FIRST-PRINCIPLES STUDY OF STRUCTURALAND RESPONSE PROPERTIES OF BARIUM

TITANATE PHASES

GOH EONG SHENG

UNIVERSITI SAINS MALAYSIA

2016

FIRST-PRINCIPLES STUDY OF STRUCTURALAND RESPONSE PROPERTIES OF BARIUM

TITANATE PHASES

by

GOH EONG SHENG

Thesis submitted in fulfilment of the requirementsfor the degree of

Master of Science

December 2016

ACKNOWLEDGEMENT

This Master thesis would not be completed successfully without the guidance and

help of many people. I am grateful for the opportunity given in this study; and through

the exposure in this research, my research skills have been greatly improved.

First and foremost, I would like to express my deepest appreciation to my supervi-

sors, Dr. Yoon Tiem Leong and Prof. Dr. Ong Lye Hock. Thanks to Dr. Yoon Tiem

Leong, my main academic supervisor, for initiating my foray into computational physics

and providing the required resources for my computations. Being a man full of the spirit

and adventurous, he allows full freedom for me to conduct my research, and bestows

me the opportunity to set up a computer cluster myself. Despite our relationship as

supervisor and students, Dr Yoon never utilize his status to strength his arguments, an

humble act that I greatly grateful of. On the other hand, Prof. Dr. Ong Lye Hock, my

co-supervisor, had been my mentor in my research journey, which I will certainly be

quite lost without his guidance. Constantly steering me towards the correct direction,

he ensures that I will be able to finish my Master on time. Despite being ready for

retirement, he readily extent his help in correcting my draft research paper and happily

shared his experiences in teaching and research. I would like to thank him again for

his meticulous efforts in explaining things to me, and for providing me with financial

assistance from his research grant.

The author also acknowledge the contributions of Universiti Sains Malaysia (USM)

and School of Physics. Thanks to USM for providing financial support to me through

Assoc. Prof. Ong’s RU grant, without which is impossible for me to continue my

study. I would also like to thank the School of Physics for providing the computer

ii

cluster resources for the computational work in order to complete this research work.

Thanks also to the government of Malaysia for sponsoring my tuition fees through the

MyMaster project.

My utmost gratitudes to all my colleagues in School of Physics, whose constant

encouragements and hard works serve as my inspirations. I appreciate the support of

my family, particularly my father and mother, for their sacrifice and patience during the

durations of my research. There are many more people I will like to thank, while not

listed here, for helping me during my difficulties.

I would to extend my thanks to the ABINIT development team, for offering such an

extensive open source software which suites my computational calculations. Thanks

also to those who had answered my queries in the forum, for helping me to solve the

various small but troublesome bugs and problems in ABINIT. Many of the structural

diagrams and graphs attached in this work were produced using the free 3D visualization

VESTA package by JP-Minerals and the Grace plotting tools.

iii

TABLE OF CONTENTS

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Abstrak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

CHAPTER 1 – GENERAL BACKGROUND AND LITERATUREREVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background of ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 First-principles ab-initio approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

CHAPTER 2 – DENSITY FUNCTIONAL THEORY . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Computation complexities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Electronic density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 The Hohenberg-Kohn theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Hohenberg-Kohn first theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Hohenberg-Kohn second theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Basic equation for DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

iv

2.6 Kohn-Sham method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.1 Non-interacting electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6.2 Kohn-Sham equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.3 Band structure energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Exchange correlation functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.1 Local density approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7.2 Generalized gradient approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 Self consistent method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

CHAPTER 3 – DENSITY FUNCTIONAL PERTURBATION THEORY . . 35

3.1 Basic background and review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Density functional perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 General formulation of perturbation theory in quantum mechanics 39

3.3.1(a) First-order perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1(b) Second-order perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Perturbation theory for Kohn-Sham formulation . . . . . . . . . . . . . . . . . . . . 46

3.3.3 Gauge freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.4 Explicit expressions of lowest order energy in DFPT . . . . . . . . . . . . . . . 52

3.3.4(a) First order energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.4(b) Second order energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Common types of perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.1 Phonons: atomic displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.2 Homogeneous macroscopic electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.3 Born effective charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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3.4.4 Static dielectric response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4.5 LO-TO splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

CHAPTER 4 – SPONTANEOUS POLARIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Polarization: microscopic perceptive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Modern theory of polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Berry phase approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.2 Wannier functions representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

CHAPTER 5 – COMPUTATIONAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1 Electronic structure package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 PAW potentials and XC functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Convergence studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4 Computation work flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4.1 Bulk structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4.2 Slab form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

CHAPTER 6 – BULK BARIUM TITANATE: GROUND STATE ANDRESPONSE PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.1 Structural properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3 Born effective charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4 Spontaneous polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.5 Phonon analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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CHAPTER 7 – BARIUM TITANATE SLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2 Computational methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3.1 Preliminary 3UC slab comparison tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3.2 Asymmetric six-unit-cell thick slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3.3 Comparison between slabs with different thickness . . . . . . . . . . . . . . . . . 131

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

CHAPTER 8 – CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

APPENDICES

Appendix A – JTH-LDA Cubic Phase Electronic Structures

Appendix B – GBRV-PBE Electronic Structures

Appendix C – Selection of input slab models

C.1 Computation complexity

C.2 Convergence studies

LIST OF PUBLICATIONS

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LIST OF TABLES

Page

Table 5.1 Valence states and matching radius of Ba, Ti and O PAWpotentials from JTH library

77

Table 5.2 Valence states and matching radius of Ba, Ti and O PAWpotentials from GBRV library

77

Table 6.1 Atomic position notations for the ferroelectric phases ofBaTiO3 in reduced coordinates.

86

Table 6.2 Lattice constants and bulk properties of BaTiO3 in cubicphase.

87

Table 6.3 Lattice constants and calculated bulk properties of BaTiO3in tetragonal phase. The notations follow that defined in Ta-ble 6.1.

88

Table 6.4 Lattice constants and calculated bulk properties of BaTiO3in orthorhombic phase. The notations follow that definedin Table 6.1.

89

Table 6.5 Lattice constants and calculated bulk properties of BaTiO3in rhombohedral phase. The notations follow that definedin Table 6.1.

89

Table 6.6 Band gap, valence band maximum (VBM) and conductionband minimum (CBM) of electronic structures of BaTiO3in cubic, tetragonal, orthorhombic and rhombohedral phaseswithin PBE XC potentials.

91

Table 6.7 Born effective charge of atoms of BaTiO3 in cubic phase. 100

Table 6.8 Born effective charge tensor of Ba and Ti of BaTiO3 in ferro-electric phase.

102

Table 6.9 Born effective charge tensor of oxygen atoms of BaTiO3 inferroelectric phase. The labels of the atoms correspond tothose defined in Table 6.1.

103

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Table 6.10 Spontatenous polarization (C/m2) of relaxed BaTiO3 intetragonal, orthorhombic and rhombohedral phase, both alongthe lattice vectors and polar directions. The polarization vec-tors for orthorhombic phase are along b and c lattice vectorswhereas rhombohedral values refer to each of the three latticevector direction. The polar axis of tetragonal, orthorhombicand rhombohedral are along [001], [011] and [111] respec-tively. For tetragonal phase the c-axis coincides with the polaraxis.

112

Table 6.11 Phonon modes and frequencies (cm−1) of BaTiO3 in cubic,tetragonal, orthorhombic and rhombohedral phases.

116

Table 6.12 Mode effective charge and contribution from each atom forthe T1u modes of the optimized cubic phase. Phonon modesare numbered according to the sequence in Table 6.11.

116

Table 6.13 Overlap matrix elements of eigenvectors of T1u TO and LOmodes of the optimized cubic phase. Phonon modes arenumbered according to sequence in Table 6.11.

117

Table 7.1 Interlayer displacements (∆di j) and rumpling (ri) of the topfour layers of a 3-unit-cell thickness slab with cubic phase andBaO terminations, in units of Å. Reference data is modified toconform to interlayer displacements and rumpling definitionsused in this work.

125

Table 7.2 Interlayer displacements (∆di j) and rumpling (ri) of the topfour layers of a 3UC slab with tetragonal phase and BaOterminations, in units of Å. Reference data is modified toconform to rumpling definitions used in this work.

126

Table 7.3 Ferroelectric distortion per layer (in units of Å) of relaxed3UC slab. Reference data is modified to conform to defini-tions used in this work.

126

Table 7.4 Interlayer displacements (∆di j) and rumpling (ri) of an asym-metric 6UC BaTiO3 slab with BaO and TiO2 terminations, inunits of Å. Layer sequence starts from outer region towards in-ner bulk region. Data corresponding to symmetric slabs withTiO2 terminations from literature is given for comparison.

129

Table 7.5 Ferroelectric distortion per layer (in units of Å) of an asym-metric 6UC BaTiO3 slab with BaO and TiO2 terminations.Layer sequence starts from outer region towards inner bulkregion. Data corresponding to symmetric slabs with TiO2terminations from literature is given for comparison.

129

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Table 7.6 Comparison of interlayer displacements (∆di j) and rumpling(ri) of the top half layers of 6UC, 8UC and 10UC slabs re-spectively, in units of Å. All the slabs are in tetragonal phaseand possess BaO (100) terminations.

132

Table 7.7 Comparison of ferroelectric distortion δFE per layer (in unitof Å) of the top half layers of 6UC, 8UC and 10UC slabsrespectively. All the slabs are in tetragonal phase and possessBaO (100) terminations.

134

Table C.1 Correspondence between k-point array (ngkpt) and numberof k-point in IBZ (nkpt)

163

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LIST OF FIGURES

Page

Figure 2.1 Comparison between direct application of Hohenberg-Kohntheorems and Kohn-Sham approach.

31

Figure 2.1(a) Direct application of Hohenberg-Kohn theorems 31

Figure 2.1(b) Kohn-Sham approach 31

Figure 2.2 Typical SCF iteration in a DFT calculation. 34

Figure 4.1 Ambiguous polarization in bulk solid (Sbyrnes321, 2011). (a)Periodic 1D charge alignment. (b) Upward polarization byselection of unit cell. (c) Downward polarization by selectionof another equivalent unit cell

68

Figure 5.1 Computational work flow of bulk phase calculations ofBaTiO3.

80

Figure 5.2 Computational work flow for BaTiO3 in slab form. 81

Figure 6.1 Unit cells of BaTiO3 in cubic phases 84

Figure 6.1(a) In 3D space 84

Figure 6.1(b) Projection onto yz plane 84

Figure 6.2 Unit cells of BaTiO3 in tetragonal phases 85

Figure 6.2(a) In 3D space 85

Figure 6.2(b) Projection onto yz plane 85

Figure 6.3 Unit cells of BaTiO3 in orthorhombic phases. The axes of thecompasses shown refer to the three lattice vectors respectively.The red, orange and blue vectors in the structure refer toCartesian x,y and z axes respectively.

85

Figure 6.3(a) In 3D space 85

Figure 6.3(b) Projection onto yz plane 85

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Figure 6.4 Unit cells of BaTiO3 in rhombohedral phases. The axes of thecompasses shown refer to the three lattice vectors respectively.The red, orange and blue vectors in the structure refer toCartesian x,y and z axes respectively.

85

Figure 6.4(a) In 3D space 85

Figure 6.4(b) Projection onto xy plane 85

Figure 6.5 Band structure (left) and density of states (right) of BaTiO3in cubic phase using JTH-PBE PAW potential.

92

Figure 6.6 Band structure (left) and density of states (right) of BaTiO3in tetragonal phase using JTH-PBE PAW potential.

93

Figure 6.7 Band structure (left) and density of states (right) of BaTiO3in orthogonal phase using JTH-PBE PAW potential. Theseparation distance between Y and X1 and A1 and T arenegligible.

94

Figure 6.8 Band structure (left) and density of states (right) of BaTiO3in rhombohedral phase using JTH-PBE PAW potential. Theseparation distance between B and Z, Q and F and P1 and Zare negligible.

95

Figure 6.9 Electron charge density contours, DOS and PDOS of BaTiO3cubic phase. The miller indices of the planes corresponds tothe structures shown in Fig. 6.1.

106

Figure 6.9(a) (200) plane. 106

Figure 6.9(b) (110) plane. 106

Figure 6.9(c) DOS and PDOS of Ti-d and O-p orbitals. 106

Figure 6.10 Electron charge density contours, DOS and PDOS of BaTiO3tetragonal phase. The miller indices of the planes correspondsto the structures shown in Fig. 6.2.

107

Figure 6.10(a) (200) plane. 107

Figure 6.10(b) (110) plane. 107

Figure 6.10(c) DOS and PDOS of Ti-d and O-p orbitals. 107

Figure 6.11 Electron charge density contours, DOS and PDOS of BaTiO3orthorhombic phase. The miller indices of the planes corre-sponds to the structures shown in Fig. 6.3.

108

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Figure 6.11(a) (200) plane. 108

Figure 6.11(b) (011) plane. 108

Figure 6.11(c) DOS and PDOS of Ti-d and O-p orbitals. 108

Figure 6.12 Electron charge density contours, DOS and PDOS of BaTiO3rhombohedral phase. The miller indices of the planes corre-sponds to the structures shown in Fig. 6.4.

109

Figure 6.12(a) (110) plane. 109

Figure 6.12(b) (111) plane. 109

Figure 6.12(c) DOS and PDOS of Ti-d and O-p orbitals. 109

Figure 7.1 BaTiO3 (100) slab supercell model with parallel in-planepolarization

121

Figure 7.2 Tetragonal BaTiO3 (100) slab surface: (left) BaO surface and(right) TiO2 surface. Polar axis is along the x-axis with z-axisnormal to the surfaces.

122

Figure A.1 Band structure (left) and density of states (right) of BaTiO3in cubic phase using JTH-LDA PAW potential.

153

Figure B.1 Band structure (left) and density of states (right) of BaTiO3in cubic phase using GBRV-PBE PAW potential.

155

Figure B.2 Band structure (left) and density of states (right) of BaTiO3in tetragonal phase using GBRV-PBE PAW potential.

156

Figure B.3 Band structure (left) and density of states (right) of BaTiO3in orthogonal phase using GBRV-PBE PAW potential. Theseparation distance between Y and X1 and A1 and T arenegligible.

157

Figure B.4 Band structure (left) and density of states (right) of BaTiO3 inrhombohedral phase using GBRV-PBE PAW potential. Theseparation distance between B and Z, Q and F and P1 and Zare negligible.

158

Figure C.1 Convergence studies with respect to ecut 162

Figure C.1(a) Convergence of total energy with respect to ecut 162

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Figure C.1(b) Convergence of surface energy with respect to ecut 162

Figure C.2 Convergence studies with respect to pawecutdg 162

Figure C.2(a) Convergence of total energy with respect to pawecutdg 162

Figure C.2(b) Convergence of surface energy with respect to pawecutdg 162

Figure C.3 Convergence studies with respect to nkpt 163

Figure C.3(a) Convergence of total energy with respect to nkpt 163

Figure C.3(b) Convergence of surface energy with respect to nkpt 163

Figure C.4 Convergence studies with respect to vacuum layer 164

Figure C.4(a) Convergence of total energy with respect to vacuum layer 164

Figure C.4(b) Convergence of surface energy with respect to vacuum layer 164

Figure C.5 Linear electron density and mean KS potential along z-direction

165

Figure C.5(a) Linear electron density along z-direction 165

Figure C.5(b) Mean KS potential along z-direction 165

xiv

LIST OF ABBREVIATIONS

1WF first order derivatives of the wavefunctions

2DTE second order derivatives of the wavefunctions

3DTE third order derivatives of the wavefunctions

BEC Born effective charge

BTO barium titanate

DFPT density functional perturbation theory

DFT density functional theory

DOS density of states

GBRV Garrity-Bennett-Rabe-Vanderbilt

GGA generalized gradient approximation

IBZ irreducible Brillouin zone

IFCs interatomic force constants

JTH Jollet-Torrent-Holzwarth

KS Kohn Sham

KDP potassium dihydrogen phosphate

LDA local density approximation

LDGT Landau-Devonshire-Ginzberg Theory

LO longitudinal optic

LTO lead titanate

xv

MEMS micro-electro-mechanical systems

NC norm-conserving

ONCVPSP optimized norm-conserving Vanderbilt pseudopotentials

PAW projector-augmented-wave

PBE Perdew-Burke-Ernzerhof

PDOS projected density of states

SCF self-consistent field

STO strontium titanate

TO transverse optic

WC Wu-Cohen

XC exchange correlation

xvi

LIST OF SYMBOLS

θ angle in degree

Ωcell unit cell volume

ao cubic lattice constant

B bulk modulus

e electron charge

Ecoh cohesive energy

E f fermi energy

P spontaneous polarization

Pα polarization in the αth direction

Pel electronic polarization

Pion ionic polarization

q wave vector

uiβ periodic displacement of the ith atom in the β th direction

Z∗ Born effective charge

xvii

KAJIAN PRINSIP-PERTAMA STRUKTUR DAN RESPONS SIFAT FASA

BARIUM TITANATE

ABSTRAK

Sifat keadaan asas BaTiO3 perovskite kristal dalam kedua-dua bentuk pukal dan

kepingan telah dikaji dengan menggunakan prinsip pertama Teori Fungsian Ketumpatan

(DFT) dan kaedah projektor gelombang imbuhan. Pengiraan kristal pukal BaTiO3

dalam semua empat fasa telah dilakukan dengan penghampiran ketumpatan tempat-

an (LDA) dan penghampiran kecerunan umum (GGA). Dua keupayaan PAW yang

berbeza berdasarkan set data yang diperkembangkan oleh Jollet-Torrent-Holzwarth

(JTH) dan Garrity-Bennett-Rabe-Vanderbilt (GBRV) telah digunakan dalam pengiraan.

Penekanan bahagian pertama diletakkan pada sifat elektronik, struktur dan sifat tindak

balas semua fasa BaTiO3 dengan menggunakan fungsian korelasi pertukaran (XC) dan

keupayaan PAW yang berbeza. Pengiraan dengan GGA didapati menghasilkan sifat

struktur yang lebih dekat dengan nilai eksperimen daripada LDA. Ciri-ciri getaran

yang diramal dengan teori fungsian ketumpatan usikan adalah konsisten dengan kajian

teori dan eksperimen terdahulu. Pengutuban spontan tiga fasa ferroelektrik yang dikira

dengan menggunakan teknik fasa Berry berbeza secara ketaranya dibandingkan dengan

nilai eksperimen tetapi selaras dengan keputusan teori lain. Fluktuasi parameter kekisi

yang dikira menggunakan fungsian XC yang berbeza secara langsung mempengaruhi

pengutuban spontan yang dikira. Kepingan BaTiO3 simetri dengan ketebalan yang

berbeza (6, 8 dan 10-unit-sel tebal) dan penamatan permukaan BaO (100) telah dikira

dengan menggunakan DFT dalam rangka fungsian PBE-GGA dan keupayaan JTH-PAW.

xviii

Kesan-kesan pengenduran permukaan kepada perubahan parameter struktur telah dikaji.

Berbeza dengan penemuan sastera dahulu, semua jarak antara lapisan individu telah ber-

kembangan kecuali jarak antara permukaan dan lapisan yang berikutnya. Perbandingan

ketiga-tiga kepingan dengan pelbagai ketebalan mendedahkan magnitud sesaran antara

lapisan bagi kepingan dengan 8-unit-sel tebal adalah rendah daripada dua kepingan

lain. Had penembusan praktikal kesan pengenduran permukaan dianggarkan pada lima

lapisan daripad permukaan BaO (100), dan sesaran ferroelektrik satar setiap lapisan

didapati tidak dipengeruhi olek ketebalan kepingan. Setara dengan penemuan sastera

dahulu, kepingan simetri BaTiO 3 bersudut empat dengan penamatan permukaan BaO

(100) diramalkan memiliki nilai pengutuban spontan satar yang lebih kecil berbanding

dengan kes pukal. Ini menunjukkan kewujudan peranan faktor dominan lain dalam

penindasan pengutuban kerana nilai yang dikira dengan teknik fasa Berry adalah terlalu

kecil berbanding dengan sesaran ferroelektrik lapisan TiO2.

xix

FIRST-PRINCIPLES STUDY OF STRUCTURAL AND RESPONSE

PROPERTIES OF BARIUM TITANATE PHASES

ABSTRACT

Ground state properties of BaTiO3 perovskite crystal, both in bulk and slab form,

were studied using first-principles density functional theory (DFT) using the projector-

augmented wave methods. The computations on bulk BaTiO3 crystals, in all four

phases, were performed within the framework of local-density approximation (LDA)

and generalized-gradient approximation (GGA). Two different PAW potentials based

on the datasets developed by Jollet-Torrent-Holzwarth (JTH) and Garrity-Bennett-Rabe-

Vanderbilt (GBRV) are employed in the calculations. The emphasis of the first part

is placed on the electronic, structural and response properties of all phases of bulk

BaTiO3 using different exchange correlation (XC) functionals and PAW potentials. It

is observed that calculations with GGA yields structural properties much closer to re-

ported experimental values than LDA. The vibrational properties predicted from density

functional perturbation theory are consistent with experimental and previous theoretical

studies. Spontaneous polarization for the three ferroelectric phases, computed using

Berry phase, differ considerably from experimental values but are consistent with other

theoretical results. The fluctuation of lattice parameters computed using different XC

functionals directly affects the computed spontaneous polarization. Symmetric BaTiO3

slabs of different thickness (6, 8 and 10-unit-cell thick) with BaO (100) surface ter-

minations were computed using DFT within PBE-GGA functionals using JTH-PAW

potentials. The effects of surface relaxation on the structural parameters changes are

xx

studied. In contrast to previous literature findings, all interlayer spacings between the

individual layers were experiencing an expansion except of the spacing between the

surface and subsequent deeper layer. A comparison across the three slabs with varying

thickness revealed the expansion in interlayer spacings in the 8-unit-cell thick slab is

smaller than the other two slabs. The practical penetration limit of the surface relaxation

effects was estimated to be five layers from the BaO (100) surface, and the computed

in-plane ferroelectric displacements of each layer were relatively independent of the

slab thickness. In agreement with previous literature findings, symmetric tetragonal

BaTiO3 slab with BaO (100) surface terminations was predicted to possess a much

smaller value of in-plane spontaneous polarization than the corresponding quantity in

the bulk case. This suggests the role of another dominant factor in suppressing the

polarization. For the computed value using Berry phase formalism, it is too small with

regard to ferroelectric displacements in the TiO2 layers.

xxi

CHAPTER 1

GENERAL BACKGROUND AND LITERATURE REVIEW

1.1 Introduction

Ferroelectricity is a classic scientific discovery that is continued to be studied to this

day since its discovery in 1921 (Valasek, 1921). With the intensive research interest

it is receiving, huge advancements have been made in the area from the identifica-

tion of numerous ferroelectric materials to novel experimental works and theoretical

understandings in various attempts to understand this phenomenon. In the simplest

terms, ferroelectricity is the existence of spontaneous polarization in a material which

can be reversed by the application of an electric field; the oscillations of electric field

vectors will result in a polarization hysteresis loop. Nowadays, ferroelectricity have

been revealed to exhibit complex interplay of dielectric and dynamical properties. In

the course of history of ferroelectricity, one of the major event is the discovery of

ferroelectricity in the perovskite crystals, where barium titanate (BTO) is the most well

known example.

The objective of this thesis is to investigate the structural and response properties

of the ferroelectric ceramic BTO from first-principles, and to make detailed studies

and comparisons between the four phases of BTO both from the theoretical and com-

putational perceptive. The thesis is mainly divided into 3 different part: theoretical,

computational and results sections, which will be further discussed later.

1

1.2 Background of ferroelectricity

The subject of ferroelectricity began with the studies of sodium potassium tartrate

tetrahydrate (Rochelle Salt) by Valasek (1921), whose works leads to the coining of the

term ”ferroelectricity”, although it should be mentioned that the associated anamolous

dielectric response had been identified by Pockels (1894) earlier as mentioned by Cross

and Newnham (1987).

Nevertheless, it is found that Rochelle Salt is, among other major obstacles, struc-

turally complex with 112 atoms per unit cell, which poses difficulties in investiga-

tion in that era. The discovery of ferroelectricity in potassium dihydrogen phos-

phate (KDP) (Busch & Scherrer, 1987) leads to the progress in understanding the

phenomena in ferroelectrics, despite having constraint that ferroelectricity only mani-

fests at temperature below −150 C. The origin of ferroelectricity in KDP lies in the

ordering of hydrogen bonds at the corners of phosphate. The presence of hydrogen

bond is then seen as a necessary condition for polar instabilities to occur in ferroelectric

materials.

When the ferroelectric properties of BTO were found, it was a significant break-

through for BTO is a simple oxide material belongs to the perovskite family. The

members of perovskite family ABO3 have the metallic A elements occupying cubic

lattice points, where each cubic unit cell in turns contains an embedded octahedron

having oxygen at its vertex and another metallic B element at its centre. Contrary

to KDP, BTO in cubic structure is paraelectric and it exhibits ferroelectric phase in

tetragonal, orthogonal and rhombohedral structures. It is natural that the then research

interests extend to other members of the perovskite family due to simplicity of the

2

structure of the ABO3 family.

However, not all the phase transition occuring in ABO3 perovskite structures are

ferroelectric in nature, with some members exhibit non-polar structural phase transi-

tions. One such perovskite structure is strontium titanate (STO), which undergoes an

antiferrodistortive transition from cubic to tetragonal structure (Cowley & Shapiro,

2006). The particular phase transition in the form of relative rotations between the

oxygen octahedrals in adjacent cells, in other words a Brillouin zone boundary type

displacement.

Following the successes of experimental works and discovery, it is inevitable that

the theoretical efforts to understand ferroelectricity were carried out. Major contribu-

tions were made by several prominent figures, including but not limited to Anderson

(1960); Cochran (1960); Slater (1950) and Devonshire (1949, 1951). The role of the

competition of long-range coulomb forces and short-range local forces in determining

the ferroelectric instability, considered as conventional theoretical explanation nowa-

days, was proposed by Slater (1950). It is theorized that the long-range dipolar force

associated with the Lorentz field is in competition with the local short-range forces

which favour the high symmetry configuration, such as the paraelectric cubic phase of

BTO. The structural phase transitions affect the balance of these two competing forces,

determining which of the two forces will prevail and thus consequently the occurrence

of ferroelectric instabilities. This explanation underlies the proposal of “displacive” type

of phase transition for the perovskites, in contrast with the “order-disorder” description

of phase transition in KDP-type. Another breakthrough occurs a decade later in 1960,

when Cochran (1960), and also Anderson (1960) in another independent research,

3

incorporate the lattice dynamics in the description of ferroelectric phase transition. The

formulation of soft-mode in the description of displacive phase transition is made by

taking one of the lattice mode as variable. This dynamical model extends the picture

of competing large-range and short-range force by considering it as the origin of the

softening of a particular lattice mode.

On the other hand, the theoretical model on the macroscopic level using thermody-

namic theory had been developed by Devonshire (1949, 1951). Devonshire’s model is a

phenomenological model built on the evaluation of free energy from the elastic, dielec-

tric, structural and thermal properties of BTO. The model is based on the earlier work

by Landau and Ginzberg and the resulting theory is now called Landau-Devonshire-

Ginzberg Theory (LDGT). With the later inclusion of crystal lattice dynamics in the

understanding of ferroelectric phase transition by Cochran (1960) and Anderson (1960),

the static phenomenological description by Devonshire was linked with atomistic de-

scriptions through atomic displacements and elementary crystal excitations.

Based on the foundational works by the pioneers, further experimental works and

theoretical models were devised, during which the field of ferroelectricity advanced

rapidly. Semi-empirical models were being used at this stage, following the realization

of the roles of lattice dynamics and soft mode in ferroelectricity. With the advent of

crystallography and spectroscopy methods and apparatus such as neutron scattering,

Raman measurement, Rayleigh scattering, and conventional X-ray diffraction and scat-

tering, the properties of soft modes of ABO3 materials were studied extensively. The

experimental data were studied and analysed by fitting into shell models. The work

of Migoni, Bilz, and Bäuerle (1976), leading to the “polarizability model” later, sug-

4

gested the non-linearity and anisotropy of polarizability of oxygen atoms as the origin of

ferroelectric behaviour of perovskites. In particular, the anisotropy of polarizability of

oxygen atoms was suggested to be associated with the hybridization between O-p states

and transition metal d-states. This explanation was supported by the first-principles

calculations, as will be discussed later.

Despite the simplicity of the perovskite structure, the complexity of ferroelectric

phase transitions were unexpectedly higher as more and more experimental data and

different theoretical models were available. One of the most distracting, while intrigu-

ing, observation was that the phase transition is not purely displacive as envisaged

by Cochran (1960), but somehow contains a mixture of displacive and order-disorder

character especially near the transition temperature. In order to further investigate

the properties of perovskite, the traditional analytical analysis are complemented and

extended by various numerical computation methods with the advent and prevalence of

more powerful and much cheaper computing machines.

1.3 First-principles ab-initio approach

First-principles method, as performed by the density functional theory (DFT) (Ho-

henberg & Kohn, 1964; Kohn & Sham, 1965), is based on the established law of physics

unlike an empirical model requiring fitting of experimental data. Since the introduction

of the Noble prize winning DFT in about 1964, the properties of condensed matter

system can be investigated at the atomistic level by solving the fundamental equations

of quantum mechanics. While the formulation of DFT is exact in itself, its practical

application is hindered by the unknown exchange and correlation terms, and the amount

5

of numerical computation is restricted by the limited efficient computing resources. The

application of DFT is observed to increase exponentially during the last two decades in

which computational power has improved dramatically leading to emergence of new

theoretical computing method.

The capability of DFT is well demonstrated in an influential work by Cohen (1992)

on the origin of ferroelectricity in perovskite oxides. The ferroelectric behaviours of

perovskites were not completely understood, where perovskites which are structurally

similar but chemically different exhibit different behaviours. One of the obvious

discrepancy is between BTO and lead titanate (LTO) where both have similar unit cell

volumes, but BTO undergoes three structural phase transitions (from cubic to tetragonal,

orthogonal and rhombohedral), whereas LTO has only one phase transition from cubic

to tetragonal phase. Using electronic structures calculations with the local density

approximation (LDA), Cohen managed to show that hybridization between Ti 3d states

and O 2p states are essential for ferroelectric, which is in line with that suggested

by Migoni et al. (1976). The different structural phase transitions between the two

materials are resolved by noting that LTO tetragonal phase contains large strains from

the hybridization of Pb and O states, whereas in BTO the Ba-O interaction is almost

ionic in nature, causing BTO to prefer a rhombohedral structure as its most stable form.

Unfortunately, the macroscopic polarization, a fundamental quantity for ferroelectric

material, cannot be directly computed using the conventional definition of polarization.

This difficulties had been resolved by King-Smith and Vanderbilt (1993); Resta (1994);

Vanderbilt and King-Smith (1993) with the modern theory of polarization. It is realized

that polarization is quantum phenomenon which cannot be expressed in the classical

6

definition of charge dipole per unit space. Rather than the absolute value of polarization,

the difference in polarization with respect to a referenced state is found to be the more

basic quantity. In the new approach, the ionic polarization is still derived using the

established electromagnetic theory, but the remaining electronic part of the polarization

is obtained from the Berry phase of the electronic wavefunctions. The Berry phase

formalism had been implemented in the framework of DFT.

The functionality of DFT is further extended to calculate additional response prop-

erties apart from the structural properties through the use of linear response theory.

Various physical properties can be computed from the second derivatives of total en-

ergy with respect to different perturbations, where the perturbations can be phonons,

static homogeneous electric field or strain. The second derivatives of total energy are

collectively called the response functions. The linear responses connected to derivatives

of energy, implemented within density functional perturbation theory (DFPT), include

phonon dynamical matrices, dielectric tensor, Born effective charges, elastic constant,

internal strain and piezoelectricity constant. Non-linear response connected to the

third-order derivatives of the energy can also be calculated, which is an extension to the

linear response method.

1.4 Thesis outline

This thesis is divided into two main portions: i) theoretical and computational

methods part and ii) results and discussion part. Excluding this chapter which is

concerned with general background and basic existing literature review, the first part

covers from Chapter 2 to Chapter 5 whereas the second part of this thesis covers two

7

chapters: Chapter 6 and Chapter 7. The last Chapter 8 is about the conclusion made

from interpretations of the obtained results.

The main computation methods used in this work, which are density functional the-

ory (DFT) and its extension, density function perturbation theory (DFPT), are discussed

in Chapter 2 and Chapter 3 respectively. The response properties are calculated by the

use of DFPT, of which basic principles are discussed in Chapter 3. Chapter 4 concerns

with the computation method for spontaneous polarization, an essential quantity in a

ferroelectric. The polarization from a microscopic perceptive is discussed, including the

limitations of conventional definition of macroscopic polarization leading the formula-

tion of modern theory of polarization. The computational flow chart is then illustrated

in Chapter 5. The detailed aspect of computations in this work including the potentials,

XC functionals and various convergence parameters are given. Both computational

work flows for the bulk and slab calculations are given.

The results of computations for the bulk BaTiO3 are presented in Chapter 6. Struc-

tural properties and electronic structure are first given for bulk BaTiO3 of all four phases

with comparisons to those of existing literatures, in addition to quantities essential for

the characterization of a ferroelectric such as the Born effective charge, spontaneous

polarization and the phonon modes and frequencies. On the other hand, Chapter 7

contains the results from structural relaxation of BaTiO3 of various thickness. The

polarization variation with respect to variation in three different slab thickness is then

presented. Finally, the last Chapter 8 is about the conclusions derived from this work

and some possible future works are suggested.

8

CHAPTER 2

DENSITY FUNCTIONAL THEORY

2.1 The Schrödinger Equation

The DFT can be understood through a review of the quantum mechanics of a many

body system. In most cases the ultimate aim of solid state physics and quantum chem-

istry is the solution of the Schrödinger equation. Consider the basic time-independent,

non-relativistic Schrödinger equation:

HΨ = EΨ, (2.1)

where Ψ is the wave function of some collection of atoms in a well-defined boundary

region. The time-independent Schrödinger equation describes the evolution of energy

as a function of position of the constituting atoms, subjected to a background potential

and inter-particles interaction described by the Hamiltonian of the system, H. In the

description of the quantum mechanics, the position of an atom is not a absolute unit,

but rather the positions of both nucleus and its electrons.

Subjected to the charge neutrality of an atom, the forces acting on the nucleus and

electrons are of the same order of magnitude. Consequently the changes in momen-

tum due to these forces must be about the same. Considering the huge differences

between the mass of nucleus and electrons, the nucleus must have much smaller velocity

compared to electrons due to its relative massive size. It follows that the motions of

nucleus and electrons have different time scales, and that it can be assumed that at

9

the time scale of the nucleus the electrons will relax to their lowest energy (ground

state) almost instantly. Hence, the solution of a quantum mechanical system can be

divided into nucleus and electrons components. The separation of motions of nuclear

and electrons into two mathematical problems is known as the Born-Oppenheimer

approximation (Born & Oppenheimer, 1927).

Born-Oppenheimer approximation thus enables us to solve the equations that de-

scribe the electron motion for fixed positions of the atomic nuclei. The eigenfunction of

the Hamiltonian is then assumed to take the following form:

Ψ(ri ,Rα) = ψ (ri ;Rα) ·ΦRα , (2.2)

where ri is the position of ith electron and Rα is the position of αth nuclei. ψ is

the electronic wave function whereas Φ is that of the nuclei part. From Eq. (2.2),

ψ (ri ;Rα) is a wave function dependent only on ri with Rα as parameters. The

ground state energy can then be expressed as a function of positions of the nuclei

E (R1, . . . ,RM), where M is the number of nuclei. The resulting energy function is

known as the adiabatic potential energy surface of the system, describing the changes

in energy with respect to positions of atoms.

Following the separation of nuclear and electronic motion, Eq. (2.1) can be simpli-

fied into

Hψ = Eψ, (2.3)

ψ is a set of quantum mechanical solutions of the electronic wave function correspond-

ing to the Hamiltonian H. Each of the solution ψn is associated to their eigenvalue En,

10

which is a real number. In the case of a system with multiple nuclei with their respec-

tive electrons interacting with each other, the minimal description of the Schrödinger

Equation, which ignores the contributions from the spins of electrons, will be:

[− h2

2m

N

∑i=1

∇2i +

N

∑i=1

Vext(ri)+N

∑i=1

∑j<i

U(ri,r j)

]ψ = Eψ, (2.4)

where m is the mass of electron. The first term in the bracket is the kinetic energy

of the electrons. V is the external potential energy, mainly due to the interaction

between electrons and the electrostatic potential of the nuclei, which are fixed in

position according to the Born-Oppenheimer approximation. The last term U refers to

the potential energy due to interaction between the electrons themselves. In the context

of Eq. (2.4), E corresponds to the ground state energy of the electrons of a particular

instant of configuration of the system.

2.2 Computation complexities

For a many-body system, the electronic wave function is shown to be a function of

the spatial coordinates of each of N electrons ψ = ψ(r1, . . . ,rN). The implication is

that for a N-electrons system there are 3N variables to be determined. The number of

electrons N is considerably larger than the number of nuclei M. The wave function is

thus a 3N dimensional function, which makes the solving of the Schrödinger equation

impractical except for a few exceptionally small systems. The ab-initio approach

assumes that the wave function of the system can be decomposed into combinations of

single individual electron wave functions. The wave function ψ is then approximated as

ψ = ψ1(r)ψ2(r) . . .ψN(r), a product of one-electron wave functions of the constituting

11

electrons. The resulting expression is known as Hartree product.

An examination of the expression of the Hamiltonian of Eq. (2.4) will reveal that

the third terms U is the main obstacle to solving the equation. The electron-electron

interactions is expressed as the summation of the pair-wise inter-electron interaction

potential. The presence of this electron-electron interaction term makes the electronic

wave function a coupled equation. In other words, the individual wave function ψi(r)

cannot be defined without considering the positions of all other electrons at the same

time. This indicates exceptional difficulties in solving the wave function of a many-body

system, due to the presence of terms corresponding to interacting electrons.

It should be noted that the wave function is dependent on the positions of electrons,

the variables themselves are not exactly defined as a consequence of the position-

momentum uncertainty theorem. This make the definition of wave function of a system

of electrons at particular exact coordinates meaningless. According to the Copenhagen

interpretation, the measurable quantity is the probability that the N electrons are at a

particular set of position r1,r2, . . . ,rN instead, and this probability is proportional to

ψ∗(r1, . . . ,rN)ψ(r1, . . . ,rN). Coupling with the notion of indistinguishable electrons

in quantum mechanics, where electrons in a system are considered as identical, the

interested quantity is the probability of a set of electrons at locations r1,r2, . . . ,rN

without considering the order of elements in the set.

2.3 Electronic density

While the wave function contains all possible information of a system, its complexity

is overwhelming; a 3N-dimensional function (4N-dimensional if spin of electron is

12

condidered). The problem shall be resolved through the use of electronic density, as

we have no direct use of the wave function. The solution can be brought back to

3-dimensional space, as will be shown later.

In quantum mechanics, electronic density is the probability of occupation of an

electron at an infinitesimal element of space surrounding a given point; a scalar quantity

depending upon three spatial variables. The definition of electron density is clear cut

for a system of one electron:

n(r) = |ψ(r)|2, (2.5)

which is known as Born’s statistical interpretation, absolute square of wave function.

This simple probabilistic interpretation does not directly hold for an ensemble of

electrons. Electron density is a 3D function about the expectation value of the density

of electrons. Using the conventional interpretation of observables from wave function,

we can write:

n(r) = 〈ψ|n(r)|ψ〉, (2.6)

where n(r) is an operator referring to electron number density.

Taken into account that electrons are point particles, the electronic density is defined

to be

n(r) =N

∑i=1

δ (ri− r), (2.7)

where the direct delta function is used and ri is the position operator for the ith electron.

The operator n(r) is then the summation of density of electron i at r over the number of

electrons.

13

The expectation value of the electronic density, without considering electron spin,

is then:

n(r) =∫· · ·∫

ψ∗(r1,r2, . . . ,rN)n(r)ψ(r1,r2, . . . ,rN)dr1 dr2 . . . drN

=N

∑i=1

∫· · ·∫

ψ∗(r1,r2, . . . ,rN)δ (ri− r)ψ(r1,r2, . . . ,rN)dr1 dr2 . . . drN

=N

∑i=1

∫· · ·∫

ψ∗(r1,r2, . . . ,rN)δ (r1− r)ψ(r1,r2, . . . ,rN)dr1 dr2 . . . drN

=N

∑i=1

∫· · ·∫

ψ∗(r,r2, . . . ,rN)ψ(r,r2, . . . ,rN)dr2 . . . drN

= N∫· · ·∫|ψ(r,r2, . . . ,rN)|2 dr2 . . . drN . (2.8)

The substitution of ri by r1 in the third step of Eq. (2.8) is a consequence of Pauli

exclusion principle; all electrons are indistinguishable that an arbitrary electron can be

selected. The presence of N is from the sum over identical numbers.

Eq. (2.8) shows that the electronic density at a particular point reduces to the density

probability of a single electron, taken into account all possible configurations of the rest

of electrons. The constant number of electrons N ensures that the electronic density

over all space is preserved, which is equal to the N itself. This can be shown by an

integration of Eq. (2.8) over all space:

∫n(r)dr = N. (2.9)

In Hatree Fock and DFT method the electronic density is typical expressed in terms

of one-electron representation. Under the assumption of non-interacting electrons, the

14

total wave function can be approximated as a Slater determinant for N electrons. In this

case, Eq. (2.8) can be simplified to, in terms of orbital wave functions and assuming

doubly occupied spatial orbitals,:

n(r) = 2N/2

∑i|ψi(r)|2, (2.10)

where the density of electron at a point is the summation of the squares of the orbital

wave functions. The presence of the factor 2 takes into account that the fact that

two electrons with different spin can occupy the same orbital, as stated by the Pauli

exclusion principle. The density of electrons thus encodes information obtainable from

the complex 3N dimensional solution of the Schrödinger equation, which is exploited

by the method of density functional theory.

2.4 The Hohenberg-Kohn theorems

The remarkable achievement of DFT lies in its ability to reduce the complexities

of many-body problem back to normal 3-dimensional space, without having to deal

with many-electron state directly. The core principles behind DFT is stated by the

Hohenberg-Kohn theorems (Hohenberg & Kohn, 1964).

2.4.1 Hohenberg-Kohn first theorem

Hohenberg-Kohn first theorem: The external potential Vext(r) is uniquely de-

termined by the corresponding ground-state electronic density, to within an additive

constant.

Before the proof is given, the definition of the external potential Vext(r) needs

15

to be first given based on the previous discussion. The problem concerned here is

a simplified system of N electrons under a static external potential. Following the

Born-Oppenheimer approximation, the static external potential Vext(r), assumed to be

mainly the electrostatic Coulomb potential imposed by the surrounding nuclei, is

Vext(r) =−∑α

Zα

|r− rα |. (2.11)

In the context of Eq. (2.4), the Hamiltonian operator H is made up of three parts:

H = T + Vext +U , where T is the kinetic energy operator, Vext is the potential energy

operator and U is the interaction energy operators. The potential energy operator is then

Vext = ∑iVext(ri).

Let n0(r) be the non-degenerate ground state density of N electrons in the potential

Vext(r), with the wave function ψ0 and energy E0. Assume a second external potential

V ′ext(r), not equal to Vext(r) + constant, giving rise to a ground state ψ ′ not different

from ψ0 with a phase factor, but having the same electron density n0(r). Then the

ground state energy can be expressed as

E0 =〈ψ0|H0|ψ0〉= 〈ψ0|T +Vext +U |ψ0〉 ,

E ′ =⟨ψ′∣∣H ′∣∣ψ ′⟩= ⟨ψ ′∣∣T +V ′ext +U

∣∣ψ ′⟩ .

The T and U are the same for all N electrons systems as evident in the formulation

of the Hamiltonian operator, so the a state ψ is completely determined by N and Vext(r).

Taking advantage of indistinguishability of electrons again, it is observed that the

16

expectation value of external potential operator is:

〈ψ|Vext|ψ〉

=∫· · ·∫

ψ∗(r1,r2, . . . ,rN)

N

∑i=1

Vext(ri)ψ(r1,r2, . . . ,rN)dr1 dr2 . . .drN

=N

∑i=1

∫· · ·∫

ψ∗(r1,r2, . . . ,rN)Vext(r1)ψ(r1,r2, . . . ,rN)dr1 dr2 . . .drN

= N∫

Vext(r1)

[∫· · ·∫

ψ∗(r1,r2, . . . ,rN)ψ(r1,r2, . . . ,rN)dr2 dr3 . . .drN

]dr1

=∫

Vext(r1)n0(r1)dr1

=∫

Vext(r)n0(r)dr .

Since ψ0 is defined to be non-degenerate, the application of the variational principle,

which E0 is the minimal energy, will give rise to the following inequality:

E0 <⟨ψ′∣∣H0

∣∣ψ ′⟩= ⟨ψ ′∣∣H ′∣∣ψ ′⟩+ ⟨ψ ′∣∣(H0− H ′)∣∣ψ ′⟩

= E ′+∫

n0(r)[Vext(r)−V ′ext(r)

]dr . (2.12)

Similarly for the second Hamiltonian H ′,

E ′ < 〈ψ0|H ′|ψ0〉= 〈ψ0|H0|ψ0〉− 〈ψ0|(H0− H ′

)|ψ0〉

= E0−∫

n0(r)[Vext(r)−V ′ext(r)

]dr . (2.13)

17

Adding Eq. (2.12) and Eq. (2.13) together results in the contradiction:

E0 +E ′ < E ′+E0.

The assumption that the second potential V ′ext(r), which is not equal to Vext(r)+constant

but have the same electron density n0(r) must be wrong.

The ground-state density hence determine, to within a constant, the external potential

term in the Schrödinger Equation. Since the Hamiltonian is determined only by the

external potential and number of electrons (which is also determined by the electronic

density, see Eq. (2.9)), the electron density n(r) implicitly determines all properties of

the system, including the wave function ψ .

Since the wave function, and hence all the properties of the system, is a functional

of electronic density, it can be asserted that the kinetic energy and electron-electron

interaction energy are functionals of the density as well: T [n] and U [n]. Collectively

these two terms are grouped together into

F = T +U , (2.14)

due to their independence of the potential and determined only by the forms of T and

U ; they are universal for a N-electrons system.

The total energy of the system can be expressed in terms of the density in principle:

EV [n] = T [n]+U [n]+V [n] = F [n]+∫

V (r)n(r)dr , (2.15)

18

where V is an arbitrary external potential in the general case.

2.4.2 Hohenberg-Kohn second theorem

Hohenberg-Kohn second theorem: The electron density that minimizes the energy

of the overall functional is the exact ground state density.

The proof of this theorem is straightforward by the use of the variational principle.

Let E0 be the ground state energy for N-electrons system in the external potential V (r)

with ground state density n0(r) in the Hamiltonian H. By the first theorem, an arbitrary

v-representable density n(r), which is a density that is ground state of an external

potential, determines its own V (r) and therefore a different wave function |ψ ′〉.

By variational principle,

⟨ψ′∣∣H∣∣ψ ′⟩> 〈ψ|H|ψ〉⟨

ψ′∣∣F∣∣ψ ′⟩+ ⟨ψ ′∣∣V ∣∣ψ ′⟩> 〈ψ|F |ψ〉+ 〈ψ|V |ψ〉F [n]+

∫V (r)n(r)dr > F [n]+

∫V (r)n0(r)dr

EV [n]> EV [n0] = E0. (2.16)

By the Hohenberg-Kohn theorems, the problem of solving the Schrödinger equation

reduces to the minimization of the energy functional EV [n]. In non-degenerate case the

ground state density corresponds to a unique ground state wave function, which is not

true for the general degenerate case. The generalization to the degenerate case can be

done, but will not covered in our scope of work.

19

2.5 Basic equation for DFT

Hohenberg-Kohn second theorem establishes a minimum principle for the energy

functional concerning electronic density. Specifically, EV [n] (see Eq. (2.15)) needs to

be minimized with the underlying constraint of constant number of electrons from the

integration of electronic density,∫

n(r)dr = N (see Eq. (2.9)).

Using the method of Lagrange multiplier, the following Lagrangian needs to be

minimized:

LV,N [n] = EV [n]−µ

[∫n(r)dr−N

], (2.17)

where µ is a Lagrange multiplier corresponding to the constraint imposed.

A routine minimization procedure will result in

0 =δLV,N [n]

δn(r)=

δEV [n]δn(r)

−µ =δF [n]δn(r)

+V (r)−µ,

or

µ =δF [n]δn(r)

+V (r). (2.18)

This is the basic equation for DFT.

It can be immediately recognised that the approximation for the universal functional

F [n] determines the accuracy of the calculation. While the formulation is exact in

principle, the accuracy of the approximation serves as a bottleneck to the accuracy of

the whole calculation, thus compromising the reliability of the results. An alternative

approach is proposed by Kohn and Sham, which overcame the necessity of finding

an approximation for the universal functional by creating a system of non-interacting

20

electrons.

2.6 Kohn-Sham method

While the theorems of Hohenberg and Kohn serves to drastically reduce the dimen-

sionality and complexity of the Schrödinger equation in principle, they did not provide

a way in which the minimization of the energy functional can be done in practical. A

method was devised by Kohn and Sham (1965), which allows the minimization to be

done in self consistence way.

Kohn Sham (KS) approach involves the mapping of a system of interacting electrons

to a non-interacting one, both having the identical electronic density. The conversion

to a non-interacting system is advantageous as the corresponding many-body wave

function is just a Slater wave function. While the original interacting system has the

real potential, the electrons in the converted fictitious non-interacting system experience

an effective single-particle ”Kohn-Sham” potential VKS(r). The KS method retains the

exact nature of the Hohenberg-Kohn theorems, in that the fictitious system has the same

ground state density as the original system, but provide a practical way to compute the

ground state electron density.

2.6.1 Non-interacting electrons

The formulation of a system where the interaction between electrons is absent is

much easier due to the lack of the U(ri,r j) term in Eq. (2.4). The implication is that

the contribution of different electrons to the overall system can be decoupled.

Due to the absence of the electron-electron interaction term, we might as well

21

encompass all the potential related contribution into one term Vs(r), which can be

decomposed into the one-electron basis. The Hamiltonian operator is modified to:

Hs =N

∑i=1− h2

2m∇

2i +

N

∑i=1

Vs(r)

=N

∑i=1

[− h2

2m∇

2i +Vs(r)

]=

N

∑i=1

hi(r). (2.19)

The overall Hamiltonian is reduced to a simple summation of the one-electron Hamilto-

nians.

The Schrödinger equation can then be rewritten for each individual one-electron

Hamiltonian:

h(ri)ψa(ri) = εaψa(ri), (2.20)

where index i is for electron and index a is for molecular orbital, the wave function for

a single electron.

While the current formulation excludes electron spin, generalization to include spin

functions can be easily done. The one-electron Schrödinger equation Eq. (2.20) is

shown to rely only on spatial coordinates, which makes the inclusion of a spin function

by multiplication straightforward.

χ(r) = ψ(r)α(ω),

where α(ω) is a spin function and χ(r) is a spin orbital.

22

Similar to the Hamiltonian of the system, the total energy is the summation of the

orbital energy εa. The overall wave function, on the other hand, is the product of orbitals

of individual electrons, which is a Hartree product. However, the Hartree product does

not respect the required antisymmetric properties of wave function for fermions. The

wave function is written instead as a determinant, known as the Slater determinant,

which obeys the antisymmetry requirement by construction:

Ψs(r1,r2, . . . ,rN) =1√N!

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

χ1(r1) χ2(r1) . . . χN(r1)

χ1(r2) χ2(r2) . . . χN(r2)

......

...

χ1(rN) χ2(rN) . . . χN(rN)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣. (2.21)

The spin orbitals in the Slater determinant are orthonormal:

⟨χi∣∣χ j⟩= δi j. (2.22)

Assuming the N/2 doubly occupied spatial orbitals and double numbers of electrons,

the density of electrons are exactly as shown in Eq. (2.10).

n(r) = 2N/2

∑i|ψi(r)|2.

Using the non-interacting kinetic energy Ts[n], the basic equation of DFT Eq. (2.18)

23

for a non-interacting electron system will become:

µ =δTs[n]δn(r)

+Vs(r). (2.23)

2.6.2 Kohn-Sham equation

In the Kohn Sham (KS) approach, the universal F [n] terms is divided into three

parts:

F [n] = Ts[n]+EH [n]+EXC[n]. (2.24)

Ts[n] is the kinetic energy of the non-interacting system of electrons with the same

electronic density n(r), a component from the separation of the total kinetic energy

after removing the correlation terms. The electron-electron interaction energy U [n]

is approximated by the classical electrostatic (Hartree) energy EH [n], the remainder

being U [n]−EH [n]. The discrepancies in energy resulting from the mapping from

an interacting to non-interacting system is compensated by the last term EXC[n], an

accumulation of all the non-classical exchange and correlation effect:

EXC = (T −Ts)+(U−EH). (2.25)

Typically, EXC is represented as EXC = EX +EC, where EX is the exchange energy and

EC is the correlation energy.

The Hartree energy EH [n] is given by:

EH [n] =e2

2

∫∫ n(r)n(r′)|r− r′|

drdr′ . (2.26)

24

From the modification of the universal functional F [n], the basic equation Eq. (2.18)

will take the following form:

µ =δTs[n]δn(r)

+δEH [n]δn(r)

+δEXC[n]δn(r)

+Vext(r)

=δTs[n]δn(r)

+VH(r)+VXC(r)+Vext(r)

=δTs[n]δn(r)

+VKS(r), (2.27)

where VKS(r) is the KS potential, expressed as:

VKS(r) =Vext(r)+VH(r)+VXC(r), (2.28)

Comparing Eq. (2.23) and Eq. (2.27), it is found that:

Vs(r) =VKS(r) =Vext(r)+VH(r)+VXC(r), (2.29)

which shows that the KS potential VKS(r) is the potential of the non-interacting electron

system. The implication is that a connection between the interacting and non-interacting

system is found.

For clarity, the Hartree potential VH(r) is

VH(r) =δEH [n(r)]

δn(r)= e2

∫ n(r′)|r− r′|

dr′ , (2.30)

25

and the exchange-correlation potential VXC(r) is

VXC(r) =δEXC[n(r)]

δn(r), (2.31)

an unknown quantity to be determined.

Following the discussion in Section 2.6.1, the overall Schrödinger equation can be

divided into a set one-electron Schrödinger equation, each corresponding to a molecular

orbital: [− h2

2m∇

2 +VKS(r)]

ψi(r) = εiψi(r), (2.32)

each molecular orbital has its corresponding eigen-energy value εi.

Similar to previous discussion, the electronic density can be constructed from the

single-particle states:

n(r) = 2∑i

ψ∗i (r)ψi(r), (2.33)

which is precisely Eq. (2.10).

For a system of non-interacting electrons, the associated kinetic energy is just

the summation of kinetic energy of individual elements. In the language of quantum

mechanics,

Ts[n] =−h2

m

N/2

∑i

∫ψ∗i (r)∇

2ψi(r)dr , (2.34)

where ψi is single-particle states. Noted that Ts[n] is an implicit functional of density n,

due to the dependence of wave function ψ on the density n.

KS formulation is only an approximate formulation in practice due to the presence

26

of the unknown exchange-correlation functional EXC[n(r)]. The implicit definition of

the functional is given in Eq. (2.25), a remainder from the extraction of non-interacting

functionals and classical effects. Kohn and Sham devise this approach with the intention

to make this unknown contribution as small as possible, but it is still a contribution if

binding energy is involved due to EXC[n(r)] and binding energy having similar order of

magnitudes. Various approximations of EXC[n(r)] are present, which will be discussed

in the following section.

2.6.3 Band structure energy

The total energy of the non-interacting electrons in the fictitious KS system is a

summation of the eigenvalues εi of one-electron Schrödinger equation Eq. (2.32). The

summation over the eigenenergies is also known as the ”band structure energy”, EBS.

Following Eq. (2.15):

EBS[n] = 2N/2

∑i=1

εi = Ts[n]+∫

VKS(r)n(r)dr

= Ts[n]+∫

Vext(r)n(r)dr+∫

VH(r)n(r)dr+∫

VXC(r)n(r)dr

= Ts[n]+∫

Vext(r)n(r)dr+2EH [n]+∫

VXC(r)n(r)dr . (2.35)

On the other hand, as the KS approach preserves the electronic density of the

interacting system, thus the energy obtained by substituting Eq. (2.24) in Eq. (2.15) is

representative of the real system:

E[n] = Ts[n]+∫

Vext(r)n(r)dr+EH [n]+EXC[n]. (2.36)

27

Comparing Eq. (2.35) and Eq. (2.36), it is shown that the sum of orbital energies

does not corresponding to the energy of the interacting system,

E[n] = 2N/2

∑i=1

εi−EH [n]−∫

VXC(r)n(r)dr+EXC[n], (2.37)

the sum of orbital energies double counts the Hartree energy and have unequal contribu-

tion from exchange and correlation part compared to the interacting system.

The overcounting of EBS can be examined from the explicit expression of the

eigenvalue ε itself:

εi = 〈ψi|hKS|ψi〉=∫

ψ∗i (r)

[− h2

2m∇

2 +Vext(r)+VH(r)+VXC(r)]

ψi(r)dr . (2.38)

The integration is over all electrons. Coulomb interaction between electrons is pair-wise

and should be be counted once only. Furthermore, examination of Hartree potential

Eq. (2.30) will reveal the inclusion of self-interaction contribution in the formula; part

of the Hartree potential includes the Coulomb interaction between an electron and itself.

2.7 Exchange correlation functionals

The results from Kohn-Sham formulation is exact, provided that the functional form

of EXC[n] is known. From its implicit definition in Eq. (2.25), it can be infer red that

the exact expression of EXC[n] is complicated due to it is the accumulation of all the

non-classical exchange and correlation effect present in a many-body problem. The

focus of the many-body problem is shifted from solving directly a 3N-dimensional

Schrödinger equation to finding a accurate form of exchange correlation (XC) functional

28

EXC[n]. Various approximations have been proposed, but only those employed in this

work will be discussed briefly due to limited scope.

2.7.1 Local density approximation

The form of a general exchange correlation functional is simply not known, except

for the simplest system: homogeneous electron gas. In a uniform electron gas system,

the electron density n(r) is a constant at all space for all r. The idea is to apply the

known exchange correlation functional form for the homogeneous electron gas to an

inhomogeneous system locally. The exchange correlation potential at each point of

space is determined according to the known value corresponding to an uniform electron

gas with electron density at that point of space:

V LDAXC (r) =V hom

XC [n(r)]. (2.39)

This approximation is called local density approximation (LDA), which takes only the

local density of electron as variable.

Despite its simple form, LDA works surprisingly well especially for a system with

slowly varying densities. In contrary, the reliability of LDA is disputable for a system

with steep density change gradient. It is the most widely used XC functional in practice,

due to both its computational simplicity and prediction of physical properties of a lot of

systems. The computation efficiency and accuracy of XC function are often inversely

proportion to each other, which makes the simple but mostly successful LDA widely

adopted.

29

2.7.2 Generalized gradient approximation

The generalized gradient approximation (GGA) is an extension to LDA, taking the

local density gradient as an extra variable in addition to the local electron density.

EGGAXC = EGGA

XC [n(r),∇n(r)]. (2.40)

The inclusion of gradient of electron density can be done in many ways, leading to

many variants of GGA. Two of most widely adopted scheme in calculations for solids

are the Perdew-Burke-Ernzerhof (PBE) functional (Perdew, Burke, & Ernzerhof, 1996)

and Perdew–Wang (PW91) functional (Perdew & Wang, 1992; Perdew & Yue, 1986).

Isolated molecules have their own specifically developed GGA schemes. GGA gives

a more reliable results in problem concerning chemical bonds than LDA, but remains

inadequate in simulating more complicated correlation effects such as Van der Waals

force interaction. Although the formulation of GGA contains the density gradient as a

extra variable, GGA is not necessary more accurate than LDA in predicting physical

properties in all system, which is one of the factors leading to the wide adoption in the

scientific community.

2.8 Self consistent method

Before the numerical scheme to solve the Kohn-Sham equations is discussed,

it is worth to summarize the algorithmic differences between the direct solution of

Schrödinger equation by the Hohenberg-Kohn theorems and the Kohn-Sham formu-

lation approach. The basic work flows involved in both approaches are depicted in

30

Fig. 2.1, with Fig. 2.1a shows the direct application of Hohenberg-Kohn theorems

whereas the Kohn-Sham method is shown in Fig. 2.1b.

Finding an approximation for F [n]

Minimization of energy functionalEV [n] = F [n] +

∫V (r)n(r)dr

for the ground state density n0(r)

Calculating the groundstate energy E0[n0]

(a) Direct application of Hohenberg-Kohn theo-rems

Construct the Kohn-Sham potential VKS

VKS(r) =Vext(r)+VH(r)+VXC(r)

Solve the one-electronSchrödinger equation foreach molecular orbitals

Construct the ground stateelectron density n0(r)

from the molecular orbital

Compute the groundstate energy E0[n0]

(b) Kohn-Sham approach

Figure 2.1: Comparison between direct application of Hohenberg-Kohn theorems andKohn-Sham approach.

Both approaches are exact in nature, but rely on approximation of different terms in

their implementation: the universal functional F [n] and exchange-correlation potential

VXC(r) for Hohenberg-Kohn and Kohn-Sham approaches respectively. Compared to the

direct application of the Hohenberg-Kohn theorems, modification by the Kohn-Sham

method is superior in that there is a direct physical estimation of the XC potential and

energy based on that of the simplest homogeneous electron gas system, rather than

the unknown form of the F [n] functional. Computational wise the implementation of

Kohn-Sham formulation is much easier due to the breakdown of the total wave function

into decoupled molecular orbital wave functions.

However, there is a circular dependency in implementation of the Kohn-Sham

31

equation. As depicted in Fig. 2.1b, the very first step is the construction of the Kohn-

Sham potential VKS(r), of which the Hartree potential VH(r) and XC potential VXC(r)

are components. The definitions of VH(r) and VXC(r) (see Eq. (2.30) and Eq. (2.31))

depend on the knowledge of electron density in space, which in turn is constructed

using molecular orbitals from the solution of one-electron Schrödinger equations, of

which the Hamiltonian comprises of VKS(r).

This circular dependency can be resolved through the use of a numerical iterative

method: self-consistent field (SCF) procedure. The basic idea behind SCF can be

outlined in the following algorithm:

1. Select a trial initial electron density n(r).

2. Construct the corresponding Kohn-Sham potential VKS(r).

3. Solve the Kohn-Sham equations (Eq. (2.32)) for the molecular orbitals ψi(r).

4. Construct a new electron density distribution from the computed molecular or-

bitals ψi(r) according to Eq. (2.10).

5. A selected convergence variable is compared for the old and new system. If

the discrepancy is within a predetermined tolerance value, the calculation is

considered completed and ground state energy is computed for the final electron

density, otherwise the whole process is repeated starting from step 2 using the

new electron density.

Noted that the starting trial quantities can be electron density or molecular orbitals

themselves, as long as the Kohn-Sham equations can be solved. The freely available

electronic structure calculation package typically starts the calculation by defining

32

random initial wave functions, in which the initial electron density is constructed from

the initial wave functions.

The typical steps involved in the SCF procedure of a DFT calculation are illustrated

in Fig. 2.2:

A solution of the Kohn-Sham equations that is self-consistent can therefore be

reached by an iterative procedure. The SCF method is a convergence process, the

accuracy of which depends on the tolerance of the convergence criteria chosen. Some of

the common criteria used are total energy, potential field, inter-particle forces and even

the wave function itself. However, most electronic packages do not follow the basic

scheme laid out in Fig. 2.2, but including some density or potential preconditioning

and advanced mixing methods such as Pulay mixing to improve the convergence of the

calculation, which will not be covered in this work.

33

Define a set of random orapproximated wave func-tions

ψ

(m)i

where m is

the number of iterationsteps, m = 0 in this case.

Construct the electron den-sity n(m)(r) from

ψ

(m)i

Into next iteration:m = m + 1

ψ(m)i

=

ψ(m+1)i

Define the Kohn-Sham potentialV (m)

KS (r) and contruct theHamiltonian operator h(m)

Solve the Kohn-Shamequations for a new set of

molecular orbitals

ψ(m+1)i

:

h(m)ψ(m+1)i = εiψ

(m+1)i

Checking the convergence ofselected criteria

DFT calculation is completed

Converged

Not converged

Figure 2.2: Typical SCF iteration in a DFT calculation.

34

CHAPTER 3

DENSITY FUNCTIONAL PERTURBATION THEORY

3.1 Basic background and review

Density functional theory (DFT), as discussed in Chapter 2, had proved to be a

hugely successfully technique in calculating the ground state properties of an electronic

system. The solution for a many-body problem is made possible by treating the electron

density as the fundamental quantity, rather than dealing with quantum mechanical wave

equation directly which involves coupled interaction in the Hamiltonian of Schrödinger

equation. The implementation of the KS method, which recast the system into individual

single-particle system, effectively reduces a N many body system to a N single-particle

system where the corresponding explicit Hamiltonian is exact and well understood.

Various physical observables or quantities that are obtained in experiment are actu-

ally the results from some form of disturbances to a particular system under investigation.

This results in well known failures of DFT in representing electronic excited states.

However, for the low level excited states close to the ground state, the Hamiltonian

of the excited system can be regarded to be close to ground state with some slight

perturbations. The quantum formalism of the perturbation theory can then be applied to

the framework of DFT, giving rise to density functional perturbation theory (DFPT).

The perturbation is usually to be included in the form of an additional small perturbing

potential. The quantities associated to the perturbed system are then expressed as

“corrections” to those corresponding to the ground state system. DFPT has produced

some good estimation of the low energy excitations on basis of some corrections to

35

the ground state, but the estimations worsen for higher energy excitations since the

deviation from the ground state properties becomes larger.

The basic implementation of DFPT can be done using the finite-difference method.

The change in the perturbed system is investigated in a series of total energy calcula-

tions at different strength of applied perturbations. The physical properties can then

be extracted thorough standard finite difference technique. For example, the perturba-

tion involved in a gamma point phonon calculation will be the shifting of the atomic

positions. The disadvantages of this simple and crude method lies in calculation of

phonon properties at arbitrary wave vector will require the construction of a supercell

instead of primary unit cell, which implies the necessity of large computational power.

Furthermore, finite-difference method cannot be readily applied in calculating electric

field perturbation of a system.

Two of the most popular implementation of DFPT are formulated by Baroni, Gian-

nozzi, and Testa (1987) and Gonze (1995a, 1995b). Baroni et al.s’ method (Baroni et al.,

1987) combines the advantage of direct supercell and dielectric matrix approaches in

studying linear response of crystals. The numerical complexity of the problem remain at

the same level as the unperturbed ground state system, as only the knowledge of states

in valance band is required. The need of a supercell in direct approach is eliminated,

yet its advantage to include non-linear effects is retained. The gist of the approach

lies in the adoption of self-consistency iteration to study the response of the overall

potential instead of inversion of dielectric matrix, where the computationally expensive

summation over conduction band in dielectric matrix approaches is replaced by Green’s

function technique.

36

On the other hand, Gonze’s formalism focuses on perturbation expansion of KS

energy functional with variational principle. Standard variational-perturbation theorems

are employed as perturbation theory is applied to the KS energy functional, which

is subjected to the variational principle. The emphasis is placed on the perturbation

expansion of wave functions by various methods such as Green’s function and sum

over states (Gonze, 1995a). Explicit formulas for the variational principles for arbitrary

order of perturbation (Gonze, 1995b) is used for obtaining general expression of higher

order derivatives of the energy. The techniques described are employed in the electronic

structure code ABINIT.

The basic concepts and theory behind DFPT will be discussed in the following

sections, without delving into the intricate details behind some derivations and theories.

The discussed concepts will serve as a basic framework for understanding the results

computed in this work. For detailed discussion behind the fundamental formulation of

DFPT, it is best to consult some review articles published by prominent figures in this

field, in particular the comprehensive review by Baroni, de Gironcoli, Dal Corso, and

Giannozzi (2001).

3.2 Response functions

The aim of DFPT is to compute the response functions, which are the physical prop-

erties from the application of perturbation to the ground state system. Mathematically,

the response functions are second, third or higher order derivatives of the total energy

with respect to the applied perturbation, although in practical most physical properties

can be computed using at most third order derivatives. The applied perturbations can

37

be mixed in the expression of the derivatives, giving different response functions for

different combinations of applied external perturbations. Three basic perturbations

include atomic displacement, homogeneous electric field and strain.

Physical properties obtainable are classified according to the order of corresponding

derivatives of total energy (non exhaustive):

1st order dipole moment, force, stress tensor component.

2nd order phonon dynamical matrices, dielectric tensor, Born effective charges, elastic

constant, internal strain, piezoelectricity.

3rd order phonon-phonon coupling, non-linear electric response, anharmonic elastic

constants, Grüneisen parameters

In ABINIT, only the first order derivatives of the wavefunctions (1WF) is explicitly

calculated; second order derivatives of the wavefunctions (2DTE) and third order deriva-

tives of the wavefunctions (3DTE) are constructed from the corresponding 1WFs. The

calculations of higher order derivatives hence reduce to relatively easy computational

task. In particular, 2DTE with respect to two separate perturbations computed from the

1WFs from the respective perturbations. 1WF is connected to the perturbation expansion

of variational principle (Gonze, 1995b), which allows an calculation algorithm based on

variational principle similar to that of ground state. Detailed derivations and subtleties

of the method are described by Gonze (1997) and Gonze and Lee (1997).

38

3.3 Density functional perturbation theory

3.3.1 General formulation of perturbation theory in quantum mechanics

Perturbation theory is commonly used in quantum mechanics to represent an oth-

erwise intractable quantum system in terms of a simpler system and “corrections” to

that simple system. The simple system is usually a system in which its mathematical

solution is known and the formulation is exact. An external additional perturbing

potential representing disturbances to the exact simple system serves as “corrections”

to the simple system in an approximate representation of the real actual system.

Only time independent perturbation theory is considered in this work where the

perturbation Hamiltonian is static. Let λ be a perturbing parameter, assumed to be

small, that characterize the perturbation. The perturbed Hamiltonian is expressed in

terms of the unperturbed Hamiltonian as:

H(λ ) = H(0)+Vper(λ ), (3.1)

where H(0) is the unperturbed Hamiltonian of the simpler system and Vper(λ ) is the

additional perturbing potential.

The unperturbed system obeys the usual time independent Schrödinger equation:

H(0)∣∣∣ψ(0)

α

⟩= ε

(0)α

∣∣∣ψ(0)α

⟩, (3.2)

in which the corresponding eigenstates∣∣∣ψ(0)

α

⟩and eigenvalues ε

(0)α are known. The

energy levels are assumed to be discrete for simplicity. Similar to the usual convention,

39

the wave functions are a set of orthonormal eigenfunctions:

⟨ψ

(0)α

∣∣∣ψ(0)β

⟩= δαβ , ∀α,β ∈ occupied orbitals, (3.3)

In terms of the perturbing terms, the Schrödinger equation is modified into:

H(λ ) |ψα(λ )〉= εα(λ ) |ψα(λ )〉 , (3.4)

where the perturbed wave functions also satisfy the normalization condition

⟨ψα(λ )

∣∣ψβ (λ )⟩= δαβ , ∀α,β ∈ occupied orbitals. (3.5)

The fundamental ansatz behind perturbation theory in quantum mechanics is that

the perturbed observables and physical quantities can be expressed in Taylor series in

terms of λ :

X(λ ) = X (0)+λX (1)+λ2X (2)+ . . . (3.6)

where X is an arbitrary quantity such as Hamiltonian, eigenfunction or eigenvalue. It

is assumed that the series is convergent and well defined, that the subsequent terms is

getting smaller in magnitude down the series. This is not a trivial assumption, but is

usually fulfilled in most cases. The αth order expansion coefficient to the X(λ ) is the

αth derivative of X(λ ) at λ = 0:

X (i) =1i!

diXdλ i

∣∣∣∣λ=0

. (3.7)

40

Putting the power series of H(λ ), ψα(λ ) and εα(λ ) into Eq. (3.4), and equating the

terms according to the order of λ will result in a hierarchical set of equations. For the

first order (λ 1):

H(0)∣∣∣ψ(1)

α

⟩+H(1)

∣∣∣ψ(0)α

⟩= ε

(0)α

∣∣∣ψ(1)α

⟩+ ε

(1)α

∣∣∣ψ(0)α

⟩. (3.8)

For the second order (λ 2):

H(0)∣∣∣ψ(2)

α

⟩+H(1)

∣∣∣ψ(1)α

⟩+H(2)

∣∣∣ψ(0)α

⟩= ε

(0)α

∣∣∣ψ(2)α

⟩+ ε

(1)α

∣∣∣ψ(1)α

⟩+ ε

(2)α

∣∣∣ψ(0)α

⟩,

(3.9)

and so on for the higher orders equations.

However, the one-state wave functions are not unique solutions, as they can be

modulated by an arbitrary phase factor

ψ(λ ) = ei f (λ )ψ(λ ), (3.10)

which also satisfy Eq. (3.4). The arbitrary phase factor f (λ ) is a real function, known

as the “gauge”. This gauge freedom needs to be taken into account in the following

derivations.

3.3.1(a) First-order perturbation

Taking the inner product of Eq. (3.8) with∣∣∣ψ(0)

α

⟩,

⟨ψ

(0)α

∣∣∣H(0)∣∣∣ψ(1)

α

⟩+⟨

ψ(0)α

∣∣∣H(1)∣∣∣ψ(0)

α

⟩= ε

(0)α

⟨ψ

(0)α

∣∣∣ψ(1)α

⟩+ ε

(1)α

⟨ψ

(0)α

∣∣∣ψ(0)α

⟩.

41

As H(0) is hermitian,⟨

ψ(0)α

∣∣∣H(0)∣∣∣ψ(1)

α

⟩= ε

(0)α

⟨ψ

(0)α

∣∣∣ψ(1)α

⟩. Due to orthonormality of

the wave functions Eq. (3.3),⟨

ψ(0)α

∣∣∣ψ(0)α

⟩= 1. The equation reduces to an expression

for the first order eigenvalues:

ε(1)α =

⟨ψ

(0)α

∣∣∣H(1)∣∣∣ψ(0)

α

⟩. (3.11)

The first order correction to the energy is hence the expectation value of first order

change in the Hamiltonian in the unperturbed state.

The result of Eq. (3.11) bears resemblance to the Hellmann-Feynman theorem (Feyn-

man, 1939; Hellmann, 1937), which equate the derivative of total energy with respect

to a parameter to the expectation value of derivative of the Hamiltonian with respect to

the same parameter.

dEdλ

= 〈ψ|dHdλ|ψ〉 . (3.12)

Eq. (3.8) can be rearranged to factorize the zeroth and first order wave functions.

The resulting equation form is known as the Sternheimer equation:

(H(0)− ε

(0)α

)∣∣∣ψ(1)α

⟩=−

(H(1)− ε

(1)α

)∣∣∣ψ(0)α

⟩, (3.13)

which has to be solved to obtain the first order correction to the wave function. As the

remaining unknown in Eq. (3.13) is only the first order wave function, Eq. (3.13) is a

nonhomogeneous linear differential equation to be solved by different methods. This is

known as Sternheimer equation approach in solving perturbations problems.

Using the method of separation of order, the orthonormality condition of wave

42

function of the first order can be obtained by inserting perturbation series of the wave

functions into Eq. (3.5):

⟨ψ

(0)α

∣∣∣ψ(1)β

⟩+⟨

ψ(1)α

∣∣∣ψ(0)β

⟩= 0, ∀α,β ∈ occupied orbitals. (3.14)

Further manipulation of Eq. (3.13) will result in an explicit expression of first order

wave function using the sum over states technique. A simple inversion of(

H(0)− ε(0)α

)in Eq. (3.13) is forbidden due to the eigenvalue of H(0) is ε

(0)α , leading to a zero in the

denominator. A workaround is to expand∣∣∣ψ(1)

α

⟩as a linear combination of unperturbed

wave functions (the set of unperturbed wave functions is complete):

∣∣∣ψ(1)α

⟩= ∑

β

c(1)αβ

∣∣∣ψ(0)β

⟩(3.15)

=occ

∑β

c(1)αβ

∣∣∣ψ(0)β

⟩+emp

∑γ

c(1)αγ

∣∣∣ψ(0)γ

⟩.

For the case of many bands, the set of complete unperturbed wave functions can be

separated into two subspaces, which are the occupied (occ) and unoccupied (empty)

subspace respectively.

Inserting Eq. (3.15) into Eq. (3.13) will result in

∑β 6=α

c(1)αβ

(ε(0)β− ε

(0)α

)∣∣∣ψ(0)β

⟩=(

ε(1)α −H(1)

)∣∣∣ψ(0)α

⟩.

43

Projecting into the subspace of other occupied orbitals∣∣∣ψ(0)

γ

⟩,

∑β 6=α

c(1)αβ

(ε(0)β− ε

(0)α

)⟨ψ

(0)γ

∣∣∣ψ(0)β

⟩= ε

(1)α

⟨ψ

(0)γ

∣∣∣ψ(0)α

⟩−⟨

ψ(0)γ

∣∣∣H(1)∣∣∣ψ(0)

α

⟩

If γ = α , the left hand side of the equation is zero, and imposition of normalization

condition will recover Eq. (3.11). However, if γ 6= α ,

c(1)αγ

(ε(0)γ − ε

(0)α

)=−

⟨ψ

(0)γ

∣∣∣H(1)∣∣∣ψ(0)

α

⟩,

c(1)αβ

=

⟨ψ

(0)β

∣∣∣H(1)∣∣∣ψ(0)

α

⟩ε(0)α − ε

(0)β

. (3.16)

The dummy variable γ is replaced by β . The coefficient c(1)αα remains unknown, which

provide us a gauge freedom.

By using the orthonormality condition Eq. (3.14), it is shown that the real part of

c(1)αβ

is zero. Due to the gauge freedom associated to the coefficient c(1)αα , we can choose

to impose its imaginary part to be zero, which is known as the parallel transport gauge.

For the parallel transport gauge, the orthonormality condition Eq. (3.14) needs to be

modified to ⟨ψ

(0)α

∣∣∣ψ(1)α

⟩= 0, ∀α ∈ occupied orbitals. (3.17)

The first order wave function is then a summation over states,

∣∣∣ψ(1)α

⟩= ∑

β

⟨ψ

(0)β

∣∣∣H(1)∣∣∣ψ(0)

α

⟩ε(0)α − ε

(0)β

∣∣∣ψ(0)β

⟩. (3.18)

44

In a practical problem, however, the summation over states is truncated, which occurs

naturally for a finite basis set.

3.3.1(b) Second-order perturbation

Taking the inner product of Eq. (3.9) with ψ(0)α , as well as exploiting the hermiticity

of H(0) and orthogonality of unperturbed wave function Eq. (3.3), we get

ε(2)α =

⟨ψ

(0)α

∣∣∣H(2)∣∣∣ψ(0)

α

⟩+⟨

ψ(0)α

∣∣∣(H− εα)(1)∣∣∣ψ(1)

α

⟩. (3.19)

Eq. (3.19) can also be rewritten in another form so as to eliminate the dependence

on first-order energy. By taking the sum of Eq. (3.19) with its own hermitian conjugate,

and using the orthogonality condition for the first-order wave function Eq. (3.14), the

alternate form can be obtained:

ε(2)α =

⟨ψ

(0)α

∣∣∣H(2)∣∣∣ψ(0)

α

⟩+

12

(⟨ψ

(0)α

∣∣∣H(1)∣∣∣ψ(1)

α

⟩+⟨

ψ(1)α

∣∣∣H(1)∣∣∣ψ(0)

α

⟩). (3.20)

The similar procedures used in deriving first-order expressions can be used to

further obtain the second-order wave function and even higher order correction, but

since the variational form second-order energy is provided in the following section, the

corrections to the wave functions are better discussed in the context of variational form.

45

3.3.2 Perturbation theory for Kohn-Sham formulation

The 2n+1 theorem of perturbation theory in DFT had been proved by Gonze and

Vigneron (1989), which essentially states that the solution for 2n+1 th derivative of

eigenenergy requires only the knowledge of perturbative eigenfunctions up to the order

n. It allows the perturbation theory in DFT to be treated in an iterative approach, where

the solution of higher order requires the knowledge of solution of all lower orders. A

generalized Sternheimer equation is to be solved iteratively, for which its first order form

is Eq. (3.13). However, for arbitrary order of perturbation, the use of a parallel transport

gauge is more convenient (Gonze, 1995a) in comparison to diagonal gauge. Although

only first order of perturbation is treated explicitly in this work, the expressions for

perturbation theory for arbitrary order are shown for completeness.

The KS energy functional may be rewritten into the form:

E[ψ] =N

∑α=1〈ψα |T +Vext |ψα〉+EHXC[n]. (3.21)

In the context of this equation N refer to the number of electons. For brevity, the

Hartree energy and XC energy are grouped into one term EHXC[n] = EH [n]+EXC[n]. It

is assumed that the operators term can be recognised from the context, so the use of the

hat symbol is omitted.

Following the variational principle, the energy of the system can be found by

minimizing Eq. (3.21), taken into account the general orthonormalization constraint:

⟨ψα

∣∣ψβ

⟩= δαβ . (3.22)

46

Using Lagrange multiplier method, Eq. (3.21) is modified into:

E[ψ] =N

∑α=1〈ψα |T +Vext |ψα〉+EHXC[n]−

N

∑α,β=1

Λβα

[⟨ψα

∣∣ψβ

⟩−δαβ

], (3.23)

in which the corresponding Lagrange-Euler equation is:

H |ψα〉=N

∑β=1

Λβα

∣∣ψβ

⟩, (3.24)

with the same KS Hamiltonian

H = T +Vext +VHXC = T +VKS. (3.25)

Taking an inner product with an occupied orbital will result in the expression for

Lagrange multipliers:

Λβα =⟨ψβ

∣∣H∣∣ψα

⟩, (3.26)

which is the matrix element of Hamiltonian between two wave functions.

Obviously, the form of Eq. (3.24) is different from that of KS equation Eq. (2.32).

This discrepancy originates from a gauge freedom due to the invariance of total energy

and density under a unitary transformation (Gonze, 1995a). At this point, it is suffice to

point out that the Lagrange parameters Eq. (3.26) are not identical to the eigenvalues of

KS equations. Eq. (3.24) is to be referred to as the generalized KS equations. Further

details about the gauge freedom will be discussed in the following section.

At this stage the formulas discussed earlier can be decomposed according to their

47

perturbation order hierarchy respectively. The electronic density can be expressed in

form of Taylor series Eq. (3.7) in terms of λ :

n(r) =N

∑α=1

ψ∗α(r)ψα(r) = ∑

i=0λ

in(i)(r) (3.27)

Inserting the series of ψα into Eq. (3.27), we get the density formula at order i:

n(i)(r) =i

∑j=0

N

∑α=1

ψ( j)∗α (r)ψ(i− j)

α (r). (3.28)

Similarly, insertion of ψα order series into the orthogonality condition Eq. (3.22) will

result ini

∑j=0

⟨ψ

( j)α

∣∣∣ψ(i− j)β

⟩= 0. (3.29)

The generalized KS equations Eq. (3.24) is modified into

i

∑j=0

H( j)∣∣∣ψ(i− j)

α

⟩=

i

∑j=0

N

∑β=1

Λ( j)βα

∣∣∣ψ(i− j)β

⟩(3.30)

with Hamiltonian at order i

H(i) = T (i)+V (i)KS (3.31)

and Lagrange multiplier at order i

Λ(i)βα

=i

∑j=0

i

∑k=0

⟨ψ

( j)β

∣∣∣H(i− j−k)∣∣∣ψ(k)

α

⟩. (3.32)

The general variational expression for energy of any arbitrary order m (m = 2n or

48

m = 2n+1) can be written as (Gonze, 1995a, 1995b):

E(m) =N

∑α=1

n

∑j=0

m−n−1

∑k=0

n

∑l=0

δ (m− j− k− l)⟨

ψ( j)α

∣∣∣H(k)∣∣∣ψ(l)

α

⟩−

N

∑α,β=1

n

∑j=0

m−n−1

∑k=0

n

∑l=0

δ (m− j− k− l)Λ(k)βα

⟨ψ

( j)α

∣∣∣ψ(l)β

⟩+

N

∑α=1

n

∑j=0

m

∑k=m−n

n

∑l=0

δ (m− j− k− l)⟨

ψ( j)α

∣∣∣(T +Vext)(k)∣∣∣ψ(l)

α

⟩+

1m!

dm

dλ m EHxc

[n

∑j=0

λjn( j)

] ∣∣∣∣λ=0

.

(3.33)

The expression follows the 2n+1 theorem, as energy of order m = 2n or m = 2n+1

only requires wave functions of order up to n. For m = 2n specifically, Eq. (3.33) is

variational, or minimal, with respect to the n-order wave function. This variational

property allows us to compute the energy of any particular order iteratively. Suppose

that we have a n− 1-order wave function, the n-order wave function can be found

from Eq. (3.33) with the constraint it minimize the function. Subsequently, the n-order

density and Hamiltonian can be found and the procedure can be repeated iteratively to

generate higher order quantities.

Apart from the variational method, the Sternheimer equation approach can be used

by solving a generalized version of Eq. (3.30) self-consistently with Eqs. (3.28), (3.29),

(3.31) and (3.32).

3.3.3 Gauge freedom

The section will discuss the issue of gauge freedom in the context of perturbation

theory in KS formulation. Briefly mentioned in Section 3.3.1(a), the gauge freedom

is associated with the zeroth order occupied orbitals, as evident in the expression of

49

Eq. (3.24) which is different from the KS equation Eq. (2.32).

The gauge freedom originates from the invariance of energy and density under a

unitary transformation of occupied orbital wave functions. Let U be a (N×N) unitary

matrix such that

U−1αβ

=U†αβ

,

and another set of wave functions are defined via an unitary transformation:

∣∣ψ ′α⟩= N

∑γ=1

Uγα

∣∣ψγ

⟩. (3.34)

Both sets of wave functions have the same corresponding electronic density and energy

as well as satisfy the same orthonormality condition Eq. (3.22).

Suppose the set of wave functions |ψα〉 satisfy KS equations Eq. (2.32) whereas

the other set of wave functions |ψ ′α〉 satisfy Eq. (3.24), then the unitary matrix U in the

case needs to diagonalize the Lagrange multipliers matrix:

N

∑β ,γ=1

[U ]αβ Λβγ [U†]γη = δαηεα (3.35)

A natural gauge choice is to use zeroth order eigenfunctions of the KS equations, which

results in

Λ(0)βα

= δβαε(0)α (3.36)

and

U (0)βα

= δβα . (3.37)

50

Consequently, Eq. (3.30) is modified to:

(H(0)− ε

(0)α

)∣∣∣ψ(i)α

⟩=

i

∑j=1

[−H( j)

∣∣∣ψ(i− j)α

⟩+

N

∑β=1

Λ( j)βα

∣∣∣ψ(i− j)β

⟩], (3.38)

which the addition of the part at left hand side is due to the constraint imposed earlier.

In addition, the orthogonality condition Eq. (3.29) takes the following form:

⟨ψ

(0)α

∣∣∣ψ(i)β

⟩+⟨

ψ(i)α

∣∣∣ψ(0)β

⟩=

−∑

i−1j=1

⟨ψ

( j)α

∣∣∣ψ(i− j)β

⟩, i > 1

0, i = 1.

(3.39)

Due to the gauge freedom, we can impose the left hand side of Eq. (3.39) to zero

regardless of the perturbation order. The resulting condition is known as the parallel

transport gauge: ⟨ψ

(0)α

∣∣∣ψ(i)β

⟩−⟨

ψ(i)α

∣∣∣ψ(0)β

⟩= 0. (3.40)

By comparing Eq. (3.40) with Eq. (3.39), we can get

⟨ψ

(0)α

∣∣∣ψ(i)β

⟩=

−1

2 ∑i−1j=1

⟨ψ

( j)α

∣∣∣ψ(i− j)β

⟩, i > 1

0, i = 1.

(3.41)

The physical implication of Eq. (3.41) is that the projections of higher order wave

functions into the subspace of unperturbed zeroth order orbitals are minimized. Particu-

larly, the first order wave function is orthogonal to the unperturbed wave function, which

is not true for order higher than one. The simplicity of the first order orthogonality

condition is reflected by the simple expression of the first order related quantities.

51

3.3.4 Explicit expressions of lowest order energy in DFPT

In this section an explicit expression of energy is given for first and second order

level of perturbation respectively, starting from the general expression of energy of

arbitrary order m Eq. (3.33). While Eq. (3.33) allows for perturbations in any variables

in the KS equations, some of the perturbations can be neglected according to the type

of perturbations considered. Particularly, in the case of this work where perturbations

take the form of atomic displacement and applications of homogeneous electric field,

the form of kinetic energy and Hartree energy terms are not affected by the perturbation

in the Hamiltonian. Such restrictions will be considered in all following equations.

3.3.4(a) First order energy

The expression for first order energy is easily derived from Eq. (3.33):

E(1) =N

∑α=1

⟨ψ

(0)α

∣∣∣(T +Vext)(1)∣∣∣ψ(0)

α

⟩+

ddλ

EHxc

[n(0)]∣∣∣∣

λ=0, (3.42)

which is similar to Eq. (3.11) with an addition of exchange correlation term from the

KS formulation. Following the 2n+1th theorem, only ground state wave function is

needed to define the first-order energy.

If we make a further simplification that the kinetic energy and XC energy terms do

not affected by the perturbation, we have

E(1) =N

∑α=1

⟨ψ

(0)α

∣∣∣V (1)ext

∣∣∣ψ(0)α

⟩, (3.43)

52

which can be decomposed into contribution from different orbitals

E(1)α =

⟨ψ

(0)α

∣∣∣V (1)ext

∣∣∣ψ(0)α

⟩, (3.44)

with the total energy

E(1) =N

∑α=1

E(1)α . (3.45)

3.3.4(b) Second order energy

The second order energy, according to the 2n+ 1th theorem, is variational with

respect to first order wave functions ψ(1)α using the general Eq. (3.33). The explicit

expression is given as:

E(2) =N

∑α=1

[⟨ψ

(1)α

∣∣∣(T +Vext)(1)∣∣∣ψ(0)

α

⟩+⟨

ψ(0)α

∣∣∣(T +Vext)(1)∣∣∣ψ(1)

α

⟩

+⟨

ψ(0)α

∣∣∣(T +Vext)(2)∣∣∣ψ(0)

α

⟩+⟨

ψ(1)α

∣∣∣(H− εα)(0)∣∣∣ψ(1)

α

⟩]+

12

∫∫δ 2EHxc

[n(0)]

δn(r)δn(r′)n(1)(r)n(1)(r′)drdr′

+

∫d

dλ

δEHxc

[n(0)]

δn(r)

∣∣∣∣∣∣λ=0

n(1)(r)dr+12

d2

dλ 2 EHxc

[n(0)]∣∣∣∣

λ=0,

(3.46)

with the constraint from Eq. (3.29):

⟨ψ

(0)α

∣∣∣ψ(1)β

⟩+⟨

ψ(1)α

∣∣∣ψ(0)β

⟩= 0. (3.47)

In the parallel-transport, the constraint simplifies to

⟨ψ

(0)α

∣∣∣ψ(1)β

⟩= 0, (3.48)

53

as alluded to in Eq. (3.41).

The first order electronic density required in Eq. (3.46) can be obtained from

Eq. (3.28):

n(1)(r) =N

∑α=1

ψ(0)∗α (r)ψ(1)

α (r)+ψ(1)∗α (r)ψ(0)

α (r). (3.49)

There are several terms in Eq. (3.46) which depend on only unperturbed variables,

which enable us to adopt the general notations:

E(i)Hxc,0 =

1i!

di

dλ i EHxc

[n(0)]∣∣∣∣

λ=0, (3.50)

V (i)Hxc,0(r) =

1i!

di

dλ i

δEHxc

[n(0)]

δn(r)

∣∣∣∣∣∣λ=0

, (3.51)

and

K(i)Hxc,0(r,r

′) =1i!

di

dλ i

δ 2EHxc

[n(0)]

δn(r)δn(r′)

∣∣∣∣∣∣λ=0

. (3.52)

In view of this, Eq. (3.53) can be rewritten as

E(2) =N

∑α=1

[⟨ψ

(1)α

∣∣∣(T +Vext)(1)∣∣∣ψ(0)

α

⟩+⟨

ψ(0)α

∣∣∣(T +Vext)(1)∣∣∣ψ(1)

α

⟩+⟨

ψ(1)α

∣∣∣(H− εα)(0)∣∣∣ψ(1)

α

⟩]+

12

∫∫K(0)

Hxc,0(r,r′)n(1)(r)n(1)(r′)drdr′

+∫

V (1)Hxc,0(r)n

(1)(r)dr+E(2)non−var,

(3.53)

where all the terms that are independent of first-order wave functions and thus unchang-

54

ing in the minimization procedure are summarized in

E(2)non−var =

N

∑α=1

⟨ψ

(0)α

∣∣∣(T +Vext)(2)∣∣∣ψ(0)

α

⟩+E(2)

Hxc,0. (3.54)

If XC functionals are not affected by the perturbations, Eqs. (3.50) to (3.52) will

vanish for i > 1. Coupled with further simplification that perturbations do not affect the

kinetic energy and Hartree energy, the expression of second-order energy will reduce to

E(2) =N

∑α=1

[⟨ψ

(1)α

∣∣∣V (1)ext

∣∣∣ψ(0)α

⟩+⟨

ψ(0)α

∣∣∣V (1)ext

∣∣∣ψ(1)α

⟩+⟨

ψ(1)α

∣∣∣(H− εα)(0)∣∣∣ψ(1)

α

⟩]+

12

∫∫K(0)

Hxc,0(r,r′)n(1)(r)n(1)(r′)drdr′+E(2)

non−var,

(3.55)

with

E(2)non−var =

N

∑α=1

⟨ψ

(0)α

∣∣∣V (2)ext

∣∣∣ψ(0)α

⟩. (3.56)

3.4 Common types of perturbation

Two of the basic perturbations will be discussed here, which is phonon atom dis-

placement and homogeneous macroscopic electric field. Both perturbations can be

represented as the second order derivative of energy with respect the type of perturba-

tions.

3.4.1 Phonons: atomic displacement

In the treatment of phonon response, the contribution of the nuclei part of total

Hamiltonian has to be taken into account in addition to the electronic energy. For

55

this purpose, the electronic energy E used before this section is redefined to be Eel

to distinguish between nuclei and electronic contribution. In the Born-Oppenheimer

approximation, the nuclei and electronic contributions can be decoupled, and the

Hamiltonian HBO depends parametrically upon the nuclear positions, with the system

of interacting electrons subjected to the electrostatic field of the fixed nuclei:

HBO(τ) =−h2

2m ∑i

∂ 2

∂r2i+

e2

2 ∑i6= j

1∣∣ri− r j∣∣ − e2

∑i,κ

Zκ

|ri− τκ |+

e2

2 ∑κ 6=µ

ZκZµ∣∣τκ − τµ

∣∣ . (3.57)

r refers to electronic coordinates whereas τ corresponding to nuclei coordinates. i, j

and κ,µ are the index labels for the electronic and nuclei part respectively. Z is the

ionic nuclear charge. Eq. (3.57) can be solved using standard DFT techniques and the

resulting energy is EBO(τ), in which the presence of nuclei coordinates as parameters

imply the adiabatic assumption of the Born-Oppenheimer approximation.

The complete Schrödinger equation incorporating the nuclei contributions is then

formed by adding the kinetic energy of nuclei in the Hamiltonian:

(− h2

2 ∑κ

1Mκ

∂ 2

∂τ2κ

+EBO(τ)

)Ψ(τ) = EΨ(τ). (3.58)

E is the total energy of the system including electronic and nuclei contributions. M is

the mass of nucleus of the constituting atoms in the system.

In a periodic system, the atoms in the system are grouped into unit cells. Let R be

the position vector of a particular unit cell with respect to a chosen origin, κ be the

index for nuclei in the unit cell and α,β be the label for Cartesian directions. Since

phonon response is to be studied, the atoms will not be stationary in their equilibrium

56

positions τrκα , but moves about some small displacements urκα around their respective

equilibrium coordinates. Due to the small displacements, EBO can be expanded in a

Taylor series:

EBO(u) = E0BO +

12 ∑

Rκα

∑R′κ ′β

∂ 2EBO

∂τRκα∂τR′κ ′βuRκαuR′κ ′β + · · · . (3.59)

E0BO is the minimum energy attainable by the system when all atoms are stationary

in their respective equilibrium positions. The first-order term is absent in the series

since it represents the forces acting to the nuclei, which must be zero when E0BO is at its

minumum and all atoms are in their equilibrium positions, FRκα = ∂EBO/

∂τRκα = 0.

A truncation at the energy at second order is called the harmonic approximation, and

higher order terms must be included if anharmonic effects are to be investigated.

In the harmonic approximation, the forces acting a nucleus due to its vibrational

motion around its equilibrium position is just a simple derivative with respect to the

small displacement:

F(Rκα) =− ∂EBO

∂uRκα

=− ∑R′κ ′β

∂ 2EBO

∂τRκα∂τR′κ ′βuR′κ ′β =− ∑

R′κ ′βΦκα,κ ′β (R,R′)uR′κ ′β .

(3.60)

Φκα,κ ′β (R,R′) is defined to be matrix of interatomic force constants (IFCs), describing

the interatomic force between two nuclei κ and κ ′.

Φκα,κ ′β (R,R′) =∂ 2EBO

∂τRκα∂τR′κ ′β. (3.61)

57

Eq. (3.60) can be rewritten as the equation of motion:

Mκ uRκα =− ∑R′κ ′β

Φκα,κ ′β (R,R′)uR′κ ′β , (3.62)

with the solution expected in the form of plane wave:

uRκα = ηmq(κα)ei(q·R−ωmqt), (3.63)

where ηmq is the phonon eigendisplacements.

Substituting Eq. (3.63) into Eq. (3.62) will result in a generalized eigenvalue prob-

lem:

Mκω2mqηmq(κα) = ∑

κ ′β

Φκα,κ ′β (q)ηmq(κ′β ), (3.64)

with ωmq is the phonon frequency and Φ is the Fourier transform of Φ:

Φκα,κ ′β (q) = ∑R′

Φκα,κ ′β (R,R′)eiq·(R′−R), (3.65)

which, using translational invariance, will reduce to

Φκα,κ ′β (q) = ∑R′

Φκα,κ ′β (0,R′)eiq·R′. (3.66)

The mass dependence of generalized eigenvalue equation Eq. (3.64) can be elimi-

nated by a change in representation of phonon eigendisplacements:

ω2mqγmq(κα) = ∑

κ ′β

Dκα,κ ′β (q)γmq(κ′β ), (3.67)

58

where the phonon eigenvector γmq(κα) is

γmq(κα) =√

Mκηmq(κα), (3.68)

and the dynamical matrix Dκα,κ ′β (q) is

Dκα,κ ′β (q) =Φκα,κ ′β (q)√

MκMκ ′, (3.69)

The phonon eigendisplacements and eigenvectors have to satisfy their own orthonor-

mality conditions

∑κα

Mκ [ηmq(κα)]∗ηmq(κα) = 1, (3.70)

∑κα

[γmq(κα)]∗γmq(κα) = 1, (3.71)

The IFC matrix and dynamical matrix are hence the second-order derivatives of Born-

Oppenheimer energy with respect to atomic displacements, which can be computed

using DFPT.

3.4.2 Homogeneous macroscopic electric field

In the description of electrostatic related scenarios, the CGS units are to be used

to instead of SI units, since the Maxwell’s equations are commonly expressed in CGS

units and presented in a more symmetrical form than in SI units (S. Venkataram, 2012).

The perturbation by macroscopic electric field is relatively simple from a theoretical

point of view, since the terms in Born-Oppenheimer Hamiltonian do not depend explic-

59

itly on the electric field and the corresponding changes only occur via induced changes

in wave functions and densities implicitly. However, there are two major problems in

the implementation of the electric field perturbation from a computational point of view.

The first problem concerns the non-periodic nature of the potential that generates

the electric field. The potential of an electron placed in a such a field is linear in space,

Vscr(r) = E · r, (3.72)

which breaks the crystalline structure periodicity. The term is also unbounded, which

is ill defined in a periodic boundary condition. Secondly, the actual electric field

concerned in the perturbation corresponds to the screened potential, as alluded to by the

“src” subscript in Eq. (3.72). It is different to the applied macroscopic electric field, due

to the internal change of field induced by the polarization of the material in response to

the applied filed.

The classical theory of electromagnetism had provided the result that, in the linear

regime, dielectric permittivity tensor relates electric displacement and polarization,

Dα = ∑β

εαβEβ = Eα +4πPα . (3.73)

The electronic part of dielectric permittivity tensor is obtained as

ε∞

αβ=

∂Dα

∂Eβ

= δαβ +4π∂Pα

∂Eβ

, (3.74)

at clamped nuclei. The corresponding contribution is denoted as ε∞

αβ, measured ex-

60

perimentally at high enough frequencies to be ignored by phonons but small enough

compared to the electronic band gap. Since polarization P is the derivative of total

energy with respect to the electric field E, the electronic dielectric tensor is related to

the second-order derivative of total energy with respect to E, which is also the dielectric

susceptibility χ .

Since a photon contains no mass, its momentum contains negligible momentum

compared to that of phonon. In that case the change in wave vector q due to the photon

approximate zero, that q→ 0. In the long wave method where q→ 0 in solving the first

problem, the potential is linear in space:

v(r) = limq→0

λ2sin(q · r)|q|

, (3.75)

in which the direction of q is along that of the homogeneous applied field. This approach

limit our treatment to the longitudinal field, since q and E are parallel to each other.

In the long wavelength limit the second order term is obtained from Taylor expansion

with respect to wave vector q. An extra auxiliary quantity is needed (Gonze, 1997),

which is the derivative of unperturbed wave functions with respect to the wave vector.

This implies that the derivatives with respect to wave vectors have to be pre-calculated.

In parallel transport gauge, the minimization procedure will result in the final expression:

EE∗αEβ

el

u(0);uEα

=

Ω0

(2π3)

∫BZ

occ

∑m

s⟨

uEα

mk

∣∣∣iukβ

mk

⟩dk , (3.76)

where the superscript on periodic part of wave function u indicates the type of perturba-

tions. Eq. (3.76) shows that the second derivative of energy with respect to electric field

61

along two directions α and β can be found from the knowledge of first-order derivative

of wave functions with respect to electric field along direction α and the first-order

change in wave functions with respect to wave vector along direction β .

3.4.3 Born effective charges

The Born effective charges Z∗κ, is a quantitative measure relating the induced polar-

ization in an insulator due to the atomic displacement at zero field. Specifically, Z∗κ,

is a proportionality constant relating the polarization of a unit cell along a direction β

and the atomic displacement of an atom κ along the direction α . It is equivalently the

change in force acting on atom by an application of electric field.

Z∗κ,βα

=Ω0

e∂Pβ

∂τκα(q = 0)=

∂Fκα

∂Eβ

. (3.77)

Both definitions are equivalent as they are both related to the mixed second order

derivative of energy with respect to atomic displacement and macroscopic electric field.

The non-stationary expression of Z∗ can be written as:

Z∗κ,βα

= Zκδβα +2Ω0

(2π)3

∫BZ

occ

∑m

s⟨

uτκα

mk,q=0

∣∣∣iukβ

mk

⟩dk , (3.78)

where the first part is the charge of ion or nucleus κ and the second part is contribution

from the electronic screening.

62

3.4.4 Static dielectric response

The relaxation of clamped ions constraint, used in calculating high frequency

dielectric tensor, is needed to investigate the interplay between applied macroscopic

electric field and polarization due to atomic displacement at q→ 0. The associated

dielectric tensor becomes the low frequency dielectric tensor ε0αβ

, which takes into

account the response of the ions. The polarization induced by a longitudinal phonon

will generate a macroscopic electric field, which in turn will exert some forces onto the

ions and altering the phonon frequencies. As a result the phonon frequencies between

transverse optical and longitudinal optical modes will different be from each other,

causing the LO–TO splitting in polar material. The corresponding phenomenon is

summarized by the Lyddane-Sachs-Teller relation (Lyddane, Sachs, & Teller, 1941).

The full response to the dielectric tensor is

ε0αβ

(ω) = ε∞

αβ+

4πe2

Ω0∑m

Sm,αβ

ω2mq=0−ω

, (3.79)

where Sm,αβ is the mode-oscillator strength tensor

Sm,αβ =

(∑κα ′

Z∗κ,αα ′η

∗mq=0(κα

′)

)×

(∑κ ′β ′

Z∗κ ′,ββ ′ηmq=0(κ

′β′)

), (3.80)

which is related to the infrared adsorption intensity.

The effective charge can also be associated to the phonon modes:

Z∗m,α =∑κβ ′ Z∗κ,αβ

ηmq=0(κβ )∑κβ

[ηmq=0(κβ )

]∗ηmq=0(κβ )

1/2 . (3.81)

63

3.4.5 LO-TO splitting

The generated macroscopic electric field due to the dynamical behaviours of atoms

will affect the long wavelength limit phonon (q→ 0), causing the LO-TO splitting

if atomic displacement along the direction of q; a longitudinal displacement. This

is reflected in Φκα,κ ′β (q → 0), which contains an additional term apart from the

Φκα,κ ′β (q = 0) derived in Section 3.4.1:

Φκα,κ ′β (q→ 0) = Φκα,κ ′β (q = 0)+ ΦNAκα,κ ′β (q→ 0), (3.82)

where ΦNAκα,κ ′β (q→ 0) is a non-analytical term which is dependent on the direction of

q. The non-analytical term is given as

ΦNAκα,κ ′β (q→ 0) =

4πe2

Ω0

(∑γ qγZ∗κ,γα

)(∑µ qµZ∗

κ ′,µβ

)∑αβ qαε∞

αβqβ

. (3.83)

Only the eigenvector of the modes longitudinal to the wave vector q will altered

from the q = 0 case in the general case, whereas the transverse modes will be identical

for both cases. For a given q, there will be one longitudinal modes and two transverse

modes, for which the mode effective charges Eq. (3.81) is perpendicular to q. The

following relation holds for a TO mode:

∑κβ

(∑α

Z∗κ,αβ

qα

)ηmq→0(κβ ) = 0. (3.84)

In some cases, the eigendisplacement of LO modes in the limit of q→ 0 is identical

to the case of q = 0 due to symmetry constraints, even though the LO phonon frequen-

64

cies still differ. The TO and LO modes can then be linked in the following approximate

form:

ω2m(q→ 0) = ω

2m(q = 0)+

4π

Ω0

∑αβ qαSm,αβ qβ

∑αβ qαε∞

αβqβ

. (3.85)

65

CHAPTER 4

SPONTANEOUS POLARIZATION

4.1 Introduction

The existence of spontaneous polarization at zero electric field is the signature

property of a ferroelectric material. In the case of BaTiO3, the spontaneous polarization

at ferroelectric phases emerges from broken symmetry of cubic phases, with the polar-

ization direction dependent on the titanium atomic shift direction. In the early stage the

phase transition of BaTiO3 is considered displacive, but later developments discovered

a combination of both displacive and order-disordered dynamics mechanisms. The

coexistence of both phase transition dynamics is not only not found experimentally in

cubic phase (Deng, 2012), but also at cubic-tetragonal and orthorhombic-rhombohedral

phase transition (Völkel & Müller, 2007). The changes in spontaneous polarization

across phase transition will only be discussed within the context of displacive transition

in this work.

A revision of basic definition of polarization will be given. Electric dipole is the

most basic interaction between multiple numbers of point charges, which consists of

two point charges with equal charges but opposite polarity in close proximity. The

dipole moment p is defined as

p = qd, (4.1)

with q is electric charge of individual point particles and d is a direction vector from

negative to positive charge of dipole. Polarization is the accumulated effect of electric

66

dipoles considered over a unit space of a material.

P = lim∆v→0

∑i pi

∆v. (4.2)

The polarization in a material can be induced by mechanisms of different levels:

• electronic polarization (atomic level)

• ionic polarization (molecular level)

• orientational polarization

While the first two types of polarization can be exhibited by any material under an

application of electric field, orientational polarization manifests in a polar material

without the need of an external field. The principle behind orientational polarization is

existence of domains with spatial separation of charges, where the direction of dipole

moments can be changed by an external field.

4.2 Polarization: microscopic perceptive

Macroscopic polarization is a well defined quantity which can be described Maxwell’s

equations under classical electrostatic. However, it is ill defined at the level of micro-

scopic model. One such example is illustrated by Fig. 4.1. Fig. 4.1 shows the non-

uniqueness of polarization of a bulk solid at the level of unit cell. While the periodic

structure of the solid remains identical, two different polarization vectors corresponding

to opposite directions can be obtained depending the selective choice of different unit

cells. From physical point of view, this behaviour is unacceptable since the physics of a

material is independent of selection of unit cells.

67

+–+–+

P

......

(a) (b)

–+

+–+–+

......

–+

P

(c)

+–+–+

......

–+

Figure 4.1: Ambiguous polarization in bulk solid (Sbyrnes321, 2011). (a) Periodic 1Dcharge alignment. (b) Upward polarization by selection of unit cell. (c) Downwardpolarization by selection of another equivalent unit cell

The standard Clausius-Mossotti (CM) model assumes identifiable polarizable units

in a material, and is inadequate for the microscopic description of polarization. Within

the picture of CM model, polarization is the sum of dipole moments per unit volume such

as that described by Eq. (4.2), which can be separated into localized contributions with

identifiable polarization centers. However, ferroelectric oxide material like BaTiO3 has

bondings with mixed ionic-covalent bonding (P. Ghosez, Gonze, Lambin, & Michenaud,

1995), with a delocalized distribution of electronic charge. In such situations, any

partition into localized polarization centers is arbitrary. According to definition of

polarization per cell in CM model which is based on localized basis, a local region for

integration must be defined. It had been shown that it is impossible to obtain polarization

from the knowledge of charge distribution alone (Resta & Vanderbilt, 2007), which can

be illustrated by several attempts described below.

1. Inspired by the definition of polarization as the total electric dipole moment per

68

unit volume, a natural attempt would be define polarization as

P =1V

∫rρ(r)dV , (4.3)

where V stands for the macroscopic volume of investigated material. A real

crystal, however, is finite and thus contains contributions from surface in addition

to bulk. The accumulated charge density at the surface, resulted by an application

of external field, will affect the total value of P, even if the interior bulk conditions

are unchanged. Hence, Eq. (4.3) cannot serve as a definition of bulk polarization.

2. Similar to the first approach but restraining the integration region to one unit cell:

P =1

Vcell

∫cell

rρ(r)dV . (4.4)

This definition suffers from the same problem illustrated in Fig. 4.1, being depen-

dent on the chosen unit cell. By averaging the result over all possible translation

shifts, the value of P easily vanishes.

3. Polarization is defined to be the average of microscopic polarization Pmic in a cell:

∇ ·Pmic =−ρ(r). (4.5)

Eq. (4.5) is not unique, as any constant vector can be added to Pmic without

affecting the result.

69

4.3 Modern theory of polarization

Following the failures to obtain polarization from charge distribution, the focus

shifted from finding absolute value of polarization to change of polarization. This

approach defines the observable value of P in parallel to that measured in experiment,

where polarization differences of a material is measured when an electric field is applied.

It was realized that polarization differences are conceptually more fundamental than the

absolute polarization. This is the basis for the modern theory of polarization, which can

be formulated with Berry phase and Wannier functions.

4.3.1 Berry phase approach

While variation in polarization is the quantity that is experimentally accessible, the

quantities of interest are often in the form of derivatives of polarization. One such

example is Born effective charge, as evident in its form in Eq. (3.77). In the case of

ferroelectric, the quantity of interest is the finite polarization difference ∆P developed

in a phase transition of state.

The modern theory of polarization lies on the proposition made by Resta (1992), that

the changes in electronic polarization due to a finite adiabatic change in KS Hamiltonian

in a crystal can be written in the form

∆Pel =∫

λ2

λ1

∂Pel

∂λdλ , (4.6)

70

where λ is a parameter characterizing the change in KS potential, with the derivative

∂Pel

∂λ=

i f |e|hNΩme

∑k

M

∑n=1

∞

∑m=M+1

⟨ψ

(λ )nk

∣∣∣p∣∣∣ψ(λ )mk

⟩ ⟨ψ

(λ )mk

∣∣∣∂V (λ )KS

∂λ

∣∣∣ψ(λ )nk

⟩(

ε(λ )nk − ε

(λ )mk

)2 + c.c. (4.7)

where N is the number of unit cells in the material, Ω is the unit cell volume, f is

occupational number of states in valence band, M is the number of occupied bands, p is

momentum operator, ε is KS eigenvalue and c.c. stands for complex conjugate.

Macroscopically, the change in bulk polarization leads to a build up of charges on

the surfaces, and a transient flow of charges in the bulk during the process, provided

that the accumulated surface charges are not allowed to be conducted. Rather than

measuring polarization directly, the focus can be shifted to the study of flow of charge

in the bulk when polarization changes. The macroscopic cell averaged current density

is given as

j(t) =dP(t)

dt, (4.8)

with the change in polarization

∆P = P(∆t)−P(0) =∫

∆t

0j(t)dt , (4.9)

which can be generalized to Eq. (4.6).

One of the conditions required for Eq. (4.6) to hold is that the system must be insu-

lating along the traversed integration path, otherwise the transient current is not uniquely

defined. The discussions onwards will be constrained to case of ferroelectric, where

λ1 refers to the centrosymmetric structure and λ2 refers to spontaneous polarization

71

structure, in the absence of an applied macroscopic electric field.

Following the works of Resta, King-Smith and Vanderbilt (1993) further developed

the theory using the Berry phase approach, culminating in the final form of modern

theory of polarization. Eq. (4.7) is recast into a form with no explicit dependence

on conduction band states ψ(λ )mk , and the polarization variation along a path can be

determined with only a knowledge of system at end points.

∆Pel = P(λ2)el −P(λ1)

el , (4.10)

with

P(λ )el =− i f |e|

(2π)3

M

∑n=1

∫BZ

⟨u(λ )nk

∣∣∣∇k

∣∣∣u(λ )nk

⟩dk , (4.11)

which is the central result of modern theory of polarization.

An examination of Eq. (4.11) reveals the term common in Berry phase theory (Resta,

2000):

A(k) = i⟨

u(λ )nk

∣∣∣∇k

∣∣∣u(λ )nk

⟩, (4.12)

known as Berry connection or gauge potential. The result of integration of Berry

connection over a closed manifold, such as the Brillouin zone, is the Berry phase.

Restoring the nuclear contribution, the total polarization is expressed as

P =eΩ

∑α

Zionα rα −

i f |e|(2π)3

M

∑n=1

∫BZ

⟨u(λ )nk

∣∣∣∇k

∣∣∣u(λ )nk

⟩dk , (4.13)

where the first term is contributions from positive ionic point charges and the second

72

term is Eq. (4.11).

4.3.2 Wannier functions representation

The physics behind Eq. (4.11) can be shown clearly when represented in terms of

Wannier functions. Wannier function, in a unit cell with position vector R, is defined in

terms of Bloch wave:

W (λ )n (r−R) =

√NΩ

(2π)3

∫BZ

e−ik·Rψ

(λ )nk (r)dk

=

√NΩ

(2π)3

∫BZ

eik·(r−R)u(λ )nk (r)dk . (4.14)

The prefactor√

N serves as the normalizing factor for Wannier function. It is obtainable

from Fourier transform of Block waves with the same band. The inverse can be also be

done through the use of inverse Fourier transform:

u(λ )nk (r) =(

1√N

)∑R

e−ik·(r−R)W (λ )n (r−R). (4.15)

In contrast to the delocalized plane wave nature of a Block wave, a Wannier function

is localized at lattice point. The Wannier functions associated with different atoms are

orthogonal to each other.

⟨Wn(r−R)

∣∣Wm(r−R′)⟩= δRR′δnm. (4.16)

Its localized nature provides information about chemical bonding, which is missing

from a pure Bloch wave description.

73

Wannier functions are non-unique, a consequence from the phase indeterminacy of

Bloch wave at every wave vector k. While phase freedom has no effect on the properties

of Bloch states, it significant affects the properties of constructed Wannier function.

This imply different Wannier functions can be constructed from the same sets of Bloch

states; a gauge freedom. By using an unitary transformation U (k) operating on a set of

Bloch states corresponding to an isolated group of bands at every k, the most general

Wannier function (Marzari, Souza, & Vanderbilt, 2003) is given by

W (λ )n (r−R) =

Ω

(2π)3

∫BZ

N

∑m=1

U (k)mn ψmke−ik·R dk , (4.17)

where N is the number of occupied Bloch orbitals. This can be regarded as a gauge

transformation of the Bloch orbitals

|ψnk〉=N

∑m=1

U (k)mn ψmk, (4.18)

before Fourier transform is performed to build Wannier function. Wannier function can

then modified into a form that concentrates around the lattice sites, called maximally-

localized Wannier functions (Marzari, Mostofi, Yates, Souza, & Vanderbilt, 2012).

In terms of Eq. (4.15), Eq. (4.11) is reduced to:

P(λ )el =− f |e|

Ω

M

∑n=1

∫r∣∣∣W λ

n (r)∣∣∣2 dr , (4.19)

which is dependent on Wannier centers, the expectation value of positions of electrons

described by Wannier functions.

74

Consider the special case where λ1 = λ2; a closed loop. The initial and final cell

periodic part of Bloch functions can then be different by only a phase factor, due to the

phase indeterminacy of Bloch function:

u(λ2)nk = eiθnku(λ1)

nk , (4.20)

where θnk is a phase factor with the same periodicity in k-space as Bloch functions.

Eq. (4.10) then takes the form

∆Pel =f |e|

(2π)3

M

∑n=1

∫BZ

∇kθnk dk . (4.21)

Under the periodic constraint, the most general form the phase factor can take is

θnk = βnk +k ·Rn, (4.22)

where βnk shares the same periodicity as θnk, which results in

∆Pel =f |e|Ω

M

∑n=1

Rn. (4.23)

Eq. (4.23) shows that polarization is quantized in units of f |e|Ω

R, when the perturbed

Hamiltonian is brought back to the initial state. This implies that the polarization

difference is only defined modulo f |e|Ω

R, which is a central result of modern theory of

polarization: the absolute polarization is multi-valued and only polarization difference

is well defined. In practice, only the part |∆P| ∣∣∣ f e

ΩR1

∣∣∣ is considered, where R1 is the

shortest lattice vector of a given system.

75

CHAPTER 5

COMPUTATIONAL METHODS

5.1 Electronic structure package

All density functional theory (DFT) calculations in this work were performed using

the ABINIT package (Gonze et al., 2009, 2002), a common project of the Université

Catholique de Louvain, Corning Incorporated, and other contributors (http://www

.abinit.org). ABINIT is an electronic structure calculation package whose primary

function is to compute the total energy, charge density, and electronic structure of a

system of electrons and nuclei using pseudopotential and a planewave or wavelet basis.

Ab initio molecular dynamics simulation capability, which is an unification of

classical molecular dynamics and DFT method, is also included in ABINIT, which

allows it to perform structural and geometry optimization using computed DFT forces

and stresses. ABINIT is also capable of calculating response properties through density

functional perturbation theory (DFPT) (Gonze, 1997; Gonze & Lee, 1997), including

atomic displacements, homogeneous electric fields which are utilized in this work.

5.2 PAW potentials and XC functionals

Our results are obtained in the context of projector-augmented-wave (PAW) ap-

proach, which is supported by ABINIT (Torrent, Jollet, Bottin, Zérah, & Gonze,

2008). PAW is an extension of augmented wave method and the pseudopotential

approach (Blöchl, 1994). Free comprehensive PAW potential dataset libraries for most

76

elements have been produced, among which are libraries produced by Jollet-Torrent-

Holzwarth (JTH) and Garrity-Bennett-Rabe-Vanderbilt (GBRV). These potentials

were designed for use in high-throughput density functional theory calculations with a

light computational cost. Their accuracy are comparable to all-electron results from

WIEN2k (Schwarz & Blaha, 2003).

For JTH PAW potentials, 5s2, 5p6, 5d0 and 6s2 levels of barium, 3s2, 3p6, 3d3 and

4s1 levels of titanium and 2s2 and 2p4 levels of oxygen are treated as valence states.

On the other hand, for GBRV PAW potentials, the chosen valence states are 5s2, 5p6,

5d0, 6s2 and 6p0 levels of barium, 3s2, 3p6, 3d1 and 4s2 levels of titanium and 2s2 and

2p4 of oxygen atom.

Table 5.1: Valence states and matching radius of Ba, Ti and O PAW potentials fromJTH library

Ba Ti O

Valence states 5s2 5p6 5d0 6s2 3s2 3p6 3d3 4s1 2s2 2p4

Matching radius (a.u.) s 2.012 2.300 1.414p 2.315 2.113 1.414d 2.315 2.113

Table 5.2: Valence states and matching radius of Ba, Ti and O PAW potentials fromGBRV library

Ba Ti O

Valence states 5s2 5p6 5d0 6s2 6p0 3s2 3p6 3d1 4s2 2s2 2p4

Matching radius (a.u.) s 2.275 1.829 1.211p 2.275 1.829 1.211d 2.275 1.829

The XC energy is evaluated within local density approximation (LDA) and general-

ized gradient approximation (GGA), formalized by Perdew-Burke-Ernzerhof (PBE), for

77

both groups of PAW potentials. The LDA version of JTH and GBRV PAW potentials

are formalized by different groups, where JTH LDA is parametrized by Perdew and

Wang (1992) whereas GBRV LDA is parametrized by Perdew and Zunger (1981) with

no spin-polarization by fitting to the data of Ceperley and Alder (1980). Both the

GGA versions of JTH and GBRV PAW potentials, for both exchange and correlation

parts, are formalized and parametrized by Perdew et al. (1996); Perdew and Yue (1986)

(abbreviated as PBE for the rest of this thesis).

5.3 Convergence studies

Convergence studies of key parameters are conducted before each calculation. The

convergence tests are done on various parameters in the sequences of:

1. Kinetic energy plane wave cut-off. This parameter controls the amount of plane

waves in the basis sets. The higher the cut-off energy, the more accurate the

calculation is, but with heavier computational cost.

2. Double grid fast Fourier transform (FFT) cut-off. This parameter defines the

energy cut-off for the fine FFT grids (also known as the double grid) that governs

the transfer of quantities (densities, potentials, · · · ) from the normal coarse FFT

grid to the spherical grid around each atom. This value needs to be equal or

greater than the plane wave cut-off. Higher magnitude of this parameter will lead

to higher required computational memory.

3. k-points grid. This is a convergence study associated with the sampling of

Brillouin zone for numerical integrations. The k-points grid in ABINIT is set up

using the Monkhorst-Pack scheme (Monkhorst & Pack, 1976), which generates

78

a uniform distribution of k-points in the Brillouin zone. The computation time

is linearly proportional to the number of treated k-points; a balance of between

numerical integration accuracy and computational cost needs to be found.

The tolerance level used in all the convergence tests is 0.0001 Ha. Although there

are some small variations between various structures of BaTiO3, the plane wave cut-

off, double grid cut-off and the k-point mesh are set at 35 Hartree, 85 Hartree and

6×6×6 Monkhorst-Pack k-points respectively for all calculations. For the calculation

of spontaneous polarization using berry phase approaches (see Chapter 4), larger

numbers of k-points are needed in the polar direction: 6×6×20 for tetragonal phase

and 20×20×20 for both orthorhombic and rhombohedral phases.

5.4 Computation work flow

5.4.1 Bulk structure

The computation procedures are summarized in Fig. 5.1. Bulk BaTiO3 unit cells

are constructed for all 4 phases. Typical of a first-principles calculation, a series of

convergence studies with respect to the kinetic energy plane wave cut-off, double grid

FFT cut-off and compactness of k-points grid are carried out. A distinction between

paraelectric and ferroelectric phases are made at this point: the lattice parameter of cubic

phase is allowed to relax, whereas lattice cell parameters for ferroelectric phases are

taken directly from available experimental values in literature. The motivation behind

this choice is that ferroelectric instability of ABO3 materials is greatly dependent on

their unit cell volumes (P. Ghosez, 1997). The atomic positions within the unit cells

of ferroelectric phases were relaxed until the residual forces acting on any particular

79

Construct bulk BaTiO3 unit cells for cubic, tetrag-onal, orthorhombic and rhombohedral phases.

Convergence tests on plane wave cut-off,double grid cut-off and k-point mesh.

Perform relaxation of latticeconstant parameter of unit cell

Perform relaxation of atomicpositions within unit cell

Total energy DFT calculations for theground state of BaTiO3 of all phases,within both LDA and PBE GGA us-ing JTH and GBRV PAW potentials

Compute the values of spontanenous po-larization within Berry phase formalism

Post-processing of results: bandstructures, density of states, . . .

Compute response properties of BaTiO3, withinLDA using both JTH and GBRV PAW potentials

paraelectric phase ferroelectric phases

Figure 5.1: Computational work flow of bulk phase calculations of BaTiO3.

Cartesian component of any atom, excluding fixed ones, were less than 5.0× 10−5

Hartree/Bohr.

Standard total energy DFT calculations are then performed on each phase of BaTiO3

within both LDA and PBE GGA using JTH and GBRV PAW potentials. Using the

computed ground state results, spontaneous polarizations are computed within the Berry

phase formalism; the calculations include paraelectric cubic phase as a control test.

Additional information such as band structures and density of states is extracted by

80

further post-processing of ground state data. Finally the response properties of BaTiO3

are calculated within DFPT within LDA using both JTH and GBRV PAW potentials.

5.4.2 Slab form

Construct tetragonal phase BaTiO3 slabswith in-plane polaization orientation and

Ba-O terminated symmetric surfaces

Convergence tests on plane wave cut-off, doublegrid cut-off, k-point mesh and vacuum thickness

Rebuild the slabs with chosen thickness alongwith the converged vacuum layer thickness

Standard DFT ground state calculationswithin PBE GGA using JTH PAW potentials

Compute the values of spontanenous po-larization within Berry phase formalism

Figure 5.2: Computational work flow for BaTiO3 in slab form.

The computation work flow to investigate BaTiO3 in slab form is described in

Fig. 5.2. Three tetragonal phase slabs of 6, 8 and 10 unit cells in thickness are prepared

and studied, each with a vacuum layer of 6 unit cells in thickness constructed using the

supercell method. The in-plane polarization orientation are chosen due to the lack of

depolarizing field associated with a out of plane orientation. Symmetric Ba-O termi-

nated surfaces are used, following the recent published results by Iles, Khodja, Kellou,

and Aubert that parallel polarization are retained for BaO but not TiO2 terminated sur-

faces (Iles et al., 2014). Standard DFT calculations are then performed with PBE GGA

81

using JTH PAW potentials. Berry phase approach is used to calculate the spontaneous

polarization in the slab, similar to the bulk BaTiO3 case.

82

CHAPTER 6

BULK BARIUM TITANATE: GROUND STATE AND RESPONSEPROPERTIES

6.1 Structural properties

The ferroelectric properties of BaTiO3 is greatly dependent on the relative positions

of atoms in the unit cell, as far as its phase transition of displacive nature is concerned.

Hence it is logical to first determine the basic structural parameters of BaTiO3 in each

of the four phases: cubic, tetragonal, orthorhombic and rhombohedral.

The schematics of BaTiO3 in all four phases are shown in Figs. 6.1 to 6.4. The atoms

in cubic phase of BaTiO3 is constrained by symmetry of a perovskite, with the sole tita-

nium(Ti) atom surrounded by six oxygen(O) atoms in an octahedral arrangement while

the octahedron itself is placed inside a cubic lattice populated by barium(Ba) atoms.

For the structures in ferroelectric phases, the atoms deviate from their symmetrized

positions in the cubic phases. The distortions in the structures give symmetric breaking

properties to BaTiO3. The structures are constructed in a way such that the polar axis

is directed along the Cartesian z-axis. For computational simplicity, the conventional

doubled orthorhombic space group structure is reduced to its basic primitive structure.

For reference purpose, the unit cell vectors, in terms of Cartesian unit vectors, for the

83

orthorhombic structure are

a = 3.984x+0y+0z,

b = 0x+2.837y+2.846z,

c = 0x−2.837y+2.846z.

whereas for rhombohedral case, the lattice vectors are:

a = 3.263x+0y+2.315z,

b =−1.632x+2.826y+2.315z,

c =−1.632x−2.826y+2.315z.

The atomic positions for the structures in ferroelectric phases are reported in Table 6.1

in reduced coordinates.

The structural optimization results and some bulk properties of the cubic phase are

reported in Table 6.2. JTH and GBRV potentials yielded essentially the same results

that compares well with other literature values. The predicted lattice constant followed

(a) In 3D space (b) Projection onto yz plane

Figure 6.1: Unit cells of BaTiO3 in cubic phases

84

(a) In 3D space (b) Projection onto yz plane

Figure 6.2: Unit cells of BaTiO3 in tetragonal phases

(a) In 3D space (b) Projection onto yz plane

Figure 6.3: Unit cells of BaTiO3 in orthorhombic phases. The axes of the compassesshown refer to the three lattice vectors respectively. The red, orange and blue vectors inthe structure refer to Cartesian x,y and z axes respectively.

(a) In 3D space (b) Projection onto xy plane

Figure 6.4: Unit cells of BaTiO3 in rhombohedral phases. The axes of the compassesshown refer to the three lattice vectors respectively. The red, orange and blue vectors inthe structure refer to Cartesian x,y and z axes respectively.

the well known trend that LDA underestimates the unit cell volume whereas GGA

overestimates it. In overall it is shown that PBE GGA gave closer prediction than LDA

to the experimental values, which is especially true for bulk modulus, where the LDA

85

Table 6.1: Atomic position notations for the ferroelectric phases of BaTiO3 in reducedcoordinates.

Phase Atom Position

Tetragonal Ba (0.0, 0.0, 0.0)Ti (0.5, 0.5, 0.5+∆T−Ti)O1 (0.0, 0.5, 0.5+∆T−O1)O2 (0.5, 0.0, 0.5+∆T−O1)O3 (0.5, 0.5, 0.0+∆T−O2)

Orthorhombic Ba (0.0, 0.0, 0.0)Ti (0.5, 0.5+∆O−Ti, 0.5+∆O−Ti)O1 (0.0, 0.5+∆O−O3, 0.5+∆O−O3)O2 (0.5, 0.5+∆O−O1, 0.0+∆O−O2)O3 (0.5, 0.0+∆O−O2, 0.5+∆O−O1)

Rhombohedral Ba (0.0, 0.0, 0.0)Ti (0.5+∆R−Ti, 0.5+∆R−Ti, 0.5+∆R−Ti)O1 (0.5+∆R−O1, 0.5+∆R−O1, 0.0+∆R−O2)O2 (0.5+∆R−O1, 0.0+∆R−O2, 0.5+∆R−O1)O3 (0.0+∆R−O2, 0.5+∆R−O1, 0.5+∆R−O1)

values deviated from the experimental value by about 30 GPa.

We report the calculated lattice constants and relaxed fractional coordinates for the

atoms in the ferroelectric phases in Table 6.3, Table 6.4 and Table 6.5; where JTH-LDA,

GBRV-LDA, JTH-PBE and GBRV-PBE refer to the potentials and XC functionals used

in the present work. The atomic shifts for Ti atom and Oi atoms (where i refer to the

position of oxygen atom) observed from the calculated relaxed fractional coordinates

in these tables are not that consistent for LDA and PBE. In particular, the deviation

in the shift of O3 atom is quite distinct between LDA and PBE results for tetragonal

phase. The tabulated results also imply that predicted atomic shifts are not only sensitive

to the choice of XC functional, but the type of basis set used in other first-principles

calculations also greatly contributes to the discrepancies in the results.

Notably, large deviations on the experimental atomic positions are observed for

86

Table 6.2: Lattice constants and bulk properties of BaTiO3 in cubic phase.

Present work Literature results

LDA PBE

Properties JTH GBRV JTH GBRV Exp. LDA GGA

Lattice constant 3.9463 3.9446 4.0308 4.0268 4.00a 3.943b 4.036c

ao(Å) 3.94d 4.033e

3.95f

Cohesive energy 36.845 36.824 32.295 32.500 31.57a 38.23b 32.7c

(eV/cell) 31.16g

Bulk modulus 192.407 195.342 160.698 162.766 162a 197b 160.84h

(GPa) 197c 166c

a Hellwege and Hellwege (1969) b P. Ghosez (1997)c Evarestov and Bandura (2012) d Cohen and Krakauer (1990) e Zhang, Cagin, andGoddard (2006) f King-smith and Vanderbilt (1992) g Weyrich and Siems (1985)h Uludogan and Cagin (2006)

the tetragonal phase, particularly on titanium (Ti) atom displacement at temperature

320 K (Kwei, Lawson, Billinge, & Cheong, 1993), which is significant to the sponta-

neous polarization of the crystal. The experimental atomic shifts for orthorhombic and

rhombohedral structures are also observed to be distributed across a range of values

(see Table 6.4 and Table 6.5), but the distribution of measured atomic shift is much

smaller in magnitude for the lowest temperature rhombohedral phases. A first-principles

study is typically assumed to be at temperature of 0 K due to no explicit temperature

dependence in the Kohn-Sham equation. This is in consistent that the calculated results

are in better agreement with experimental values for rhombohedral phases than the

higher temperature tetragonal case.

The consistent huge difference between LDA and PBE bulk modulus (B) values is a

clear manifestation of the inherent difference in unit cell volume estimation between the

two XC functionals. In the present calculation of bulk modulus, the constraint on unit

cell volumes of ferroelectric phases are relieved, and relaxations of unit cell volumes

87

Table 6.3: Lattice constants and calculated bulk properties of BaTiO3 in tetragonalphase. The notations follow that defined in Table 6.1.

a(Å) c(Å) ∆T−Ti ∆T−O1 ∆T−O2 Ecoh(eV) B(GPa) Ref.

3.994 4.036 0.0156 -0.0185 -0.0319 36.799 147.611 JTH-LDA3.994 4.036 0.0150 -0.0183 -0.0308 36.776 151.984 GBRV-LDA3.994 4.036 0.0157 -0.0161 -0.0266 32.309 79.741 JTH-PBE3.994 4.036 0.0153 -0.0161 -0.0262 32.515 82.777 GBRV-PBE3.986 4.026 0.015 -0.014 -0.023 Expa

3.986 4.026 0.0135 -0.0150 -0.0250 Exp(301K)b

3.9938 4.0361 0.0215 -0.0095 -0.0233 Exp(320K)c

3.994 4.036 0.0143 -0.0186 -0.0307 LDAd

3.991 4.035 0.0165 -0.0156 -0.0272 37.92 167.64 PW91e

4.007 4.186 0.0193 -0.0226 -0.0431 32.75 100 PBEf

141 Expg

a Shirane, Danner, and Pepinsky (1957) b Harada, Pedersen, and Barnea (1970)c Kwei et al. (1993) d P. Ghosez (1997) e Uludogan, Cagin, and Goddard (2002)f Evarestov and Bandura (2012) g Schaefer, Schmitt, and Dorr (1986)

of BaTiO3 are done at five different values of pressure at −1.0 GPa, −0.5 GPa, 0 GPa,

0.5 GPa, and 1.0 GPa respectively. Bulk modules is then derived from the fitting of

optimized unit cell volumes at different pressures according to the equation,

B =−VdPdV

, (6.1)

where V in this context refers to the optimized stress-free unit cell volume of each phase

and P here refers to pressure (the notation is limited to this equation).

An interesting trend is observed where the bulk modulus drops significantly from

paraelectric cubic phase to ferroelectric tetragonal phase, but increasing slightly from

tetragonal phase to lower temperature orthorhombic and rhombohedral phases. On the

other hand, cohesive energy is almost constant across the three successive structural

transitions, which is in stark contrast with bulk modulus.

88

Table 6.4: Lattice constants and calculated bulk properties of BaTiO3 in orthorhombicphase. The notations follow that defined in Table 6.1.

a(Å) b(Å) c(Å) ∆O−Ti ∆O−O1 ∆O−O2 ∆O−O3 Ecoh(eV) B(GPa) Ref.

3.984 5.674 5.692 0.0138 -0.0146 -0.0256 -0.0159 36.817 145.531 JTH-LDA3.984 5.674 5.692 0.0134 -0.0145 -0.0253 -0.0159 36.789 145.821 GBRV-LDA3.984 5.674 5.692 0.0137 -0.0127 -0.0215 -0.0140 32.327 84.645 JTH-PBE3.984 5.674 5.692 0.0134 -0.0125 -0.0212 -0.0139 32.523 85.953 GBRV-PBE3.9841 5.6741 5.6916 0.0079 -0.0145 -0.0233 -0.0146 Exp(230K)a

3.990 5.669 5.682 0.010 -0.010 -0.016 -0.010 Exp(263K)b

3.984 5.674 5.692 0.0127 -0.0144 -0.0230 -0.0162 LDAc

3.9914 5.7830 5.8223 0.0159 -0.0075 -0.0206 -0.0234 40.26 87.39 PBEd

3.995 5.792 5.837 0.0172 -0.0154 -0.0314 -0.0203 32.76 96 PBEe

a Kwei et al. (1993) b Shirane et al. (1957) c P. Ghosez (1997) d Uludogan and Cagin (2006)e Evarestov and Bandura (2012)

Table 6.5: Lattice constants and calculated bulk properties of BaTiO3 in rhombohedralphase. The notations follow that defined in Table 6.1.

a(Å) deg(°) ∆R−Ti ∆R−O1 ∆R−O2 Ecoh(eV) B(GPa) Ref.

4.001 89.87 -0.01247 0.01307 0.02173 36.829 140.380 JTH-LDA4.001 89.87 -0.01187 0.01276 0.02079 36.800 143.712 GBRV-LDA4.001 89.87 -0.01137 0.01058 0.01675 32.347 93.293 JTH-PBE4.001 89.87 -0.01155 0.01091 0.01717 32.516 93.641 GBRV-PBE4.001 89.868 -0.013 0.011 0.018 Exp(77.4K)a

4.0036 89.839 -0.0128 0.0109 0.0193 Exp(15K)b

4.003 89.84 -0.0105 0.0116 0.0183 LDAc

4.073 89.710 -0.0150 0.0143 0.0249 32.77 101 PBEd

a Hewat (1973) b Kwei et al. (1993) c Hermet, Veithen, and Ghosez (2009)d Evarestov and Bandura (2012)

89

6.2 Electronic structure

The band structures and corresponding density of states within KS formalism are

calculated and given in Figs. B.1 to B.4 of appendix for JTH-PBE PAW potentials. The

respective diagrams for GBRV-PBE potentials and LDA XC potential are attached in

the Appendix. The traversed k-vector path in plotting the eigenvalues of KS equations

at high symmetry paths or points followed that of the path suggested by Setyawan and

Curtarolo (2010), which give all information available in the first Brillouin zone.

The correct estimation of band gap is a weakness of DFT within the KS formal-

ism, for the energies of the KS eigenstates have no real physical meaning. The only

redeeming point is that the eigenvalue of the highest occupied state will be the first

ionization energy of the system if the XC functional used is exact, which is not the

case in actual implementation of DFT. However, KS eigenenergy spectrum is known

to have qualitative agreement with experiment, (without considering highly correlated

electronic system) that reasonable prediction of trends of electronic band structures can

be studied.

The KS energy eigenvalues are shifted to have a zero Fermi energy in each of cubic,

tetragonal, orthorhombic and rhombohedral structures, together with the accompanying

plots of density of states. All the shortest transitions path from valence to conduction

bands are shown to be indirect, where the highest occupied valence state and the lowest

unoccupied conduction state are not the same point in the Brillouin zone. The basic

information extracted from the calculated electronic structures of each phase (within

JTH-PBE) is summarized in Table 6.6.

90

Table 6.6: Band gap, valence band maximum (VBM) and conduction band minimum(CBM) of electronic structures of BaTiO3 in cubic, tetragonal, orthorhombic andrhombohedral phases within PBE XC potentials.

Phase Potentials Band gap (eV) VBM CBM

Cubic JTH-PBE 1.712 R Γ

GBRV-PBE 1.760JTH-LDA 1.756

Tetragonal JTH-PBE 1.801 A Γ

GBRV-PBE 1.843

Orthorhombic JTH-PBE 2.116 T Γ

GBRV-PBE 2.141

Rhombohedral JTH-PBE 2.253 Z Γ

GBRV-PBE 2.309

91

Γ X M Γ R XE

G (indirect) = 1.71194 eV

-60

-50

-40

-30

-20

-10

0

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-60

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure 6.5: Band structure (left) and density of states (right) of BaTiO3 in cubic phase using JTH-PBE PAW potential.

92

Γ X M Γ Z R A Z X R M AE

G (indirect) = 1.80086 eV

-50

-40

-30

-20

-10

0

10E

ner

gy (

eV)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure 6.6: Band structure (left) and density of states (right) of BaTiO3 in tetragonal phase using JTH-PBE PAW potential.

93

Γ X S R A Z Γ YX1 A1T Y Z TE

G (indirect) = 2.11617 eV

-50

-40

-30

-20

-10

0

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure 6.7: Band structure (left) and density of states (right) of BaTiO3 in orthogonal phase using JTH-PBE PAW potential. The separation distancebetween Y and X1 and A1 and T are negligible.

94

Γ L B1 B Z Γ X Q F P1 Z L P

EG (indirect)

= 2.25291 eV

-50

-40

-30

-20

-10

0

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure 6.8: Band structure (left) and density of states (right) of BaTiO3 in rhombohedral phase using JTH-PBE PAW potential. The separationdistance between B and Z, Q and F and P1 and Z are negligible.

95

The valence band maximum (VBM) and conduction band minimum (CBM) are

consistent with previous theoretical studies. Cubic BaTiO3 is found to have a indi-

rect Γ−R gap within PBE, consistent with that found by Bilc et al. (2008); Sanna,

Thierfelder, Wippermann, Sinha, and Schmidt (2011). In contrast with the result of

Saha, Sinha, and Mookerjee (2000) that suggested a direct band gap of 1.2eV using

first-principles tight-binding linear muffin-tin methods within LDA, LDA calculation in

this work yields an indirect Γ−R energy gap similar to the PBE case. The DFT band

energy gap of ∼ 1.7eV is in agreement with the calculated result of Sanna et al. (2011)

of 1.63eV and Seo and Ahn (2013) of 1.9eV. Expectedly, the cubic band gap is much

lower than the experimental value of 3.3eV(Wemple, 1970). It was reported by Seo

and Ahn (2013) that the effect of more advanced GW calculations with the inclusion

of quasiparticle effect have only a small effect on the band dispersion, but widen the

interband separation to more closely approximate experimental value; the trend of band

dispersion within DFT single particle approximation is acceptable.

Qualitative analysis of electronic structure within DFT KS formalism is verified

with a further comparison of ferroelectric tetragonal BaTiO3 phase. The experimental

value of the band gap for tetragonal phase is 3.4eV(Wemple, 1970). In comparison with

that of cubic phase, the band gap of tetragonal phase increases by a small 0.1eV, which

is reflected by the values of band gap of tetragonal BaTiO3 in this work which are also

0.1eV higher that their cubic counterparts for all PAW potentials and XC functionals.

For the orthorhombic and rhombohedral structures, an experimental energy band

gap value is unavailable to the extent of our knowledge. In the contrast to GGA

PBE functional calculations by Evarestov and Bandura (2012) which obtained band

96

gaps of 2.5eV and 2.7eV for orthorhombic and rhombohedral phases respectively, the

corresponding results obtained in this work are lower by approximately 0.4eV. The

change in band gap of about 0.15eV due to structural transition from orthorhombic to

rhombohedral phase, however, is consistent with that of Evarestov and Bandura (2012).

Despite the variation in KS energy eigenvalue within various XC functionals, a

comparatively large increase in band gap of about 0.3eV is observed for the structural

transition from tetragonal to orthorhombic phase. Evarestov and Bandura (2012)

reported an even higher change in band gap of 0.6eV for this particular structural

transition. Nevertheless, experimental difficulties with regard to characterization of the

orthorhombic and rhombohedral phase of BaTiO3 lead to the lack of information of

electronic structures of these two phases.

Across all four structures down the temperature, it is noticed that the bondings

in BaTi3 exhibit a mixture of ionic and covelent characters. A examination of the

DOS plots in Figs. B.1 to B.4 reveals that deep valence band states are mainly ionic in

character, in that their occupancies of states manifest as sharp peaks around each energy

eigenvalue in the band diagram. However, bondings of covalent nature are noticed

around the vicinity of Fermi energy level, with a spread in density of states for the

topmost valence band and the lowest conduction band. This represents a hybridization

of states with a mixture of ionic and covalent bonding characters, which is essential

to the formation of large anomalous effective charge in BaTiO3. The exact nature of

the hybridization can be further investigated with a projected density of states (PDOS)

plots, discussed in Section 6.3.

97

Another distinct feature of DOS plots is the sharp valence and conduction band

edges near Fermi energy, which is supported by theoretical (Salehi, H., Shahtahmasebi,

N., & Hosseini, S. M., 2003) and experimental findings of sharp adsorption and binding

energy edges in photoemission experiment (Chen et al., 2011; Hudson, Kurtz, Robey,

Temple, & Stockbauer, 1993). The electronic structures were found to be not changing

significantly during cubic to tetragonal phase transitions (Hudson et al., 1993), where

the results in this work show that it is also true for tetragonal to orthorhombic and

orthorhombic to rhombohedral transitions.

6.3 Born effective charge

Born effective charge (BEC) Z∗, a fundamental quantity relating polarization to

atomic displacement, is essentially used to analyse the contribution of various atoms to

the ferroelectricity of BaTiO3. Ferroelectric materials are well known for exhibiting

large effective charges compared to the usual ionic charges of the constituent elements.

This large effective charge is dubbed as the anomalous charge in literature (Rabe, Ahn,

& Triscone, 2007). The BEC tensor is a measure of the interaction between polarization

and sublattice displacement in the absence of an applied electric field. The BEC tensor

element of the ith atom is defined as

Z∗i,αβ=

Ωcell

e∂Pα

∂uiβ, (6.2)

where α and β denote two directions and i denotes an atom. Pα is the component of

the polarization in the αth direction and uiβ is the periodic displacement of the ith

atom in the β th direction. Ωcell is the unit cell volume and e is the electron charge.

98

From the equation, it is obvious to see that BEC tensor connects the polarization and

displacement. Therefore, BEC is a second-rank tensor whose elements are restricted by

the symmetry of the ionic site. Recent first-principles calculations reported anomalous

values of the elements of BEC tensors of the constituent ions of ferroelectric materials,

which is related to the increasing covalent character of cation-anion bonds in these

materials. In the following, we will examine the BEC of BaTiO3 based on the JTH and

GBRV PAW potentials.

Born effective charges of all four phases of BaTiO3 were computed in the framework

of DPFT within the XC functional LDA. GGA calculation is not included due to the

current limitation of ABINIT in implementing DFPT with GGA in PAW potential.

The charges of Ba, Ti and O in the cubic phase of BaTiO3 are presented in Table 6.7.

For oxygen atom, its effective charge is divided into Z∗O‖ and Z∗O⊥ , which refer to the

charge induced with displacement of oxygen atom parallel or perpendicular to the

Ti-O bond direction respectively. In general, it is seen from Table 6.7 that the BEC

for Ti and some O ions are much larger compared to their nominal charges. This is

due to the strong hybridization between O 2p and Ti 3d orbitals which will be further

analyzed by examining the total density of states (DOS) and projected density of

states (PDOS). Comparing to the nominal ionic charges of Ba and Ti of ZBa =+2 and

ZTi =+4 respectively, the magnitudes of the anomalous charges of Ba and Ti increase

by a factor of 1.441 and 1.986 in JTH PAW potential, but the increment factors for

those of GBRV PAW potentials are 1.439 and 1.770 respectively. The ratio Z∗Ba/ZBa is

almost similar in magnitude for both JTH and GBRV potentials, but the ratio Z∗Ti/ZTi

differs significantly between JTH and GBRV potentials. This indicates considerable

differences in construction of both Ti PAW potentials in simulating response properties

99

of Ti element.

Table 6.7: Born effective charge of atoms of BaTiO3 in cubic phase.

Z∗Ba Z∗Ti Z∗O⊥ Z∗O‖ Z∗Ba/ZBa Z∗Ti/ZTi Ref.

2.8828 7.9447 -2.4142 -5.9991 1.441 1.986 JTH-LDA2.8787 7.0787 -2.1338 -5.6899 1.439 1.770 GBRV-LDA2.9 6.7 -2.4 -4.8 1.450 1.675 Phenomenological theorya

2.77 7.25 -2.15 -5.71 1.385 1.813 LDAb

2.74 7.32 -2.14 -5.78 1.370 1.830 LDAc

2.75 7.16 -2.11 -5.69 1.375 1.790 LDAd

2.71 7.80 -2.15 -6.21 1.355 1.950 PBEe

2.69 7.41 -2.14 -5.82 1.345 1.853 PBEe(expt. volume)a Axe (1967) b P. Ghosez et al. (1995) c P. H. Ghosez, Gonze, and Michenaud(1998) d Zhong, King-Smith, and Vanderbilt (1994) e Uludogan and Cagin (2006)

Our results show a similar trend compared to results of other first-principles cal-

culations (P. Ghosez et al., 1995; P. H. Ghosez et al., 1998; Uludogan & Cagin, 2006;

Zhong et al., 1994); except there is a consistently high effective charges for all atoms

obtained with JTH potential compared to GBRV potential in our work. In particular,

Z∗Ti(JTH) is 0.866 higher in magnitude than Z∗Ti(GBRV). The estimation of Z∗ by Axe

(1967) using a phenomenological theory is often used as a comparative benchmark in

other works (P. Ghosez et al., 1995; Uludogan & Cagin, 2006). Our estimations for Z∗Ba

and Z∗O⊥ agree very well with those of Axe whereas the charges of Ti(Z∗Ti) and O(Z∗O‖)

are overestimated (Axe, 1967). The large anomalous effective charges in Ti and O are

implicit indication of covalence effects contributing to the hybridization between O-2p

and Ti-3p orbitals, as discussed by Ghosez et al. using a band-by-band decomposition

method (P. Ghosez et al., 1995). Hybridization between O orbital and unoccupied

metal orbitals had long been suggested as a factor controlling ferroelectricity in ABO3

compounds (Cohen, 1992).

100

The BEC tensors of Ba, Ti and O atoms in the ferroelectric phase of BaTiO3 are

presented in Table 6.8 and Table 6.9. Due to the presence of off diagonal elements in

the tensors corresponding to O atoms, charge tensors of O are separated from those

of Ba and Ti. There is a non-negligible difference between charge tensors obtained

from JTH and GBRV potentials for all ferroelectric phases, and those obtained from

JTH are consistently higher in magnitude, as we have seen in the cubic phase. These

differences are very distinct in magnitude of tensor elements for Ti atom than those

of Ba atom. Using other LDA calculations for the tetragonal phase (P. Ghosez et al.,

1995) as benchmarks, the results shows that JTH has overestimated the Ba and Ti

charge tensors, whereas GBRV has given closer estimates. From table Table 6.8, it is

noticed that the magnitude of the charge tensor elements of Ba are relatively stable, but

those of Ti reduce steadily in the process of structural phase transition from the higher

symmetry phase to low symmetry phase. This is consistent with theoretical finding in

the literature that the magnitude of Z∗Ti is dependent on the Ti-O bond length (P. Ghosez

et al., 1995). From Table 6.9, the comparison between Z∗O for both JTH and GBRV

shows that the differences in the off-diagonal terms of the charge tensor are insignificant;

however, the contrast in diagonal terms are greatly dependent on the potentials used.

The presence of off-diagonal terms starting with orthorhombic phase indicates the

shifting of polarization axis from the direction of lattice vectors, where the ferroelectric

polar axis is kept at Cartesian c-axis for all phases respectively.

101

Table 6.8: Born effective charge tensor of Ba and Ti of BaTiO3 in ferroelectric phase.

Phase Z∗Ba Z∗Ti Ref.

Z∗Ba,11 Z∗Ba,22 Z∗Ba,33 Z∗Ti,11 Z∗Ti,22 Z∗Ti,33

Tetragonal 2.995 2.995 3.117 7.437 7.438 6.130 JTH-LDA2.829 2.829 2.939 6.719 6.719 5.531 GBRV-LDA2.72 2.72 2.83 6.94 6.94 5.81 LDAa

2.74 2.74 2.80 7.19 7.19 6.17 LDAb

2.720 2.720 2.818 7.033 7.033 5.687 LDAc

Orthorhombic 2.833 2.934 2.885 7.354 6.931 6.027 JTH-LDA2.828 2.925 2.879 6.555 6.154 5.267 GBRV-LDA2.72 2.81 2.77 6.80 6.43 5.59 LDAa

Rhombohedral 2.9113 2.9113 2.8551 6.9756 6.9756 5.9009 JTH-LDA2.9013 2.9012 2.8505 6.2352 6.2352 5.2222 GBRV-LDA2.79 2.79 2.74 6.54 6.54 5.61 LDAd

2.783 2.783 2.737 6.608 6.608 5.765 LDAe

a P. Ghosez et al. (1995) b Shah, Bristowe, Kolpak, and Rappe (2008)c Siraji and Alam (2014) d P. Ghosez (1997) e Hermet et al. (2009)

102

Table 6.9: Born effective charge tensor of oxygen atoms of BaTiO3 in ferroelectric phase. The labels of the atoms correspond to those definedin Table 6.1.

Phase Potential O1 O2 O3

Tetragonal JTH

−5.788 0 00 −2.407 00 0 −2.187

−2.407 0 00 −5.788 00 0 −2.187

−2.238 0 00 −2.238 00 0 −4.873

GBRV

−5.483 0 00 −2.100 00 0 −1.919

−2.100 0 00 −5.483 00 0 −1.919

−1.965 0 00 −1.965 00 0 −4.633

Orthorhombic JTH

−5.644 0 00 −2.213 00 0 −2.245

−2.272 0 00 −3.826 1.5290 1.329 −3.334

−2.272 0 00 −3.826 −1.5290 −1.329 −3.334

GBRV

−5.366 0 00 −1.947 00 0 −1.978

−2.008 0 00 −3.566 1.5300 1.332 −3.084

−2.008 0 00 −3.566 −1.5300 −1.332 −3.084

Rhombohedral JTH

−2.739 −0.964 0.689−0.964 −3.852 1.193

0.582 1.008 −2.919

−2.739 0.964 0.6890.964 −3.852 −1.1930.582 −1.008 −2.919

−4.408 0 −1.3780 −2.183 0−1.164 0 −2.919

GBRV

−2.488 −0.965 0.698−0.965 −3.603 1.208

0.596 1.032 −2.691

−2.488 0.965 0.6980.965 −3.603 −1.2080.596 −1.032 −2.691

−4.160 0 −1.3950 −1.931 0−1.192 0 −2.691

103

Hybridizations between Ti 3d and O 2p-states in all four phases are confirmed

by charge density contour plots, DOS and PDOS in Fig. 6.9,Fig. 6.10, Fig. 6.11 and

Fig. 6.12 respectively. For brevity only valence and conduction bands close to the

Fermi level are shown in the PDOS plots. The top valence band is dominated by the

O 2p-orbitals, in combination with a significant mixture of Ti 3d-orbitals indicating

the hybridization between the orbitals. On the other hand, the majority of bottom of

conduction band is formed by Ti 3d-orbitals with a small mixture of O 2p-orbitals. In

overall, a strong hybridization between Ti 3d and O 2p-orbitals is shown in the vicinity

of Fermi level for all the four phases.

In the charge density contour plots, the high intensity contour lines in Ti and O atoms

are omitted for clarity. The charge density overlap along various crystalline planes

in paraelectric and ferroelectric phases provides information on the nature of atomic

bonding in the BaTiO3 unit cell. The significant overlap of charge density contour

lines near Ti and O atoms in all phases indicates the existence of covalent bonding

between the atoms, in contrast with the mainly ionic nature of Ba atoms shown by the

accumulation of electron charge density around them. The paraelectric and ferroelectric

phases are clearly distinguished by the charge density distribution along the crystalline

planes; symmetrical charge distribution occurs in the cubic phase whereas ferroelectric

phases exhibits asymmetrical charge distribution with different directional preferences.

This is clearly due to the displacements of Ti and O ions.

The asymmetry behavior in charge density gives rise to spontaneous polarization in

the ferroelectric phases, which will be discussed in the following section. Despite the

large overlap between Ti and O orbitals, the cubic phase does not possess ferroelectric

104

properties due to its symmetrical atomic arrangements. The practically non-existing

covalent bond between Ba and O provide an explanation to the anomalous charges of

Ba and Ti atoms, where charge of Ti has a huge increment from its nominal charge

whereas Ba charge remains relatively close to its formal charge.

105

Ti

O

O

O O

(a) (200) plane.

Ti

O

OBa Ba

Ba Ba

(b) (110) plane.

-5 0 5 10E-E

F (eV)

0

5

10

15

elec

trons/

eV/c

ell

totalTi-dO-p

(c) DOS and PDOS of Ti-d and O-p orbitals.

Figure 6.9: Electron charge density contours, DOS and PDOS of BaTiO3 cubic phase.The miller indices of the planes corresponds to the structures shown in Fig. 6.1.

106

Ti

O

O

O O

(a) (200) plane.

Ti

O

OBa Ba

Ba Ba

(b) (110) plane.

-5 0 5 10E-E

F (eV)

0

5

10

15

elec

tro

ns/

eV/c

ell

totalTi-dO-p

(c) DOS and PDOS of Ti-d and O-p orbitals.

Figure 6.10: Electron charge density contours, DOS and PDOS of BaTiO3 tetragonalphase. The miller indices of the planes corresponds to the structures shown in Fig. 6.2.

107

Ti

O O

O O

(a) (200) plane.

TiO O

Ba

Ba

Ba

Ba

(b) (011) plane.

-5 0 5 10E-E

F (eV)

0

5

10

15

20

elec

tro

ns/

eV/c

ell

totalTi-dO-p

(c) DOS and PDOS of Ti-d and O-p orbitals.

Figure 6.11: Electron charge density contours, DOS and PDOS of BaTiO3 orthorhombicphase. The miller indices of the planes corresponds to the structures shown in Fig. 6.3.

108

Ti

O

O

Ba

Ba

Ba

Ba

(a) (110) plane.

O

O

O

Ba Ba

Ba

(b) (111) plane.

-5 0 5 10E-E

F (eV)

0

5

10

15

elec

tro

ns/

eV/c

ell

totalTi-dO-p

(c) DOS and PDOS of Ti-d and O-p orbitals.

Figure 6.12: Electron charge density contours, DOS and PDOS of BaTiO3 rhombohedralphase. The miller indices of the planes corresponds to the structures shown in Fig. 6.4.

109

6.4 Spontaneous polarization

The modern theory of polarization (Resta & Vanderbilt, 2007), namely the Berry

phase approach, has overcome the limitation in the first-principles calculation of spon-

taneous polarization (P). According to the Berry phase approach, spontaneous po-

larization arises from both ionic polarization (Pion) and electronic polarization (Pel).

While the ionic component is a well defined quantity from electromagnetic theory,

the electronic part of polarization cannot be directly evaluated on the basis of usual

localized contributions.

The values of P in the ferroelectric tetragonal, orthorhombic and rhombohedral

phase of BaTiO3 are calculated using Berry’s phase formulation as shown in Table 6.10.

Verification of Berry phase approach is done by proving the absence of net P in the cubic

phase of BaTiO3; apart from some commonly encountered inherent numerical noises.

In our work, values of P obtained from both LDA and PBE calculations are considerably

higher than the experimental values, and those from the LDA show bigger deviation. It

is noticed that values of P for tetragonal phase BaTiO3 reported by some theoretical

groups (Bilc et al., 2008; Iles et al., 2014) are closer to the experimental values; such as

0.26 C/m2 (Bilc et al., 2008) and 0.29 C/m2 (Iles et al., 2014). These calculations are

performed using DFT within the XC functional of GGA with Wu and Cohen potential

(GGA-WC) (Wu & Cohen, 2006). Using ABINIT and the same Berry phase approach

similar to our work, Wang et al. (Wang, Meng, Ma, Xu, & Chen, 2010) obtained values

of P closer to the experimental data, especially for the rhombohedral phase. In their

calculations, both lattice cell lengths and atomic positions are optimized, while fixing

the cell at the equilibrium volume of the cubic phase. Similar to the structural properties

110

calculations, we obtained a more accurate value of P for rhombohedral phase than the

other two ferroelectric phases.

Another independent investigation of the values of P is done by fixing both the

lattice cell lengths and atomic positions with experimental values (Kwei et al., 1993) to

isolate the effect of numerical structural relaxations. Contrary to the results for relaxed

structures, consistent values of P are obtained in both LDA and PBE calculations. The

values of P for tetragonal, orthorhombic and rhombohedral phases are 0.311 C/m2,

0.342 C/m2 and 0.397 C/m2 respectively. It is implied that the discrepancies between

values of P of BaTiO3 for LDA and PBE can be mainly traced back to accuracy of

structural properties prediction. By predicting more accurate structural parameters than

LDA, PBE approach is expected to give closer results to the experimental data.

There is a contrary trend comparing our calculated values of P for the three fer-

roelectric phases of BaTiO3 using LDA and PBE approaches with the experimental

results (Merz, 1949; Wieder, 1955). We observe the increasing trend in our calculated

values of P with the structural change from tetragonal to orthorhombic and rhombo-

hedral. This trend of variation of P with structural phase transition, in tandem with

temperature phase transition, is contrary to the temperature variation of P shown in Fig-

ure 3 of the article by Wieder (Wieder, 1955), which were obtained from the hysteresis

loop measurements. Similar trend of temperature variation of P is also reported earlier

by Merz (Merz, 1949).

111

Table 6.10: Spontatenous polarization (C/m2) of relaxed BaTiO3 in tetragonal, or-thorhombic and rhombohedral phase, both along the lattice vectors and polar directions.The polarization vectors for orthorhombic phase are along b and c lattice vectorswhereas rhombohedral values refer to each of the three lattice vector direction. Thepolar axis of tetragonal, orthorhombic and rhombohedral are along [001], [011] and[111] respectively. For tetragonal phase the c-axis coincides with the polar axis.

Phase Present work Exp Literature results

lattice polar Method LDA GGA

Tetragonal 0.351 JTH-LDA 0.263a 0.3402b 0.29c

0.341 GBRV-LDA 0.265d 0.26e

0.317 JTH-PBE 0.30f

0.313 GBRV-PBE 0.243g

Orthorhombic 0.288 0.408 JTH-LDA 0.307a 0.397b

0.285 0.4037 GBRV-LDA 0.330h

0.263 0.373 JTH-PBE0.259 0.367 GBRV-PBE

Rhombehedral 0.250 0.434 JTH-LDA 0.335a 0.402b

0.241 0.419 GBRV-LDA 0.34i 0.350h

0.212 0.368 JTH-PBE0.216 0.376 GBRV-PBE

a Wieder (1955) b P. Ghosez (1997) c Iles et al. (2014) d Shah et al.(2008) e Bilc et al. (2008) f Zhong et al. (1994) g Nakhmanson, Rabe,and Vanderbilt (2005) h Wang et al. (2010) i Hewat (1973)

6.5 Phonon analysis

A summary of phonon modes and frequencies of BaTiO3 in cubic, tetragonal, or-

thorhombic and rhombohedral phases is presented in Table 6.11. Long range coulomb

interaction associated with longitudinal phonon modes are taken into account by con-

sidering non-analytical contribution to the dynamical matrix as q→ 0 (Gonze & Lee,

1997). Experimental measured phonon frequencies (Laabidi, Fontana, & Jannot, 1990;

Luspin, Servoin, & Gervais, 1980; Tenne et al., 2004; Venkateswaran, Naik, & Naik,

1998), although not available for all modes, are used to gauge the accuracy of our works.

The zone center structural instability of the cubic phase via soft modes was obtained

for both XC functionals, which is indicated by the imaginary T1u transverse optical

112

(TO) mode frequency. All the calculated phonon frequencies exhibit only a small

deviation from the experimental values, except for the largest T1u longitudinal optical

(LO) frequency produced by GBRV LDA functional, which is 45 cm−1 lower than

experimental value of 710 cm−1. On the other hand, JTH LDA calculation produces

results on par with values calculated using computationally more expensive hybrid XC

PBE0 functional (Evarestov & Bandura, 2012), although it should be mentioned that

the PBE0 calculation was done using lower DFT integration precision.

Phase transition of BaTiO3 from high temperature cubic phase to lower temperature

tetragonal, orthorhombic and rhombohedral phases are accompanied with branching of

the triply degenerate T1u and silent T2u modes of the cubic phase. The transitions of

phonon symmetry modes from cubic phase are as follow:

T1u→ E+A1→ A1 +B1 +B2→ E+A1

T2u→ E+B1→ A1 +A2 +B2→ E+A2

The silent (infrared and raman inactive) T2u mode through transition path produces

the silent B1, A2 and A2 modes in tetragonal, orthorhombic and rhombohedral phase

respectively. The direction of the perturbation wave vector q determines the occurrence

of LO-TO splitting of the doubly degenerate E modes, where the splitting occurs for q

vectors perpendicular to the z axis whereas the modes remains degenerate for q vectors

parallel to the z axis. The non-degenerate modes A1 are polarized along the optical

Cartesian z axis for all three ferroelectric phases.

The observed deviations of phonon mode frequencies from experimental values are

113

much smaller than 10%, but are consistence with other values in the literature (Choud-

hury, Walter, Kolesnikov, & Loong, 2008; Evarestov & Bandura, 2012; Hermet et al.,

2009; Seo & Ahn, 2013). Nevertheless, our overall findings agree sufficiently well

with the corresponding experimental values and other LDA results. For the phonon

modes of the highest frequency, calculations from JTH LDA functional again pro-

duce values closer to the experimental data (Laabidi et al., 1990; Venkateswaran et al.,

1998). Contrary to the usual perception, LDA phonon frequency calculations are shown

to produce results of the same quality as other more expensive calculations such as

PBE0 (Evarestov & Bandura, 2012).

The imaginary frequencies occur for T1u(TO) mode in cubic phase, E(TO) mode in

tetragonal phase and B1(TO) orthorhombic phase, while it is absent in rhombohedral

phase. Being the lowest temperature phase, the rhombohedral phase has the most stable

structure among the all four phases.

For the sake of completeness the mode effective charge of T1u phonon modes of the

optimized cubic phase are computed (Table 6.12). The unstable TO1 mode (numbered

according to the sequence in Table 6.11), as expected, has an effective charge much

higher than the remaining two T1u TO modes. The displacement of titanium and oxygen

atoms along the Ti-O bonds have particularly high partial contributions to the total mode

effective charge; signifying that this coulomb interaction plays a significant contribution

to polarization. While the similar partial contribution is also observed for TO2, its

overall total mode effective charge is much lower than that of TO1 due to the opposing

sign of Ti and O‖.

114

TO and LO modes generally cannot be coupled directly, however, it has been long

reported that TO and LO T1u phonon modes of cubic phase exhibit a close one to one

correspondence (P. Ghosez, 1997; Zhong et al., 1994). The overlap matrix elements

of eigenvectors of T1u TO and LO modes are direct indications that mixing multiple

modes are small in magnitude, (see Table 6.13). The most probable correspondences

occur between TO1 and LO3 modes, TO2 and LO1 modes as well as TO3 and LO2

modes. The coupling between the unstable TO1 mode with imaginary frequencies and

LO3 modes with the highest frequency shows the reported giant LO-TO splitting of

perovskite ferroelectrics (Zhong et al., 1994).

A simplification is made with the assumption that the LO eigendisplacements when

q→ 0 is the same as those at q = 0. An approximate LO phonon frequencies can then

be obtained using symmetry constraints, mode-oscillator strength tensor and electronic

dielectric constant according to Eq. (62) in the paper of Gonze and Lee (1997). The

dielectric tensor, similar to BEC, is one of the physical properties connected to second

derivatives of total energy with respect to phonons and static homogeneous electric field

perturbations. Since LDA is well known of producing inaccurate dielectric constant,

scissor method (Levine & Allan, 1989) was used to reduce the overestimated dielectric

constant (in atomic unit) of 6.746 to 5.645 by fixing the band gap to the experimental

value of 3.2 eV. The resulting values are 722 cm−1, 218 cm−1 and 488 cm−1, which

are consistent with calculated DFPT values of 707 cm−1, 179 cm−1 and 464 cm−1.

115

Table 6.11: Phonon modes and frequencies (cm−1) of BaTiO3 in cubic, tetragonal,orthorhombic and rhombohedral phases.

Mode JTH-LDA GBRV-LDA Experiment(This work) (This work)

CubicT1u (TO) 173i, 185, 475 153i, 184, 478 softa, 182a, 482a

T1u (LO) 180, 464, 707 178, 456, 665 180a, 465a, 710a

T2u (silent) 290 288 306a

TetragonalA1 (TO) 162, 329, 518 160, 322, 515 170b, 270b, 520b

A1 (LO) 185, 461, 743 183, 450, 703 185b, 475b, 720b

E (TO) 196i, 168, 287, 451 186i, 166, 285, 452 softb, 180b, 305b, 486b

E (LO) 161, 287, 446, 663 159, 285, 441, 617 180b, 305b, 463b, 715b

B1 (silent) 290 288 305b

OrthorhombicA1 (TO) 167, 298, 306, 522 163, 292, 309, 520 -, -, -, 532c

A1 (LO) 183, 298, 469, 719 180, 297, 463, 676 -, 320c, -, -B1 (TO) 154i, 169, 454 150i, 167, 454 -, 193c, 490c

B1 (LO) 164, 447, 670 161, 440, 624B2 (TO) 163, 266, 283, 470 160, 260, 284, 470 -, 270c, -, -B2 (LO) 177, 282, 445, 730 176, 282, 430, 691 -, -, -, 720c

A2 (silent) 289 287 320c

RhombohedralA1 (TO) 170, 291, 528 166, 282, 521 173d, 242d, 522d

A1 (LO) 184, 473, 709 180, 469, 662 187d, 485d, 714d

E (TO) 166, 242, 297, 469 163, 227, 297, 467E (LO) 177, 297, 446, 724 175, 296, 434, 679 -, 310d, -, -A2 (silent) 280 278

a Reference Luspin et al. (1980). b Reference Venkateswaran et al. (1998).c Reference Laabidi et al. (1990). d Reference Tenne et al. (2004).

Table 6.12: Mode effective charge and contribution from each atom for the T1u modesof the optimized cubic phase. Phonon modes are numbered according to the sequencein Table 6.11.

Modes Contribution of constituting elements Mode effective charge

Ba Ti O⊥ O‖

TO1 0.016 3.759 0.929 4.157 9.789TO2 0.959 −3.873 1.150 2.669 2.054TO3 0.008 −0.719 −1.170 4.336 1.286

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Table 6.13: Overlap matrix elements of eigenvectors of T1u TO and LO modes of the op-timized cubic phase. Phonon modes are numbered according to sequence in Table 6.11.

LO1 LO2 LO3

TO1 0.188 −0.279 −0.942TO2 −0.982 −0.063 −0.178TO3 −0.009 0.958 −0.285

6.6 Summary

In this chapter, comparisons of structural properties and polarization of all four

structural phases (cubic, tetragonal, orthorhombic and rhombohedral) of BaTiO3 is made

using both LDA and GGA-PBE XC functionals from two source of PAW potentials:

JTH and GBRV. Zone center phonon mode properties are also reproduced with PAW

potentials as a test of transferability of the potentials to predict response properties

of a material. LDA and PBE calculations are shown to provide sufficiently accurate

structural information for BaTiO3, with PBE produces closer results of derived structural

properties such as cohesive energy and bulk modulus to experimental data than those

from LDA. Spontaneous polarization in various structural phases of ferroelectric BaTiO3

depends heavily on the accuracy of predicted structural parameters, which contribute to

better results with PBE as XC functionals.

Nevertheless, one important point to note is that the increasing trend of spontaneous

polarization with decreasing temperature contradicts the experimental trend of decreas-

ing spontaneous polarization with the transition from tetragonal to rhombohedral phase.

This discrepancy can be traced to the imposition of artificial symmetries to each phase

to mimic the real BaTiO3 structure at each phase. A full computation will require

some runs of molecular simulations where the forces between particles are calculated

through ab-initio method, which is not feasible for both the limited time span and

117

required computational power. For the current work, however, the simple imposition of

symmetries to each phase of BaTiO3 is an acceptable approximation, for the explicit

dynamics of BaTiO3 is not the emphasis of this work. Besides, Born effective charges

of all four phases are computed, including the less reported orthorhombic phase. In

summary JTH PAW potentials produce born effective charges with higher magnitude

than GBRV potentials. Although LDA calculation suffers from its lower accuracy in

computing structural properties, the qualities of phonon mode calculations under the

framework of LDA are unusually good, as proven by the current work.

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CHAPTER 7

BARIUM TITANATE SLAB

7.1 Motivation

The continued miniaturization of electronic devices, of which BaTiO3 is an usual

component, makes nanoscale material investigation a necessity. The investigations

of nanostructures often involve ab-initio calculations of the studied material in finite

forms, one of which is in the shape of a slab. A slab is formed by a truncation of a bulk

structure, where the geometry remains infinite in a two-dimensional plane but finite

in the direction orthogonal to the plane accompanied by a vacuum gap. Usually, the

motivation behind the study of a material in slab form is to study the properties of a

thin film, which involves the growth of the film on top of a supporting substrate. For

the purpose of simulating the finite size effect of a material, however, it is sufficient to

study a slab as a free standing structure without including the supporting substrate. In

this work DFT calculations are performed for BaTiO3 in slab form. The free standing

BaTiO3 slab model is adopted for DFT calculations.

One of the complication of DFT calculation of a slab is the applicability of the data

compared to real experimental results. The geometry, or the thickness particularly, of a

slab studied using DFT is usually limited in size, where the input model structure is often

limited to an order of ten unit cells in thickness due to steep scaling of computational

costs and constraints. In contrast, the thickness of a typical thin film grown in an

experimental is of several hundred nanometres. While the theoretical results cannot be

compared with the experimental result directly, finite changes such as the changes in

119

interlayer distance and roughness of surface compared to bulk structure are best studied

using an electronic structure code. In this sense, the results of DFT calculations can

serve as supplementary information to those obtainable from experimental works.

The focus in this chapter lies in the investigation of BaTiO3 tetragonal (100) surface

structures via DFT calculation within the framework of PAW method. The structural

modification for three slabs with different thickness are investigated, along with the

in-plane spontaneous polarization using the same Berry phase formalism as in bulk

calculation. The possibility surface reconstruction is omitted in the consideration of

this work, with only the perpendicular and parallel atomic displacements are studied in

the form of interlayer spacing and rumpling.

7.2 Computational methodology

The general computational flow is as described in Chapter 5. The general supercell

slab model is illustrated in Fig. 7.1, with the Cartesian x-axis in direction of polar axis

and z-axis normal to the plane.

The slabs can be separated into two major groups depending on the top and bottom

layer structures: symmetric (non-stoichiometric) and asymmetric (stoichiometric) slabs.

A symmetric (asymmetric) slab possesses similar (different) surface layers for both of

its top and bottom layer. In the case of BaTiO3 slab with tetragonal (100) surface, two

type of layers are present: (i) BaO and (ii) TiO2 layers (Fig. 7.2).

The symmetric type of slab is the focus of this work. The thickness of slab is

quantified in terms of number of unit cell present, where the term "unit cell" is exactly

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Figure 7.1: BaTiO3 (100) slab supercell model with parallel in-plane polarization

the same as that defined in a bulk calculation. For ease of reference, a naming convention

is established, in which a slab of x-unit-cell thick is to be referred to a xUC slab (6UC

slab for a slab with 6-unit-cell thick etc). Symmetric slabs with BaO surface layer

are constructed for three thickness of 6, 8 and 10 unit cells respectively. An extra

asymmetric slab with BaO and TiO2 layers as top and bottom surfaces respectively

is built for the case of 6-unit-cell thick slab to investigate the effect of perpendicular

symmetry in the slab. The motivation behind the choice of BaO symmetric slab is due

to the findings of Iles et al. (2014), who reported the existence of in-plane polarization

for the BaO symmetric slab.

The initial lattice parameters of unit cell blocks in the slabs are obtained from

further relaxation of lattice constants of the primitive cell of bulk BaTiO3 in tetragonal

structural phase in Chapter 6. Under the constrain of constant unit cell volume, the

relaxed bulk lattice parameters are estimated to be a = 3.976Å and c = 4.072Å. Similar

121

Figure 7.2: Tetragonal BaTiO3 (100) slab surface: (left) BaO surface and (right) TiO2surface. Polar axis is along the x-axis with z-axis normal to the surfaces.

to the structural relaxation procedure in Chapter 6, the atomic positions in the slabs are

relaxed with the convergence criteria of residual forces of less than 5.0×10−5 Ha/Bohr

acting on any atom in each Cartesian direction, excluding the atoms fixed in positions

by symmetry (the y-coordinate of the atoms).

The same DFT calculation parameters are used for all slabs. The wave function is

expanded at each point in reciprocal space in terms of plane wave with a kinetic energy

cutoff of 32 Ha; and an array of 6×6×1 Monkhorst-Pack k-point mesh is chosen. For

the spontaneous polarization computation within the Berry phase approach, the number

of k-point along the polarization direction is increased to 20, which results in 20×6×1

k-point array. The convergence studies conducted to determine the implementation

parameters are described in detail in Appendix C.

7.3 Results and discussion

7.3.1 Preliminary 3UC slab comparison tests

A systematic quantification method of the atomic relaxation at different atomic

layers is needed. In this work, the quantitative atomic relaxation estimations followed

122

that used by Padilla and Vanderbilt (1997) and Iles et al. (2014). Two parameters,

charactering the out of plane atomic displacements are defined for each atomic layer i,

are introduced:

1. average displacement: βi = [δz(Mi)+δz(Oi)]/2,

2. rumpling: ri = [δz(Mi)−δz(Oi)]/2,

where M refers to the metal elements in each atomic layer (Ba or Ti), O refers to oxygen

elements in same layer. δz refer to displacement of Ba, Ti or O elements from their

ideal unrelaxed positions along the out-of-plane z-axis (Fig. 7.2). For the TiO2 layer,

δz(O) is the average displacements of two oxygen in the same plane.

Due to the relative abundance of information in regard to BaTiO3 slab with cubic

(001) surface, a preliminary test concerning the accuracy of the current method is

performed. A symmetric 3UC slab with cubic (001) surface (7 atomic layers) is used as

a test case. The initial lattice constant of the BaTiO3 unit cells comprising the BaTiO3

slab in cubic phase follows that obtained in Chapter 6, which is 4.030Å. For the cubic

slab surface, the only degree of freedom subjected to atomic relaxation is the z-direction

perpendicular to the slab lateral plane.

The interlayer displacements and rumpling of each layer obtained are presented in

Table 7.1. Due to the symmetric in the perpendicular direction, only the data corre-

sponding to the top half layers are provided. The positive or negative signs presiding

the rumpling value is dependent on the whether the top of bottom half of the slab is

chosen for analysis. The physical quantity ∆di j, the interlayer spacing displacement, is

defined to be the difference of average displacements β of the successive layer i and j.

123

The outermost interlayer spacing, which is between the surface and second top layer,

contracts by about 0.11Å, which is most significant distortion among all the layers.

This is expected due to lack of atomic coordination number at the surface layer, and the

accuracy of this value is supported by all the cited references in Table 7.1 except Cai

et al. (2007), who report a value of −0.0924Å. A trend of alternating contraction and

expansion of the interlayer spacing down the surface is noticed and supported by the

cited literatures, with the displacement magnitudes decrease towards the bulk region.

As expected, the surface has the highest value of rumpling, which is about 2.5

times that of the second layer. The magnitude of rumpling r1 value of 0.0260Å is

consistent with reported value of 0.27Å in literature, except Iles et al. (2014) who report

a much higher value with magnitude of 0.0340Å. The rumpling values alternate in

direction towards the bulk region with decreasing magnitude, where the central layer

is observed to have no rumpling as in the bulk case. A point of interest is that the

interlayer displacements and rumpling values is fairly insensitive to the thickness of the

slab, as reference values cited in Table 7.1 are for slabs of different layers. The atomic

relaxations in a symmetric slab are hence assumed to be limited to the first few layers at

the surface.

In contrast with the use of experimental lattice constant of tetragonal phase of

BaTiO3 in Chapter 6, relaxation of bulk unit cell lattice constant is done to obtain

an initial estimate of the size of the tetragonal slab supercell model. This serves to

isolate the effects of stress-related variables from the results, so that the consequences

of surface relaxations can be emphasised and studied. Under a constant volume unit

cell optimization using the bulk tetragonal BaTiO3 unit cell of Chapter 6 as the input

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Table 7.1: Interlayer displacements (∆di j) and rumpling (ri) of the top four layers of a 3-unit-cell thickness slab with cubic phase and BaO terminations, in units of Å. Referencedata is modified to conform to interlayer displacements and rumpling definitions usedin this work.

This work GGA-WCa LDAb LDAc,* GGA-PBEd B3PWe

∆d12 −0.1155 −0.1140 −0.1103 −0.1106 −0.0924 −0.1155∆d23 0.0640 0.0610 0.0432 0.0435 0.0731 0.0630∆d34 −0.0145 −0.0230 −0.0156 −0.0158 −0.0036

r1 −0.0260 0.0340 −0.0274 0.0277 −0.0273 −0.0273r2 0.0101 −0.0100 0.0087 0.0079 0.0102 0.0068r3 −0.0087 0.0110 −0.0053 0.0059 −0.0108r4 0.0000 −0.0020 0.0000 0.0076

a Iles et al. (2014) b Padilla and Vanderbilt (1997) c Meyer, Padilla, andVanderbilt (1999) d Cai et al. (2007) e Eglitis, Borstel, Heifets, Piskunov,and Kotomin (2006) * Rumpling for each layer is in absolute value.

geometry, the lattice constants are relaxed to values of a = 3.976Å and c = 4.072Å.

Similar to the cubic surface, a symmetric 3UC BaTiO3 slab with tetragonal BaO (100)

surfaces is relaxed and the results are again compared to existing literature findings.

Similar to the cubic slab case, the slab model is made up of 7 alternating BaO and TiO2

layers, with BaO layers as both terminated outer surfaces and TiO2 layer at the center

of slab.

One of the differences with the previous literature findings is that the sign of ∆d34 is

positive instead of negative as found by Iles et al. (2014); Meyer et al. (1999); Padilla

and Vanderbilt (1997), which is also in contrast with that found for the slab with cubic

unit cells. This implies that the two innermost layers are expanding. While the top

two interlayer spacings shrinks and expands respectively, in agreement with literature

findings, the magnitude of ∆d12 is smaller than other theoretical results whereas that of

∆d23 is about twice larger. Plane wave basis sets were employed in the referenced works:

Iles et al. (2014) uses norm-conserving plane wave pseudopotentials whereas ultrasoft

125

Table 7.2: Interlayer displacements (∆di j) and rumpling (ri) of the top four layers of a3UC slab with tetragonal phase and BaO terminations, in units of Å. Reference data ismodified to conform to rumpling definitions used in this work.

This work GGA-WCa LDAb LDAc,*

PAW NC

∆d12 −0.0775 −0.0849 −0.1120 −0.1089 −0.1103∆d23 0.0886 0.0868 0.0580 0.0451 0.0433∆d34 0.0214 0.0173 −0.0120 −0.0156 −0.0158

r1 −0.0208 −0.0228 0.0350 −0.0297 0.02955r2 0.0090 0.0088 −0.0100 0.0091 0.00985r3 −0.0073 −0.0075 0.0100 −0.0077 0.00788r4 0 0 −0.0020 0

a Iles et al. (2014) b Padilla and Vanderbilt (1997)c Meyer et al. (1999) * Rumpling for each layer is in absolutevalue.

Table 7.3: Ferroelectric distortion per layer (in units of Å) of relaxed 3UC slab. Refer-ence data is modified to conform to definitions used in this work.

Layer This work GGA-WCa,* LDAb LDAc

δFE(BaO) δFE(TiO2) δFE(BaO) δFE(TiO2) δFE(BaO) δFE(TiO2) δFE(BaO) δFE(TiO2)

1 0.0749 −0.0056 0.0623 0.06382 0.1391 −0.0476 0.0727 0.07183 0.0654 −0.0100 0.0523 0.05194 0.1665 −0.0016 0.1326 0.1037

Bulk 0.0752 0.1698a Iles et al. (2014) b Padilla and Vanderbilt (1997) c Meyer et al. (1999) * Estimations from graph.

pseudopotentials (Vanderbilt, 1990) are employed by both Meyer et al. (1999); Padilla

and Vanderbilt (1997). On the other hand, PAW potentials, which is a combination of

all electrons method and plane wave implementation are used in this work.

An attempt to resolve the discrepancies with the literature results is made by re-

peating the calculations using norm-conserving (NC) pseudopotential instead of PAW

potentials. For this purpose, the optimized norm-conserving Vanderbilt pseudopo-

tentials (ONCVPSP) designed by Hamann (2013) are used. The resulting values are

tabulated alongside their corresponding PAW values for comparison in Table 7.3. Apart

from some numerical deviations, no significant change is noticed between PAW and

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ONCVPSP results. It is particular evident by observing that δ23 is still almost twice

the magnitude of the literature results, and δ34 bears the same positive sign as the PAW

value. At the very least, the differences in the basis sets are hence not the dominant

contributor to the deviations of computed results in Table 7.3. The relative accuracy

with respect to literature results is still cannot be ascertained, due to the absence of a

decisive experimental work and the possibility of surface reconstruction in an actual

surface.

Values of rumpling of top four layers are in agreement with literature results, except

for r1 which is about 0.01 Å less in magnitude compared to the others. r4 is found to be

zero, implying a bulk-like flat geometry for the central layer similar to the cubic case.

A brief inspection reveals a decrement in magnitude of rumpling from the surface to

the central layers, in resemblance to the case of cubic surface. In accordance with the

definition of rumpling ri = [δz(Mi)−δz(Oi)]/2, positive values implies a higher z-axis

position of metal elements compared to oxygen, and vice versa for negative values. An

alternating positive and negative values of rumpling, as shown in Table 7.2 signifies the

presence of oscillating dipole moments in the direction normal to plane. The total dipole

moment perpendicular to surface is the net summation of individual dipole moments

of each layer, and is dominated by the outer layers due to atomic relaxations. The

remaining dipole moment determines the magnitude of the depolarizing field, which

is negligible in this case due to the cancellation of local moments by each successive

layer. This also holds true for the case of slab with cubic unit cells.

Compared to the cubic case, the tetragonal BaTiO3 slab with in-plane polarization

orientation has atomic relaxation along the parallel direction; the orthogonal Cartesian

127

directions on the plane are not equivalent. The contributions of the surface effects and

atomic relaxations to the in-plane ferroelectric distortion are tabulated in Table 7.3. The

ferroelectric distortion δFE per layer is defined to be

δFE = δx(Mi)−δx(Oi), (7.1)

with the naming convention similar to the previous perpendicular case. i is a numbering

notation of the layers and δx refers to the displacement along the x-axis. Only half of

the slab is explicitly discussed due to symmetry, with layer index 1 corresponds to the

outermost surface layer. BaO layers are found to contain less ferroelectric distortion

than TiO2 layers, which is natural corresponding the bulk distortion of 0.0752 Å and

0.1698 Å for BaO and TiO2 layers respectively. Compared to the bulk distortion,

δFE(BaO) is larger for the surface layer, but it is smaller for the second BaO layer. In

contrast, both the first TiO2 layer below the surface and central TiO2 layer have smaller

values of δFE compared to the bulk distortion, with δFE increases towards the bulk

region of the slab. The ferroelectric displacement of the central fourth layer is hence

affected by the surface effect, compared to the perpendicular case in which the rumpling

is zero similar to the bulk structure.

7.3.2 Asymmetric six-unit-cell thick slab

The results from the relaxation of an asymmetric six-unit-cell thick tetragonal

slab with in-plane polarization orientation, with BaO and TiO2 surface terminations at

opposing ends, are tabulated in Table 7.4. For the purpose of a systematic presentation

of data, the top and bottom half of the asymmetric slab will be regards as two entities,

128

each representing the BaO and TiO2 ends of terminations. The layer naming follows

the convention that index 1 refers to the outermost surface layer and increasing indexes

describes the deeper layers. The existing literature results for symmetric slab with TiO2

(100) terminations are included for references.

Table 7.4: Interlayer displacements (∆di j) and rumpling (ri) of an asymmetric 6UCBaTiO3 slab with BaO and TiO2 terminations, in units of Å. Layer sequence starts fromouter region towards inner bulk region. Data corresponding to symmetric slabs withTiO2 terminations from literature is given for comparison.

Asymmetric slab Symmetric TiO2 slab ref.

BaO Termination TiO2 Termination GGA-WCa LDAb LDAc

∆d12 −0.07848 −0.07961 −0.109 −0.114 −0.1143∆d23 0.10116 0.10627 0.056 0.048 0.0473∆d34 0.01222 0.00854 −0.014 −0.016 −0.0158∆d45 0.04806 0.04945∆d56 0.02599 0.02529

r1 −0.01634 −0.06725 0.053 −0.048 0.0985r2 0.01131 0.04550 −0.045 0.041 0.0827r3 −0.00552 −0.01317 0.008 −0.007 0.0158r4 0.00557 0.00426 −0.006 0r5 0.00089 −0.00482r6 0.00388 −0.00137

a Iles et al. (2014) b Padilla and Vanderbilt (1997) c Meyer et al. (1999)

Table 7.5: Ferroelectric distortion per layer (in units of Å) of an asymmetric 6UCBaTiO3 slab with BaO and TiO2 terminations. Layer sequence starts from outerregion towards inner bulk region. Data corresponding to symmetric slabs with TiO2terminations from literature is given for comparison.

Asymmetric slab Symmetric TiO2 slab ref.

BaO Termination TiO2 Termination LDAa LDAb

Layer δFE Layer δFE

BaO 0.0760 TiO2 0.2268 0.1749 0.17556TiO2 0.1399 BaO 0.0641 0.0575 0.05586BaO 0.0663 TiO2 0.1902 0.1374 0.13566TiO2 0.1723 BaO 0.0723 0.0659 0.06783BaO 0.0699 TiO2 0.1791TiO2 0.1765 BaO 0.0709a Padilla and Vanderbilt (1997) b Meyer et al. (1999)

129

While there are no significant difference between interlayer relaxations for both

BaO and TiO2 terminations, there are some clear distinctions in the values of rumpling

for each layer. This is evident for the cases of the first three outer layers. Particularly,

the rumpling of the TiO2 surface is about four times larger in magnitude than BaO

surface rumpling. Following the rumpling definition in Section 7.3.1, Ti surface atoms

relax deeper into the bulk region than Ba atoms, while the oxygen atoms in both cases

relax out from the surface. The same trend had also been reported by Iles et al. (2014)

and Padilla and Vanderbilt (1997). The larger magnitude of surface layer rumpling

for TiO2 terminations indicates a more uneven surface. However, the possibility of

surface reconstruction is not explored by this work, so that the results cannot be directly

indicative of a actual surface in an experiment.

The similar difference in rumpling is also observed for the second outer layer, albeit

this time the metal elements are observed to relax out of the layer, or a movement of

oxygen elements towards the inner bulk region. The surface relaxation effect starts

to wear off for the third layer, as the ratio of rumpling of TiO2 to BaO terminations

is reduced to about two compared to four for the first and second layers. While the

rumpling for inner layers has some differences in terms of the direction sign, their

magnitudes are considerably smaller. One distinguishing distinct of the asymmetric

slab is that while the rumpling values are smaller towards the inner regions, the values

do not actually reach zero for the case of BaO and TiO2 terminations, compared to the

case of symmetric slab which has zero rumpling for the fourth layer (see Table 7.2).

In comparison to the symmetric slab, the ferroelectric distortions of BaO half of the

asymmetric slab have only some small deviations, particularly for the outer three layers.

130

This occurrence is expected, as the different surface terminations in the asymmetric

slab only mainly affects the symmetry in the perpendicular direction. With references

to bulk BaO layer distortion of 0.0752Å and bulk TiO2 layer distortion of 0.1698Å,

some comments about the effect of surface on the in-plane ferroelectric distortion can

be made. There are not distinct changes in the BaO in-plane distortion regardless of

the surface or inner layers for both ends of BaO and TiO2 terminations compared to

the bulk value. Considering the relatively minor role of Ba atoms in the ferroelectricity

of BaTiO3, as explored in Chapter 6, the corresponding small change in ferroelectric

distortion can be understood. On the other hand, the in-plane distortions of TiO2 layer

are observed to be higher than the bulk value, except for the outer TiO2 layer closest to

the BaO surface of slab. Particularly, the increment in the distortion is highest for the

surface layer of TiO2, in which the increment is about one-third of the bulk value. The

in-plane ferroelectric distortions decrease gradually toward the inner region, but still

remain higher than the corresponding bulk value. The enhanced dipole moment due

to surface contribution contradicts with the greatly suppressed in-plane spontaneous

polarization of BaTiO3 slab reported in experiment (Li et al., 2015), suggesting the

additional contributions apart from the surface effects.

7.3.3 Comparison between slabs with different thickness

The interlayer spacing displacement and rumpling values of symmetric 6UC, 8UC

and 10UC slabs respectively with tetragonal (001) surface orientation are compared

in Table 7.6. Similar to the asymmetric 6UC slab in Section 7.3.2, only the outermost

interlayer spacing is observed to shrink whereas all the deeper layers moves away from

each other. The greatest distortion occurs between second and third layer from the

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Table 7.6: Comparison of interlayer displacements (∆di j) and rumpling (ri) of the tophalf layers of 6UC, 8UC and 10UC slabs respectively, in units of Å. All the slabs are intetragonal phase and possess BaO (100) terminations.

Thickness of slab (in terms of unit cell)

6 8 10

∆d12 −0.0788 −0.0789 −0.0786∆d23 0.0971 0.0928 0.0962∆d34 0.0182 0.0132 0.0178∆d45 0.0435 0.0420 0.0429∆d56 0.0329 0.0305 0.0322∆d67 0.0360 0.0331 0.0355∆d78 0.0326 0.0340∆d89 0.0317 0.0343∆d910 0.0341∆d1011 0.0340

r1 −0.0212 −0.0211 −0.0211r2 0.0087 0.0091 0.0088r3 −0.0090 −0.0087 −0.0090r4 0.0026 0.0020 0.0027r5 −0.0019 −0.0018 −0.0021r6 0.0005 0.0007 0.0007r7 0.0000 −0.0006 −0.0005r8 0.0003 0.0002r9 0.0000 −0.0001r10 0.0000r11 0.0000

surface, in which ∆d23 is about five times greater in magnitude than the change in

spacings directly below it, ∆d34.

All the ∆di j possess positive signs except ∆d12, where the positive sign indicates

an expansion of the spacing between two consecutive layers. The contraction in the

outermost interlayer spacing is largely off-set by the great expansion between second

and third layer from the surface. Coupled with separation of consecutive inner layers

from each other due to relaxation, a great change in the perpendicular dimension of the

slabs is shown compared to the initial structure.

132

While the respective interlayer displacements for 6UC and 10UC slabs are observed

to be almost the same, those of 10UC slab are consistently smaller than that of 6UC slab.

Comparatively, the interlayer displacements for the 8UC slab show a stark difference

from their counterpart in 6UC and 10UC slabs. Apart from the outermost spacing, the

expansions of spacings due to relaxation are noticeably smaller in the 8UC slab. The

reduction in expansion of interlayer spacings is especially evident for ∆d23 and ∆d34,

where ∆d34 in the 8UC slab is approximately 30% smaller than the same quantity in the

6UC slab. However, the discrepancies in interlayer spacings of 8UC slab are getting

smaller towards the central bulk region.

On the other hands, the rumpling values do not vary considerably across the variation

in slab thickness including the 8UC slab, in contrast to the interlayer displacements.

This signifies an insensitivity of relative displacements between the metallic elements

and oxygen in each layer towards to the thickness of slab. Similar to previous obtained

results for asymmetrical slab, the greatest distortion occurs in the surface layer, leading

to an uneven surface if the possibility of surface reconstruction is not accounted for.

The magnitude of rumpling drops drastically from the third to fourth layer, in which

the magnitude of r4 is only about one fourth that of r3. The directions of the rumpling,

for which a positive sign indicate outwards displacement of metal elements and inwards

displacement oxygen atoms, alternate for each consecutive layer again. This leads to

a small net dipole moment in the direction perpendicular to the surface, similar to the

case of 3UC slab. In tandem with the interlayer displacement, the rumpling approaches

to its limit of zero from the sixth layer onwards. This signifies that the influence of

surface effects on the atomic structural arrangement of BaTiO3 is greatly limited to the

133

five outermost layers, with only some small interlayer displacements are shown for the

sixth layers onwards.

Table 7.7: Comparison of ferroelectric distortion δFE per layer (in unit of Å) of the tophalf layers of 6UC, 8UC and 10UC slabs respectively. All the slabs are in tetragonalphase and possess BaO (100) terminations.

Layer Thickness of slab (in terms of unit cell)

6 8 10

1 0.0750 0.0758 0.07542 0.1392 0.1391 0.13953 0.0652 0.0666 0.06624 0.1722 0.1714 0.17235 0.0689 0.0706 0.06996 0.1763 0.1753 0.17627 0.0691 0.0709 0.07038 0.1765 0.17689 0.0711 0.0704

10 0.176911 0.0704

In contrast to displacements perpendicular to the surface, there is almost no variation

in δFE per layer for the slabs with different thickness. The distortion of the TiO2 layer

is about twice much larger than BaO layer, which is expected expected due to the role

of Ti in the asymmetric charge distribution of BaTiO3 in ferroelectric phase. Compared

to the bulk distortion of 0.0752Å and 0.1698Å for BaO and TiO2 again, the presence

of surface terminations only contributes a meagre increase of about 0.01Å to 0.015Å.

If the in-plane polarization along the x-direction is assumed to be solely dependent

on the dipole moments created due to ferroelectric distortions, then the insensitivity in

the values of δFE over three slabs should indicate a similar magnitude of the generated

spontaneous polarization. However, through the use of Berry phase formalism, the

magnitude of in-plane spontaneous polarizations along the bulk polar axis in the are

134

computed to be 0.0821C/m2, 0.0183C/m2 and 0.0299C/m2 for the 6UC, 8UC and

10UC slabs respectively; it is found that the formation of polarization is suppressed

even in the thickest 10UC slab. On the other hand, the asymmetric 6-unit-cell thick

slab is found to possess an even smaller magnitude of spontaneous polarization of

0.0175C/m2

The magnitude of polarization is not the only aspect that is different from the

bulk case; the directions of polarization along the x-axis for the 6UC and 8UC slabs

are reverse of that in the bulk case. Contrary to the bulk tetragonal structure where

polarization vector is restricted to one axis, there are also some components of in-plane

polarization vector orthogonal to the bulk polar axis, whose magnitude is even greater

than that along the polar axis. However, these deviations to the bulk case is not further

investigated as they do not fall under the current scope of study, which is limited to the

analysis of structural relaxation of the slabs at the atomic level.

The finding that BaO terminations retain an in-plane polarization of 0.09C/m2

by Iles et al. (2014) is hence reproduced and confirmed. Using a simple method of

estimating unit polarization by the knowledge of relative displacement between Ti

and the surrounding oxygen octahedral, which is found to be between 0.005nm and

0.010nm, Li et al. (2015) estimated that the in-plane spontaneous polarization falls

within the range of 5µC/m2 to 10µC/m2. By following the same method and taking

the value of δFE in the TiO2 layer as the relative displacement of Ti within the oxygen

octahedral, it is found that the in-plane polarizations are present in the slabs, with

the thinnest 6UC slab possess a value of 16µC/m2. Since the result contradicts with

the small polarization value found by Berry phase formulation, it is suggested that

135

additional factors such as domain formation and ordering as well as reconstruction of

surface are necessary to be considered in the study of the spontaneous polarization in a

slab, apart from the contribution of atomic displacement.

7.4 Summary

Symmetric BaO surface terminated slab models of different thickness (3, 6 ,8,

10-unit-cell-thick) are constructed and structural relaxations are performed. Another

asymmetric slab consisting 6UC slab as well as both BaO and TiO2 terminations are

constructed and studied. The interlayer displacements are all positive except for the

first interlayer spacing from the surface which shrinks. This is distinct from literature

findings that report alternate signs in interlayer displacement down the surface. The

greatest distortion originates from ∆d23, which indicates a large expansion of interlayer

spacing between the second and third layers from the surface. The expansion in

interlayer spacing is practically identical for the 6UC and 10UC slabs. However, the

interlayer displacements of the 8UC slab are consistently smaller than that in 6UC

and 10UC slabs. In contrast to the interlayer spacing, the variations of rumpling

values is comparatively much more smaller across the different slab thickness, but form

perpendicular dipole moments of opposite directions for each layer of the slabs. The

net dipole moments perpendicular to the surface is hence greatly reduced, leading to a

smaller depolarization field across the vacuum above and below the slab supercell. The

rumpling almost vanishes for layers deeper than five layers from the surface, indicating

the first five layers closest to the surfaces are subjected the most to the surface effect.

Regarding the parallel ferroelectric displacement of each layer, the distortions in

136

the TiO2 layers are much larger, about twice to thrice in magnitude, than the BaO

layers. The dipole moment in the TiO2 layer closest to surface is suppressed to a small

degree, whereas the deeper TiO2 layers have enhanced parallel dipole moment. The

ferroelectric distortions are found to be quite constant across the different slabs with

varying thickness. By using the Berry’s phase formalism, the BaO terminated slabs

are found to possess a small value of in-plane polarization, even for the thinnest 6UC

slab. This is in contrast with the relatively much larger value found using only the

ferroelectric displacement in the TiO2 layers, which indicates the need for consideration

of contribution of additional factors in the future work.

137

CHAPTER 8

CONCLUSION

Barium titanate, BaTiO3, both in bulk and slab forms had been investigated in this

work. First-principles density functional theory (DFT) had been employed to study the

basic ground state properties and the corresponding response properties were studied

density functional perturbation theory (DFPT), which is an extension of DFT with the

application of perturbation theory. The first part of the work specifically concerned

with the applicability of DFT in studying the properties of BaTiO3 within the frame-

work of local-density approximation (LDA) and generalized-gradient approximation

(GGA) using the projector-augmented wave (PAW) potentials. Concurrently, various

aspects of DFT method were explored from its application on the classic ferroelectric

material BaTiO3 for all of its structural phases: cubic, tetragonal, orthorhombic and

rhombohedral, in the order of descending temperature. A major drawback of DFT

lies in its direct inapplicability within the thermodynamic domain, due to the lack of

temperature related macroscopic description in the Hohenberg-Kohn formulations at the

root of DFT method. In describing the properties of a material which has its properties

changed haphazardly with temperature such as BaTiO3, the ideal procedure of con-

ducting an ab-initio molecular dynamics (MD) simulations coupled with DFT studies

at particular interested simulation time points, is undoubtedly impractical with the

exception of availability of state-of-the-art equipments. Instead, a compromise is made

by forcing an artificial symmetric constraint to the initial model, which corresponds to

a predetermined structure of the material to be studied. This preliminary knowledge

138

of geometry usually stems from previous experimental discovery, which in the case of

this work refers to the space group symmetry of the four phases of BaTiO3 at different

temperature. It is important that this subtle approximation should be taken into account

when evaluating the accuracy of predicted structural and response properties of BaTiO3.

The accuracy of structural properties of bulk BaTiO3 were shown to be of great

importance to the accuracy of subsequently derived quantities and response proper-

ties. While both LDA and PBE-GGA yielded comparable "ground state" structural

parameters, the PBE calculations were shown to produce more accurate derived struc-

tural properties such as cohesive energy and bulk modulus, if the experimental results

were used as the benchmark. Although the computed spontaneous polarizations in

various structural phases of ferroelectric BaTiO3 were comparatively more accurate

for PBE calculations than LDA, both XC functionals were shown to have no effects

on the Berry phase computation of the spontaneous polarization if the same structural

information was used. It can be said that the accuracy of computed polarization is

merely a reflection of previous structural properties calculations. The section about the

response properties concerns the Born effective charges (BEC) and zone center phonon

mode properties, computed within the framework of LDA. BEC played an essential

role in the ferroelectricity of BaTiO3, linking the interplay of dynamical and electronic

properties. Unusually high values BEC were computed for titanium and oxygen for

displacements along the Ti-O bond, which is expected due to occurrence of anomalous

charges in ferroelectric materials. Surprisingly the results varied greatly for both JTH

and GBRV PAW potentials, especially for the BEC charge of titanium, which concerns

with the transferability of PAW potentials themselves. The computation of phonon

modes symmetry at gamma point, on the other hands, was better than expected, despite

139

the inferior performance of LDA in predicting the structural properties. The results

demonstrate a need of caution to accept the result of a response properties calculation

without conducting additional comparative studies using different XC functionals and

potentials, for the construction of the PAW potentials themselves were predominantly

based on only the correct prediction of structural properties of elements.

The second part of this work touches upon the study of ferroelectric tetragonal

BaTiO3 slabs in various thickness. The effect on the shift of atomic positions due to

surface relaxation are explicitly investigated. It is found that only the outermost five

layers of BaTiO3 slabs, regardless of having 6-unit-cell, 8-unit-cell or 10-unit-cell slab

thickness, are greatly affected by the surface relaxations, indicating the penetrating

limit of surface effect. In particular, the rumpling of individual layers vanish for layers

beyond the penetrating limit, although the interlayer displacements still experienced

some lingering influences from the surface relaxations. Interestingly, the ferroelectric

displacements of the individual layers were noticed to be independent of the slab

thickness, where the in-plane distortion of the 6-unit-cell thick slab is same as that of

the 10-unit-cell thick slab. In agreement with literature findings, symmetric tetragonal

BaTiO3 slab with BaO (100) surface terminations was predicted to possess a much

smaller value of in-plane spontaneous polarization than the corresponding quantity in

the bulk case, although the finding of spontaneous polarization in the thinnest 6-unit-cell

thick slab would warrant additional future research with indications of additional factors

not investigated in this work.

140

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APPENDICES

APPENDIX A

JTH-LDA CUBIC PHASE ELECTRONIC STRUCTURES

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Γ X M Γ R XE

G (indirect) = 1.75642 eV

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure A.1: Band structure (left) and density of states (right) of BaTiO3 in cubic phase using JTH-LDA PAW potential.

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APPENDIX B

GBRV-PBE ELECTRONIC STRUCTURES

154

Γ X M Γ R XE

G (indirect) = 1.75997 eV

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure B.1: Band structure (left) and density of states (right) of BaTiO3 in cubic phase using GBRV-PBE PAW potential.

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Γ X M Γ Z R A Z X R M AE

G (indirect) = 1.8426 eV

-50

-40

-30

-20

-10

0

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure B.2: Band structure (left) and density of states (right) of BaTiO3 in tetragonal phase using GBRV-PBE PAW potential.

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Γ X S R A Z Γ YX1 A1T Y Z TE

G (indirect) = 2.14099 eV

-50

-40

-30

-20

-10

0

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure B.3: Band structure (left) and density of states (right) of BaTiO3 in orthogonal phase using GBRV-PBE PAW potential. The separationdistance between Y and X1 and A1 and T are negligible.

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Γ L B1 B Z Γ X Q F P1 Z L P

EG (indirect)

= 2.30937 eV

-50

-40

-30

-20

-10

0

10E

-EF

(eV

)

εF

0 5 10 15 20electrons/eV/cell

-50

-40

-30

-20

-10

0

10

E-E

F (e

V)

Figure B.4: Band structure (left) and density of states (right) of BaTiO3 in rhombohedral phase using GBRV-PBE PAW potential. The separationdistance between B and Z, Q and F and P1 and Z are negligible.

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APPENDIX C

SELECTION OF INPUT SLAB MODELS

C.1 Computation complexity

The selection of input slab models for DFT computations contains some subtleties

on the implementation of a DFT calculation, which are not present for a bulk calculation.

It is worth to make a short discussion on computational complexities of a slab calculation

in view of this.

The study of a slab under plane wave implementation of DFT necessitates the

construction of a supercell model, in contrast to the use of primitive unit cell for bulk

calculation. The periodic boundary condition, utilized by a plane wave electronic

structure package to simplify the calculation of a bulk structure, has the undesired effect

of creating unwanted interactions of a slab with its periodic images. The interaction

between the periodic images is usually avoided in a supercell approach by including

layers of vacuum of sufficient thickness. While the localised basis set implementation

avoids the problem of periodicity, the periodic supercell method has the advantage of

implementation of efficient fast Fourier transform (FFT) algorithms. Some examples

include the FFTW library (Frigo & Johnson, 2005; Johnson & Frigo, 2007, 2008)

developed at MIT as well as FFT routines developed by Goedecker (1993, 1997);

Goedecker, Boulet, and Deutsch (2003), which greatly facilitates the conversions of a

physical quantity from real space to reciprocal space and vice versa.

The scaling of DFT computational cost with respect to number of electrons treated

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in a system is of the order of O(N3e ), due to the diagonalization of the Hamiltonian of

the system. This places a limit on the system size, which is the thickness of the slabs in

this work. On the other hand, parallelization of the computing processes over multiple

computing nodes is actually limited due to the actual implementation of parallelization

scheme in a typical plane wave based DFT. The most efficient parallelization scheme

in ABINIT is over the number of k-points in reciprocal space, where the calculations

for each point in reciprocal space are independent from each other. There exists an

inverse proportionality relationship between the volume of a system in real space and

reciprocal space, which means a small reciprocal space is to be expected if a large slab

supercell model is to be adopted. The k-points in the system is hence limited in number,

which places a bottleneck to the efficiency of the parallelization scheme. On the other

hand, the advanced KGB parallelization scheme (Bottin, Leroux, Knyazev, & Zérah,

2008) implemented in ABINIT enables additional computation parallelizations over

electron bands and wave vectors of plane waves. However, the efficiency of the KGB

parallelization is heavily dependent upon a system of fast interconnections between

compute nodes, as the parallelization over bands and plane waves is not embarrassingly

parallel, unlike the parallelization over k-points.

Taken into account the total computation resources of 19 compute nodes, with 4

compute cores and 4GB of RAM memory per node, the thickness of the BaTiO3 slabs

studied is limited to 6, 8 and 10 unit cells respectively. The selection of slabs with the

mentioned thickness enables the calculations to be completed within reasonable amount

of time, with the calculation of the slab of 10 unit cells in thickness took at least 1.5

months of time to complete. 3-unit-cell thick slabs in both cubic and tetragonal phases

are used for preliminary comparison with existing literature results.

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Taken into account the total computation resources of 19 compute nodes, with 4

compute cores and 4GB of RAM memory per node, the thickness of the BaTiO3 slabs

studied is limited to 6, 8 and 10 unit cells respectively. The selection of slabs with the

mentioned thickness enables the calculations to be completed within reasonable amount

of time, with the calculation of the slab of 10 unit cells in thickness took at least 1.5

months of time to complete.

C.2 Convergence studies

The 10-unit-cells model is used as a test case for convergence study, where the

converged results serve as the upper bound for models with smaller size . There are

a total of four parameters of which the optimal values have to be determined through

convergence study:

1. the number of vacuum layers,

2. ecut (kinetic energy cut-off controlling the number of planewaves in the basis),

3. pawecutdg (energy cut-off for double grid or fine FFT grid),

4. ngkpt (arrays of grid points for k-points generation).

The energy related parameters mainly depend upon the design of PAW potentials

used in a computation. The convergence studies on ecut and pawecutdg can be done

without specifically considering the system size, as there is little cross-influence between

the convergences of energetic parameters and the size of system. The number of vacuum

layer is quantified in terms of lattice constants, and a minimal of 2 vacuum layers is

used for the convergence studies of ecut and pawecutdg. The k-point grid array is set to

6×6×1, where the number of grid points in z-direction is 1 to prevent neighbouring

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slabs from interacting with each other. Based on Fig. C.1, the converged ecut value is

chosen to be 32 Ha, which have a convergence of 0.006 Eh.

(a) Convergence of total energy with respect toecut

(b) Convergence of surface energy with respect toecut

Figure C.1: Convergence studies with respect to ecut

The double grid cut-off variable pawecutdg is dependent on the chosen ecut value,

where the value pawecutdg must be equal or greater then ecut. Following the same

procedure as the convergence tests of ecut by investigating the total system energy and

surface energy, the results are summarized in Fig. C.2. An optimal value of 75 Ha is

chosen.

(a) Convergence of total energy with respect topawecutdg

(b) Convergence of surface energy with respect topawecutdg

Figure C.2: Convergence studies with respect to pawecutdg

The converged ecut and pawecutdg values of 32 Ha and 75 Ha respectively are used

in the convergence studies for the k-point array variable, ngkpt. However to be precise,

it is the number of k-points in the irreducible Brillouin zone (IBZ) that is needed to

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be converged, since most electronic codes including ABINIT automatically reduce the

amount of treated k-points by taking advantage of symmetry of the system. Table C.1

lists the k-point array candidates and corresponding number of k-points in IBZ. The

summarized results in Fig. C.3 shows that the differences in total energy and surface

energy is negligible starting from nkpt value of 9, which corresponding to the ngkpt

array of 6×6×1. The chosen converged value of ngkpt is thus 6×6×1.

Table C.1: Correspondence between k-point array (ngkpt) and number of k-point inIBZ (nkpt)

k-point array (ngkpt) number of k-point in IBZ (nkpt)

4×4×1 46×6×1 98×8×1 16

10×10×1 3612×12×1 49

(a) Convergence of total energy with respect tonkpt

(b) Convergence of surface energy with respect tonkpt

Figure C.3: Convergence studies with respect to nkpt

The final parameter subjected to convergence study is the number of vacuum layers,

quantified in terms of lattice constants. The total energy and surface energy of the 10-

unit-cells model are oscillating in magnitude with respect to the vacuum layer thickness,

as shown in Fig. C.4. This is in part due to the same ngkpt of 6×6×1 is used for all

computations, despite different system sizes with each vacuum layer thickness. Taking

163

into account the dramatical increase of needed computational power with the system

size, a compromising value of 6 vacuum layers is chosen. Electron density and mean KS

potential along the z-direction are recorded to ensure that 6 vacuum layers are enough

to prevent neighbouring slabs from interacting each other. Fig. C.5 shows that the

slabs are well separated from each other. The trailing edges of Appendix C.2, however,

shows that there are still minimal interactions across the vacuum gaps, as evident in

the non-zero KS potential in the vacuum region. The potential in the vacuum can be

minimized by further increasing the vacuum layer thickness, but the current chosen

vacuum layer thickness is enough for the purpose of this work.

(a) Convergence of total energy with respect tovacuum layer

(b) Convergence of surface energy with respect tovacuum layer

Figure C.4: Convergence studies with respect to vacuum layer

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0 500 1000 1500 2000Linear density (electron/reduced unit)

0

0.2

0.4

0.6

0.8

1

Red

uce

d z

-coord

inat

e

(a) Linear electron density along z-direction

-1 -0.8 -0.6 -0.4 -0.2 0Mean KS potential (Ha)

0

0.2

0.4

0.6

0.8

1

Red

uce

d z

-coord

inat

e

(b) Mean KS potential along z-direction

Figure C.5: Linear electron density and mean KS potential along z-direction

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LIST OF PUBLICATIONS

Goh, E. S., Ong, L. H., Yoon, T. L., & Chew, K. H. (2016a). Structural and re-sponse properties of all BaTiO3 phases from density functional theory using theprojector-augmented-wave methods. Computational Materials Science, 117, 306- 314. Retrieved from http://www.sciencedirect.com/science/article/pii/S0927025616300106 doi: http://dx.doi.org/10.1016/j.commatsci.2016.01.037

Goh, E. S., Ong, L. H., Yoon, T. L., & Chew, K. H. (2016b). Structural relaxationof BaTiO3 slab with tetragonal (100) surface: Ab-initio comparison of differentthickness. Current Applied Physics, 16(11), 1491 - 1497. Retrieved from http://www.sciencedirect.com/science/article/pii/S1567173916302346 doi:http://dx.doi.org/10.1016/j.cap.2016.08.024

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