
系統識別號 
U00020706200714563900 
中文論文名稱

過渡金屬氧化物激發態之第一原理研究 
英文論文名稱

Firstprinciples study of excitation on transition metal oxides by linear response method 
校院名稱 
淡江大學 
系所名稱(中) 
物理學系博士班 
系所名稱(英) 
Department of Physics 
學年度 
95 
學期 
2 
出版年 
96 
研究生中文姓名 
李啟正 
研究生英文姓名 
ChiCheng Lee 
學號 
889180013 
學位類別 
博士 
語文別 
英文 
口試日期 
20070531 
論文頁數 
117頁 
口試委員 
指導教授薛宏中 委員錢凡之 委員彭維鋒 委員郭光宇 委員魏金明

中文關鍵字 
線性響應
Wannier函數
激子
聲子
氧化鎳
氧化鈷
鈦酸鉛

英文關鍵字 
linear response
Wannier function
exciton
phonon
NiO
CoO
PbTiO3

學科別分類 

中文摘要 
氧化鎳(NiO)與氧化鈷(CoO)是一強關聯的系統且為標準的Mott絕緣體。非共振非彈性X光散射實驗中發現氧化鎳與氧化鈷的dd軌域之躍遷呈現很強之動量轉換相依(qdependence)的特性，且此激子(exciton)的能量分布在準粒子能隙(quasiparticle band gap)之中。為了研究此局域化的躍遷行為，我們發展了局域化、有對稱性、能量可解析之Wannier函數。經由對Wannier電子電洞對的Fourier轉換之解析，我們了解此強動量轉換相依的起源且發現一新的選擇定則。進一步，我們提出一個稱為TDLDA+U的方法來解BetheSalpeter方程式用以描述電子電洞對間的吸引力來得出此能量分布在能隙中的激子。經由此方法，我們發現氧化鎳中的激子強烈混成在Mott能隙中。在HartreeFock近似下，一個有趣的動量轉換相依現象將被討論。另一方面，鈦酸鉛(PbTiO3)為一鐵電性材料，我們利用密度泛函微擾理論(DFPT)研究其晶體振動之特性。經由計算零溫下鈦酸鉛的聲子頻譜與彈性波速，我們確定了鈦酸鉛在低溫下的鐵電結構。更重要的，我們詳細分析鈦酸鉛在立方(cubic)與四方(tetragonal)晶體結構下的橫向光頻(TO)與縱向光頻(LO)振動模式間的偶合。不像立方結構中所發現的橫向光頻與縱向光頻間之巨大分裂，四方結構呈現正常的橫向與縱向振動模式之偶合關係。根據此發現，我們重新解釋了在鈦酸鉛摻雜鈣原子(Pb1xCaxTiO3)下所發現的橫向與縱向之光頻分裂行為。最後我們將討論實驗上所觀測到鈦酸鉛摻雜鈣原子所發生之結構相變的成因，並預測低溫下新結構相變之發生。 
英文摘要 
We investigate the Frenkel excitons in NiO and CoO, the standard prototype of Mott insulator, within the diagrammatic picture of densitydensity response function. Recently immobile excitons showing up in the Mott gap from dd excitations of NiO and CoO found by nonresonant inelastic Xray scattering possess strong qdependence and large binding energies. To study the local excitations, a new localized, symmetryrespecting, energyresolved Wannier function is developed to perform the calculation. A novel TDLDA+U method is proposed to solve the relevant BetheSalpeter equation and the attraction of electronhole pairs of corresponding realspace Wannier functions is well included. Strongly hybridized Frenkel excitons of NiO are found in the Mott gap via this approach and the local HartreeFock interaction reveals an interesting qdependence in NiO and CoO due to the effect of quantum interference of excitons.
Electronic structure calculations from first principles in density functional perturbation theory are made to elucidate the lattice dynamic properties of tetragonal PbTiO3. At low temperatures, the mechanical stability of ferroelectric PbTiO3 is identified by obtaining full calculated phonon dispersion curves and sound velocities. Most importantly, a normal coupling characteristic between transverse optical (TO) and longitudinal optical (LO) phonon modes are found in the stable tetragonal PbTiO3, unlike the giant LOTO splitting in cubic perovskites, as determined by studying correlation matrices. Based on this information, the essentials of the analysis of LOTO splitting are emphasized to examine the effect of doping in mixed Pb1xCaxTiO3. Finally, we discuss the mechanism of structural phase transition observed by experiment in Pb1xCaxTiO3 and predict a new phase transition (cubicorthorhombic) for cubic Pb0.5Ca0.5TiO3 at low temperatures.

論文目次 
Chapter 0 Introduction 1
Chapter 1 Linear Density Response 3
1.1 NonRelativistic Hamiltonian 3
1.1.1 The Lagrangian for manybody systems 3
1.1.2 Canonical momentum 3
1.1.3 Nonrelativistic Hamiltonian 4
1.1.4 Quantization of the electromagnetic field 4
1.1.5 Intrinsic spin in the magnetic field 5
1.1.6 BornOppenheimer approximation 5
1.1.7 Model Hamiltonian for electrons 6
1.2 NonResonant Inelastic Photon Scattering 7
1.2.1 Time dependent perturbation theory (TDPT) 7
1.2.2 Inelastic photon scattering 8
1.3 Linear Density Response 12
1.3.1 Densitydensity response function 12
1.3.2 Excitation energy of Nparticle system 14
1.3.3 Connection with dynamic structure factor 16
1.4 TimeDependent DensityFunctional Theory 18
1.4.1 Densityfunctional theory (DFT) 18
1.4.2 KohnSham response function 21
1.4.3 Meaning of KohnSham response function 23
1.4.4 Application to phonon calculation 25
Chapter 2 Excitons in NiO and CoO 28
2.1 Introduction 28
2.2 Geometric structure 30
2.3 Electronic structure 31
2.3.1 LDA+U method 31
2.3.2 Band structure 33
2.3.3 Wannier function 42
2.4 Excitons 45
2.4.1 Experimental results 45
2.4.2 Property of excitons 48
2.4.3 Angular dependence 50
2.4.4 Energy of exciton in NiO 54
2.4.5 Energy of exciton in CoO 69
2.5 Summary 83
Chapter 3 Phonons in PbTiO3 and Related Mixing Material 84
3.1 Introduction 84
3.2 Geometric structure 86
3.3 Computational method 88
3.4 Results and Discussion 89
3.4.1 Structural detail 89
3.4.2 Lattice dynamics of tetragonal PbTiO3 90
3.4.3 Dynamical effective charge analysis of PbTiO3 93
3.4.4 Doping effect in Pb1xCaxTiO3 103
3.4.5 Phase transitions in Pb1xCaxTiO3 105
3.5 Summary 107
Chapter 4 Conclusion 108
Bibliography 110
List of Tables
Table 2.1 Fractional coordinates of selected k points for bandstructure plots of NiO and
CoO 42
Table 3.1 Calculated (Type I, Type II (see in the text) and GGA[82]) and observed
(Expt.[82]) lattice constants (in a.u.), c/a ratios, and internal parameters (in
fractional coordinates along the z direction (u)) for tetragonal PTO. O1, O2 and
O3 denotes the oxygen atom on the yz, xz, and xy face of the unit cell,
respectively. 90
Table 3.2 Optimized internal parameters for tetragonal Pb0.75Ca0.25TiO3 corresponding to
experimental lattice constants. O1, O2 and O3 denotes the oxygen atom on the
yz, xz, and xy face of the unit cell, respectively. 90
Table 3.3 Elastic constants cμν in GPa; piezoelectric constants i eν in C/m2, and
dielectric constants ij ε (with phonon contribution) for tetragonal PTO.
Corresponding calculated data with respect to theoretical optimized lattice
constants are presented in parentheses. 92
Table 3.4 Born effective charges, the shortest TiO bondlength dTiO (in a.u.) and the
enthalpy difference (in meV) with respect to the cubic (C) phase for the
tetragonal (T) phase of tetragonal PTO at different lattice volumes. The
calculated value of hydrostatic pressure corresponding to each smaller cell
volume is also given for reference. 95
Table 3.5 Correlation matrix and mode effective charges for cubic and tetragonal PTO at
experimental lattice constants. The data in parentheses for cubic and tetragonal
phases are obtained from the literature[80] and theoretical optimized lattice
constants, respectively. 96
Table 3.6 Dielectric constant along caxis (ε 33(∞)) and Born effective charges ( Z* ) for
Pb1xCaxTiO3 and a fictitious tetragonal CaTiO3 (see in the text). All data are
obtained at experimental lattice constants. Effective charges Z* refer to atom
displacements parallel to caxis. The oxygen at the bottom of TiO5 in tetragonal
PTO is represented as O3 and the others are denoted by O1. 96
List of Figures
Fig. 1.1 KohnSham response function KS(x,x'; ) σσ χ ω , ' KS 0
σ σ χ ≠ = . 21
Fig. 1.2 Connection between manybody and KohnSham response functions. 24
Fig. 2.1 Rhombohedral structure of NiO (CoO). Antiferromagnetic order is constructed
from (111) planes composed of Ni (Co) atoms along the [111] direction in rocksalt
structure denoted schematically by small arrows. 30
Fig. 2.2 LDA+U band structure of NiO. The bands with sorbital character of (a) spinup
Ni1 (b) spindown Ni1 (c) spinup Ni2 (d) spindown Ni2 are highlighted by open
circles. 34
Fig. 2.3 LDA+U band structure of NiO. The bands with porbital character of (a) spinup
Ni1 (b) spindown Ni1 (c) spinup Ni2 (d) spindown Ni2 are highlighted by open
circles. 35
Fig. 2.4 LDA+U band structure of NiO. The bands with dorbital character of (a) spinup
Ni1 (b) spindown Ni1 (c) spinup Ni2 (d) spindown Ni2 are highlighted by open
circles. 36
Fig. 2.5 LDA+U band structure of NiO. The bands with sorbital character of (a) spinup O
(b) spindown O and porbital character of (c) spinup O (d) spindown O are
highlighted by open circles. 37
Fig. 2.6 LDA+U band structure of CoO. The bands with sorbital character of (a) spinup
Co1 (b) spindown Co1 (c) spinup Co2 (d) spindown Co2 are highlighted by
open circles. 38
Fig. 2.7 LDA+U band structure of CoO. The bands with porbital character of (a) spinup
Co1 (b) spindown Co1 (c) spinup Co2 (d) spindown Co2 are highlighted by
open circles. 39
Fig. 2.8 LDA+U band structure of CoO. The bands with dorbital character of (a) spinup
Co1 (b) spindown Co1 (c) spinup Co2 (d) spindown Co2 are highlighted by
open circles. 40
Fig. 2.9 LDA+U band structure of CoO. The bands with sorbital character of (a) spinup O
(b) spindown O and porbital character of (c) spinup O (d) spindown O are
highlighted by open circles. 41
Fig. 2.10 Selected k paths for bandstructure plots of NiO. 42
Fig. 2.11 Energy levels of dcharacter Wannier functions in NiO and CoO. 43
Fig. 2.12 Comparison between LDA+U within RPA and nonresonant inelastic Xray
scattering experiment for NiO at small momentum transfer (q~0.7Å1) at absolute
unit. The low energy part in panel (a) is enlarged in panel (b). 45
Fig. 2.13 Dynamical structure factors found by nonresonant inelastic Xray scattering
experiment for NiO and CoO. Highresolution (0.3 eV) measurements of (a) the
qdependence of the dd peak intensities for NiO in the [111] direction; (b) the
qorientation dependence in CoO at q=7 Å1. Panel (c) and panel (d) represent the
qorientation dependence for q=3.5 Å1 in NiO and q=3.75 Å1 in CoO,
respectively. 46
Fig. 2.14 Lowresolution (1.1 eV) nonresonant inelastic Xray scattering measurements
and LDA+U/RPA calculations of the dynamical structure factor for NiO and CoO:
(a) measurements along the [100] and [111] directions for NiO; (b) similar
measurements on CoO; (c) calculations for NiO along the [100] and [111]
directions; (d) similar calculations for CoO. 47
Fig. 2.15 Density fluctuation propagator in the manybody electron system. 48
Fig. 2.16 BetheSalpeter equation. 49
Fig. 2.17 A twoband ladder example for the BetheSalpeter equation. 49
Fig. 2.18 (a) qdependence of transition probability of e’g to eg for NiO (3.5Å1 at white
line). Averaging over cubic equivalent antiferromagnetic domain is imposed. (b)
and (c) are the polar plots of the calculated and measured dd spectral weights,
respectively. 50
Fig. 2.19 qdependence of transition probability of e’g to eg for pure cubic symmetry.
Averaging over cubic equivalent antiferromagnetic domain is imposed. 51
Fig. 2.20 qdependence of transition probability of ag to eg for NiO. Averaging over cubic
equivalent antiferromagnetic domain is imposed. 51
Fig. 2.21 (a) qdependence of transition probability of e’g to eg for CoO (3.5Å1 at white
line). Averaging over cubic equivalent antiferromagnetic domain is imposed. (b)
and (c) are the polar plots of the calculated and measured dd spectral weights,
respectively. 52
Fig. 2.22 qdependence of transition probability of e’g to ag for CoO. Averaging over cubic
equivalent antiferromagnetic domain is imposed. 52
Fig. 2.23 Wannier functions of eg states in (a) NiO and (b) CoO. Note the bulge distortion
for CoO (at arrow) compared to the nearly cubicsymmetry shape for NiO. 53
Fig. 2.24 Wannier functions of e’g states in (a) NiO and (b) CoO. Note the slightly less
nodal shape for CoO (at arrow) compared to the nearly cubicsymmetry shape for
NiO. 53
Fig. 2.25 Illustration of (a) Fock and (b) Hartree interaction diagrams in BetheSalpeter
equation. 55
Fig. 2.26 2.27 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the L0 case in NiO. The corresponding diagram is presented
inside the plot. 56 57
Fig. 2.28 2.30 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the Local case in NiO. The corresponding diagram is presented
inside the plot. 57 58
Fig. 2.31 2.35 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the Fock case in NiO. The corresponding diagram is presented
inside the plot. 59 61
Fig. 2.36 2.40 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the HartreeFock case in NiO. The corresponding diagram is
presented inside the plot. 61 63
Fig. 2.41 χ (q~7Å1,q~7Å1;ω ), the results of the L0 case weighting by the Fourier
transforms of Wannier functions at q~7Å1 for NiO. The zero intensity along
<100> direction reflects the nodal direction. 65
Fig. 2.42 χ (q~7Å1,q~7Å1;ω ), the results of the Local case weighting by the Fourier
transforms of Wannier functions at q~7Å1 for NiO. The zero intensity along
<100> direction reflects the nodal direction. 65
Fig. 2.43 χ (q~7Å1,q~7Å1;ω ), the results of the Fock case weighting by the Fourier
transforms of Wannier functions at q~7Å1 for NiO. The zero intensity along
<100> direction reflects the nodal direction. 66
Fig. 2.44 χ (q~7Å1,q~7Å1;ω ), the results of the HartreeFock case weighting by the
Fourier transforms of Wannier functions at q~7Å1 for NiO. The zero intensity
along <100> direction reflects the nodal direction. 66
Fig. 2.45 qdependence of the imaginary parts of χ for the L0 case in NiO with the
energies at (a) 6.81eV and (b) 7.74eV. Averaging over cubic equivalent
antiferromagnetic domain is imposed. 67
Fig. 2.46 qdependence of the imaginary parts of χ for the Local case in NiO with the
energies at (a) 1.65eV and (b) 1.95eV. Averaging over cubic equivalent
antiferromagnetic domain is imposed. 67
Fig. 2.47 qdependence of the imaginary parts of χ for the Fock case in NiO with the
energies at (a) 1.43eV and (b) 1.95eV. Averaging over cubic equivalent
antiferromagnetic domain is imposed. 68
Fig. 2.48 qdependence of the imaginary parts of χ for the HartreeFock case in NiO with
the energies at (a) 1.72eV and (b) 2.61eV. Averaging over cubic equivalent
antiferromagnetic domain is imposed. 68
Fig. 2.49 2.50 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the L0 case in CoO. The corresponding diagram is presented
inside the plot. 70
Fig. 2.51 2.53 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the Local case in CoO. The corresponding diagram is presented
inside the plot. 71 72
Fig. 2.54 2.61 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the Fock case in CoO. The corresponding diagram is presented
inside the plot. 72 76
Fig. 2.62 2.68 The real (dashed line) and imaginary (solid line) parts of polarization
propagator L for the HartreeFock case in CoO. The corresponding diagram is
presented inside the plot. 76 79
Fig. 2.69 χ (q~7Å1,q~7Å1;ω ), the results of the L0 case weighting by the Fourier
transforms of Wannier functions at q~7Å1 for CoO. 80
Fig. 2.70 χ (q~7Å1,q~7Å1;ω ), the results of the Local case weighting by the Fourier
transforms of Wannier functions at q~7Å1 for CoO. 80
Fig. 2.71 χ (q~7Å1,q~7Å1;ω ), the results of the Fock case weighting by the Fourier
transforms of Wannier functions at q~7Å1 for CoO. 81
Fig. 2.72 χ (q~7Å1,q~7Å1;ω ), the results of the HartreeFock case weighting by the
Fourier transforms of Wannier functions at q~7Å1 for CoO. 81
Fig. 2.73 qdependence of the imaginary parts of χ for the HartreeFock case in CoO
with the energies at (a) 1.37eV , (b) 2.32eV, (c) 2.81eV, and (d) 3.17eV. Averaging
over cubic equivalent antiferromagnetic domain is imposed. 82
Fig. 3.1 Cubic structure of PbTiO3. 86
Fig. 3.2 Tetragonal structure of PbTiO3. 86
Fig. 3.3 Tetragonal structure of Pb0.75Ca0.25TiO3. 87
Fig. 3.4 Primitive unit cell of cubic Pb0.5Ca0.5TiO3. 87
Fig. 3.5 Phonon dispersion curves of tetragonal PTO corresponding to Type I and Type II
calculations (see in the text) are shown as solid and dotted curves, respectively.
Open symbols (circles, triangles)[85] and solid symbols (circles[71],
diamonds[83], squares[84]) denote various experimental data. 91
Fig. 3.6 Sound velocities of longitudinal (L) and two shear waves (S1 and S2) of tetragonal
PTO in (a) ab and (b) ac planes. Shear wave polarizes nearly perpendicular
(parallel) to the plane is represented as S1 (S2). Brillouin scattering experiment
data[72] and calculations based on experimental and theoretical relaxed
geometries are plotted as dashed, solid and dotted lines, respectively. 93
Fig. 3.7 TO1 mode of cubic PbTiO3 with frequency 215i cm1. The direction of wave vector
q is pointed out on the righthand side of the plot. 97
Fig. 3.8 LO3 mode of cubic PbTiO3 with frequency 600.9 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 97
Fig. 3.9 TO2 mode of cubic PbTiO3 with frequency 63.0 cm1. The direction of wave vector
q is pointed out on the righthand side of the plot. 98
Fig. 3.10 LO1 mode of cubic PbTiO3 with frequency 62.7 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 98
Fig. 3.11 TO3 mode of cubic PbTiO3 with frequency 442.5 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 99
Fig. 3.12 LO2 mode of cubic PbTiO3 with frequency 414.1 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 99
Fig. 3.13 TO1 mode of tetragonal PbTiO3 with frequency 145.8 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 100
Fig. 3.14 LO1 mode of tetragonal PbTiO3 with frequency 188.7 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 100
Fig. 3.15 TO2 mode of tetragonal PbTiO3 with frequency 360.2 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 101
Fig. 3.16 LO2 mode of tetragonal PbTiO3 with frequency 443.2 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 101
Fig. 3.17 TO3 mode of tetragonal PbTiO3 with frequency 646.4 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 102
Fig. 3.18 LO3 mode of tetragonal PbTiO3 with frequency 784.8 cm1. The direction of wave
vector q is pointed out on the righthand side of the plot. 102
Fig. 3.19 Difference between square of A1(3LO) and A1(3TO) phonon frequencies for
tetragonal Pb1xCaxTiO3 from Raman experiment[68]. 103
Fig. 3.20 Eigendisplacement of the softest mode in cubic Pb0.5Ca0.5TiO3. 106 
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