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研究生:周美雲
論文名稱:二氧化鈦的電子特性和光學特性之探討
論文名稱(外文):Electronic and Optical Properties of TiO2: Rutile, Anatase and Brookite.
指導教授:許貞雄
指導教授(外文):Chen-Shiung Hsue
學位類別:碩士
校院名稱:國立清華大學
系所名稱:物理學系
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:英文
論文頁數:85
中文關鍵詞:二氧化鈦電子特性光學特性
外文關鍵詞:Titanium DioxideElectronic propertiesOptical propertiesRutileAnataseBrookite
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摘 要
我們利用ab-initio pseudopotentials with plane-wave basis set within the local-density approximation 這個方法來對二氧化鈦的三種不同型態:Rutile, Anatase 和Brookite做理論計算。理論計算的結果總共分成兩大部分:其一是二氧化鈦的電子性質,另一部份是二氧化鈦的光學性質。在電子性質方面,我們計算了能帶結構、狀態密度以及其結構參數,而光學性質方面則計算其介電函數包括實部和虛部。
將我們所得到的能帶結構圖及狀態密度圖和實驗結果以及其他理論計算做比較均頗吻合。Rutile的能隙最小而Anatase的能隙和Brookite 很接近。至於能隙的型態,Rutile和Brookite是直接能隙,而Anatase是間接能隙。這三種型態的總狀態密度圖均很相近,其導帶均可分成兩個部分:能量較低的是由鈦的 t-2g 所形成,而鈦的 e-g 則形成能量較高的部分。我們也將總狀態密度圖再分成 s 和 p 和 d 三部分,由這些圖我們可以分析氧原子和鈦原子之間鍵結的情形。對這三種不同型態物質,我們都有對其結構參數作計算,並得到其晶格常數和晶格比例常數以及internal coordinate 的最小值。這些值和實驗值以及其他理論計算所得的值做比較,其誤差均小於百分之二,此結果甚合理。另外亦得到Bulk modulus,這個值亦和實驗值接近。
在光學性質方面,我們發現這三種型態物質均有anisotropic 結構。我們也將所計算的曲線和實驗以及其他理論計算的結果做比較,其結果也在合理的範圍。我們得到的介電常數,Rutile 和實驗值較接近、Anatase 比實驗值小而Brookite我們得到的值是 5.4左右。
Abstract
Calculations have been performed by using ab-initio pseudo-potentials with plane-wave basis set within the local-density approximation for three phases of titanium dioxide. The band structure, density of states, structure optimization and optical properties of three phases of titanium dioxide have been studied. The general features of the band structure for rutile and anatase are quite similar and both in agreement with experimental data and other calculations.
The band structure of brookite is also close to other calculations. Rutile has the smallest band gap among these phases and brookite has the largest one. Rutile and brookite have direct band gap at Γ but anatase has an indirect one. The general features of three phases of TiO2 are quite similar. From the analysis of partial density of states, we find the hybridization of 2p states of O and 3d states of Ti.
We minimize the total energy of the three phases of TiO2 with respect to the structure parameters a, c/a and u. Our calculated lattice constant a, lattice constant ratio c/a and the internal coordinate u are in good agreement with experi-mental results and close to other theoretical calculations. Bulk modulus is also obtained by fitting the data with Murnaghan’s equation of states. Our data of bulk modulus of rutile and anatase are consistent with experimental values. Brookite has the smallest value of these polymorphs this may be attributed to its structure. We also studied the optical properties of TiO2 by calculating the dielectric functions. All three phases of TiO2 show anisotropic structures in the dielectric functions. Static dielectric constants are obtained from the real part of the dielectric function. Our calculations show subtle differences in the optical properties of three phases of TiO2.
Contents
Chapter 1 Introduction………..………………………….............1
Chapter 2 Basic Principle and Methods of Calculation
2.1 Basic Principle……….………………………….........4
2.1.1 Overview…………...………………….…………….….4
2.1.2 Hohenberg — Kohn theorem…………...……………..5
2.1.3 Local Density Approximation………..……………...7
2.1.4 Local-Spin-Density Approximation……….………...9
2.2 Methods of Calculation……………..…………........10
Chapter 3 Crystal Structures
3.1 The structure of Rutile………..………………….....13
3.2 The structure of Anatase…………..………….….....15
3.3 The structure of Brookite…………..………...…....17
Chapter 4 Results and discussion
4.1 Electronic properties…………………..…..…….....19
4.1.1 Band structure of TiO2…………………....……..19
[A] The band structure of rutile…………………..…..19
[B] The band structure of anatase……………….………21
[C] The band structure of brookite………………..…..23
4.1.2 Density of states of TiO2…………….…………….25
[A] Density of states of Rutile…………..…………….25
[B] Density of states of Anatase……………..…………27
[C] Density of states of Brookite…………..………...29
4.1.3 Structural parameters of TiO2…………...….…..31
4.2 Optical properties…….………………………...…....34
4.2.1 Optical properties of Rutile…...…………………35
4.2.2 Optical properties of Anatase…………….....….38
4.2.3 Optical properties of Brookite…………...……..40
Chapter 5 Summery and conclusion………………...….............42
Reference……………………………………………….................44
List of Tables
Table (1): Calculated band structure of TiO2………………………47
Table (2): Calculated structural parameters of TiO2…………….48
List of Figures
Figure 3.1:.Crystal structure of TiO2…………….....……………49
Figure 4.1: Band structure of TiO2 (Rutile)……………………….50
Figure 4.2: Band structure of TiO2 (Anatase)………………………51
Figure 4.3: Band structure of TiO2 (Brookite)…………………….52
Figure 4.4: Total density of states of TiO2 (Rutile)……………53
Figure 4.5: Integrated density of states of TiO2 (Rutile)…….53
Figure 4.6: Partial density of states of TiO2 (Rutile)…………54
Figure 4.7: Total density of states of TiO2 (Anatase)………….55
Figure 4.8: Integrated density of states of TiO2 (Anatase)……55
Figure 4.9: Partial density of states of TiO2 (Anatase)……….56
Figure 4.10: Total density of states of TiO2 (Brookite)….……57
Figure 4.11: Integrated density of states of TiO2 (brookite)…57
Figure 4.12: Partial density of states of TiO2 (Brookite)…….58
Figure 4.13: Cohesive energy vs lattice constant of TiO2 (Rutile)59
Figure 4.14: Cohesive energy vs internal coordinate of TiO2(Rutile)59
Figure 4.15: Cohesive energy vs lattice constant ratio of TiO2 (Rutile)……..60
Figure 4.16: Cohesive energy vs lattice constant of TiO2 (Anatase)……60
Figure 4.17: Cohesive energy vs internal coordinate of TiO2 (Anatase)…….61
Figure 4.18: Cohesive energy vs lattice constant ratio of TiO2 (Anatase)…...61
Figure 4.19: Cohesive energy vs lattice constant of TiO2 (Brookite)………..62
Figure 4.20: Cohesive energy vs b/a of TiO2 (Brookite)……….62
Figure 4.21: Cohesive energy vs c/a of TiO2 (Brookite)……….63
Figure 4.22: Imaginary part of TiO2 (Rutile)…………………..64
Figure 4.23: Imaginary part of TiO2 (Rutile)…………………..65
Figure 4.24: Real part of TiO2 (Rutile)………………………...67
Figure 4.25: Real part of TiO2 (Rutile)………………………….68
Figure 4.26: Imaginary part of TiO2 (Anatase)………………...70
Figure 4.27: Imaginary part of TiO2 (Anatase)………………….71
Figure 4.28: Real part of TiO2 (Anatase)………………………..73
Figure 4.29: Real part of TiO2 (Anatase)………………………..74
Figure 4.30: Imaginary part of TiO2 (Brookite)………………..76
Figure 4.31: Imaginary part of TiO2 (Brookite)………………..76
Figure 4.32: Imaginary part of TiO2 (Brookite)………………..77
Figure 4.33: Real part of TiO2 (Brookite)……………………….77
Figure 4.34: Real part of TiO2 (Brookite)……………………….78
Figure 4.35: Real part of TiO2 (Brookite)……………………….78
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