4.3-4 pNA  SHG HOMO-1 orbital density  66
圖4.3-5 pNA SHG HOMO orbital density  66
圖4.3-6 pNA SHG LUMO orbital density  67
圖4.3-7 pNA SHG LUMO+1 orbital density  67
圖4.3-8由左到右依序是對位 para（pNA）、鄰位 ortho（oNA）、間位 meta（mNA）硝基苯胺 68
表目錄
表2.4-1 SiO2_quartz 經過核修補之後的折射率在能隙修正後得到的結果與實驗值相當符合 26
表3.2-1 SiO2_quartz Energy cutoff測試設定 32
表3.2-2 SiO2_quartz  k-Points 取樣測試的參數設定 33
表3.2-3 SiO2_quartz  k-Points 測試的結果 33
表3.2-4 SiO2_quartz 空 bands 測試參數設定 33
表3.2-5 SiO2_quartz empty bands測試結果 33
表3.2-6 SiO2 系列參數設定 35
表3.2-7 HfO2 Energy cutoff 測試設定 36
表3.2-9 HfO2空 bands 數目測試的設定 36
表3.2-10 HfO2 空 bands 數目測試介電常數的比較 36
表3.2-11 HfO2 參數設定1  37
表3.2-12 HfO2 參數設定2  38
表3.2-13其他氧化物參數設定1  41
表3.2-14其他氧化物參數設定2  42
表3.3-1 SiO2 結構密度大小影響著介電常數的大小  53
表3.5-1材料依其介電常數之大小排列  57
表3.5-2材料大致依其介電常數之大小排列  58
表4.3-1 pNA Energy cutoff 測試參數設定  63
表4.3-2 pNA Empty bands 測試參數設定  64
表4.3-3 pNA Empty bands 測試結果空 bands 不夠會影響β 的精確性  64
表4.3-4 pNA 分子不同電場下的β值  64
表4.3-5 pNA 分子不同電場下的能隙  65
表4.3-6 pNA 分子不同電場下能階互相取代之後，若是使用相同的能階則β無明顯改變（橫向來看）；若是在同樣電場下則β會因不同的能階取代而跟者改變（縱向來看），所以β的確受到能階的影響  68
表4.3-7硝基苯胺 3 種同分異構物，結構不同能階也不同，但是 β 改變的趨勢是隨著波函數，而不是能階  68

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