defect behaviors in zinc oxide and zinc titanates ceramics...
TRANSCRIPT
APPROVED: Jincheng Du, Major Professor Richard F. Reidy, Committee Member Thomas Scharf, Committee Member Zhenhai Xia, Committee Member Nigel Shepherd, Committee Member and the
Chair of the Department of Materials Science and Engineering
Costas Tsatsoulis, Dean of the College of Engineering
Victor Prybutok, Vice Provost of the Toulouse Graduate School
DEFECT BEHAVIORS IN ZINC OXIDE AND ZINC TITANATES CERAMICS FROM
FIRST PRINCIPLES COMPTUER SIMULATIONS
Wei Sun
Dissertation Prepared for the Degree of
DOCTOR OF PHILOSOPHY
UNIVERSITY OF NORTH TEXAS
December 2016
Sun, Wei. Defect Behaviors in Zinc Oxide and Zinc Titanates Ceramics from First
Principles Computer Simulations. Doctor of Philosophy (Material Science and Engineering),
December 2016, 121 pp., 16 tables, 34 figures, 164 numbered references.
ZnO and ZnO-TiO2 ceramics have intriguing electronic and mechanical properties and
find applications in many fields. Many of these properties and applications rely on the
understanding of defects and defect processes in these oxides as these defects control the
electronic, catalytic and mechanical behaviors. The goal of this dissertation is to systematically
study the defects and defects behaviors in Wurtzite ZnO and Ilmenite ZnTiO3 by using first
principles calculations and classical simulations employing empirical potentials. Firstly, the
behavior of intrinsic and extrinsic point defects in ZnO and ZnTiO3 ceramics were investigated.
Secondly, the effect of different surface absorbents and surface defects on the workfunction of
ZnO were studied using DFT calculations. The results show that increasing the surface coverage
of hydrocarbons decreased the workfunction. Lastly, the stacking fault behaviors on ilmenite
ZnTiO3 were investigated by calculating the generalized stacking fault (GSF) energies using
density functional theory based first principles calculations and classical calculations employing
effective partial charge inter-atomic potentials. The gamma-surfaces of two low energy surfaces,
(110) and (104), of ZnTiO3 were fully mapped and, together with other analysis such as ideal
shear stress calculations.
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Copyright 2016
By
Wei Sun
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ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to Dr. Jincheng Du for providing me the
opportunity to work with him. His approach towards research has always inspired me and
motivated me to learn more. His discourses on various aspect of computational material science
will always help me think critically.
I would also like to thank my committee members, Dr. Narendra Dahotre, Dr. Richard F.
Reidy, Dr. Thomas Scharf, Dr. Nigel Shepherd, and Dr. Zhenhai Xia for their help, comments,
and suggestions during the course of my PhD.
In addition, a thank you to Dr. Donghai Mei, Dr. Jitendra Kumar Jha, and Dr. Victor
Ageh, for their invaluable help and support while carrying out some part of the work reported in
this dissertation.
I would also take this opportunity to thank my group members: Mrunal Kumar
Chaudhari, Leopold Kokou, Ye Xiang, Yun Li, Jessica Rimsza, Lu Deng, Chao-hsu Chen,
Xiaonan Lu, and PoHsuen Kuo for all their support and encouragement.
Last but not the least, I am very grateful for all the love, support and encouragement that
I received from my family and friends without which I would never have come so far in my life.
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TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ........................................................................................................... iii LIST OF TABLES ........................................................................................................................ vii LIST OF FIGURES ..................................................................................................................... viii CHAPTER 1 INTRODUCTION .................................................................................................... 1
1.1 Motivation and Background ................................................................................... 1
1.2 Contribution of the Dissertation.............................................................................. 3 CHAPTER 2 LITERATURE REVIEW ......................................................................................... 5
2.1 ZnO-TiO2 Binary Systems ...................................................................................... 5
2.1.1 History and Phase Diagram ........................................................................ 5
2.1.2 Microstructure ........................................................................................... 11
2.2 Computational Modeling Study on Defect Properties of ZnO/ZnTiO3 ................ 13 CHAPTER 3 COMPUTATIONAL METHODOLOGY .............................................................. 16
3.1 Computational Methodology ................................................................................ 16
3.2 Classic Simulation with Empirical Potentials ....................................................... 17
3.2.1 Two-Body Pair Interaction ....................................................................... 17
3.2.2 Potential Forms ......................................................................................... 18
3.3 First Principle Calculations ................................................................................... 19
3.3.1 The Schrodinger Equation and Born-Oppenheimer Approximation ........ 19
3.3.2 Hartree-Fock Method ................................................................................ 21
3.3.3 The Variational Principle .......................................................................... 22
3.4 Density Functional Theory ................................................................................... 22
3.4.1 The Hohenberg-Kohn Theorem ................................................................ 23
3.4.2 The Kohn-Sham Equations ....................................................................... 23
3.4.3 Exchange-Correlation Functional ............................................................. 25
3.4.4 Vienna ab initio Simulation Package (VASP) .......................................... 27 CHAPTER 4 POINT DEFECTS CALCULATIONS IN ZINC OXIDE AND ZINC TITANATE....................................................................................................................................................... 28
4.1 Abstract ................................................................................................................. 28
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4.2 Introduction ........................................................................................................... 28
4.3 Computational Detail ............................................................................................ 31
4.3.1 Computational Methods ............................................................................ 31
4.3.2 Defect Formation Energy .......................................................................... 32
4.3.3 Defect Transition Energy .......................................................................... 34
4.3.4 Band Gap Corrections ............................................................................... 34
4.4 Geometric Optimization........................................................................................ 35
4.4.1 Geometric Optimization of Wurtzite ZnO ................................................ 36
4.4.2 Geometric Optimization of ZnTiO3 .......................................................... 37
4.5 Defect Formation Energy Calculations for ZnO ................................................... 38
4.5.1 Effect of Supercell Size ............................................................................ 38
4.5.2 Primary Point Defects ............................................................................... 39
4.5.3 Defect Structure and Formation Energies of [SbZn-2VZn] Complex ......... 40
4.6 Defect Formation Energy Calculations for ZnTiO3 .............................................. 43
4.6.1 Bulk Properties and Defect Structures of ZnTiO3 .................................... 43
4.6.2 Formation Energy ..................................................................................... 46
4.6.3 Defects Binding Energy ............................................................................ 50
4.6.4 Transition Level ........................................................................................ 51
4.7 Conclusions ........................................................................................................... 52 CHAPTER 5 SURFACE ENERGY AND WORKFUNCTION STUDY ON ZINC OXIDE ..... 54
5.1 Abstract ................................................................................................................. 54
5.2 Introduction ........................................................................................................... 54
5.3 Simulation Detail .................................................................................................. 57
5.4 Geometric Optimization........................................................................................ 58
5.5 Surface Relaxation and Energy Calculations ........................................................ 59
5.5.1 Surface Relaxation of Non-Polar 𝟏𝟏 𝟎𝟎 𝟏𝟏 𝟎𝟎 Surface .................................... 59
5.5.2 Surface Relaxation of Polar (𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏)/(𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏) Surface ....................... 62
5.5.3 Surface Energy Calculations ..................................................................... 66
5.6 Effect of Surface Adsorption on the Workfunction of ZnO Surfaces .................. 67
5.6.1 Effect of Methyl Adsorption on Workfunction ........................................ 69
5.6.2 Effect of –CF3 and –F Adsorptions on Workfunction .............................. 71
5.6.3 Effect of Surface Non-Stoichiometry on Workfunction ........................... 74
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5.7 Dipole Moment and Workfuntion ......................................................................... 75
5.8 Charge Density Difference ................................................................................... 78
5.9 Conclusion ............................................................................................................ 80 CHAPTER 6 STACKING FAULTS STUDY ON ZINC TITANATE ........................................ 82
6.1 Abstract ................................................................................................................. 82
6.2 Introduction ........................................................................................................... 82
6.3 Computational Details .......................................................................................... 85
6.4 Geometric Optimization........................................................................................ 87
6.5 Surface Energy Calculation .................................................................................. 88
6.5.1 Surface Construction of (110) ................................................................... 89
6.5.2 Surface Construction of (104) ................................................................... 90
6.5.3 Surface Energy Calculation of (110) and (104) Planes ............................ 90
6.6 Stacking Faults Behavior for the (110) and (104) Planes ..................................... 92
6.6.1 Stacking Faults on (110) ........................................................................... 92
6.6.2 Stacking Faults on (104) ........................................................................... 95
6.7 Structure Relaxation and Electronic Structures of the Low Energy Stacking Faults............................................................................................................................... 97
6.7.1 Idea Shear Stress on (104) and (110) ........................................................ 97
6.7.2 Structure Relaxation on {104}<4 5 𝟏𝟏> System ........................................ 99
6.7.3 Density of States on {104}<4 5 𝟏𝟏> System ............................................ 101
6.7.4 Statistical Analysis of Atom Distances on (104)𝟒𝟒 𝟓𝟓 𝟏𝟏 System .............. 102
6.8 Conclusion .......................................................................................................... 103 CHAPTER 7 SUMMARY AND FUTURE WORK .................................................................. 105 APPENDIX: PUBLICATIONS RESULTING FROM THIS DISSERTATION ....................... 107 REFERENCES ........................................................................................................................... 109
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LIST OF TABLES
Page
Table 4.1. Calculated structural parameters for Wurtzite ZnO ..................................................... 37
Table 4.2. Comparison of observed and calculated properties for ZnTiO3 .................................. 37
Table 4.3. Formation energies of elementary defects in ZnO. ...................................................... 39
Table 4.4. Defect formation energies of 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 − 2𝑉𝑉𝑆𝑆𝑆𝑆 and Zni under O-rich conditions .......... 42
Table 4.5. Comparison of observed and calculated properties for ZnTiO3 .................................. 44
Table 4.6. Formation energies of mono- and di-vacancies in ilmenite ZnTiO3 under metal-rich and O-rich conditions. The Fermi level at the valence band maximum. ...................................... 46
Table 5.1. Calculated structural parameters for Wurtzite ZnO ..................................................... 59
Table 5.2. Summary of atom relaxation perpendicular to the surface for ZnO (1 0 1 0) surfaces (atom numbers shown in Figure 5.3) ............................................................................................ 62
Table 5.3. Summary of relaxation (without) perpendicular to the surface of ZnO (0 0 0 1)/(0 0 0 1) surfaces ......................................................................................................... 64
Table 5.4. Calculated adsorption energy and workfunction due to -CH3 adsorption ................... 70
Table 5.5. Calculated adsorption energy and workfunction due to -CF3 adsorption .................... 72
Table 5.6. Calculated Adsorption energy and workfunction due to –F adsorption ...................... 73
Table 6.1. Comparison of observed and calculated properties for ZnTiO3 .................................. 88
Table 6.2. Surface energies for the two faces of ilmenite ZnTiO3 ............................................... 91
Table 6.3. Values of unstable stacking fault energy (γusf), stacking fault energy (γsf), and the ratio range γsf/γusf for empirical potential and DFT calculation results ................................................. 98
Table 6.4. Structural relaxation of two upper and lower layers around slip plane at 1/2[4 5 1] along surface normal ................................................................................................................... 100
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LIST OF FIGURES
Page
Figure 2.1. The ZnO-TiO2 phase diagram published by Dulin and Rase[15] ................................ 6
Figure 2.2. Infrared spectra for (A) cubic and (B) hexagonal forms of ZnTiO3 presented by Yamaguchi et al[19]. ....................................................................................................................... 8
Figure 2.3. The system ZnO-TiO2 phase diagram published by Yang and Swisher[20] ................ 9
Figure 2.4. ZnO-TiO2 phase diagram including solid solution published by Kim et al[17]......... 10
Figure 2.5. Schematic structure of inverse spinel Zn2TiO4 .......................................................... 11
Figure 2.6. Crystal structures of ilmenite-type ZnTiO3 and LiNbO3-type ZnTiO3[23] ................ 12
Figure 2.7. Schematic structures of ZnO crystal structures, Zinc blende and Wurtzite ............... 13
Figure 4.1. Energy versus Volume curve for Wurtzite ZnO......................................................... 36
Figure 4.2. DFT calculated formation energy of Vo under Zn-rich condition as a function of different supercell sizes................................................................................................................. 39
Figure 4.3. Schematic of the 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 − 2𝑉𝑉𝑆𝑆𝑆𝑆 defect complex (left). Bond length change for the Sb-O relative to regular Zn-O bonds for neutral (right top), 1- (right middle), and 3- (right bottom) charge states after geometry optimization. The red spheres are O, the gray spheres are Zn, the brown spheres are Sb and the dotted spheres are Zn vacancies. ...................................... 40
Figure 4.4. Schematic structure of hexagonal ZnTiO3 and possible vacancy sites. ..................... 43
Figure 4.5. Variation of calculated defect formation energies of the lowest energy vacancy types in ilmenite ZnTiO3 with Fermi level energy (EF). ........................................................................ 47
Figure 4.6. Transition levels for mono- and di-vacancies in ZnTiO3 ........................................... 51
Figure 5.1. Energy versus Volume curve for Wurtzite ZnO......................................................... 59
Figure 5.2. Schematic structure of Wurtzite ZnO. Red large ball: O; Grey small ball: Zn .......... 61
Figure 5.3. (a) Unrelaxed and (b) relaxed (1 0 1 0) structure of the four double-layer surface model of ZnO. (c) First double-layer of (1 0 1 0) structure, the arrows denote the available adsorption sites on the surface. Red small ball: O; Grey large ball: Zn ..................................... 61
Figure 5.4. (a) Unrelaxed (0 0 0 1)/(0 0 0 1) surface structure of the five double-layer surface model of ZnO. (b) Relaxed structure without dipole correction. (c) Relaxed structure with dipole correction. (d) Relaxed structure with pseudo hydrogen and dipole correction. (e) First double-
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layer of (0 0 0 1)/(0 0 0 1) tructure, the arrows denote the available adsorption sites on the surface. Red small ball: O; Grey large ball: Zn ............................................................................ 63
Figure 5.5. Electrostatic potential of z direction for polar (0 0 0 1)/(0 0 0 1) slabs, (a) as-cleaved without pseudo hydrogen correction; (b) Zn-terminated with pseudo hydrogen correction on dangling oxygen atoms at the bottom of the slab (see inset), with (dashed) and without (solid) dipole correction; and (c) O-terminated with pseudo hydrogen correction on dangling Zn atoms at the bottom of the slab (see inset), with (dashed) and without (solid) dipole correction; (d), calculated quantitative displacements along z direction of surface Zn and O atoms as a function of number of sub-layers for (0 0 0 1)/(0 0 0 1) slabs ................................................................. 65
Figure 5.6. Diagram of electrostatic potential for ZnO (1 0 1 0) surfaces, a, symmetric slabs; b, asymmetric slabs with F adsorption. Φ is workfunction, Φ1 and Φ2 are workfunctions of top and bottom surfaces. ............................................................................................................................ 68
Figure 5.7. Workfunctions as a function of surface -CH3 coverage. a, (1 0 1 0), surface; b, (0 0 0 1)/(0 0 0 1) surface .......................................................................................................... 71
Figure 5.8. Workfunction change as a function of surface -CF3 coverage. a, (1 0 1 0) surface; b, (0 0 0 1)/(0 0 0 1) surface .......................................................................................................... 73
Figure 5.9. Workfunctions at different surface F coverage for (1 0 1 0) (black) and (0 0 0 1) (red) surfaces; F atoms bonded with surface Zn atoms ................................................................ 73
Figure 5.10. Work function of the (1 0 1 0) surface with different surface O/Zn ratios for oxygen deficient (a) and zinc deficient (b) surfaces .................................................................................. 75
Figure 5.11. Change in work function as a function of change in dipole moment density of the (1 0 1 0) surface. (a) induced by -CH3, (b) induced by -CF3, (c) induced by surface non-stoichiometry................................................................................................................................. 77
Figure 5.12. Cross section of surface schematic structures (top) and charge density difference Δρ(r) (bottom) at the (1 0 1 0) surface with selected adsorbates, a, with F; b, with -CH3. Red (Blue) region represents region of electron buildup (depletion) ................................................... 79
Figure 6.1. (110) structure of six-layer surface model of ZnTiO3 with (a) O termination, and (b) Zn/Ti termination .......................................................................................................................... 89
Figure 6.2. (104) structure of six-layer surface model of ZnTiO3 with (a) O/Ti/Zn termination, and (b) Zn/Ti termination.............................................................................................................. 90
Figure 6.3. Calculated surface energies as a function of slab thickness; (a), (110) surface, and (b), (104) surface ................................................................................................................................. 91
Figure 6.4. The complete (110) γ-surface (a) and corresponding contour plot (b) of ZnTiO3 ..... 93
Figure 6.5. The generalized stacking fault energies of (110) as a function of shear displacement along [1 1 0] and [0 1 0] directions. The triangle, round, and square dots are the calculated
x
structures of corresponding displacement vectors, while the lines are fitted results by using the Full Width at Half Maximum (FWHM) Gaussian function ......................................................... 95
Figure 6.6. The complete (110) γ-surface (a) and corresponding contour plot (b) of ZnTiO3 ..... 96
Figure 6.7. The generalized stacking fault energies of (104) as a function of shear displacement along [4 5 1], [0 1 0] directions. The triangle, round, and square dots are the calculated structures of corresponding displacement vectors, while the lines are fitted results by using the Full Width at Half Maximum (FWHM) Gaussian function ............................................................................ 97
Figure 6.8. (a) Schematic structure of saddle point at 1/2[4 5 1] at (104) surface normal; (b) atom labels for two layers with different cation sequences ................................................................. 100
Figure 6.9. Total and partial DOS for initial, local maximum and local minimum structures of (104)4 5 1 system ....................................................................................................................... 102
Figure 6.10. Cation-cation and cation-anion distance distribution for initial, local maximum and local minimum structures of (104)[4 5 1] system ...................................................................... 103
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CHAPTER 1
INTRODUCTION
1.1 Motivation and Background
Metal oxide ceramics [1] possess a variety of great properties and have wide availability
always attract great interest of research in science and technology due to their potential large
economic impact. Titanium dioxide, zinc oxide and binary compounds of the two are such kind of
ceramic materials. They are wideband semiconductor (with bandgap around 3.2eV) with great
properties. TiO2 has been widely used as pigment in sunscreens[2], paints[3], ointments,
toothpaste[4]. Since the photocatalytic phenomenon of TiO2 under ultraviolet (UV) light was
discovered by Fujishima and Honda[5], intensive efforts have been made to the study of TiO2 and
related materials. ZnO material has also been a subject of research over decades for its properties
and wide applications such as transparent electronics[6], ultraviolet light emitters, piezoelectric
devices and chemical sensors[7]. ZnO has a similar band gap with TiO2 and large exciton binding
energy (60 meV)[8], which is much larger than other materials commonly used as semiconductors
for blue green light-emmiters devices, such as ZnSe and GaN[9]. Therefore, the combination of
these two oxides has been extensively studied, and shows a promising prospect on affore-
mentioned and other novel applications. The ABO3 complex oxides that are rich in polymorph
structures have been intensively investigated. For example, the ZnTiO3 oxides find applications in
catalysts, nanofibers, white pigments, microwave dielectrics, phosphors, nonlinear optical and
luminescent materials, antimicrobial stone coating, and gas sensors.
Technological progress of devices using these materials depend on fundamental
understanding of the bonding and structure, especially the defect structures and their impact on
properties. Because in materials, it is not often the overall structure or bonding that control the
2
properties, but the types of defects present and their concentration and distribution. It is well known
that the defects in a material greatly affect the properties of materials. For example, point defects,
intrinsic or extrinsic, in semiconducting material control their electrical properties[10], [11]. It is
thus vital not only to understand the bulk property and behaviors but also defects in the metal
oxides for a variety of applications.
One great property for ZnO/ZnTiO3 compounds is their high temperature endurance, which
contribute to solid lubricant. Recent studies showed that metal oxides such as TiO2, V2O5 and WO3
can achieve low friction coefficients[12] due to the close relation to two dimensional defects such
as Megéli phase or stacking fault formations. In addition, significant reduction in the sliding wear
factor and friction coefficient was achieved with ZnO/Al2O3/ZrO2 nanolaminates coatings, which
was attributed to the {0 0 0 2} basal stacking faults in nanocrystalline ZnO, in carbon-carbon
composite[12]. Therefore, it becomes very important to understand the tribological behavior for
hexagonal ilmenite ZnTiO3, which has a similar structure to Wurtzite ZnO. The dependence of
these materials on the mechanical properties can be well understood by studying the stacking fault,
which is considered to be a one dimensional or linear defect in crystals.
Free surfaces, which denote two-dimensional or planar defects in material science, come
from the external surfaces at which the solid terminates at a vapor or liquid. The surface conditions,
such as energies, structures, and contamination, can significantly affect the workfunction, which
may contribute to the electrical conductivity. Fundamental understanding of the effect of surface
properties on the ZnO workfunction is therefore vital on the purpose of seeking its potential
applications.
Although some experimental studies have been done to understand those defect behavior,
the information is still lacking because of the limits of experimental methods. With the rapid
3
expansion of the computing resources in the past few decades, modeling and simulation have
become effective material research tools to overcome the challenges of experimental techniques
and behaviors under harsh conditions. In this dissertation, both classical simulations that employ
empirical potentials and first principles calculations are used to study the atomic and electronic
structures, defect and associated behaviors in metal oxide ceramics at atomic scale. These
computational methods are tightly coupled with experimental investigations through close
collaborations with groups with relevant expertise.
1.2 Contribution of the Dissertation
In this dissertation, systematic studies on the bulk and defects behavior were performed by
using first principles density functional theory and classical simulation methods. The following
are the main contributions of this work:
(1). Despite their importance, understanding of defects in complex ceramics is very limited.
This dissertation provides a systematic study of point defects, surfaces, and stacking faults in ZnO-
TiO2 ceramics by using first principles calculations, and the results have provided insights to the
electronic, mechanical, catalytic and other properties.
(2). The intrinsic and extrinsic defects of ZnO and ZnTiO3 were determined, some for the
first time in the literature from first principles calculations. In addition to Zn, Ti, and O mono-
vacancies, the formation energies and binding energies for [VTi-VO], [VZn-VO], and [ZnTi-TiZn]
complexes were also investigated. In addition, for wurtzite ZnO, the formation and ionization
energies of defects in [SbZn-2VZn] and [AsZn-2VZn] pairs were discussed.
(3). The effect of different surface absorbents and surface defects on the workfuntion of
wurtzite ZnO were investigated by using first principle calculations. A method to treat dipolar
4
surfaces to ensure accurate calculations of workfunction and to analyze contributions from dipole
moment was proposed. For polar ZnO surfaces, (0 0 0 1)/(0 0 0 1�), pseudo-hydrogen were
introduced to saturate the dangling bonds of atoms at the bottom of the surface slab to balance the
net dipole moment and mimic the bonding in bulk. Therefore, accurate workfunction values were
achieved from the electrostatics studies.
(4). The generalized Stacking Fault (GSF) method was used to investigate the stacking
fault behavior on ilmenite ZnTiO3. The γ-surfaces for (110) and (104) surfaces were fully mapped
for the first time. In addition, energy profiles along certain directions, as well as ideal shear stress,
were studied to determine the favorable glide direction. The simulation results were compared,
agreeing well with TEM results.
5
CHAPTER 2
LITERATURE REVIEW
2.1 ZnO-TiO2 Binary Systems
Materials with a variety of properties and relatively low cost are essentials requirement in
industries. Zinc titanates based oxides have variety polymorphs and can be sintered below
1373K, which are potential candidates for a large amount of applications[13]. Zinc titanates have
been studied for applications in many fields such as paint pigment, gas sensor and catalytic
sorbent. In recent years, great interest in zinc titanates has been revealed again mainly because of
the potentiality of ZnTiO3 as a promising candidate for microwave dielectric ceramics, especially
for low temperature co-fired ceramics (LTCC).
2.1.1 History and Phase Diagram
Zinc titanate is in view of the possible formation of 3 double-oxides of different
stoichiometries and different TiO2-modifications as well. Actually, zinc titanate is not really a
newly discovered material. The earliest study of the ZnO-TiO2 system dates back to 1937. Cole
and Nelson[14] successfully synthetized Zn2TiO4 by a conventional solid state reaction using
ZnO and TiO2 powders (ZnO:TiO2 = 2 : 1). They found that the solid solution of Zn2TiO4 and
TiO2 would be formed when the calcinated below 1218K. As the calcination temperature
increases, ZnTiO4 and rutile TiO2 would be found.
Due to its potential in many applications, fundamental studies regarding the phase
diagram of ZnO-TiO2 and the crystallographic characterization of the titanium-zinc-oxides began
in the 1960s. Because solid state reactions are to be described not only as a function of the
thermodynamic parameters, but as a function of the possibility of movement of the component
6
towards the phase boundary[13], contradictory data concerning the conditions of preparation of
the inverse spinel compounds Zn2TiO4 (ZnO : TiO2 = 2 : 1), ilmenite ZnTiO3 (ZnO : TiO2 = 1 :
1) and Zn2Ti3O8 (ZnO : TiO2 = 2 : 3) exists in literature. Dulin and Rase[15] created a phase
diagram of ZnO-TiO2 system under relatively high temperature. This phase diagram indicates
that from approximately 870 K, the spinel Zn2TiO4 and hexagonal ZnTiO3 are
thermodynamically stable while ZnTiO3 decomposes into Zn2TiO4 and rutile TiO2 as the
temperature increases to 1220 K.
Figure 2.1. The ZnO-TiO2 phase diagram published by Dulin and Rase[15]
7
Later in 1961, the existence of the meta-stable compound Zn2Ti3O8 with a structure
derived from the spinel type was shown for the first time by Bartram and Slepetys[16]. In their
study, Zn2Ti3O8 with a cubic structure was obtained when sulfate-containing hydrous TiO2 of
anatase structure was reacted with ZnO at 973 to 1173 K. But when anatase TiO2 was replaced
by rutile, ZnTiO3 could be easily formed. Nevertheless, the experimental results of many
references demonstrated that the source of TiO2, no matter anatase or rutile, is not the decisive
factor. Zn2TiO4 and ZnTiO3 could be formed simultaneously under many conditions, especially
under solid-state conditions[17]. Liu et al[18] proposed in 2008, that anatase and rutile TiO2
crystalline powders mixed with ZnO powder could synthesize zinc titanates by using a
traditional solid-state reaction method, a low cost and a favorite of the ceramics industry. They
concluded that the content of ZnTiO3 was dependent on the grain size of the original anatase in
anatase-derived zinc titanates. Using a combination of ZnO and rutile TiO2, only ZnTiO3 existed
at lower calcining temperatures, while Zn2TiO4 appeared at higher calcining temperatures as a
result of decomposition of ZnTiO3[18].
Interestingly, Yamaguchi et al.[19] reported that a compound tentatively denoted as
Zn2Ti3O8 was found to be a low-temperature form of ZnTiO3 by using an amorphous material
prepared by the hydrolysis of zinc acetylacetonate Zn(C5H7O2)2 and titanium isopropoxide
Ti(OC3H7)4. The experimental data also indicated that a cubic-to-hexagonal transformation
occurred at 1020 K and the decomposition of ZnTiO3 in Zn2TiO4 and rutile TiO2. This was
verified by Dulin and Rase[15].
8
Figure 2.2. Infrared spectra for (A) cubic and (B) hexagonal forms of ZnTiO3 presented by Yamaguchi et al[19].
Further investigations on the meta-stable phase Zn2Ti3O8 are under taken by material
scientists in following years. Yang and Swisher[20] proposed a new ZnO-TiO2 phase diagram
containing additional information based on the Dulin and Rase. They reported that traditional
thermal equilibrium experiments were not as valuable as chemical transformation experiments,
because of the sluggishness of the phase changes accompanying temperature changes. They also
pointed out that Zn2Ti3O8 is a thermodynamically stable compound below 1073K, above which
the ZnTiO3 and Zn2TiO4 exist. Kim et al. demonstrate in their paper the controversial results of
the previous phase equilibria studies.[17] They point out that the most widely known phase
diagram does not indicate the presence of a Zn2TiO4-TiO2 solid-solution phase. The solubility of
TiO2 in Zn2TiO4 spinel ceramics was 33% at temperatures higher than 1145 K and decreased
9
when temperature lower than 1145 K. Also their study confirmed the precipitate phase, Zn2TiO8,
which was formed during cooling at low temperature. Based on the experimental data and
previous work, Kim proposed a new phase diagram of the ZnO-TiO2 system.
Figure 2.3. The system ZnO-TiO2 phase diagram published by Yang and Swisher[20]
10
Figure 2.4. ZnO-TiO2 phase diagram including solid solution published by Kim et al[17]
Li et al. [21] prepared a Zn2TiO4 sample by annealing raw materials of ZnO and TiO2 in a
molar ratio of 3 : 2 at 1632 K and observed a precipitate. The size of the precipitate is about 40
nm, and its composition is approximatly ZnTiO3. Besides, the results of electron diffraction
pattern indicate that the precipitate has cubic crystal symmetry with the same lattice constant as
that of the spinel-type Zn2TiO4.
Regarding all the above fundamental studies concerning the phase diagram of ZnO-TiO2
system, such system contains three compounds, Zn2TiO4, ZnTiO3 and Zn2Ti3O8. Zn2Ti3O8 phase
is a low-temperature form of ZnTiO3 and ZnTiO3 can decompose into Zn2TiO4 and rutile TiO2
when the temperature reaches 1020 K.
11
2.1.2 Microstructure
Figure 2.5. Schematic structure of inverse spinel Zn2TiO4
Zn2TiO4 have an inverse spinel structure with space group Fd-3m. Bartram and Slepetys
proposed one of the few systematic experimental studies on the bulk structure of Zn2TiO4.[16]
32 oxygen atoms comprise an fcc lattice with associated interstitial tetrahedral and octahedral
sites. In this unit cell, one-half of the divalent Zn cations occupy the T sites and the O sites are
filled by a stoichiometric mix of the Zn and Ti cations. This is quite different from the normal
spinel, in which O and T sites are filled with one-half and one-eighth occupancy, respectively.
The O sites are filled with the trivalent cation while the T sites are filled with the divalent cation.
ZnTiO3, on the other hand, is more complicated in structure properties. The most stable
phase of ZnTiO3 is ilmenite. It was reported that nanocrystalline ilmenite ZnTiO3 can be
successfully synthesized by using the Solution Combustion Synthesis (SCS) method[22]. The
12
ilmenite structure is considered to be derived from structure of corundum. Ti4+ and Zn2+ occupy
2/3 octahedral voids leaving the rest of the 1/3 octahedral voids empty and the columbic
repulsion between Zn2+ and Ti4+ ions causes each to move slightly toward the adjacent
unoccupied octahedral site. The oxygen ions also shift slightly from idealized hexagonal closed-
packed positions.
Figure 2.6. Crystal structures of ilmenite-type ZnTiO3 and LiNbO3-type ZnTiO3[23]
Other than ilmenite structure, no other polymorphs of ZnTiO3 were reported until 2014.
Yoshiyuki Inaguma et al.[23] described another polymorph of ZnTiO3 with a LiNbO3-type (LN-
type) structure, which was synthesized under a pressure of 16-17 GPa and 1373-1473 K. The
LN-type structure can be described as a derivative of the perovskite structure[24], which is the
most common structure for ABO3 ceramics. Similar to perovskite compounds, LN-type also
contains three-dimensional corner-sharing BO6 octahedra (TiO6 in this case), and the cooperative
13
cation shift along the hexagonal c-direction against close-packed anions (the oxygen in this
case).
Zinc oxide crystals, like most of the group II-VI binary compounds, exhibit two major
polymorphs, cubic zincblende and hexagonal wurtzite. The ZnO4 tetrahedral coordination is
typical of sp3 covalent bonding, where each Zn atom is surrounded by four cations at the corners
of a tetrahedron.
Figure 2.7. Schematic structures of ZnO crystal structures, Zinc blende and Wurtzite
2.2 Computational Modeling Study on Defect Properties of ZnO/ZnTiO3
Bulk properties of crystalline ZnO have been theoretically studied by using first principle
calculations. In addition, point defects of wurtzite ZnO have been extensively studied by using
first principle calculations. In 2000, Chris G. Van de Walle[25] reported the study of native
defects of ZnO using a first principles pseudopotential method and concluded that the most
14
abundant defects in ZnO are Zn and O vacancies[25]. He then conducted a theoretical
investigation on the behavior of hydrogen as a shallow donor in ZnO[26] was done by him. It
was found that hydrogen can be a donor in ZnO and controlling the conductivity requires
controlling the exposure to hydrogen very carefully[26]. The local density approximation (LDA)
method was used for defect calculations by S. B. Zhang et al. in 2001[27] while Fumiyasu Oba
et al[28]. reported points defect calculations with plan-wave pseudopotential method within the
generalized gradient approximation (GGA). In Oba’s paper, the formation energies of the donor-
type defects are very low under p-type conditions[28].
However, a shortcome of density functional theory is the underestimation of band gap of
some semiconductors, especially for those oxides containing transition metals, such as ZnO. The
occupied states within the band gap induced by native defects show the ratio of conduction band
to valence band. Therefore, the underestimation of band gap results incorrect defect formation
energies[29]. Several approaches have been introduced to correct the underestimation of band
gaps, such as LDA+U method[30][31][32], extrapolation scheme, and the B3LYP[33] and HSE
hybrid functional[34]. For intrinsic defects, it was found that oxygen vacancies and zinc
vacancies are the most preferred defects, followed by the Zn interstitial and ZnO anti-site[29].
Because of the donor levels associated with oxygen vacancies and zinc interstitials,
undoped ZnO is considered to be a n-type semiconductor. To develop a p-type ZnO, this material
with large-size-mismatched dopants such as P, As, and Sb were introduced. Limpijumnong et
al[35]. provide first principle calculations on [AsZn-2VZn] and [SbZn-2VZn] models in As- and Sb-
doped ZnO and found that the formation energies of the compensating native defects can be
maximized while the formation energies of the dopant can be minimized[35].
15
For free surface defects, there are four major Miller index surfaces existing in Wurtzite
ZnO, the non-polar (1 0 1� 0), non-polar (1 1 2� 0), polar (0 0 0 1) with Zn termination and polar
(0 0 0 1�) with O termination. Meyer and Marx[36] provided first-principles calculations with
Local Density (LDA) approximation on these major surfaces and the results showed a strong
contraction of the outermost double-layer spacing after relaxation. Meyer[37] provided a phase
diagram of the O terminated (0 0 0 1�) surface of ZnO in thermal equilibrium with O2 and H2
gases by combining first-principles calculations with thermodynamic formalism. Breedon et
al.[38] calculated the adsorption of NO and NO2 on the ZnO surfaces. In addition, glycine
adsorption on the Zn-terminated (0 0 0 1) ZnO surface has also been studied by using first
principle calculations[39]. Generalized gradient approximation (GGA) method with Perdew-
Burke-Ernzerhof (PBE) was used to investigate the O2 adsorption on the Al-doped ZnO unpolar
(1 0 1� 0) by Ma et al.[40].
The basal-plane stacking faults of Wurtzite ZnO are the main type of extended
defects[41]. Yan et al. [41] studied the energetic and electronic structure of basal plane stacking
faults in wurtzite ZnO by using first-principle calculations and concluded the high density of
stacking faults from mismatched substrates of ZnO are to the low formation energies.
Unlike ZnO, the electronic and mechanical properties on ZnTiO3 are much less studied
theoretically. Little is known of the behaviors of point defects, as well as the surface and stacking
faults properties. This work provides comprehensive defect studies on ilmenite ZnTiO3,
including point defects, surface properties, and stacking faults behaviors.
16
CHAPTER 3
COMPUTATIONAL METHODOLOGY
3.1 Computational Methodology
Traditionally, simulation programs were written for serial computations due to the
limitation of CPU performance. As the substantial development of microprocessor technology
during past decades, processers are now capable of executing multiple instructions in the same
cycle, and thus can solve problems with multi-CPUs while in a reasonable time. For material
science study, computational simulation have became a significant part since larger system and
more complicated properties were able to be solved by using advanced parallel computing
resources.
Two kinds of computational methods have been applied in this work: Classic simulation
and First Principle calculation. The classic simulation, requiring lower computing resources, can
be adopted to relatively larger material systems. It involves empirical potential with fitted
parameters to a model of forces between atoms. The disadvantage of this approach is the empirical
potentials have to be fitted for different systems for variety properties, i.e. mechanical, or
electronic behaviors, due to changes of chemical environment. Therefore, fitting of empirical
parameters was found to draw great interest to plenty of researchers.
The first principle method, on the other hand, starting from fundamental of condensed
matter systems, provides predictions on macroscopic properties of materials, by solving the
interactions between positive atom nucleus and negative electrons. Since all the physical and
chemical behaviors of material systems can eventually described by these basic interactions, the
First Principle methods usually have a consistency and accuracy over different systems on one
hand but computationally expensive on the other hand. Currently, system involving First Principle
17
approaches are usually limited to hundreds of atoms while it can be increase to thousands even ten
thousands when applying classic simulation.
In this section, two programs and related background theories are briefly discussed. First,
General Utility Lattice Program (GULP) is based on the empirical potentials with fitted Zn-O
parameters. Second, plane-wave pseudopotential method within Density Functional Theory
carried out by Vienna ab initio Simulation Pachage (VASP) is discussed.
3.2 Classic Simulation with Empirical Potentials
3.2.1 Two-Body Pair Interaction
According to the Born model[42] of ionic solids, Two kinds of interactions contribute to
the internal energy of ionic material; the long-range electrostatic interaction and the short-range
between atoms when they are bonded. The long-range interaction between ion pair, Coulomb
interaction, is by far the majority part of the total energy. The Coulomb’s law is given by:
𝑈𝑈𝑖𝑖𝑖𝑖𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑞𝑞𝑖𝑖𝑞𝑞𝑗𝑗4𝜋𝜋𝜖𝜖0𝑟𝑟𝑖𝑖𝑗𝑗
(3.1)
Where the i and j represent two ions, with a separation distance of 𝑟𝑟𝑖𝑖𝑖𝑖, charges of 𝑞𝑞𝑖𝑖 and 𝑞𝑞𝑖𝑖, while
the 𝜖𝜖0 is the permittivity of free space.
Compared to the long-range term, the short range interactions are relatively more
complicated. It represents the energy contribution of two bonded bodies. The short range
interaction often contains sufficient repulsive potential, combining with attractive components,
which are highly dependent on the interacting ions. Due to the Pauli Exclusion Principle, the
repulsive interaction arises from the overlap of the electron clouds when the distance between two
atoms is very small. The attractive interaction, the minority part within the short range interaction,
18
is usually known as van der Waals force, which originates from the dipoles on each paired
interacting ions. The potentials used in this study are discussed below.
3.2.2 Potential Forms
As we discussed in 3.2.1, the contributions to the energy of chosen potential forms must
be included both repulsive and attractive terms, despite the fact that later is a very small component
of total energy. Some pair functional developed in early days only considered repulsive part along
with the long-range Coulomb interaction, such as the potential developed by Born and Lande in
1918[43], and the one introduced by Born and Mayer in 1932[44]. On the other hand, the Lennard-
Jones potential[45] with an attractive part of C6 was used in this work:
𝑈𝑈𝑖𝑖𝑖𝑖𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝑟𝑟𝐿𝐿−𝐽𝐽𝐶𝐶𝐿𝐿𝐿𝐿𝐽𝐽 = 𝐶𝐶12
𝑟𝑟𝑖𝑖𝑗𝑗12 −
𝐶𝐶6𝑟𝑟𝑖𝑖𝑗𝑗6 (3.2)
Where rij is the atomic distance and variables C and m can be chosen to determine the equilibrium
inter-atomic separation, in which the m value is normally 12.
In addition, Buckingham potential, introduced by Richard Buckingham in 1938[46],
contains a same attractive term with Lennard-Jones form, but with a two parameter exponential
version of repulsive term. The form is described as:
𝑈𝑈𝑖𝑖𝑖𝑖𝐵𝐵𝐶𝐶𝐵𝐵𝐵𝐵𝑖𝑖𝐿𝐿𝐵𝐵ℎ𝐿𝐿𝐶𝐶 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(−𝑟𝑟𝑖𝑖𝑗𝑗
𝜌𝜌) − 𝐶𝐶6
𝑟𝑟𝑖𝑖𝑗𝑗6 (3.3)
It can be seen from equation 3.3 that the introduction of three parameters, 𝐴𝐴, 𝜌𝜌, and C, makes the
Buckingham potential more flexible to predict the energy.
It should be noted that the cut-off value need to be determined carefully due to the short
range nature of these two potentials. In order to well reproduce the experimental data, we combine
the variables from literatures and the ones we fitted, which are thoroughly discussed in following
19
section. Normally, the value of cut-off range is determined by running a series of identical
calculations with increasing values of the short range cut-off, following by comparing the lattice
energies from each calculation. When the energy reaches the convergence criterion, further
increase of the cut-off value will be a waste of computing resources.
3.3 First Principle Calculations
3.3.1 The Schrodinger Equation and Born-Oppenheimer Approximation
The development of Quantum mechanics built by Heisenberg, Schrodinger, and many great
scientists provides the key tools to deal with the physical chemistry for micro systems. The time
independent Schrodinger equation is applied to describe the given non-relativistic physical system.
It has the following form:
𝐸𝐸Ψ = 𝐻𝐻�Ψ (3.4)
where 𝛹𝛹 and E are the wavefunction and energy of the particular state of the system, respectively,
and 𝐻𝐻� represents the Hamiltonian operator, which describes the way that particles interact with
one another. The complete 𝐻𝐻� can be described as following:
𝐻𝐻(𝑟𝑟,𝑅𝑅) = 𝑇𝑇𝐿𝐿(𝑟𝑟𝑖𝑖) + 𝑉𝑉𝐿𝐿𝐿𝐿(𝑟𝑟𝑖𝑖) + 𝑇𝑇𝑁𝑁(𝑅𝑅𝐼𝐼) + 𝑉𝑉𝑁𝑁𝑁𝑁(𝑅𝑅𝐼𝐼) + 𝑉𝑉𝑁𝑁𝐿𝐿(𝑟𝑟𝑖𝑖,𝑅𝑅𝐼𝐼)
= −∑ ℏ2
2𝐶𝐶𝑒𝑒𝑖𝑖 ∇𝑟𝑟𝑖𝑖
2 + 12∑ 𝐿𝐿2
�𝑟𝑟𝑖𝑖−𝑟𝑟𝑗𝑗�𝑖𝑖,𝑖𝑖 − ∑ ℏ2
2𝑀𝑀𝐼𝐼∇𝑅𝑅𝐼𝐼2
𝑖𝑖 + 12∑ 𝑍𝑍𝐼𝐼𝑍𝑍𝐽𝐽𝐿𝐿2
�𝑅𝑅𝐼𝐼−𝑅𝑅𝐽𝐽�𝐼𝐼,𝐽𝐽 − ∑ 𝑍𝑍𝑖𝑖𝐿𝐿2
|𝑟𝑟𝑖𝑖−𝑅𝑅𝐼𝐼|𝐼𝐼,𝐽𝐽 (3.5)
where e and N represent the electron and nuclei in the system. It can be seen in Equation 3.5 that
five terms exist. From left to right, these terms denote the kinetic energies of electrons, the
Coulombic interations between electrons, kinetic energies of atomic nucleus, the interactions
between atomic nucleus, and the electrons interactions between electrons and atomic nucleus,
respectively.
20
For many body system, solving the Schrodinger equation accurately has been a great
challenge for decades. The so called First-principles calculations, are the sums of methods to
describe plenty of physical and chemical properties by solving the Schrodinger equation. The first
principle calculations provide the valuable information of electron distributions in the system,
which can describe the bonds forming or breaking.
It is known that the atomic nuclei can move much slower than the electrons, while the mass
of nuclei is much larger than those of electrons. Therefore, the motions of electrons and atomic
nuclei can be treated separately, that is to say, the nuclei being immobile when dealing with
electrons. Under Born-Oppenheimer approximation[47], the wave functions of electrons are only
determined by the positions of atomic nuclei, therefore the kinetic terms in Equation 3.5 can be
treated separately. The equation of electronic motion can be described as:
𝐻𝐻𝐿𝐿Ψe(𝑟𝑟,𝑅𝑅) = 𝐸𝐸𝐿𝐿Ψe(𝑟𝑟,𝑅𝑅) (3.6)
where Ψe is the wavefunctions of electrons, and 𝐻𝐻𝐿𝐿 represents the Hamiltonian of electrons. The
later can be further described as:
𝐻𝐻𝐿𝐿 = 𝑇𝑇𝐿𝐿(𝑟𝑟𝑖𝑖) + 𝑉𝑉𝐿𝐿𝐿𝐿(𝑟𝑟𝑖𝑖,𝑅𝑅𝐼𝐼) + 𝑉𝑉𝐿𝐿𝐿𝐿(𝑟𝑟𝑖𝑖) (3.7)
In addition, the equation of atomic nuclei motion is described as:
𝐻𝐻𝑁𝑁ΨN(𝑟𝑟,𝑅𝑅) = 𝐸𝐸𝑁𝑁ΨN(𝑟𝑟,𝑅𝑅) (3.8)
where ΨN is the wavefunctions of atomic nuclei, and 𝐻𝐻𝐿𝐿 represents the Hamiltonian of atomic
nuclei. The later can be further described as:
𝐻𝐻𝑁𝑁 = 𝑇𝑇𝑁𝑁(𝑅𝑅𝐼𝐼) + 𝑉𝑉𝑁𝑁𝑁𝑁(𝑅𝑅𝐼𝐼) + 𝐸𝐸𝐿𝐿 (3.9)
The Born-Oppenheimer approximation does not describe the motion of the atoms
straightforward, which is quite different with the classical Newtonian mechanics. The time
independent non-relativistic Schrodinger equation can be solved accurately for one-electron
21
system. While for complex systems containing more than one electrons, the electrostatic
interactions between those electrons are more complicated.
3.3.2 Hartree-Fock Method
The Hamiltonian of electrons under Born-Oppenheimer approximation is given by
Equation 3.7. The key feature of this equation is the correlation, the Coulombic interactions,
between electrons in the system. Hartree[48] treated every electron with a single electronic
wavefunctions, which did not obey the Pauli Exclusion Principle. As a result, the wavefunctions
of the system with multi-electrons can be described as product of wavefunctions of single
electron, which is shown below:
Ψe(𝑟𝑟) = ∏ ∅𝑖𝑖(𝑟𝑟𝑖𝑖)𝑖𝑖 (3.10)
However, this method does not consider Pauli Exclusion Principle. Later in 1930,
Fock[49] improved the Hatree equation by using the Slater matrix, thus the electrons could obey
Pauli Exclusion Principle:
Ψ = 1√𝑁𝑁!
�∅1(𝐴𝐴1) ⋯ ∅𝑁𝑁(𝐴𝐴1)
⋮ ⋱ ⋮∅1(𝐴𝐴𝑁𝑁) ⋯ ∅𝑁𝑁(𝐴𝐴𝑁𝑁)
� (3.11)
where 𝐴𝐴𝑖𝑖 = (𝑟𝑟𝑖𝑖,𝜎𝜎𝑖𝑖), and represent the coordinates of electron and spin orbitals.
The full Hartree-Fock equations are given by
ϵ𝑖𝑖∅𝑖𝑖(𝑟𝑟) = �− 12∇2 + 𝑉𝑉𝑖𝑖𝐶𝐶𝐿𝐿(𝑟𝑟)�∅𝑖𝑖(𝑟𝑟) + ∑ ∫𝑑𝑑𝑟𝑟′
�∅𝑗𝑗(𝑟𝑟′)�2
|𝑟𝑟−𝑟𝑟′|∅𝑖𝑖(𝑟𝑟)𝑖𝑖 − ∑ 𝛿𝛿𝜎𝜎𝑖𝑖𝜎𝜎𝑗𝑗 ∫ 𝑑𝑑𝑟𝑟′
∅𝑗𝑗∗(𝑟𝑟′)∅𝑖𝑖(𝑟𝑟′)
|𝑟𝑟−𝑟𝑟′|∅𝑖𝑖(𝑟𝑟)𝑖𝑖
(3.12)
We can see from Equation 3.12 that there are four terms in the right hand side. The first two
terms denote to the kinetic energies and the electron-ion potential, respectively. The third term,
22
so called Hartree term, represents the electrostatic potential. The fourth term is called exchange
term, which arising from the Pauli Exclusion Principle.
The Hartree-Fock method does not consider the correlation between electrons, and makes
an assumption of a single-determinate form for wavefunctions. The electrons depend on an
average non-local potential of the other electrons.
3.3.3 The Variational Principle
The variational principle, which indicates that the calculated energy for an arbitrary
wavefunction is always larger than the ground-state energy, can be used to determine the ground-
state wavefunction, Ψ0. In the solution of the Schrodinger equation, the number of electrons, N,
and the potential of nuclei, Vn-e, for certain material are the system dependent parameters. Hence
the wavefunctions, Ψ, and the corresponding eigenvalues of the electronic Hamiltonian, H, can be
determined.
3.4 Density Functional Theory
One shortcoming of the First principle calculations based on Hartree-Fock equation is the
high cost of computer resources, especially then dealing with large ionic systems. Significant
progress has been made to developing approaches for ab initio calculations of material properties
within the past three decades. The discovery of Density Functional Theory (DFT) provided an
effective solution. The DFT is merely a functional of the electron density, providing one-electron
potential instead of the uncontrollable complexity of the interactions between electrons[12].
23
3.4.1 The Hohenberg-Kohn Theorem
Tremendous studies have been done in the development of methods for solving the
Schrodinger equation[50], [51]. In 1964, Hohenberg and Kohn reported a theorems, which claim
that the ground state energy of the system can be achieved by a variation treatment of electron
density[52]. The idea of Hohenberg-Kohn theorem is to calculated the Hamiltonian according to
the electron density and the ground state energy, 𝐸𝐸𝐶𝐶 = 𝐸𝐸𝐶𝐶[𝜌𝜌𝐶𝐶]. It proved that the ground state
energy, as well as the molecules properties can be determined by using the Density Functional
Theory. The Hamiltonian of electrons in the system can be described as:
𝐻𝐻� = −12∑ ∇𝑖𝑖2𝐿𝐿𝑖𝑖=1 + ∑ v𝐿𝐿𝑒𝑒𝑒𝑒𝐿𝐿
𝑖𝑖=1 (𝑟𝑟𝑖𝑖) + ∑ ∑ 1𝑟𝑟𝑖𝑖𝑗𝑗𝑖𝑖≫𝑖𝑖𝑖𝑖 (3.13)
where v𝐿𝐿𝑒𝑒𝑒𝑒(𝑟𝑟𝑖𝑖) = −∑ 𝑍𝑍𝛼𝛼𝑟𝑟𝑖𝑖𝛼𝛼𝛼𝛼 , 𝐻𝐻Ψ𝐶𝐶 = E𝐶𝐶Ψ𝐶𝐶, representing the external potential of electron in the
external potential field. Therefore, the ground state properties, such as wave functions and total
energies, can be determined by the Schordinger Equation since the external potential field and the
total number of electrons are fixed. Therefore the Hamiltonian of the system under Hohenberg-
Kohn theorem is described as:
𝐻𝐻 = − ℏ2
2𝐶𝐶∑ ∇𝑖𝑖2𝑖𝑖 + ∑ v𝐿𝐿𝑒𝑒𝑒𝑒𝑖𝑖 (𝑟𝑟𝑖𝑖) + 1
2∑ 𝐿𝐿2
�𝑟𝑟𝑖𝑖−𝑟𝑟𝑗𝑗�𝑖𝑖≠𝑖𝑖 (3.14)
Where the first term in Equation 3.14 denotes to the kinetic and potential energies, and v𝐿𝐿𝑒𝑒𝑒𝑒(𝑟𝑟𝑖𝑖)
represents the interactions between nuclei and external potential. The last term represents the
nuclei-nuclei interaction in the system. According to the Hohenberg-Kohn theorem, the ground
state properties of a system is determined by the functional of electronic density.
3.4.2 The Kohn-Sham Equations
According to the Hohenberg-Kohn theorem, the key feature in the Density Functional
24
Theory is expression the energy functional, the Hohenberg-Kohn functional, which is calculated
by Kohn and Sham[53]. Generally, the wavefunction, as well as the density, of interacting
electrons are quite different from that of non-interacting electrons. The Kohn-Sham Equations
constructed a fictitious non-interacting system, whose density is the same as the one of interacting
electrons[53], and the wavefunction of the system is constructed by Slater determinant.
Starting from Hartree equation, the minimum density of electrons can be described after
using the variations calculations as:
�− 12∇2 + 𝑣𝑣𝐿𝐿𝑒𝑒𝑒𝑒(𝑟𝑟) − 𝜖𝜖𝑖𝑖�𝜑𝜑𝑖𝑖(𝑟𝑟) = 0 (3.15)
and the density functional is described as:
𝑆𝑆(𝑟𝑟) = ∑ �𝜑𝜑𝑖𝑖(𝑟𝑟)�2𝑁𝑁
𝑖𝑖=1 (3.16)
𝑣𝑣𝐿𝐿𝑒𝑒𝑒𝑒(𝑟𝑟) = 𝑣𝑣𝑒𝑒𝐵𝐵(𝑟𝑟) + ∫ 𝐿𝐿(𝑟𝑟′)|𝑟𝑟−𝑟𝑟′|
𝑑𝑑𝑟𝑟′ (3.17)
𝑣𝑣𝑒𝑒𝐵𝐵(𝑟𝑟) = 𝛿𝛿𝐸𝐸𝑥𝑥𝑥𝑥[𝐿𝐿(𝑟𝑟)]𝛿𝛿𝐿𝐿(𝑟𝑟)
�𝐿𝐿𝑣𝑣(𝑟𝑟)=𝐿𝐿(𝑟𝑟)
(3.18)
where the density of electrons is the function of the position of electrons. The kinetic energy,
(K.E.) for this is known exactly and is described as:
𝐾𝐾.𝐸𝐸. = −12∑ �𝜑𝜑𝑖𝑖�∇2�𝜑𝜑𝑖𝑖�𝑁𝑁𝑖𝑖=1 = −1
2∑ �∇𝜑𝜑�
2𝑁𝑁𝑖𝑖=1 (3.19)
Meanwhile the ground state energy is described as:
𝐸𝐸 = ∑ 𝜖𝜖𝑖𝑖𝑖𝑖 + 𝐸𝐸𝑒𝑒𝐵𝐵[𝑆𝑆(𝑟𝑟)] − ∫𝑣𝑣𝑒𝑒𝐵𝐵(𝑟𝑟)𝑆𝑆(𝑟𝑟)𝑑𝑑𝑟𝑟 − 12 ∫𝑑𝑑𝑟𝑟𝑑𝑑𝑟𝑟′
𝐿𝐿(𝑟𝑟)𝐿𝐿(𝑟𝑟′)|𝑟𝑟−𝑟𝑟′|
(3.20)
The term 𝐸𝐸𝑒𝑒𝐵𝐵 in Kohn-Sham equation describes the non-classic interactions, in this case
the exchange and correlation, within the electrons.
25
3.4.3 Exchange-Correlation Functional
In the section 3.5.2 we know that every term in the Kohn-Sham DFT energy can be
calculated except for the 𝐸𝐸𝑒𝑒𝐵𝐵, the exchange-correlation energy. The accuracy of the DFT-based
calculations highly depends on the choice of Exchange-correlation functional. However, the
computational determination of the exchange- correlation functional is complicated and would
not be useful. Therefore, the approximate exchange correlation functionals were used. Among
the various choices, the most commonly used are Local Density Approximation (LDA)[54],
Generalized Gradient Approximation (GGA), and hybrid functions.
3.4.3.1 Local Density Approximation (LDA)
The local density approximation (LDA) is considered to be the relatively simple one and
has wide applications. The concept of “local density” was firstly reported in Thomas-Fermi
model and further studied by Kohn and Sham. They simulate the exchange-correlation functional
by using the same electron density within homogeneous electron gas, in which the density of
electron can be calculated by using the quantum Monte Carlo method accurately. In addition, it
is called Local Spin Density Approximation (LSDA) if the spin of electron is involved.
The expression of LSDA can be described as:
𝐸𝐸𝑒𝑒𝐵𝐵𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿�𝑆𝑆↑ + 𝑆𝑆↓� = ∫𝑑𝑑3𝑟𝑟𝑆𝑆(𝑟𝑟)𝜖𝜖𝑒𝑒𝐵𝐵ℎ𝐶𝐶𝐶𝐶𝐶𝐶(𝑆𝑆↑ + 𝑆𝑆↓) (3.20)
where, 𝑆𝑆↑ and 𝑆𝑆↓ denote to the spin up/down of the electrons.
In LDA, the exchange-correlation energy of each element only depends on the local
electron density at that element[54]. However, the exchange-correlation energy does not include
the non-local contributions resulted from the in-homogeneities in the real electron density at a
distance from the element.
26
Despite the accuracy approximation on kinetic and Coulombic energies in LDA, the
correlation is relatively inaccurate, resulting the overestimation of total energy of the system.
This is the disadvantage of LDA.
3.4.3.2 Generalized Gradient Approximation (GGA)
Different from the LDA, researchers introduced the gradient in the electron density,
called generalized gradient approximation (GGA). The exchange-correlation functional is
described as:
𝐸𝐸𝑒𝑒𝐵𝐵𝐺𝐺𝐺𝐺𝐿𝐿�𝑆𝑆↑,𝑆𝑆↓� = ∫𝑑𝑑3𝑟𝑟𝑆𝑆(𝑟𝑟)𝜖𝜖𝑒𝑒𝐵𝐵(𝑆𝑆↑,𝑆𝑆↓, �∇𝑆𝑆↑�, �∇𝑆𝑆↓�) (3.21)
In GGA, the the electron density, as well as the gradient of density, are found in the exchange-
correlation functional. The GGA is perfect for a system contains very high electron density, in
which the exchange-correlation energy is the key feature[55].
Compared to LDA, the deficiencies of the LDA and LSDA were corrected on a broader
basis by GGA. GGA tends to over-correct the overestimation effect conducted by LDA, and it
leads to under-estimation to some extends[55]. The commonly used GGA methods include
Becke88[56], PW-91[57], and PBE[55], in which the PBE functional is used in our work.
3.4.3.3 Hybrid Functional
The exchange-correlation energy, EXC, can be divided into two parts, the exchange
energy, Ex, and the correlation energy, Ec, since the Exc demonstrates separate parameterization
of each component. The hybrid functional was proved to be an effective way to pair those two
components. The approximate exchange and correlation energies can be calculated by using
DFT, while the exact exchange energy can be determined by Hatree-Fock (HF). The
27
combination of those two can be done by hybrid functional, such as B3LYP functional[58]. It
can be described as:
𝐸𝐸𝑒𝑒𝐵𝐵𝐵𝐵3𝐿𝐿𝐿𝐿𝐿𝐿 = 𝐶𝐶1𝐸𝐸𝑒𝑒𝐿𝐿𝐶𝐶𝐿𝐿𝑒𝑒𝐿𝐿𝑟𝑟 + (1 − 𝐶𝐶1)𝐸𝐸𝑒𝑒𝐻𝐻𝐻𝐻 + 𝐶𝐶1𝐸𝐸𝑒𝑒𝐵𝐵88 + 𝐶𝐶3𝐸𝐸𝐵𝐵𝐿𝐿𝐿𝐿𝐿𝐿 + (1 − 𝐶𝐶3)𝐸𝐸𝑒𝑒𝑉𝑉𝑉𝑉𝑁𝑁 (3.22)
3.4.4 Vienna ab initio Simulation Package (VASP)
In this work, we use Vienna ab initio simulation package (VASP) code, which has been
proven to be effective for the solid materials calculations, to carry all the DFT calculations. By
using pseudopotentials and plane wave basis set, VASP code is able to find the solution of Kohn-
Sham equations and in order to perform the ab initio quantum mechanical calculations.
The basic methodology also allows use of post-DFT corrections such as hybrid
functionals mixing DFT and Hartree-Fock exchange, many-body perturbation theory and
dynamical electronic correlations within the random phase approximation. VASP makes use of
efficient iterative matrix diagonalization techniques, like the residual minimization method with
direct inversion of the iterative subspace (RMM-DIIS) or blocked Davidson algorithms in order
to determine the electronic ground state. This work utilizes VASP, which is described in more
details by Kresse and Furthmüller[59], [60].
28
CHAPTER 4
POINT DEFECTS CALCULATIONS IN ZINC OXIDE AND ZINC TITANATE*
4.1 Abstract
The behavior of intrinsic defects in ilmenite zinc titanates have been studied by using
Density Functional Theory (DFT) calculations. The band gap correction, as well as three
experimental conditions are introduced for defect formation energies calculations with all
possible charge states. Unlike the BaTiO3 system, the differences of formation energies between
O1 and O2, Zn1 and Zn2, as well as Ti1 and Ti2 can be neglected. We proved that both mono-
and di- vacancies exist in their nominal charge states over the majority of the band gap. The
dominant defects in ilmenite ZnTiO3 system are oxygen vacancies under metal-rich conditions
while the preferred vacancy highly depends on the Fermi level under oxygen-rich conditions.
The calculated binding energies show a preference of Zn/Ti-oxygen vacancy complexes over the
mono-vacancies. The transition levels for mono- and di-vacancies in ZnTiO3 have also been
investigated. For wurtzite ZnO, the DFT calculations were used to investigate the local
environment of the Sb doped ZnO structure which shows p-type electrical conductivity. The
calculated formation and ionization energies of [SbZn-2VZn]’ acceptor complexes show that these
defects are responsible for the p-type conuctivity.
4.2 Introduction
ZnO is a promising material for wide bandgap optoelectronic applications such as light-
emitting diodes (LEDs) and lasers because of its 3.4 eV bandgap, and large exciton binding
*Some results of this chapter are presented in Jitendra Kumar Jha, Wei Sun, Reinaldo Santos-Ortiz, Jincheng Du, Nigel D Shepherd, "Electro-optical performance of molybdenum oxide modified aluminum doped zinc oxide anodes in organic light emitting diodes: A comparison to indium tin oxide" Mater. Express, Vol. 6, No. 3, 2016 with permission from IOP Science
29
energy of 60 meV[61]. The significantly lower costs make ZnO an attractive potential
replacement for gallium nitride (GaN, bandgap and exciton binding energies of 3.4 eV and 24
meV, respectively) which is widely used in short-wavelength optoelectronic devices. However,
to exploit its potential for optoelectronic applications such as LEDs and diode lasers, p-type ZnO
must be developed[62]–[64].
Undoped ZnO is intrinsically n-type due to donor levels associated with hydrogen
impurities[65]–[67] with native point defects such as oxygen vacancies (VO) and Zinc
interstitials (Zni). From a materials standpoint, the basic requirement for engineering p-type ZnO
is creating a larger number of acceptor states relative to donor states. However, its tendency to
form self-compenstating, native donor defects makes this a significant challenge. It was reported
that P-type ZnO can be achieved by doping with arsenic[68] and antimony[69] where the p-type
behavior has been attributed to the formation of point defect complexes associated with As or Sb
substitution on the zinc sub-lattice. Others have proposed that As[70]–[73] and Sb[74]–[76]
occupy oxygen lattice positions (VI sub-lattice) in ZnO[77].
The ilmenite structured ZnTiO3 has received considerable attention and has been widely
used as pigments[78], catalysts[79], [80], microwave dielectrics[81]–[83], gas sensors[84], and
solid lubricants[85]. Extensive studies have been used to investigate the application on low-
temperature co-fired ceramics (LTCC) due to its low sintering temperature and outstanding
dielectric properties[86]–[88]. The ZnTiO3 films with pure hexagonal phase can be successfully
synthesized by using RF magnetron sputtering between 700 and 800°C[89]. The further
investigation of the technological applications of ilmenite ZnTiO3 highly depends on the basic
understanding of the physical properties. The defects present and related properties greatly affect
30
the materials. It is widely accepted that the intrinsic defects in semiconducting material are
significant in controlling the electrical properties[10], [11].
For ZnTiO3, despite the attention in recent years, very few studies have been done on the
theoretical side compared to the experimental studies, especially on defects behaviors. While for
p-type ZnO, computational studies are also limited. One disadvantage of experimental
investigation is that the information provided by studies such as conductivity and diffusivity give
a macroscopic average over the sample and are ambiguous to a specific defect[90]. Another
shortcoming of experimental studies is the restriction on certain charge states or types of defects
due to the extremely localized information (such as positron annihilation spectroscopy) [90].
Therefore, experimental data are inconclusive in determining specific defects[91]. On the other
hand, computational simulations can obtain detailed information of individual defects with
variety of configurations and charge states.
Atomistic computer simulations have been used to study point defects in a number of
systems. The Zhang-Northrup formalism[92] is among the most well-known approaches, in
which the defect formation energy is defined as the difference between the Gibbs free energies of
a defective and perfect cell with regard to specific chemical potentials and the contribution of
electrons being added and removed. For p-type ZnO, the first principles theoretical studies
carried by Limpijumnong et al suggest that Sb and As occupy Zn sites, resulting in the formation
of AsZn-2VZn or SbZn-2VZn complexes which act as acceptors[35]. They found from their
calculations[35] that the formation energy was also low. They calculated that the AsZn-2VZn
complex had a formation energy of 1.59 eV and an ionization energy of 0.15 eV by using the
Zhang-Northrup formalism[92], which are contrary to some of the experimental reports. In
addition, the Zhang-Northrup formalism has been applied on a variety of ABO3 ternary
31
compounds, such as LaGaO3[93] and SrTiO3[94], in which the intrinsic defects have been
studied as well as the transition metal doping. For other ABO3 compounds, the Zhang-Northrup
formalism has been applied to BaTiO3 with a cubic perovskite structure by Erhart et al[91] using
first-principle calculations. Dawson et al[95]. reported the DFT study on BaTiO3 compounds but
with a hexagonal structure and calculated all possible mono-vacancies as well as the di-vacancies
in variety of charge states. Despite these progresses, the specific mechanisms of p-type doping in
ZnO is a subject of active debate in the scientific community, and DFT study of intrinsic defect
in ilmenite ZnTiO3 is still lacking to the best of our knowledge.
One drawback of conventional DFT calculations is the underestimation of the band gap
for compounds containing transition metals[96]. The ZnTiO3 system, a significant issue
calculations defect formation energies, which depend on the Fermi level. This problem can be
corrected by using the band gap correction method introduced by Erhart and Albe[91]. In this
chapter, plane wave basis set density functional theory (DFT) calculations were carried out to
study the formation and ionization energies of defects in ZnO:Sb. In addition, a systematic study
on intrinsic vacancies, including mono- and di-vacancies by combining first principles DFT and
band gap corrections has also been studied. The defect formation energies and transition levels of
variety charges were calculated.
4.3 Computational Detail
4.3.1 Computational Methods
Computational studies based on plane wave basis set Density Functional Theory (DFT)
calculations were carried out to study the formation and ionization energies of defects in ZnO by
using the Vienna abinitio simulation package (VASP) [97]. The generalized gradient
32
approximation (GGA) exchange correlation functional was used with the PBE parameterization.
Projected augmented wave pseudo-potential (PAW) was used to describe core electrons[98]. For
geometric optimization, the initial structures were fully relaxed until the forces acting on each of
the atoms were less than 0.05 eV/Å.
For defect calculations, a cutoff energy of 400eV for plane wave expansion and 2x2x1K-
point meshing were chosen after convergence tests. To study the substitutional defects and defect
complexes, a 3x3x4 supercell based on fully optimized wurtzite structured ZnO with 142 atoms
were utilized to avoid interactions of defects from adjacent cells. The defect formation energy
can be calculated by DFT.
4.3.2 Defect Formation Energy
In this work, we calculated the formation energy for all the defects involved. By using the
Zhang-Northrup formalism[92], the formation energy of a defect with charge q is given by:
𝐸𝐸𝑒𝑒(𝑞𝑞) = 𝐸𝐸𝑒𝑒𝐶𝐶𝑒𝑒𝐿𝐿 (𝑞𝑞) − 𝐸𝐸𝑒𝑒𝐶𝐶𝑒𝑒𝐵𝐵𝐶𝐶𝐶𝐶𝐵𝐵 − ∑ 𝑆𝑆𝑖𝑖𝜇𝜇𝑖𝑖𝑖𝑖 + 𝑞𝑞(𝐸𝐸𝑉𝑉𝐵𝐵𝑀𝑀 + 𝐸𝐸𝐻𝐻) (4.1)
Where 𝐸𝐸𝑒𝑒𝐶𝐶𝑒𝑒𝐿𝐿 (𝑞𝑞) is the total energy of the supercell with defects, and 𝐸𝐸𝑒𝑒𝐶𝐶𝑒𝑒𝐵𝐵𝐶𝐶𝐶𝐶𝐵𝐵 is the energy of a
perfect equivalent supercell. 𝐸𝐸𝐻𝐻 is the Fermi energy and 𝐸𝐸𝑉𝑉𝐵𝐵𝑀𝑀 is the valence-band maximum
(VBM) of the ideal crystal. In this work, it has the value between 0 and Eg, where Eg is the band
gap of the hexagonal ZnTiO3 and wurtzite ZnO. In the last term, ni indicates the number of atoms
of type I that have been added to (ni>0) or removed from (ni<0) the constructed cell while
creating the defect, and 𝜇𝜇𝑖𝑖 is the chemical potential of the chosen chemical environment of
corresponding element. The chemical potential of the reference state for an element is equivalent
to its calculated total free energy per atom. For defect formation energy calculations, the
chemical potentials depend on the experimental growth conditions, and should be calculated to
33
establish the thermodynamic boundaries of the system[99]. Here we use the total energy
calculated under T=0 K for all the atomic species.
For ZnO, the chemical potential terms, 𝜇𝜇𝑂𝑂 and 𝜇𝜇𝑍𝑍𝐿𝐿, represent the flow of atoms between
the atomic reservoirs and the host crystal, which are determined by the experimental conditions;
Zn-rich environment and O-rich environment. Depending on different Zn/O ratios during
experiments, the 𝜇𝜇𝑂𝑂 and 𝜇𝜇𝑍𝑍𝐿𝐿 are different. Here we only consider the extreme conditions, in
which the Zn and O are maximum rich. In Zn rich condition, the upper bound of 𝜇𝜇𝑍𝑍𝐿𝐿 leads to a
lower bound of 𝜇𝜇𝑂𝑂[100], therefore, the chemical potential of Zn equals to the energy of the
metallic Zn, while the chemical potential of O, 𝜇𝜇𝑂𝑂, is calculated as 𝜇𝜇𝑂𝑂 = 𝜇𝜇𝑍𝑍𝐿𝐿𝑂𝑂 − 𝜇𝜇𝑍𝑍𝐿𝐿, where the
𝜇𝜇𝑍𝑍𝐿𝐿𝑂𝑂 term is equivalent to the energy of bulk ZnO. While for O rich condition, the chemical
potential of O ( 𝜇𝜇𝑂𝑂) equals to half of 𝜇𝜇𝑂𝑂2, which is the energy of isolated O2 molecule, thus
resulting in the lower bound of 𝜇𝜇𝑍𝑍𝐿𝐿. Under this condition, 𝜇𝜇𝑍𝑍𝐿𝐿 = 𝜇𝜇𝑍𝑍𝐿𝐿𝑂𝑂 − 𝜇𝜇𝑂𝑂.
For ZnTiO3, since it can be experimentally grown under varying metal/O ratios, the
chemical potentials of individual atoms must be placed with bounds in thermodynamic
equilibrium, which denotes to the formation energy of the ZnTiO3 crystal. This condition is
described as:
𝜇𝜇𝑍𝑍𝐿𝐿 + 𝜇𝜇𝑇𝑇𝑖𝑖 + 3𝜇𝜇𝑂𝑂 = 𝐻𝐻𝑒𝑒[𝑆𝑆𝑆𝑆𝑇𝑇𝑍𝑍𝑂𝑂3] (4.2)
According to Equation 4.2, 𝜇𝜇𝑍𝑍𝐿𝐿 and 𝜇𝜇𝑇𝑇𝑖𝑖 are equivalent to the energies of the metallic Zn and Ti
under maximum metal-rich conditions, respectively. Hence 𝜇𝜇𝑂𝑂 reaches its lower bound and has a
value of 𝜇𝜇𝑂𝑂 = [𝜇𝜇𝑍𝑍𝐿𝐿𝑇𝑇𝑖𝑖𝑂𝑂3 − 𝜇𝜇𝑍𝑍𝐿𝐿 − 𝜇𝜇𝑇𝑇𝑖𝑖]/3 under metal-rich conditions. While under O-rich
condition, the chemical potential of oxygen is bound by the energy of O in an isolated O2
molecule and leads to a lower bound of the chemical potential of metal. However, further
constraints are added under maximum O-rich conditions in addition to equation 4.2,
34
𝜇𝜇𝑍𝑍𝐿𝐿 + 𝜇𝜇𝑂𝑂 ≤ 𝐻𝐻𝑒𝑒[𝑆𝑆𝑆𝑆𝑂𝑂] (4.3)
𝜇𝜇𝑇𝑇𝑖𝑖 + 2𝜇𝜇𝑂𝑂 ≤ 𝐻𝐻𝑒𝑒[𝑇𝑇𝑍𝑍𝑂𝑂2] (4.4)
where 𝐻𝐻𝑒𝑒[𝑆𝑆𝑆𝑆𝑂𝑂] represents the total energy of wurtzite ZnO and 𝐻𝐻𝑒𝑒[𝑇𝑇𝑍𝑍𝑂𝑂2] is the total energy of
rutile TiO2. When 𝜇𝜇𝑍𝑍𝐿𝐿 reaches its maximum negative value in equation 4.3, 𝜇𝜇𝑍𝑍𝐿𝐿 has the value of
𝜇𝜇𝑍𝑍𝐿𝐿 = 𝐻𝐻𝑒𝑒[𝑆𝑆𝑆𝑆𝑂𝑂] − 𝜇𝜇𝑂𝑂, thus 𝜇𝜇𝑇𝑇𝑖𝑖 is calculated as 𝜇𝜇𝑇𝑇𝑖𝑖 = 𝐻𝐻𝑒𝑒[𝑆𝑆𝑆𝑆𝑇𝑇𝑍𝑍𝑂𝑂3] − 𝜇𝜇𝑍𝑍𝐿𝐿 − 3𝜇𝜇𝑂𝑂. Similarly,
under maximum 𝜇𝜇𝑇𝑇𝑖𝑖 negative value in equation (4), the chemical potential of Ti is calculated as
𝜇𝜇𝑇𝑇𝑖𝑖 = 𝐻𝐻𝑒𝑒[𝑇𝑇𝑍𝑍𝑂𝑂2] − 2𝜇𝜇𝑂𝑂 and 𝜇𝜇𝑍𝑍𝐿𝐿 = 𝐻𝐻𝑒𝑒[𝑆𝑆𝑆𝑆𝑇𝑇𝑍𝑍𝑂𝑂3]− 𝜇𝜇𝑇𝑇𝑖𝑖 − 3𝜇𝜇𝑂𝑂.
4.3.3 Defect Transition Energy
The energy required to change the charge state of a defect, known as defect transition
energy, is defined as
𝜖𝜖(𝑞𝑞 𝑞𝑞′⁄ ) = [𝐻𝐻(𝑞𝑞) − 𝐻𝐻(𝑞𝑞′)]/(𝑞𝑞′ − 𝑞𝑞) (4.5)
Where q and q’ are two different charge states of the defect. Owing to this transition, there are
often transition levels induced in the band gap of semiconductors that correspond to the thermal
ionization energies[100].
4.3.4 Band Gap Corrections
A disadvantage of conventional DFT method is the under estimation of the band gap[96].
In this work, the calculated band gap (Eg) of 2.1 eV is considerably smaller than the
experimental value. In this work, a 3.4 eV[22] band gap value derived by UV-vis spectrum was
used to correct the calculated band gap. Since the defect formation energy highly depend on EF,
the underestimation of band gap can significantly affect the calculated formation energy. Here
we adopt the approach that was introduced by Erhart and Albe in the case of cubic BaTiO3[91],
35
which gives a correction of the band structure obtained by rigidly shifting the valence band and
conduction band with respect to each other. The correction energy is given as
∆𝐸𝐸𝐶𝐶𝐵𝐵 = 𝑆𝑆𝐿𝐿∆𝐸𝐸𝐶𝐶𝐵𝐵 + 𝑆𝑆ℎ∆𝐸𝐸𝑉𝑉𝐵𝐵 (4.6)
Where 𝑆𝑆𝐿𝐿 and 𝑆𝑆ℎ are the number of electrons occupying conduction band states and the number
of holes occupying valance band states, respectively. This correction approach only considers the
effect of the band gap error on the band energy and assumes rigid level[91]. In addition, we
further assumed the offset of the calculated band structure to be restricted to the conduction
band. That is, ∆𝐸𝐸𝑉𝑉𝐵𝐵 = 0 and ∆𝐸𝐸𝐶𝐶𝐵𝐵 = 𝐸𝐸𝐺𝐺𝐿𝐿𝑒𝑒𝑒𝑒𝑒𝑒 − 𝐸𝐸𝐺𝐺𝐵𝐵𝐿𝐿𝐶𝐶𝐵𝐵, where the 𝐸𝐸𝐺𝐺
𝐿𝐿𝑒𝑒𝑒𝑒𝑒𝑒 and 𝐸𝐸𝐺𝐺𝐵𝐵𝐿𝐿𝐶𝐶𝐵𝐵 represent the
experimental derived and calculated band gap, respectively. It should be noted that such
corrections only affect oxygen vacancies and di-vacancies in charge states. Furthermore, these
defects already have significantly higher formation energies than the equivalent defect in its
nominal charge state over the majority of the band gap and the band gap corrections succeed in
only further increasing these defect formation energies[91].
4.4 Geometric Optimization
In order to find optimized structures for further calculations and check the reliability of
computational methods, the total energies as a function of volume around the experimental
volume were calculated for both ZnO and ZnTiO3. At each volume, the cell shape and atom
positions were allowed to relax while the cell volume was maintained constant. The obtained
total energy versus volume curve was fitted to the Birch Equation of State, the equation form is
shown as:
𝐸𝐸 = 𝐸𝐸0 + 98𝐵𝐵0𝑉𝑉0((𝑉𝑉0
𝑉𝑉)23 − 1)2 + 9
16𝐵𝐵0𝑉𝑉0(𝐵𝐵0′ − 4)((𝑉𝑉0
𝑉𝑉)23 − 1)3 (4.7)
36
Where 𝑉𝑉0 and 𝐸𝐸0 are equilibrium volume and energy, respectively. 𝐵𝐵0 and 𝐵𝐵0′ are the bulk
modulus and its pressure derivative at the equilibrium volume.
4.4.1 Geometric Optimization of Wurtzite ZnO
A fitted energy versus volume curve for wurtzite ZnO is shown in Figure 4.1 while the
obtained equilibrium lattice parameters and those from earlier studies and experimental values
are listed in Table 4.1. It can be seen that the c/a ratio and c parameter results agree well with the
experimental c/a ratio, and are consistent with earlier GGA calculations[101]. The calculated c
parameter is slightly higher than the experimental values[102], consistent with the results from
literature: the GGA functional slightly overestimates the c parameter and the cell volume, while
the LAD functional does the opposite.
Figure 4.1. Energy versus Volume curve for Wurtzite ZnO
37
Table 4.1. Calculated structural parameters for Wurtzite ZnO
Wurtzite ZnO c/a V (Å3)
Previous GGA17 1.61 49.19
Previous LDA17 1.62 45.01
This work 1.60 49.79
Experiment18 1.60 47.56
4.4.2 Geometric Optimization of ZnTiO3
The equilibrium lattice parameters and mechanical properties for hexagonal ZnTiO3 were
calculated by using both DFT with LDA and GGA functional. The calculated lattice parameters
and elastic constants are summarized in Table 4.2, together with comparison of experimental and
previous calculations. The bulk, shear, and Young’s moduli were obtained according to the
Voigt-Ruess-Hill approximation[103], [104].
Table 4.2. Comparison of observed and calculated properties for ZnTiO3
This Work References
Exp. VASP-GGA
VASP-LDA LDA1 CASTEP-
GGA2 V(Å3) 321.6 295.6
2.72 298.8 324.0 311.93
c/a 2.74 2.73 2.77 2.683 C11 (GPa) 292.9 395.9
217.3 148.75 13.1 0.85 292.0 61.3
317.3 C12 145.7 142.0 C13 81.0 90.0 C14 16.7 11.7 C15 C33
-5.75 216.5 -0.8
236.3
C44 49.6 47.6 B (GPa) 154.4 229.7 164.9 G (GPa) 64.6 77.7 68.6 E (GPa) 170.1 209.4 180.7 202~1554
It can be seen that our calculated values are good agreement with the previous DFT
38
calculations and experimental results[85],[105]. As it was proved in previous studes, the LDA
calculations underestimate the lattice parameters and volume, but overestimating the mechanical
properties while GGA results, on the other hand, would overestimate the lattice parameters and
underestimate the mechanical properties[106]. For further calculations, DFT with GGA
exchange and correlation functional was used on both ZnO and ZnTiO3 structures.
4.5 Defect Formation Energy Calculations for ZnO
4.5.1 Effect of Supercell Size
Due to the 3-D periodic boundary condition, the supercell size needs to be large enough
to minimize the effect among the defects. In this work, 2×2×1, 2×2×2, 3×2×2, 3×3×2, and 3×3×3
supercells were constructed, with the number of atoms of 16, 32, 48, 72, and 108, respectively.
To investigate the point defects in ZnO, the VO under zinc rich condition was chosen to calculate
the supercell size effect formation energy. The calculated formation energy of Vo with various
charges as a function of logarithm volume is shown in Figure 4.2.
It can be seen in Figure 4.2 that the formation energy of Vo barely changes with the
increase of volume. In addition, for Vo1+, the energy differences among the listed volumes are
relatively small while for Vo2+, the energies differ by less than 0.1 eV, except for the 16 atom-
supercell. Therefore, the 3×3×2 supercell with 72 atoms was adopted for the simple defect
calculations. Larger 3×3×4 supercell was used for the complex defect calculations, such as
XZn − 2VZn defect.
39
Figure 4.2. DFT calculated formation energy of Vo under Zn-rich condition as a function of
different supercell sizes.
4.5.2 Primary Point Defects
Table 4.3 lists the formation energies of primary isolated point defects under Zn-rich and
O-rich conditions. For Zn-rich environment, the defect formation energies of Vo in wurtzite ZnO
for 0, 1+, and 2+ charge states are 0.84, 0.57, and -2.27 eV, respectively, while for O-rich
condition, the calculated values are 3.93, 3.67, 0.82 eV, respectively. It can be seen that the
defect formation energy values of Vo under Zn-rich condition are much lower than those under
O-rich condition, indicating Vo defects are more stable under Zn-rich condition. In addition, for
both Zn and O-rich conditions, Vo2+ is the most stable defect.
Table 4.3. Formation energies of elementary defects in ZnO. Defect type
Charge state
Zn-rich (eV) O-rich (eV) This work Ref This work Ref
VO 0 0.84 0.73 3.93 4.11
1+ 0.57 -0.11 3.67 3.27 2+ -2.27 -1.19 0.82 2.19
VZn 0 5.34 5.21 1.86 1.83 1- 5.59 7.05 2.10 1.90 2- 6.05 8.84 2.56 1.93
Zni 0 2.85 2.07 6.34 5.45
1+ 0.40 1.27 3.89 4.65 2+ -1.54 0.61 1.94 3.99
40
4.5.3 Defect Structure and Formation Energies of [SbZn-2VZn] Complex
In this work, the following defect equation is proposed to describe the structural origins
of the p-type electrical conductivity:
𝑆𝑆𝑆𝑆2𝑂𝑂5𝑍𝑍𝐿𝐿𝑂𝑂�⎯� 2[𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿]′ + 2𝑆𝑆𝑆𝑆𝑖𝑖 + 5𝑂𝑂𝐶𝐶𝑒𝑒
where [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿]′ is an acceptor complex, 𝑆𝑆𝑆𝑆𝑖𝑖 is a donor defect, and 𝑂𝑂𝐶𝐶𝑒𝑒 represents neutral
O sites. The above equation is representative of ‘pose’ ionization where the [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿]𝑒𝑒
complex has donated a hole to become [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿]′.
Figure 4.3. Schematic of the [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿] defect complex (left). Bond length change for the Sb-O relative to regular Zn-O bonds for neutral (right top), 1- (right middle), and 3- (right
bottom) charge states after geometry optimization. The red spheres are O, the gray spheres are Zn, the brown spheres are Sb and the dotted spheres are Zn vacancies.
In order to evaluate the feasibility of the reaction described by equation above,
computations were performed to investigate [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿]′ defect complexes with neutral, 1-,
41
and 3- charged states, and their formation energies. We began by constructing defect models,
showed in Figure 4.3, which were then fully relaxed using DFT calculations in large super cells
(144 atoms). In this work, oxygen rich conditions are more relevant since p-type behavior was
only obtained by post deposition annealing in oxygen. Therefore, the chemical potential of
oxygen was obtained from the energy of the O2 molecule, and μZn was defined as μZn= μZnO- μO.
Similarly, μSb is assumed under O-rich conditions to be μSb = ( μSb2O3 - 3μO)/2. In the last term of
formation energy equation 4.1, q and EF are the charge of the defect and its Fermi level position
with respect to the valence band maximum (VBM)[100], respectively. We focused our DFT
calculations on the [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿] defect complex, which is proposed as the major contributor to
the p-type behavior. The defect association is due to Coulombic attraction between positively
charged SbZn and negatively charged VZn defects. A schematic of the [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿]′ defect
complex and the corresponding Sb-O bond length charges from fully relaxed structures are
shown in figure 4.3. It was found that the complex is stable, and Sb remains four-fold
coordinated with considerable bond length variations in all three states.
Similar bond length changes were observed in earlier DFT calculations by Limpijumnong
et al for the 3- charge state[35]; however, a relaxation to trigonal bi-pyramidal (five-fold
coordination) claimed therein for the neutral complex was not observed here. The larger
supercell used in our work and norm conserving PAW pseudopotential could contribute to the
difference. The bond length changes of the computed 3- charge state (corresponding to Sb3+)
show two elongations (6.23 and 11.57%) and two minor contractions (-0.62 and -1.12%) which
are consistent with the measured increase in tensile stress (compared to relaxed bulk ZnO) in the
as-deposited film. For the 1- charge state (corresponding to Sb5+ and thus annealed film), all the
Sb-O bonds show minor contractions (-0.92 to -4.31%), which is not surprising as Sb5+ has a
42
smaller ionic radius compared to Sb3+. But this latter result is in conflict with the observed
tensile stress measured in the annealed, p-type films. Other structural aspects might also
influence the cell parameters and hence the tensile stress of the annealed films, but further
investigation is needed to fully understand this discrepancy.
Table 4.4. Defect formation energies of [𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿] and Zni under O-rich conditions
Species Charge Energy (Ev)
This work (GGA) Literature
[𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿
− 2𝑉𝑉𝑍𝑍𝐿𝐿]
0 1.87 2.00 (LDA)
1- 2.04 2.16 (LDA)
3- 5.45 5.33 (LDA)
Zni
0 6.34 5.45 (GGA) 4.25 (GGA+U) 3.4 (LDA)
1+ 3.89 4.65 (GGA) 1.69 (GGA+U) 1.5 (LDA)
2+ 1.94 3.99 (GGA) 0.02 (GGA+U) -0.2 (LDA)
The defect formation energies for neutral, 1-, and 3- charged states of this complex are
summarized in table 4.4. The calculated formation energies of the acceptor complex
[𝑆𝑆𝑆𝑆𝑍𝑍𝐿𝐿 − 2𝑉𝑉𝑍𝑍𝐿𝐿]′ and donor defect (Zni) are 2.04 eV and 3.89 eV, respectively. These results and
those from literature are compared in table 4.4. The ionization energies of the Zn interstitial and
acceptor complex are 1.95 eV and 0.17 eV, respectively. Therefore from both ionization and
formation energies, it is feasible that these defect complexes may be responsible for the p-type
behavior observed in the annealed films and the reaction indicates the need for excess oxygen as
is experimentally observed. The incorporation of oxygen with annealing may also explain the
measured increase in tensile stresses as shown in table 2, since more of the larger O has to be
accommodated.
43
4.6 Defect Formation Energy Calculations for ZnTiO3
4.6.1 Bulk Properties and Defect Structures of ZnTiO3
Figure 4.4. Schematic structure of hexagonal ZnTiO3 and possible vacancy sites.
The equilibrium lattice parameters and mechanical properties for hexagonal ZnTiO3 were
calculated by using both DFT with LDA and GGA functionals. Figure 1 shows the schematic
structure of hexagonal ZnTiO3 (space group: R3�) after structure relaxation. It can be seen that in
this typical ilmenite cell, the Zn and Ti ions are coordinated octahedrally to the nearest O ions.
The same types of octahedra are connected with edge sharing, while each type of octahedron
shares one face with the other type in the adjacent layer[107], stacking alternatively along c-axis.
The calculated lattice parameters and elastic constants are summarized in Table 4.5, together
with comparison of experimental and previous calculations. The equilibrium lattice parameters
44
were obtained from energy versus volume curve, which was fitted to the Birch equation of
state[108].
Table 4.5. Comparison of observed and calculated properties for ZnTiO3
This Work References
Exp. VASP-GGA
VASP-LDA GULP LDA1 CASTEP-
GGA2 V(Å3) 321.6 295.6 301.1 298.8 324.0 311.93
c/a 2.74 2.72 2.73 2.73 2.77 2.683 B (GPa) 154.4 229.7 186.2 164.9 G (GPa) 64.6 77.7 80.7 68.6 E (GPa) 170.1 209.4 211.5 180.7 202~1554
The bulk, shear, and Young’s moduli were obtained according to the Voigt-Ruess-Hill
approximation[103], [104]. It can be seen that our calculated values are in good agreement with
the previous DFT calculations and experimental results[85],[105]. As it was proved in previous
studies, the LDA calculations underestimate the lattice parameters and volume while
overestimating the mechanical properties[106]. For the rest of the calculations, DFT with GGA
exchange and correlation functionals was used to calculate the stacking fault energies along
certain directions and the surface energies.
We considered all types of mono-vacancies in ilmenite ZnTiO3 as well as the possible
combinations for the di-vacancies complex. As can be seen in Figure 4.4, six sites of mono-
vacancies are available in this structure, face-sharing O1, edge-sharing O2, Zn1, Zn2, Ti1, and
Ti2 vacancies. However, very small differences on formation energies are found for the two sites
of the same kind vacancies according to our calculations. For example, the face-sharing O1
vacancy without charges under a O-rich condition has a formation energy of 6.7965 eV while for
the corner-sharing O2 vacancy under same conditions, the formation energy is 6.7964 eV. For
the rest types of defects, the differences of formation energies are less than 0.01 eV, which is
quite different from the case of hexagonal BaTiO3 with a similar structure of ZnTiO3. Dawson et
45
al[95] reported a DFT calculated results with a difference of 0.8 eV between the chosen O
vacancies. Therefore, only one value for each type of vacancy is discussed following.
For vacancy complexes, the simultaneous presence of (VO + VZn) and (VO + VTi) are
considered in two types. One is the nearest-neighbor vacant sites (Type I) with distances of 2.05
Å and 1.89Å for (VO + VZn) and (VO + VTi), respectively. Another is vacant sites with distance
apart (Type II) with distances larger than 6Å for both of the di-vacancies. Type I vacancies for
both pairs were chosen in this work since the preference for the formation of these complexes
over Type I, which results from the strong binding behavior between there metallic atoms and
their nearest-neighbor O atoms. This result is in close agreement with that in BaTiO3 system[95].
In addition, multiple charge states were calculated for all types of vacancies listed above.
In the purpose of chemical potential calculations, Wurtzite ZnO and Rutile TiO2 were
calculated by the GGA method. ZnO has the Wurtzite structure with lattice constants of a=3.24
Å and c=5.21 Å. DFT calculations produce lattice constants of a=3.30 Å and c=5.28 Å and an
enthalpy of formation of -18.23 eV for ZnO. Rutile TiO2 has lattice constants of a=4.58 Å and
c=2.96 Å, in comparison to calculated values of a=4.59 Å and c=2.95 Å and enthalpy of
formation of -52.85 eV. For metal, a lattice constant of a=2.66 Å and c=4.95 Å and total energy
of -2.53 were calculated for hcp structural Zn; these values agree reasonable well with the
experimental equivalents of a=2.66 Å and c=4.94 Å. Ti also has a hcp structure with calculated
lattice constants of a=2.95 Å and c=4.67 Å and a total energy of -15.52 eV, in comparison to
experimental lattice constants of a=2.95 Å and c=4.68 Å. For O, the chemical potential is
calculated by putting an isolated O2 dimmer in a large enough cell (15×15×15 Å in this work).
46
4.6.2 Formation Energy
The formation energies of the isolated point defects under the limiting chemical
potentials of O-rich (Ti deficient), O-rich (Zn deficient), and metal-rich conditions with a Fermi
level at the valence band maximum (VBM) are presented in Table 4.6. It is worth mentioning
that there are no clear differences between difference vacancy sites of same type. Only slight
differences are observed between two O-rich conditions. It can be seen from Table 4.6 that VTi in
all charged states under O-rich (Ti deficient) condition have higher energies over VTi under O-
rich (Zn deficient) condition, while for VZn the trend is clearly different.
Table 4.6. Formation energies of mono- and di-vacancies in ilmenite ZnTiO3 under metal-rich and O-rich conditions. The Fermi level at the valence band maximum.
q Ef (O-rich)a Ef (O-rich)
b Ef (metal-rich)
VO 0 6.7965 6.7965 2.3151
+1 3.4218 3.4218 -1.0596 +2 -0.4752 -0.4752 -4.9566
VZn 0 3.5096 3.5305 7.0174 -1 4.6500 4.6710 8.1578 -2 5.8479 5.8689 9.3557
VTi
0 5.6063 5.5854 15.5427 -1 6.8410 6.8201 16.7774 -2 8.1672 8.1463 18.1036 -3 9.6123 9.5914 19.5487 -4 11.2384 11.2175 21.1748
VZn -VO +1 2.4964 2.5173 1.5227 0 3.6573 3.6783 2.6837 -1 7.5538 7.5748 6.5802
VTi -VO
+1 4.4669 4.4459 9.9219 0 5.6566 5.6357 11.1116 -1 6.9529 6.9320 12.4080 -2 8.4396 8.4186 13.8946 -3 12.3391 12.3181 17.7941
ZnTi –TiZn 0 2.57 2.57 2.57
47
Obviously, VO2+ is the most dominant defect under metal-rich and p-type (EF at VBM)
conditions. Under metal-rich conditions, the calculated values for VO0, VO
1+, and VO2+ are 2.32, -
1.05, and -4.96 eV, respectively. In addition, the VZn and VTi have much higher formation
energies than VO for all charge states. While under O-rich conditions, the calculated values are
6.79, 3.42, and -0.47 eV for VO0, VO
1+, VO2+, respectively, representing VO
2+ remain dominant
under O-rich conditions. Above VO2+ and VO
1+, VZn0 has the next lower energies, followed by
VTi.
Figure 4.5. Variation of calculated defect formation energies of the lowest energy vacancy types in ilmenite ZnTiO3 with Fermi level energy (EF).
48
In addition of mono-defects, Table 4.6 also shows the formation energies of two Schottky
pairs in h-ZnTiO3, [VO+VZn] and [VO+VTi]. Under O-rich conditions, the second most dominant
defect after VO2+ is the 1+ charged [VO+VZn] complex. As Schottky pairs involve the
simultaneous presence of VO and VZn and VO and VTi pairs, their formations are independent of
the chemical potential of Zn, Ti, and O[100]. While under metal-rich conditions, [VO+VZn] in 1+
charge state becomes the third stable defects, after the VO2+ and VO
1+.
It is worth to mention that the q parameter in the second term in equation (1) determines
the charge that a defect carries in a hexagonal ZnTiO3 crystal. To further understand the
electronic behavior of defects, the formation energy of defects as a function of Fermi energy EF,
which varies from 0 to the band gap (Eg). The formation energies of charged defects highly
depend on the Fermi level position and they are given as a function of EF under three
experimental conditions, O-rich with Ti deficient, O-rich with Zn deficient, and metal-rich
conditions, which are plotted in Figure 4.5. The estimate position of defect levels within the band
gap can be estimate by the kinks in the curves in Figure 4.5, which represent the transitions
between different charge states of a particular defect. For both Ti and Zn deficient conditions,
VO2+ is the dominant defect when EF under 1.55 eV, while for the metal-rich condition, oxygen
vacancies dominate over the entire band gap with negative formation energies and therefore are
the most abundant defect. It is noteworthy that only at EF =2.5 eV the formation energy reaches 0
eV. This suggests that under metal-rich conditions the non-stoichiometric hexagonal ZnTiO3 is
promoted by the low formation-energy VO. This observation is slightly different from the h-
BaTiO3 under metal-rich condition, in which the formation energy of VO reaches 0 eV at the
conduction band minimum (CBM)[95].
49
Unlike O vacancies, formation energies of both VTi and VZn vacancies exhibit positive
values under all the combination of chemical potentials at the valance band maximum (VBM)
due to the difficulty in removing an ion with a high charge density such as Ti4+ from an ionic
system[95]. Interestingly, the DFT calculated results on cubic BaTiO3 and hexagonal BaTiO3
showed that Ti vacancies are not observed in any chemical environment although this is almost
certainly a result of the fact that only neutral defects were considered[95]. In addition, similar
results have been reported using potential-based methods. Regardless of this, the VTi4- dominate
in both O-rich conditions in top area of band gap (EF>2.56 eV for Ti deficient, and EF>2.64 for
Zn deficient), representing n-type materials.
In Figure 4.5, either under metal-rich or O-rich conditions, the formation energies of Zn
vacancies suggest that for the majority of the band gap the VZn is not the dominant species.
Figure 4.5 also shows that for only a small part of the band gap within the range from 1.6 to 2.7
eV in O-rich condition with deficient Zn and from 1.65 to 2.8 eV in O-rich condition with
deficient Ti the Zn vacancy species becomes the dominant defect in the material. This is different
from the hexagonal BaTiO3 case, in which Zn vacancies only dominate in O-rich conditions with
deficient Ba. Unlike VTi4- species, Zn vacancy does not have such a strong dependence on the
Fermi level since it shows a 2- charge compared to 4- charge of Ti. Therefore, it allows Ti
vacancies to dominate in the top range of the band gap for O-rich conditions.
For the simultaneous presence of [VO+VZn] and [VO+VTi] complexes, two kinds of
combinations were taken in to consideration in this work: the nearest-neighbor vacant sites (Type
I) and vacant sites with distance apart (Type II). For [VO+VZn] pairs, the distances between the
two species are 9.34Å and 2.05Å for Type I and Type II, respectively. For [VO+VTi] pairs, the
distances between the two species are 9.50Å and 2.10Å for Type I and Type II, respectively. Our
50
calculated results show that the Type I complexes for both [VO+VZn] and [VO+VTi] have the
lower formation energies compared to Type II. The deep charge transition levels of these di-
vacancies defects locate about 2 eV. In addition, the [VO+VTi] complexes have relatively low
formation energies under O-rich conditions, and becomes lower than [VO+VZn] at CBM, seen in
Figure 4.5. However, under metal-rich condition, the [VO+VTi] complex exhibits high formation
energies, reaching 11.8 eV at its maximum, while the formation energies of [VO+VZn] showed
this pair is the second preferred defect within most part of the band gap, following the VO.
Moreover, all these defect complexes have charge transitions occurring inside the band gap and
do not give rise to shallow states.
4.6.3 Defects Binding Energy
The formation of defect pairs between negatively charged metal vacancies and positively
charged oxygen vacancies need to be considered because of the strong Coulombic interaction
between the point defects in ZnTiO3. The binding energy (Eb) of a certain defect pair provide the
information of defect pair preference. It can be described as:
𝐸𝐸𝐶𝐶 = 𝐸𝐸𝑒𝑒(𝑋𝑋) + 𝐸𝐸𝑒𝑒(𝑌𝑌) − 𝐸𝐸𝑒𝑒(𝑋𝑋𝑌𝑌)
where Ef(XY) is the formation energy of the defect pairs and Ef(X) and Ef(Y) are the formation
energies of a single defect. It is worth mentioning that the binding energy is independent of
chemical potentials. A positive binding energy represents a preference for the cluster over its
components[95], therefore the defect pair would readily form. Binding energies of 1.72 eV and
1.37 eV were calculated for type I [VO+VZn]0 and type II [VO+VZn]0, respectively. For the Ti-O
vacancy pairs, binding energies of 2.32 ev and1.37 ev were calculated for type I [VO+VTi]2- and
type II [VO+VTi]2-, respectively. Obviously, these results indicate that defect pairs will readily
51
form where possible, and type I [VO+VZn] and [VO+VTi] di-vacancies are energetically preferred
over type II pairs.
4.6.4 Transition Level
The equilibrium defect transition levels, presented in a band scheme and depicted in
Figure 4.6, can be calculated from the formation energies according to Equation (5). As
vacancies occur in their nominal charge states (VTi4-, VZn
2-, and VO2+) almost over the entire band
gap, only the band edges are shown.
Figure 4.6. Transition levels for mono- and di-vacancies in ZnTiO3
52
It can be seen that charge transition of VO vacancies take place close to the conduction
band, which is in good agreement with the density function theory study of hexagonal ZnTiO3
using a LDA method.[95] While for binary ZnO, a transition from 2+ to 0 of VO is reported at
1.4 eV (below CBM of ZnO), representing a deep double donor[100]. The +1/0 transition of
[VZn-VO] takes place at 1.2 eV above the VBM while the 0/- transition of this complex occurs
close to the CBM. For a [VTi-VO] complex, the -2/-3 transition occurs near the edge of CBM and
the subsequent -2/-1, -1/0, and 0/+1transitions occurs far from the CBM. Those two mono-
vacancies with transitions close to the edge of CBM denotes to shallow donor and could
contribute to n-type conductivity in hexagonal ZnTiO3 system. This result is in agreement with
the BaTiO3 system in which the transition level of VO and [VTi-VO] with same charges appear at
similar locations within the band structure.
4.7 Conclusions
Density functional theory calculations have been used to investigate the point defects in
Wurtzite ZnO and ilmenite hexagonal ZnTiO3. For ZnO, we suggest a 3×3×4 supercell for the
complex defect calculations, such as [XZn − 2VZn] complexes, according to the test of size effect
on suppercells. The results show that the formation energies of [SbZn − 2VZn]′ complex and the
Zni defects are consistent with those from the literature (Table 4.4). In addition, both formation
energies and ionization energies indicate that these defect complexes may be responsible for the
p-type behavior, which is observed in the annealed films conducted experimentally.
As for ZnTiO3, the intrinsic defects in hexagonal ilmenite ZnTiO3 were successfully studied
by using density functional theory. All the three mono-vacancies have been taken into
consideration under a range of possible charges. In addition, metal oxygen di-vacancies, as well
53
as ZnTi and TiZn di-antisites have also been considered in this work. Formation and binding
energies were calculated within the DFT method, in which the defect formation energies were
derived using the Zhang-Northrup formalism[92] in a range between Valance Band Maximum
(VBM) and Conduction Band Minimum (CBM). A band gap correction method[91] was adopted
in this work in order to correct the band structure by rigidly shifting the valence band and
conduction band with respect to each other.
The oxygen vacancy is proved to be the dominant defect under metal-rich conditions
within the whole range of band gap, while for O-rich conditions. The negative binding energies
of di-vacancies represent the Zn/Ti and O vacancies are bound in di-vacancy clusters under all
three conditions.
54
CHAPTER 5
SURFACE ENERGY AND WORKFUNCTION STUDY ON ZINC OXIDE*
5.1 Abstract
ZnO has been actively studied for potential usage as a transparent conducting oxide (TCO)
for a variety of applications including organic light emitting diodes and solar cells. In these
applications, fine-tuning the workfunction of ZnO is critical for controlling interfacial barriers and
improving the charge injection (or outcoupling) efficiencies. We have performed plane wave
periodic Density Functional Theory (DFT) calculations to investigate the effect of different surface
absorbents and surface defects (including surface non-stoichiometry) on the workfunction of ZnO.
The aim was to understand the underlying mechanism of workfunction changes, in order to
engineer specific workfunction modifications. Accurate calculations of workfuncitons of polar
surfaces were achieved by introducing balancing pseudo charges on one side of the surface to
remove the dipolar effect. It was found that increasing the surface coverage of hydrocarbons (-
CH3) decreased the workfunction, while adsorption of highly electronegative -F and -CF3 groups
and increases in surface O/Zn ratio increased the workfunction of ZnO. The increase of
workfunction was found to be directly correlated to the enhancement variation of surface dipole
moment due to adsorptions or other surface modifications. Introducing surface absorbents that
increase surface dipole moment can be an effective way to increase workfunction in ZnO TCOs.
5.2 Introduction
Zinc oxide is a wide band gap (3.27eV) semiconductor with potential applications in
* This chapter is reproduced from Wei Sun, Yun Li, Jitendra Kumar Jha, Nigel D Shepherd, Jincheng Du “Effect of surface adsorption and non-stoichiometry on the workfunction of ZnO surfaces: A first principles study" J. Appl. Phys. 117, 165304 (2015) with permission from AIP Scitation
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optoelectronic and microelectronic devices such as short wavelength light-emitting diodes
(LED)[109] and transparent thin-film transistors[110], [111]. ZnO in its intrinsic or doped forms
can have high electrical conductivity, which combined with its high optical transparency makes it
a potential alternative to commonly used indium-tin-oxide (ITO)[112], in applications such as
solar cells and organic light emitting diodes (OLEDs)[113], [114]. The need for ITO alternatives
is driven by the Earth’s limited reserve of indium and commercial considerations (ZnO is much
cheaper than ITO). In addition to high electrical conductivity, the TCO anode in OLEDs must
possess a suitably large workfunction to minimize the interfacial barrier for hole injection. This in
turn facilitates lower threshold voltages and higher device power efficiency. Fundamental
understanding of the effect of surface orientation, defects, nonstoichiometry, surface absorbents
and contamination on ZnO workfunction is therefore critical to exploit its potential as a TCO.
Workfunction is sensitive to surface conditions including atomic arrangement, crystal
orientation and termination, and contamination by adventitious hydrocarbon, hydroxyl
species[115], [116] and other surface contamination[117]. This sensitivity therefore provides
pathways to “engineer” specific workfunction changes. Theoretically, the contributions to
workfunction are twofold[118], [119] as shown in equation (1). The first component is the bulk
chemical potential (μ) measured with respect to a suitable reference potential or zero, and the
second component is a surface dipole barrier (𝐷𝐷), which depends on the surface charge distribution.
Both the bulk and surface contributions are measured with respect to the same reference potential.
𝛷𝛷 = 𝐷𝐷 − 𝜇𝜇 (5.1)
For metals, the surface dipole barrier plays a more significant role of those two components[120].
It was reported that the workfunction change of W surfaces is dominated by the surface[121] dipole
moment change, and that adsorbate-induced changes in dipole density and workfunction follows
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a linear relationship. It was also proven that when the surface was covered by adsorbates with
higher electronegativity than the terminating atoms, an increase in workfunction was obtained;
conversely, workfunction decreased when the surface species had lower electronegtivity[122].
Unlike metals, ionic crystal surfaces exhibit more complicated atomic arrangements and properties
due to the variation of surface dipole moment, surface constructions, and orientations. In addition,
the intrinsic dipole moment existing in the bulk due to the alternating positively and negatively
charged layers cannot be ignored. Tasker classified ionic crystal surfaces into three types[123].
Type I and II comprised of neutral and charged planes, respectively, with no net dipole moment
along the surface normal. Type III surfaces not only consist of charged planes, but also have a net
dipole moment along surface normal. The intrinsic dipole moment generated by cations and anions
of ionic crystal surfaces results in more complicated behavior related to surface contamination and
reconstruction.
Wurtzite ZnO exists with three major thermodynamically stable surfaces: type I non-
polar(1 0 1� 0), type III polar Zn terminated (0 0 0 1) and O terminated (0 0 0 1�) surfaces. Kuo
et al[124] reported that surface treatments such as oxygen plasma, leading to a change of surface
chemistry and stoichiometry and possible surface relaxation, results in workfunction increases. M.
Breedon et al[117] have studied the adsorption of nitrogen and oxygen onto ZnO surfaces, and
found that the workfunction increases when the adsorbate accepts electrons from the surface, and
a decrease of workfunction resulted from a donation of electrons from adsorbate to the surface. In
this study, first principle calculations based on density functional theory (DFT) were performed to
investigate the surface relaxation, as well as the mechanisms of workfunction change on non-
polar (1 0 1� 0), polar (0 0 0 1) and polar (0 0 0 1�) surfaces of wurtzite ZnO induced by adsorbed
methyl groups, as well as more electronegative trifluoromethyl (-CF3) groups and fluorine.
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5.3 Simulation Detail
Vienna ab inito package (VASP) was used to carry out density functional theory
calculations. The exchange correlation function is treated within the generalized gradient
approximation (GGA) with the PBE form, and the electronic structure such as the interaction
between ions and electrons is described by projected augmented wave pseudopotentials. The
electronic wave functions are expanded in a plane wave up to a kinetic energy cutoff of 400 eV.
The first Brillouin zone was sampled using a 6×6×3 Monkhorst-Pack k-point mesh for bulk
properties calculations, 6×4×1 for calculations of (1 0 1� 0) surface models, and 4×4×1 for the
(0 0 0 1) and (0 0 0 1�) surface models. For the purpose of investigating the effect of surface
modification on the workfunction, adsorbates or vancancies on the surface, dipole correction were
introduced and the dipole moment was calculated.
Slab models with 3-D periodic boundary conditions were employed for the surface energy
and workfunction calculations. A 15Å vacuum separation was introduced to avoid interactions
between two adjacent layers and the number of atomic layers used for each surface was increased
for each type of surfaces until the surface energy was converged. Methyl, trifluoromethyl, and
fluorine were used as surface hydrocarbon and electronegative contamination. They were attached
to the surface oxygen or zinc atoms on relaxed structures with different surface coverage, and
subsequently geometry optimization was performed. To understand workfunction changes related
to non-stoichiometric surface effects that could result from surface Ar sputtering or oxygen plasma
treatments, oxygen or zinc deficient surfaces were generated.
For workfunction calculations of asymmetric surfaces, due to the applied periodic
boundary condition, the electrostatic potentials differ at two sides of slab. A dipole correction
introduced by Neugebauer and Scheffler[125] was added to compensate the unbalanced
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electrostatic potential[126]. For surfaces with permanent dipole moments, such as ZnO
(0 0 0 1) and (0 0 0 1�) surfaces, hydrogen pseudo atoms were added to saturate the dangling
bonds of atoms at the bottom of the surface slab to balance the permanent dipole moment and
mimic the bonding in the bulk. This method was used to obtain accurate absorption energies on
polar surfaces of ZnO[127], and we found it to be effective to enable accurate calculations of
workfunction of polar surfaces.
5.4 Geometric Optimization
In order to check the reliability of the computational methods, we calculated total energies
as a function of the volume around the experimental volume. At each volume the cell shape and
atom positions were allowed to relax while the cell volume was maintained constant. The obtained
total energy versus volume curve was fitted to the Birtch-Murnaghan equation of state, based on
which the equilibrium volume was obtained, fitted energy versus volume curve is shown in Figure
5.1. The obtained equilibrium lattice parameters and those from earlier studies and experimental
values are compared in Table 5.1. The c/a ratio and c parameter results agree well with the
experimental c/a ratio, and are consistent with earlier GGA calculations[101]. The calculated c
parameter is slightly higher than the experimental values[102], consistent with the results from
literature: the GGA functional slightly overestimates the c parameter and the cell volume, while
the LAD functional does the opposite.
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Figure 5.1. Energy versus Volume curve for Wurtzite ZnO
Table 5.1. Calculated structural parameters for Wurtzite ZnO
Wurtzite ZnO c/a V (Å3)
Previous GGA17 1.61 49.19
Previous LDA17 1.62 45.01
This work 1.60 49.79
Experiment18 1.60 47.56
5.5 Surface Relaxation and Energy Calculations
5.5.1 Surface Relaxation of Non-Polar (𝟏𝟏 𝟎𝟎 𝟏𝟏� 𝟎𝟎) Surface
Wurtzite ZnO has three major planes with low Miller indices: non-polar (1 0 1� 0), polar
(0 0 0 1) with Zn termination, and polar (0 0 0 1�) with O termination, shown in Figure 5.2. The
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slabs for all Miller faces were cleaved from the fully relaxed bulk structure. After initial cleavage
of surfaces, a vacuum slab of 15 Angstrom is inserted between the top and bottom surfaces. The
vacuum slab is large enough to prevent interactions between the two surfaces. The atom positions
of the created surface were then allowed to fully relax until the convergence criterion is satisfied,
while the cell shape and volume remains fixed in the process. For the (1 0 1� 0) sufaces, a slab with
four double layers (8 sub-layers) was used, as is shown in Figure 5.3. The change of atom
arrangements is relatively small after relaxation as shown in Fig. 5.3b. Using the Tasker notation
for ionic surfaces[123], the (1 0 1� 0) slab before relaxation can be classified as a Type I surface
since each layer is charge neutral and zero dipole moment exists. However, after relaxation a finite
dipole moment emerges on the (1 0 1� 0) surface as a result of surface reconstruction. Figure 5.3b
shows the relaxed (1 0 1� 0) surface of a slab with four layers, in which the O and Zn atoms are no
longer in the same line after relaxation.
We calculated the relaxation displacements in the z direction of atoms in each sub-layer
marked in Figure 5.3, and present the results in Table 5.2. The positive and negative values in
angstroms in this table represent movement inward or outward with respect to the surface plane of
atoms (dotted line in Figure 5.3), respectively. The surface tends to relax to decrease its area and
the number of dangling bonds to minimize the total surface energy. It can be seen from Table 5.2
that the atom designated as Zn in the first layer moved downward into the bulk, leading to a change
in the Zn-O bond length and misalignment of Zn and O atoms in the direction perpendicular to the
surface. The Zn-O bond lengths in the first layer decreased from around 2.0 to 1.87 Å after
relaxation. An angle of 10.9° formed between O and Zn in the in-plane direction, compared to 0°
before relaxation. This is in good agreement with the findings of Meyer and Marx[36]. Both
surface Zn atoms moved inward after relaxation, while the Zn atoms in layer two and seven moved
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about 0.14 Å more inward compared to the surface Zn atoms. The displacements of O atoms are
much smaller compared to Zn, and decreased with increasing slab thickness. In general, the
relaxation resulted in a decrease of spacing between layers, and also the formation of a dipole
moment directed from O atoms to inward moving Zn atoms i.e., along direction perpendicular to
the surface. This surface reconstruction after relaxation indicates that polarization may exist even
on the non-polarized ZnO cleavage planes.
Figure 5.2. Schematic structure of Wurtzite ZnO. Red large ball: O; Grey small ball: Zn
Figure 5.3. (a) Unrelaxed and (b) relaxed (1 0 1� 0) structure of the four double-layer surface model of ZnO. (c) First double-layer of (1 0 1� 0) structure, the arrows denote the available
adsorption sites on the surface. Red small ball: O; Grey large ball: Zn
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Table 5.2. Summary of atom relaxation perpendicular to the surface for ZnO (1 0 1� 0) surfaces (atom numbers shown in Figure 5.3)
Displacement (Å) Displacement (Å) Zn1 -0.35 O1 0 Zn2 0.12 O2 -0.03 Zn3 -0.07 O3 0.01 Zn4 0.04 O4 -0.02 Zn5 -0.04 O5 0.02 Zn6 0.08 O6 -0.01 Zn7 -0.12 O7 0.03 Zn8 0.35 O8 0
5.5.2 Surface Relaxation of Polar (𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏)/(𝟎𝟎 𝟎𝟎 𝟎𝟎 𝟏𝟏�) Surface
Differing from (1 0 1� 0) surface, the (0 0 0 1)/(0 0 0 1�) surfaces have dipole moments
due to the asymmetric arrangement, which experienced substantial rearrangements after
relaxation. As shown in Figure 5.4a, a five double layer slab model with Zn terminated (0 0 0 1)
on one side and O terminated (0 0 0 1�) on the other side was created by cleaving a relaxed wurtzite
ZnO crystal along the z direction. Dipole moments perpendicular to the surface with the direction
from O sub-layer to Zn atom sub-layer exist for every Zn-O double layer, consistent with the III
surface classification[123]. As a result, the net dipole moment increases with the slab thickness
and the polar surfaces are unstable[36]. Surface reconstruction after relaxation leads to a decrease
of the dipole moment and thus more thermodynamically stable structures. This change can be
clearly seen in Figure 5.4b. The two separate zinc and oxygen sub-layers merged together into one
layer consisting both Zn and O atoms, reducing the polarity of the surfaces. To be more specific,
the calculated displacements of Zn and O atoms marked in Figure 5.4e are shown in Table 5.3. the
zinc atoms from the first sub-layer move downward along the z direction, while the oxygen atoms
from the second sub-layer (same double layer with first sub-layer) move upward and as a result
the Zn and O atoms in the relaxed structure are located in same plane. Figure 5.4e shows the
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relaxed structure with dipole correction included during DFT calculations. However, the resulting
surface reconstruction is very similar to that without dipole correction (Figure 5.4b), indicating
that dipole correction does not have a strong effect on surface relaxation. It will be shown in that
the introduction of pseudo-hydrogen on the bottom surface can lead to correct surface structure
during relaxation for polar surfaces (Figure 5.4d)
Figure 5.4. (a) Unrelaxed (0 0 0 1�)/(0 0 0 1) surface structure of the five double-layer surface
model of ZnO. (b) Relaxed structure without dipole correction. (c) Relaxed structure with dipole correction. (d) Relaxed structure with pseudo hydrogen and dipole correction. (e) First double-
layer of (0 0 0 1�)/(0 0 0 1) tructure, the arrows denote the available adsorption sites on the surface. Red small ball: O; Grey large ball: Zn
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Table 5.3. Summary of relaxation (without) perpendicular to the surface of ZnO (0 0 0 1�)/(0 0 0 1) surfaces
Displacement (Å) Displacement (Å) Zn1 -0.61 O1 0.05 Zn2 -0.40 O2 0.20 Zn3 -0.29 O3 0.32 Zn4 -0.19 O4 0.43 Zn5 -0.04 O5 0.52
DFT calculations of polar metal oxide surfaces are usually complicated by long-range
electrostatic and dipolar interactions of the surface layers. Our relaxed (0 0 0 1)/(0 0 0 1�) surface
slabs, either with or without dipole correction, showed a rearrangement of the first layer from a
double Zn-O layer to a non-polar flat layer with Zn and O atoms relaxed into the same plane to
lower the surface dipole moment.
Workfunctions is usually calculated by averaging the electrostatic potential along the z-
direction. The large surface reconstruction for the polar surfaces can significantly modify the
electrostatic potential and thus change the calculated workfunction. It has been proven
experimentally that ZnO crystal can be grown along [0001] direction and exists with polarized
termination[128]. Based on these considerations, we employed a compensation method by
saturating the surface O/Zn atoms with pseudo-hydrogen atoms at the bottom of the slab to ensure
accurate calculation of the workfunction[129], [130]. For each O atom at the bottom of the Zn-
terminated slab, a pseudo-hydrogen atom with a positive charge of +0.5|e| acting as ¼ Zn saturated
the dangling bond. Similarly, for each Zn atom at the bottom of O-terminated slab, a pseudo-
hydrogen with a negative charge of -0.5|e| acting as ¼ O saturated the dangling bond. As a result,
the internal net dipole moment in the slab was compensated. Figure 5.5a shows the electrostatic
potential decrease along the z-direction and that the electrostatic potential relative to vacuum is
also slanted. These variations of the electrostatic potential of both the slab and vacuum layer make
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it difficult to determine the workfunction of the surface. By introducing the compensation pseudo-
atoms, as shown in Figure 5.5b and 5.5c, the curve of calculated electrostatic potential becomes
flat during the vacuum region which indicates that the workfunctions of the top and bottom layers
of the two given slabs remain at same level after compensation. Figure 5.5d shows the
displacements along the z direction of surface Zn and O atoms for an as-cleaved surface (solid
lines), and pseudo-hydrogen saturated surfaces (dot lines). Clearly, the compensated surface has a
much lower level of reconstruction as reflected by the smaller displacements of the Zn and O
atoms.
Figure 5.5. Electrostatic potential of z direction for polar (0 0 0 1�)/(0 0 0 1) slabs, (a) as-cleaved without pseudo hydrogen correction; (b) Zn-terminated with pseudo hydrogen correction on dangling oxygen atoms at the bottom of the slab (see inset), with (dashed) and without (solid) dipole correction; and (c) O-terminated with pseudo hydrogen correction on dangling Zn atoms
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at the bottom of the slab (see inset), with (dashed) and without (solid) dipole correction; (d), calculated quantitative displacements along z direction of surface Zn and O atoms as a function
of number of sub-layers for (0 0 0 1�)/(0 0 0 1) slabs
Therefore, introduction of pseudo-hydrogen to the bottom surface to balance the dipoles
result in much improved vacuum electron static potential (close to horizontal), due to minimized
surface dipoles. This is shown in a relatively small change of electrostatic potential after dipole
correction (Figure 5.5b). Additionally, inclusion of pseudo-hydrogen enables the polar surface to
relax to reasonable surface structures (Figure 5.4d). Hence combining pseudo-hydrogen during
relaxation and dipole correction gave the most accurate workfunction calculations of polar surfaces
with or without surface adsorption. The following workfunction calculations performed in this
work for (0 0 0 1)/(0 0 0 1�) surfaces include both with compensating pseudo-hydrogen atoms
and dipole correction.
5.5.3 Surface Energy Calculations
For semiconductors with contacting surfaces, stability is represented by surface energy,
Esurf, which is defined as
𝐸𝐸𝐽𝐽𝐶𝐶𝑟𝑟𝑒𝑒 = 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡−𝐸𝐸𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝐿𝐿𝐿𝐿
(5.2)
where Etot is the total energy of the slab model, Ebulk is energy of the bulk ZnO containing the same
number of molecular units as the slab, A is the total surface area, and n is the number of fully
relaxed surfaces. For a surface slab that has bottom fraction to be frozen to represent the bulk, n
equals 1 and for a surface slab with two freely relaxing surfaces n equals 2. For polar surfaces,
dipole corrections are included in the surface energy calculations to minimize the effect of surface
dipoles on the convergence of total energy as a function of system size.
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Convergence tests of surface energy as a function of the number of layers in the model were
performed, and it was found that four double layer slab models for nonpolar surfaces and five
double layer slab models for polar surfaces were sufficient to obtain converged energy values. The
surface energy of (1 0 1� 0) is found to be 0.86 J/m2, 0.27 lower than that of the (0 0 0 1)/(0 0 0 1�)
surfaces. The lower surface energy of the (1 0 1� 0) surface can be explained by fewer broken
bonds and relatively smaller surface rearrangement of the (1 0 1� 0) surface compared the
(0 0 0 1)/(0 0 0 1�) surfaces. This result is in agreement with experimental observations that the
(1 0 1� 0) surface is commonly found in samples with slower cooling rate during preparation hence
from more thermodynamically stable surfaces[131], [132]
5.6 Effect of Surface Adsorption on the Workfunction of ZnO Surfaces
The work function is defined as the minimum energy required removing an electron from
the bulk of a material through a surface to a point in vacuum immediately outside the surface. In
the calculation of workfunction, the surface is assumed to be in its ground state both before and
after removal of the electron[133]. At 0K and a perfect vacuum, the workfunction is defined as the
energy difference between the Fermi level and the vacuum level, which can be written as
𝑊𝑊𝑒𝑒 = 𝐸𝐸𝑣𝑣 − 𝐸𝐸𝑒𝑒 (5.3)
Figure 5.6a and b exhibit the schematic energy diagram of symmetric and asymmetric surface slab
models, respectively. For the symmetric surface, the vacuum level (Ev) is defined as the converged
electrostatic poetential outside the slab surface. Due to the periodic condition, the vacuum potential
is taken as the halfway point of the vacuum slab above the surface where the influence from the
adjacent cell is minimum. The Fermi energy, Ef, refers to the energy of the highest occupied
electronic state of the system. In Figure 5.6b, the electrostatic potential of asymmetric surface
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demostatrates a “potential jump” in vacuum region with the dipole correction applied, representing
the workfunction of the upper and lower surfaces[125]. This “potential jump” cancels the artificial
dipole moment in the vacuum region caused by periodic boundary conditions[126], [134].
Figure 5.6. Diagram of electrostatic potential for ZnO (1 0 1� 0) surfaces, a, symmetric slabs; b, asymmetric slabs with F adsorption. Φ is workfunction, Φ1 and Φ2 are workfunctions of top and
bottom surfaces.
The workfunction is calculated by using the slab supercell approach that we used for
surface energy calculations. Extremely clean surfaces are difficult to realize experimentally as
samples become contaminated when in contact with low vacuum or atmospheric conditions. The
chemisorbed and physisorbed contaminates affect the surface structure and electronic properties,
including workfunction. In this study, methyl groups and trifluoromethyl groups were chosen to
represent electron neutral and electron negative groups, respectively. They were studied on both
non-polar and polar surfaces.
The adsorption energy of the reaction intermediates on the ZnO surfaces is defined as
𝐸𝐸𝐿𝐿𝐿𝐿𝐽𝐽 = 𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑎𝑎𝑏𝑏𝑎𝑎𝑡𝑡𝑒𝑒+𝑎𝑎𝑏𝑏𝑎𝑎𝑏𝑏−𝑁𝑁×𝐸𝐸𝑎𝑎𝑎𝑎𝑎𝑎𝑡𝑡𝑎𝑎𝑏𝑏𝑎𝑎𝑡𝑡𝑒𝑒−𝐸𝐸𝑎𝑎𝑏𝑏𝑎𝑎𝑏𝑏𝑁𝑁
(5.4)
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where 𝐸𝐸𝐿𝐿𝐿𝐿𝐽𝐽𝐶𝐶𝑟𝑟𝐶𝐶𝐿𝐿𝑒𝑒𝐿𝐿+𝐽𝐽𝐶𝐶𝐿𝐿𝐶𝐶 refers to the total energy of the adsorbate with the ZnO surface slabs; 𝐸𝐸𝐽𝐽𝐶𝐶𝐿𝐿𝐶𝐶
is the total energy of the corresponding contaminant free slabs, N is the number of molecules on
the surface, and 𝐸𝐸𝐿𝐿𝐿𝐿𝐽𝐽𝐶𝐶𝑟𝑟𝐶𝐶𝐿𝐿𝑒𝑒𝐿𝐿 is the total energy of the adsorbates, -CH3/-CF3 in this work, in vacuum.
For adsorption on the (1 0 1� 0) surfaces, the C atoms in methyl and trifluoromethyl groups
can adsorb onto O sites to form O-C bonds, or onto Zn sites to form Zn-C bonds. There are four
available sites on both polar and non-polar surfaces that are available for the adsorption of methyl
and trifluoromethyl groups. For the 50% adsorption (half monolayer) situation, there are two
combination of adsorption sites: diagonal or on one side for the two -CH3 or -CF3 groups, which
results in different adsorption energies. Due to the large molecular size of the –CF3 groups, surface
coverage over 50% became unstable and led to dissociation of the –CF3 structures. Hence
adsorption energy and workfunction were only calculated up to 50% coverage for –CF3 adsorption.
5.6.1 Effect of Methyl Adsorption on Workfunction
Table 5.4 summarizes the calculated adsorption energies, as well as the workfunctions of
all possible adsorption configurations for -CH3 adsorbates to represent hydrocarbon contaminants
on ZnO surfaces. The term monolayer (ML) used to describe the surface coverage is defined as
the number of -CH3 groups per surface Zn or O site. The possible surface coverages are ¼, ½, ¾
and 1. The adsorption energy of unit -CH3 decreases as the surface coverage increases for all
surface configurations (Table 5.4). For the non-polar surface, methyl groups adsorbed onto O sites
by forming C-O bonds were found to have 0.2 to 1 eV lower adsorption energy than those adsorbed
onto Zn sites. Thus oxygen sites are considered to be the preferred adsorption sites of methyl
groups and likely other hydrocarbons. A similar trend was observed for polar surfaces, where the
adsorption energy per methyl group is 0.5 to 1 eV lower for adsorption onto oxygen sites than on
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zinc site on the surfaces and the adsorption energy per methyl group also increases with increasing
methyl coverage. The adsorption energies on polar surfaces are considerably (0.6 to 2 eV for each
type of adsorption site) lower than non-polar surfaces. For example, the adsorption energy on
oxygen site of ¼ coverage is -1.91 eV for the (1 0 1� 0) surface while that for the (0 0 0 1) surface
is -3.96 eV, about 2 eV lower than the non-polar surface, suggesting the adsorption on polar surface
is stronger.
Table 5.4. Calculated adsorption energy and workfunction due to -CH3 adsorption
Surface coverage
Adsorption energy (eV/-CH3) Work Function (eV)
(1 0 1� 0) (0 0 0 1�)/(0 0 01) (1 0 1� 0) (0 0 0 1�)
/(0 0 01) Zn-C O-C Zn-C O-C Zn-C O-C Zn-C O-C
Clean N/A N/A N/A N/A 5.81 3.25 6.96 0.25 -0.96 -1.91 -2.88 -3.96 5.67 3.98 3.32 5.68 0.5 -0.95 -1.67 -2.68 -3.64 5.56 3.59 6.08 3.72 0.75 -0.85 -1.36 -1.91 -2.70 5.52 3.50 6.18 1.79 1.0 -0.84 -1.18 -1.43 -2.06 5.41 2.52 6.20 1.94
The workfunctions of methyl covered ZnO surfaces are shown in Figure 5.7. For the non-
polar (1 0 1� 0) surface, the workfunction decreases considerably with methyl adsorption. There is
significant decrease in workfunction when adsorption is on an oxygen site. Workfunction
experience initial rapid decrease from 5.81 eV for the pristine surface to 3.98 eV for ¼ coverage
to 3.59 eV with ½ coverage. Further increases in coverage do not considerably change the work
function. The workfunction change shows a similar trend as the adsorption energy as a function of
coverage on the nonpolar surface, suggesting higher adsorption energy corresponds to higher
workfunction[124]. In the case of adsorption on zinc sites, workfunction does not change much
with surface coverage. As the adsorption energy is much lower for methyl adsorbed onto oxygen,
the trend associated with O sites is more relevant to observed workfunction changes. This result is
in good agreement with experimental observation that shows ZnO workfunction decreased when
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the surface was contaminated with hydrocarbon. The workfunction can be increased by oxygen
plasma treatment to remove the hydrocarbon contaminants[124]. We will show later that -CH3
adsorption generates a surface dipole moment which is opposite in direction to the intrinsic surface
dipole moment, resulting in the reduction of the net surface dipole moment and workfunction.
Compared to adsorption on oxygen sites, the workfunction change with coverage showed
a very different behavior for methyl adsorption on zinc sites. Workfunctions remained almost
constant with a slight decrease for the (1 0 1� 0) surface. However, for the polar (0 0 0 1) surfaces,
the workfunction increased with increasing surface coverage on Zn sites. This difference can be
explained by the direction of induced dipole moment: carbon has a higher electronegativity than
zinc, and the induced dipole moment is opposite to the case when adsorption is on an oxygen site.
This enhances the original surface dipole moment and leads to an increase of workfunction.
Figure 5.7. Workfunctions as a function of surface -CH3 coverage. a, (1 0 1� 0), surface; b,
(0 0 0 1�)/(0 0 0 1) surface
5.6.2 Effect of –CF3 and –F Adsorptions on Workfunction
F has the highest electronegativity of 3.98 among all elements, larger than that of C (2.55),
Zn (1.65), and O (3.44) atoms. Consequently, trifluoromethyl or fluorine is expected to induce a
much larger dipole moment compared to methyl groups. The focus of this section is to understand
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the effect of these functional groups with large induced surface dipole moment caused by –CF3
and –F adsorption on the adsorption energy, surface relaxation and the workfunction of ZnO
surfaces.
The calculated adsorption energies and workfunction of trifluoromethyl covered ZnO
surfaces are listed in Table 5.5. We only reported the results for coverage up to 0.5 ML as higher
coverage caused dissociation of –CF3 groups possibly due to large electrostatic interactions and
the steric effect of the adsorbed groups. It is interesting to point out that for –CH3 adsorption even
with 1ML (100%) coverage the adsorbed molecules were stable on ZnO surfaces. Similar to
methyl group adsorption, the adsorption energy is lower on oxygen sites than on zinc sites for both
polar and non-polar surfaces. This is not surprising as both –CF3 and –CH3 groups bond to ZnO
surface through the carbon atom, which is electropositive in nature and prefers to bond to the
electronegative oxygen site rather than electropositive zinc site on ZnO surfaces.
The workfunction as a function of surface coverage with –CF3 adsorption are summarized
in Figure 5.8 and show interesting trends. On (1 0 1� 0) surfaces, the workfunction change
associated with Zn and O bonding sites show opposite trends as a function of coverage, increasing
with coverage on Zn sites and doing the opposite for O sites. For (0 0 0 1)/(0 0 0 1�) surfaces, an
increase of workfunction was observed for bonding with Zn site while the workfunction maintain
essentially constant for adsorption on O sites.
Table 5.5. Calculated adsorption energy and workfunction due to -CF3 adsorption
Surface coverage
Adsorption energy (eV/-CF3) Work Function (eV)
(1 0 1� 0) (0 0 0 1�)/(0 0 01) (1 0 1� 0) (0 0 0 1�)
/(0 0 01) Zn-C O-C Zn-C O-C Zn-C O-C Zn-C O-C
Clean N/A N/A N/A N/A 5.81 3.25 6.96 0.25 -0.85 -2.42 -3.08 -4.23 6.75 4.19 8.75 7.32 0.5 -0.81 -2.15 -2.73 -3.08 7.12 4.01 8.17 6.39
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Table 5.6. Calculated Adsorption energy and workfunction due to –F adsorption Surface
Coverage Work Function (eV)
(1 0 1� 0)/Zn-F (0 0 0 1)/Zn-F Clean 5.82 3.21 0.25 7.36 5.50 0.5 8.41 9.88 0.75 8.55 10.42 1.0 8.94 12.98
Figure 5.8. Workfunction change as a function of surface -CF3 coverage. a, (1 0 1� 0) surface; b,
(0 0 0 1�)/(0 0 0 1) surface
Figure 5.9. Workfunctions at different surface F coverage for (1 0 1� 0) (black) and (0 0 0 1)
(red) surfaces; F atoms bonded with surface Zn atoms
Fluorine atoms represent highly electronegative adsorbates, and their effect on the
workfunction of ZnO was studied. Only adsorption of fluorine on zinc sites i.e. Zn-F bonding
74
structures were studied because a highly electronegative F atom would not favor bonding to
surface oxygen atoms that are already negatively charged. In addition, Fig. 5.12 clearly shows
strong charge transfer between F and Zn atom but not with oxygen on the surface. The
workfunctions for F on the two surfaces corresponding to different concentrations are shown Table
5.8 and summarized in Figure 5.9. On both polar and non-polar surfaces, F adsorption leads to an
increase of workfunction. The increase of workfunction is much higher for polar (0001) surfaces
than non-polar(1 0 1� 0) surfaces. For example, for 1ML coverage the increase of workfunction is
more than 9 eV for the (0001) surface while it is only about 3 eV for the (1 0 1� 0) surface. These
results clearly show the distinctive impact of surface adsorbates on workfunction depending on the
surface polarity, and suggest that the direction of intrinsic and induced surface dipole moment
influences workfunction. The result also suggest that large workfunction increases and more
generally changes can be achieved, but it requires careful matching of the surface with specific
adsorbates.
5.6.3 Effect of Surface Non-Stoichiometry on Workfunction
It has been shown experimentally that an increase of workfunction can be achieved by
oxygen plasma treatment of ZnO surfaces[124]. Surface slabs with non-stoichiometry were
therefore constructed to study the effect of O/Zn ratio on the workfunction. Two possible scenarios
were taken into account: removing surface oxygen atoms to form zinc rich surfaces (ZnO1-x) and
removing zinc atoms to form oxygen rich surfaces (ZnO1+x). Figure 5.10 shows the calculated
work function with respect to the O/Zn ratio. The workfunction increases with increasing oxygen
to zinc ratio, suggesting that surface treatments that increase the oxygen concentration can increase
the workfunction which is supported by the experimental data[124] The underlying mechanism
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can be explained by enhanced surface dipole moments which will be detailed in the discussion
section. In general, a higher surface O/Zn ratio results in a more negative (1 0 1� 0) surface. This
can be due to the induced dipole moment that enhances the inherent surface dipole moment and
hence increases the workfunction (explained in detail in the discussion section).
Figure 5.10. Work function of the (1 0 1� 0) surface with different surface O/Zn ratios for oxygen
deficient (a) and zinc deficient (b) surfaces
5.7 Dipole Moment and Workfuntion
All of the results we obtained so far suggest that the change of workfunction is associated
with the change of surface dipole moment induced by a change in chemical makeup of the surface.
Accurate calculation of surface dipole moments was therefore needed to determine if there was a
direct correlation of the two. The density of dipole moment p along the surface normal (z) direction
can be defined as[121], [135]:
𝐴𝐴 = ∫ 𝒛𝒛 ∙ 𝜌𝜌𝑖𝑖𝐶𝐶𝐿𝐿𝐽𝐽+𝐿𝐿𝐶𝐶𝐿𝐿𝐵𝐵𝑒𝑒𝑟𝑟𝐶𝐶𝐿𝐿𝐽𝐽(𝒛𝒛)𝑑𝑑𝒛𝒛𝐵𝐵/2𝑍𝑍0 (5.5)
where 𝜌𝜌𝑖𝑖𝐶𝐶𝐿𝐿𝐽𝐽+𝐿𝐿𝐶𝐶𝐿𝐿𝐵𝐵𝑒𝑒𝑟𝑟𝐶𝐶𝐿𝐿𝐽𝐽 is the total charge density of the XY plane, 𝑆𝑆0 refers to the center of the
surface slab to represent the bulk electron density, and c/2 refers to the center of the vacuum slab.
The relationships between calculated dipole moment and workfunction of non-polar
(1 0 1� 0) surfaces are shown in Figure 5.11. Here, Δϕ and Δp are work function change and surface
76
dipole density change with respect to those of a clean (1 0 1� 0) surface, respectively. Δp is the
change in surface dipole moment density, defined as the difference between the dipole density of
contaminated or non-stoichiometric surfaces and pristine surfaces. Positive values of Δp in Figure
5.11 indicate that the induced dipole has a direction perpendicular and pointing into the surface
while the negative values refer to dipole moment with direction pointing out of the surface. It was
found in Figure 5.11 that the change of work function exhibits a linear relationship with the change
of surface dipole moment. Lewng, Kao and Su[121] also found a linear relationship in their work
regarding the adsorption on surfaces of tungsten. However, their results showed a consistent slope
of 180.95, while in this work, according to Figure 5.11, the slopes are quite different. For methyl
group induced change, the line has a slope of 235.31, lower than the one in trifluoromethyl group
case (Figure 5.11a and b). For non-stoichiometric surfaces (Fig. 5.11c), the slope is 253.85. Ionic
crystals consist of planes with both anions and cations therefore their surfaces may have dipole
moments with a direction normal to surface may exist.
According to equation (5.5), the net dipole moment along z direction of the surface slab
mainly comes from the integral from center of the slab to the center point of vacuum, which
depends on the electrostatic potential at the two points. For metals, the starting point 𝑆𝑆𝐵𝐵𝐿𝐿𝐿𝐿𝑒𝑒𝐿𝐿𝑟𝑟 is
unrelated to the contamination induced change in surface dipole density since the internal
electrostatic potential remains unchanged after surface absorption[121].
As a typical type I surface, (1 0 1� 0) shows no net dipole moment perpendicular to the
surface as it consists of charge neutral planes and there is no net dipole moment. On the other hand,
due to the existence of induced dipole moment in each layer in ZnO (1 0 1� 0) surfaces, absorption
on the surface, including contamination and removal of surface atoms, has a significant impact on
the electrostatic potential in the slab. However, calculations of dipole moment using equation 5.5
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for type III (0 0 0 1)/ (0 0 0 1�) surfaces with net dipole moments and charged planes is unreliable.
This is because adsorption on the surface leads to significant changes of the internal and net dipole
moments, and eventually the total charge density. This effect is further amplified in the integral of
the charge density starting from the 𝑆𝑆𝐵𝐵𝐿𝐿𝐿𝐿𝑒𝑒𝐿𝐿𝑟𝑟, which leads to an unpredictable estimation of surface
dipole density and is therefore not applicable in the case of polar surfaces.
Figure 5.11. Change in work function as a function of change in dipole moment density of the (1 0 1� 0) surface. (a) induced by -CH3, (b) induced by -CF3, (c) induced by surface non-
stoichiometry
Surface adsorption induced dipole moments depend both on the adsorption species and the
surface atom to which the adsorbate bonds. The induced surface dipole moment either enhances
or diminishes the inherent dipole moment of the surface, and results in a change of the
78
workfunction. This correlation is clearly shown in Fig. 5.11. Alternatively, this can be understood
by the direction of induced surface dipole moment due to adsorption. Adsorption energy results
show that the -CH3 groups prefer bonding with surface O atoms, which leads to a dipole moment
pointing outward from the surface and opposite the inherent surface dipole moment. As a result, a
decrease in work function with the increase of surface -CH3 coverage was observed on (1 0 1� 0)
and (0 0 0 1�) surfaces. F atoms on the other hand has a high electronegativity, even higher than
that of O, hence F atoms prefer to bond with surface Zn atoms, as indicated from the adsorption
energies. As a result of Zn-F bonding, electrons are transferred to the F, and the dipole moment
due to F adsorption has a direction pointing inward to the surface, enhancing the inherent surface
dipole moment. As shown in Figure 8, the work function increases with the increase in adsorbed
concentration of F.
5.8 Charge Density Difference
The origin of surface dipole moment change is due to the surface electron density
redistribution after adsorption, which leads to bond formation between surface atoms and
adsorbates. Charge density difference is defined as
∆𝜌𝜌 = 𝜌𝜌𝑍𝑍𝐿𝐿𝑂𝑂+𝑋𝑋 − 𝜌𝜌𝑍𝑍𝐿𝐿𝑂𝑂 − 𝜌𝜌𝑋𝑋 (5.6)
where 𝜌𝜌𝑍𝑍𝐿𝐿𝑂𝑂+𝑋𝑋, 𝜌𝜌𝑍𝑍𝐿𝐿𝑂𝑂, and 𝜌𝜌𝑋𝑋 are charge density distributions of the surface plus adsorbates, a
pristine ZnO surface, and the adsorbates alone, respectively. Fig. 5.12 shows the charge density
difference plot of (1 0 1� 0) with 0.25 ML -CH3 and F as an example. These represent two typical
changes of workfunction : -CH3 adsorption leads to a decrease of workfunction and –F adsorption
an increase of workfunction. Strong electronic rearrangement can be observed in Figure 5.12
where the charge density difference of a planar cut through absorbates and corresponding
79
adsorption sites was plotted. In Figure 5.12a, the stronger electronegativity of fluorine leads to a
charge buildup region around it and a charge depletion around Zn. The Zn-F bond formation is
characterized by a large charge transfer from surface Zn to F yielding a dominant bond in F/ZnO
system and therefore promotes the increase of workfunction. In addition to the large
electronegativity value of F, the observed large charge transfer is also due to the under coordinated
nature of Zn on the surface. In the case of -CH3 adsorption, charge transfer is much smaller. When
surface oxygen atoms bond to carbon, certain charge transfer happens from the negatively charged
oxygen and the hydrogen atoms to carbon, which leads to a dipole moment pointing out of the
surface in a direction perpendicular to the surface. This is supported by the charge depletion region
observed around O atom in Figure 5.12b. There are also two charge accumulation regions observed
around C/H and C/O, which is due to covalent bond formation. In summary, methyl group
adsorption on surface O leads to a small charge buildup region which are surrounded by electron
depletion regions, resulting in a dipole moment pointing outside the surface and a decrease in
workfunction[136].
Figure 5.12. Cross section of surface schematic structures (top) and charge density difference Δρ(r) (bottom) at the (1 0 1� 0) surface with selected adsorbates, a, with F; b, with -CH3. Red
(Blue) region represents region of electron buildup (depletion)
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In general, the results of these calculations show that the variation of workfunction due to
surface adsorption on ZnO surfaces is closely related to the change of surface dipole moment,
which agrees well with earlier first principles calculations[124]. It is suggested that methods that
can modify surface dipole moment such as oxygen plasma treatments, Ar sputtering cleaning, or
selected surface adsorption the surface dipole moment can be modified and hence the workfunction
be change. For ZnO surfaces, addition of strongly electronegative species such as –F or –CF3 is an
effective way of increasing the workfunction.
5.9 Conclusion
First principles density functional theory (DFT) calculations have been performed to study
the change of workfunction of low energy polar and nonpolar surfaces of ZnO as a function of
surface adsorbents coverage including methyl, trifluoromethyl, and fluorine groups. The surface
structures were represented by slabs with both sides free to relax, and convergence tests were
performed to find the minimum number of layers needed by studying the surface energy as a
function of layers in the surface model. Surface relaxation was observed on both the pristine and
adsorbed surfaces and it was found that Zn atoms were found to move inward on non-polar
(1 0 1� 0) surfaces. Dipole correction was introduced in workfunction calculations in order to
cancel the unbalanced electrostatic potential between asymmetric slabs caused by periodic
boundary conditions. In addition, as for polar (0 0 0 1)/(0 0 0 1�) surfaces, pseudo hydrogen atoms
were added to saturate the dangling bonds on the one side of the slabs, which was found to be
successful in avoiding unreasonable surface reconstruction after relaxation. Hence a combination
of pseudo hydrogen introduction and dipole correction enable the accurate structure relaxation and
workfunction calculations of both polar oxides.
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The results show that adsorption of methyl (-CH3) groups on oxygen sites of ZnO surfaces
is energetically more favorable for both polar and non-polar surfaces. Methyl adsorption on
oxygen site leads to large monotonic workfunction decrease on both types of surfaces. The
decrease of workfunction is initially large (over 3 eV) for small coverages and eventually levels
off. For trifluoromethyl groups (-CF3), stable adsorption was only observed for coverage up to
0.5ML, with adsorption on oxygen sites again being more favorable on both types of surfaces.
Adsorption of –CF3 on oxygen sites was also energetically more favorable than on surface zinc
atoms. The adsorption trifluoromethyl groups on oxygen site leads to a decrease of workfunction
on non-polar surfaces but polar surface remains essentially unchanged. Fluorine was found to
stably bond only with surface Zn atoms due to its high electronegativity. Fluorine adsorption leads
to a large increase of workfunction for both polar and nonpolar surfaces. These results suggest
surface cleaning by removing hydrocarbons or surface modification by adsorption of
electronegative species can substantially increase the workfunction of ZnO.
The change of workfunction was found to correlate linearly with the surface dipole
moment. Increasing the surface dipole moments such as introducing large electronegative
adsorbents (such as fluorine) can increase the workfunction. Charge density plots of adsorbed
surfaces confirmed the charge transfer and surface dipole moment changes due to adsorption.
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CHAPTER 6
STACKING FAULTS STUDY ON ZINC TITANATE*
6.1 Abstract
The stacking fault behaviors on ilmenite ZnTiO3 were investigated by calculating the
Generalized Stacking Fault (GSF) energies using density functional theory (DFT) based first
principles calculations and classical calculations employing effective partial charge interatomic
potentials. The γ-surfaces of two low energy surfaces, (110) and (104), of ZnTiO3 were fully
mapped and, together with other analysis such as ideal shear stress calculations, it was found
that the {104}<4 5 1�> slip system has the lowest energy and most favorable. The simulation
results were compared and discussed with high resolution TEM results and it showed that
computational prediction agreed well with experimental observations. The atomic structures of
the low energy stacking faults were analyzed and their electronic structures calculated and
compared with bulk ZnTiO3 structures. It was found that stacking fault formation led to
narrowing of the band gap and creating inter bandgap states, mainly due to the dangling bonds
and bonding defects, as compared to the bulk structures.
6.2 Introduction
Metal oxides such as TiO2, V2O5 and WO3 are found to have low friction coefficients
and are potential solid lubricants[12]. The lubricious properties were found to be closely related
to two dimensional defects such as Megéli phase or stacking fault formations. These transition
metal oxides also have applications in catalysis[80], electronics[137], and spintronics[138] due
* This chapter is reproduced from Wei Sun, Jincheng Du. Structure, energetics, and electronic properties of stacking fault defects in ilmenite-structured ZnTiO3. Modelling Simul. Mater. Sci. Eng. 24 (2016) 065015 with permission from IOP Science
83
to their wide band gap semiconductor behavior and peculiar electronic structures. Either
tribological or electronic properties are impacted by existence of defects and defect processes in
these oxides and understanding of the defects plays an important role in understanding both
mechanical and electronic properties. This chapter provides insight of the stacking fault defects
in ZnTiO3 ceramics by performing atomistic computer simulations using classical force field
based energy minimization and first principles density functional theory calculations.
Solid lubricants have been increasingly applied to control friction and wear under severe
conditions, such as high temperature, vacuum, and radiation[139]. Conventional solid lubricants,
such as graphite[140], MoS2[141], or WS2[139], have been investigated extensively, while metal
oxides have attracted growing interest in the past few decades due to their higher thermal and
chemical stability. Studies have shown that metal oxides are capable of providing complex
mechanical behavior from the changes in sub-surface regions[142]. Mohseni and Scharf[12]
recently reported that significant reduction in the sliding wear factor and friction coefficient was
achieved with ZnO/Al2O3/ZrO2 nanolaminates coatings, which was attributed to the {0 0 0 2}
basal stacking faults in nanocrystalline ZnO, in carbon-carbon composite. Significant ductility in
perovskite strontium titanate at ambient temperatures has been reported in the last decade[143],
[144]. Interestingly, Hirel et al.[145] reported that the unusual mechanical behaviors in strontium
titanate are closely related to the behavior of dislocation motion along the stacking fault parallel
to the {1 -1 0} planes. Although stacking faults and tribological behaviors of perovskite
structured ABO3 oxides have been intensively studied recently, little is known of the ilmenite
structured ABO3 structures. Since ilmenite ABO3 has a hexagonal structure, which is similar to
wurtzite ZnO, oxides with such structure could be promising solid lubricants.
84
Ilmentite structured ZnTiO3 has been widely used as pigments[78], catalysts[79], [80],
microwave dielectrics[81]–[83], gas sensors[84], and low-temperature co-fired ceramics
(LTCC)[86]–[88]. Compared to electronic and optical properties of ilmenite ZnTiO3, its
mechanical properties are much less studied. Furthermore, little is known of the defects,
especially surface and stacking fault defects, in this oxide. However, information on stacking
faults and related dislocation motion is critical to understand friction and lubricious
behaviors[85]. Sun et al. recently reported a combined computational and experimental study of
ZnTiO3 and found that {104}<4 5 1�> stacking faults exist in ZnTiO3 thin films made from
atomic layer depositions (ALD) that result in sliding-induced ductility and low friction
coefficient films. This kind of behavior can exist in other ilmenite structured oxides. Stacking
fault in this glide system has been predicted theoretically by using Density Functional theory and
confirmed experimentally using high resolution TEM. It was found that the ALD films with the
stacking fault on {104} planes showed reduced friction due to interfacial sliding.
Atomistic computer simulations have been used to study the two dimensional defects such as
stacking fault formation in a number of systems. The most well-known approach is the
calculation of generalized stacking faults (GSF) introduced by Vitek[146],[147] to investigate
crystal plasticity. The GSF has been applied on a wide variety of metals and alloys[148]–[150].
For metal oxides, the GSF method has been applied to MgSiO3 with a post-perovskite structure
by Carrez et al.[151] using first-principle calculations. Goryaeva et al.[152] reported the
generalized stacking faults on the same structure by using both DFT and empirical potentials and
resulting in a reasonable agreement on slip systems with previous theoretical and experimental
study. Meanwhile, the GSF approach applied on perovskite ABO3 oxides has been studied by
Hirel et al.[145], in which three pervoskite oxides, SrTiO3, BaTiO3, and PbTiO3, were
85
investigated by DFT and shell-model empirical potentials. Despite this progress, understanding
of stacking fault defects in complex metal oxide is still lacking.
In the present research, we provide a systematic study on two-dimensional defects,
including low energy (104) and (110) surfaces, and stacking fault defects along these surfaces, as
well as their structural and electronic properties, of ilmenite ZnTiO3 by combining first
principles DFT and empirical potentials based calculations. The atomic structure relaxations and
electronic properties on stacking faults and surfaces were also investigated. These results from
simulations were compared with experimental data and those from the literature to provide
insights of the two-dimensional defects in ZnTiO3.
6.3 Computational Details
Classical atomistic calculations were performed by using the General Utility Lattice
Program (GULP) package[153]. The interatomic interactions were modeled with short-range
Buckingham potentials and long-range Coulombic interactions. The Buckingham potential has
the expression: V(r) = Aexp(-r/ρ) – C/r6. The magnitudes of the point charges, in this case, are
reduced, partial charges. The potential parameters[85] for Zn-O are obtained by fitting to known
structural and mechanical properties of Wurtzite zinc oxide, while the Ti-O and O-O parameters
were from Ref [154].
In order to test the reliability of empirical parameters, Vienna ab initio package
(VASP)[97] was used to carry out the density functional theory (DFT) calculations. The energy
was calculated by using the PAW-PBE[98] generalized gradient approximation (GGA)
implanted in VASP code. The electronic wave functions were expanded in a plane wave up to a
kinetic energy cutoff of 400 eV. A Gaussian smearing of 0.05 eV was applied. The first Brillouin
86
zone was sampled using a 6×6×2 Monkhorst-Pack k-point mesh for bulk properties calculation
and a 3×4×1 for stacking faults calculation.
To model the stacking faults behavior, we employed the generalized stacking fault (GSF),
known as energy-displacement curve or surface, which was introduced by Vitek in the
1960s[146],[147]. The GSF curve and surface (γ-surface) are calculated based on the evaluation
of energy at every displacement when given crystal planes glide past one another, representing
the energy cost of rigid shifts of two blocks of crystal on a given plane[155]. The γ-surface and
GSF energy profiles deducted from the γ-surface provide valuable information of most likely
dislocation reaction and favorable glide system[155], [156].
The stacking faults models were generated by shifting half of the supercell relative to the
other half. The ionic position was allowed to relax in the direction normal to the stacking fault at
every displacement. For 3-D periodic boundary conditions, the model contains two stacking
faults, which were constructed rigidly equivalent during supercell building process. Because of
such, the supercell model used for (104) stacking faults calculation contains 26 layers. The
formation energy of the stacking faults is obtained as:
𝑬𝑬𝒇𝒇 = [𝑬𝑬𝒕𝒕𝒕𝒕𝒕𝒕(𝑑𝑑𝐴𝐴𝑑𝑑𝐴𝐴𝑑𝑑𝑑𝑑) − 𝑬𝑬𝒕𝒕𝒕𝒕𝒕𝒕(𝑆𝑆𝑏𝑏𝑏𝑏𝑏𝑏)]/𝒏𝒏𝒏𝒏 (6.1)
Where Etot(defect) is the total energy of the supercell containing stacking faults. Etot(bulk) is the
total energy of the reference bulk supercell. S and n represent the stacking fault area and number
of stacking faults within one supercell, respectively.
The ideal shear stress (ISS), derived from the GSF energies, was introduced to describe
the resistance of shearing, denoting the minimum energy that a system requires to deform
plastically[157]. It is defined as: 𝜎𝜎𝐼𝐼𝐿𝐿𝐿𝐿 = 𝜕𝜕𝛾𝛾𝐺𝐺𝐿𝐿𝐻𝐻(𝑏𝑏)/𝜕𝜕𝐶𝐶, where 𝜕𝜕𝛾𝛾𝐺𝐺𝐿𝐿𝐻𝐻(𝑏𝑏) is the non-linear
generalized stacking faults energy function, and u denotes the displacement[158].
87
6.4 Geometric Optimization
The equilibrium lattice parameters and mechanical properties for hexagonal ZnTiO3 were
calculated by using both DFT with LDA and GGA functionals and empirical potentials with
parameters recently developed[85]. Figure 4.4 shows the schematic structure of hexagonal
ZnTiO3 (space group: R3�) after structure relaxation. It can be seen that in this typical ilmenite
cell, the Zn and Ti ions are coordinated octahedrally to the nearest O ions. The same types of
octahedra are connected with edge sharing while each type of octahedron shares one face with
the other type in the adjacent layer[107], stacking alternatively along c-axis. The calculated
lattice parameters and elastic constants are summarized in Table 6.1, together with comparison
of experimental and previous calculations. The equilibrium lattice parameters were obtained
from energy versus volume curves, which were fitted to the Birch equation of state[108]. The
bulk, shear, and Young’s moduli were obtained according to the Voigt-Ruess-Hill
approximation[103], [104].
It can be seen that our calculated values are in good agreement with the previous DFT
calculations and experimental results[85],[105]. As proven in previous studies, the LDA
calculations underestimate the lattice parameters and volume while overestimating the
mechanical properties[106]. For the rest of the calculations, DFT with GGA exchange and
correlation functionals were used to calculate the stacking fault energies along certain directions
and the surface energies. The DFT data also provides a benchmark data of classical calculations
where mapping of stacking fault energies in two-dimensional space was performed using
empirical potentials with parameters in ref [85].
88
Table 6.1. Comparison of observed and calculated properties for ZnTiO3
This Work References
Exp. VASP-GGA
VASP-LDA GULP LDA1 CASTEP-
GGA2 V(Å3) 321.6 295.6 301.1 298.8 324.0 311.93
c/a 2.74 2.72 2.73 2.73 2.77 2.683 C11 (GPa) 292.9 395.9 364.9 317.3
C12 145.7 217.3 140.4 142.0 C13 81.0 148.75 93.4 90.0 C14 16.7 13.1 21.2 11.7 C15 C33
-5.75 216.5
0.85 292.0
-12.8 292.0 -0.8
236.3
C44 49.6 61.3 53.4 47.6 B (GPa) 154.4 229.7 186.2 164.9 G (GPa) 64.6 77.7 80.7 68.6 E (GPa) 170.1 209.4 211.5 180.7 202~1554
6.5 Surface Energy Calculation
Surface structures and energies of oxides determine their equilibrium crystal morphology
and many properties such as workfunction and adsorption energies. In addition, stable surfaces
are also significant in determining the stacking fault formations. Thus, we first calculated the
surface energies of ZnTiO3 by using periodic DFT calculations with surface slabs cut from
optimized bulk ZnTiO3 structure. These slab models contained integer number of ZnTiO3 unit so
that they are all stoichiometric, and both the top and bottom surfaces were allowed to relax.
Convergence tests of the surface energy versus the slab thickness were performed to ensure the
slab was thick enough to remove the interaction of the bottom and the top surfaces. Since
ilmenite ZnTiO3 has a relatively complex structure compared to other ZnTiO3 polymorphs, for
each of the surface indexes, different surface constructions could exist, and the surface energies
varied by the choice of how the surfaces were terminated. Various terminations were tested and
compared, and only the lowest energies were reported. In this work, we only considered two
surfaces: (104) and (110), both of which were main surfaces determined by experimental XRD
89
and TEM studies[85]. We adopted the criterion by Christensen and Carter[159], where the
surface slab models were constructed to be symmetric, non-polar, and thus stable due to no long-
range electrostatic potential. The unrelaxed slabs were cut from the fully optimized bulk crystal.
All atoms in the slab were then allowed to relax in DFT calculations to obtain fully relaxed
surface structure models.
6.5.1 Surface Construction of (110)
Figure 6.1 shows the O-terminated and Zn/Ti-terminated (110) surfaces of six-layered
slab models. As can be seen in Figure 6.1(a), two nonequivalent surface oxygen atoms, as well as
surface Zn and Ti atoms, can be observed on O-rich surfaces. The inequivalent O atoms, both
with a coordination of 3, are located at different z planes in the surface. It can be seen from the
figure that O1 bonds with two Zn atoms and one Ti atom, while O2 bonds with two Ti atoms and
one Zn atom. Besides, the Zn and Ti atoms (marked in Figure 6.1) at a lower z plane only have a
coordination number of 5. Due to the coordination number loss compared to the ones in the bulk,
this cation layer has a high impact on surface properties. Different from the O-rich surface, only
cations exhibit a coordination number loss on the Zn/Ti terminated surfaces. As indicated in
Figure 6.1b, the surface Zn and Ti atoms in a same z plane only have a coordination number of 3,
much lower than that in bulk structures.
Figure 6.1. (110) structure of six-layer surface model of ZnTiO3 with (a) O termination, and (b)
Zn/Ti termination
90
6.5.2 Surface Construction of (104)
The (104) surface slabs are shown in Figure 6.2a and 6.2b. There are two different
surface constructions, both of which have mixed cation and anion terminated surfaces. Figure
6.2a indicates two nonequivalent surface oxygen atoms differed by z planes. O1, with a lower z
value, has a coordination number of 3, connecting with two Zn atoms and one Ti atom. However,
O2, which is located in higher z plane, suffers from a higher coordination number loss. Every O
atom at this z plane only connects to two Ti atoms, with a coordination number of 2. On the
other hand, the surface construction shown in Figure 6.2b has two nonequivalent O atoms
located in the same z plane, both with a coordination number of 3, in which O1 connects with
two Zn atoms and one Ti atom while O2 connects with two Ti atoms and one Zn atom.
Figure 6.2. (104) structure of six-layer surface model of ZnTiO3 with (a) O/Ti/Zn termination,
and (b) Zn/Ti termination
6.5.3 Surface Energy Calculation of (110) and (104) Planes
The surface energy for a slab is defined as[160],[161]:
𝐸𝐸𝐽𝐽𝐶𝐶𝑟𝑟𝑒𝑒 = 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡−𝐸𝐸𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏2𝐿𝐿
(6.2)
where Etot is the total energy of the surface slab, while Ebulk refers to the energy of the bulk
ZnTiO3 system with the same number of ZnTiO3 units. A refers to the total surface area, which is
considered to be twice the area since the slab has two free relaxing surfaces. Figure 6.3a and b
exhibits the surface energies as a function of the layer thickness of relaxed cell models. The slab
91
with a thickness of 9 layers were converged, which hence were used in subsequent calculations.
Table 6.2 summarizes the surface energies for both (110) and (104) faces. Here we calculated
both relaxed and unrelaxed surface energies. In this table, the values in the unrelaxed column
denote the energies of rigid surface models cut from bulk structures, while values in the relaxed
column represent the energies of the former models after atomic relaxations. It can be seen from
Table 6.2 the consistency of the ordering of the unrelaxed and relaxed surface stabilities. And for
both (110) and (104), cation terminated surfaces show a higher relaxation range than the O
terminated ones, and the relaxation energy of the surfaces ranges from 37.0% to 56.4% of the
total energy. Consequently, the (104) surface is more stable than the (110) surface, which shows
the same trend as we reported in previous work. It is worth noting these surface energies are
slightly lower than the ones reported earlier[85]. This is mainly due to new surface terminations
adopted to remove any surface dipole moment according to Christensen and Carter[159]
Table 6.2. Surface energies for the two faces of ilmenite ZnTiO3 Orientation
Surface Energy (J/m2) Relaxation Planar density
(#/Å2) Unrelaxed Relaxed (104)-O
(104)-Zn/Ti (110)-O
(110)-Zn/Ti
1.08 2.73 1.66 4.32
0.68 1.19 1.04 2.04
37.03% 56.41% 37.34% 52.78%
0.26 0.26 0.24 0.24
(a) (b) Figure 6.3. Calculated surface energies as a function of slab thickness; (a), (110) surface, and (b),
(104) surface
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6.6 Stacking Faults Behavior for the (110) and (104) Planes
6.6.1 Stacking Faults on (110)
Both (110) and (104) planes were observed in deposited ZnTiO3 thin films
experimentally. DFT calculations (Table 6.2) show that the surface energies for oxygen and
Zn/Ti terminated (110) films are 1.04 and 2.04 J/m2, respectively. These surface energies are
higher than (104) surfaces by 0.5-0.8 eV. A fully relaxed (110) surface model is shown in Figure
6.1, in which we can see that one (110) surface layer contains one cation sub-layer mixed with
Zn and Ti atoms and two sub-layers of O atoms. To generate stacking faults, cut planes between
two successive anion sub-layers rather than between cation and anion sub-layers were used
because the former provides a lower energy.
The γ-surface and corresponding contour plot of a (110) plane calculated by GULP are
shown in Figure 6.4, in which the shapes and peaks of (110) γ-surface indicate a complicated
energy distribution. The local maximum energy area on γ-surface are the unstable stacking fault
energies, representing the energy barrier along the dislocation nucleation direction[162]. The
local minimum points correspond to the meta-stable stacking faults structures, exhibiting
different configurations along different directions. The γ-surface shows several local energy
maxima (dark red region) and local minimum (light yellow region), these “mountains” and
“valleys” are marked on the contour plot (Figure 6.4b), providing the detailed information and
clear view on energy distribution of corresponding γ-surface. Note that plenty of sharp peaks
appear on the γ-surface, indicating unusual high energies due to one-half of super-cell shifts
rigidly along the directions passing through these areas. Since the repulsive force between two
atoms increases exponentially when the distance of them decreases, the ultra-high energy areas
on the γ-surface come from those unreasonable structures during the shifting.
93
(a)
(b)
Figure 6.4. The complete (110) γ-surface (a) and corresponding contour plot (b) of ZnTiO3
Apart from the general information of possible metastable stacking faults acquired from
the γ-surface and contour plot, the energy profiles along specific directions are used for
analyzing the dislocation behavior[145]. All the curves of energy profiles in this work were
taken from the γ-surface and then fitted by using the full width at half maximum (FWHM)
version of the Gaussian function[163]. According to the energy distribution in contour plot, the
94
stacking faults behavior of (110) plane could be analyzed by studying the energy profiles along
[1 1� 0] and [0 0 1] directions, as shown in Figure 6.5. When the cell shifts along these typical
directions, it passes through the energy barriers, known as γusf, and then reaches the saddle
points, known as γsf. These GSF curves were used by James R. Rice to describe the process of
full dislocation nucleation from a crack tip, and the unstable stacking fault energy γusf denotes the
barrier for a partial dislocation to overcome[162]. And then a stacking fault with an energy of γsf
will be formed after the partial dislocation begins to propagate[155]. It can be seen in Figure 6.5
that for both the chosen directions on (110) plane, three humps as well as two saddle points exist.
Although the GSF energy curve comes with a saddle point along both [1 1� 0] and [0 1 0]
directions, the one on [0 1 0] direction shows higher γusf and γsf , Those factors indicate a
corresponding high energy barrier and less-stable local minimum on [0 1 0] direction.
Consequently, according to both Figure 6.4 and 6.5 the most favorable path lies along the [1 1� 0]
direction with a displacement vector 1/3[1 1� 0] and 2/3[1 1� 0].
Figure 6.5 also demonstrates the GSF energy for (110) along the minimum energy path [1
1� 0] calculated by using VASP. The shape of energy profile of DFT is similar to that of GULP,
with a slight difference in the saddle point and minimum locations. The slight mismatch is
caused by equilibrium lattice parameter and cell volume difference within the two methods.
However the presence of a saddle point at the curve along 1/3 and 2/3[1 1� 0] direction performed
by DFT confirm the metastable stacking faults on (110) plane in ZnTiO3.
95
Figure 6.5. The generalized stacking fault energies of (110) as a function of shear displacement along [1 1� 0] and [0 1 0] directions. The triangle, round, and square dots are the calculated
structures of corresponding displacement vectors, while the lines are fitted results by using the Full Width at Half Maximum (FWHM) Gaussian function
6.6.2 Stacking Faults on (104)
In the (104) surface models, shown in Figure 6.2, a (104) layer contains one cation sub-
layer mixed with Zn and Ti atoms and one oxygen sub-layer. The chosen cut plane was located
between the two sub-layers.
The complete (104) γ-surface and corresponding contour plot of ZnTiO3 computed by GULP
are shown in Figure 6.6. The shapes and peaks of the (104) γ-surface indicate a smoother energy
distribution compared to (110). For one thing, the γ-surface of the (104) plane shows a smaller
area than that of the (110) plane, indicating shorter shift vectors of slips on (104). For another,
the γ-surface only demonstrates three local energy maxima and one local minimum, as marked in
Figure 6.6. Unlike the favorable path of (110) plane on the side, the energy distribution in
96
contour plot of (104) indicates that the most reasonable displacement directions are along
diagonal path of the rectangle.
(a)
(b)
Figure 6.6. The complete (110) γ-surface (a) and corresponding contour plot (b) of ZnTiO3
As shown in Figure 6.7, the GSF energy as a function of displacement along the [4 2 1�],
[0 1 0] and [4 5 1�] were calculated by GULP. For [0 1 0], there is no saddle point but one energy
barrier in the GSF energy curve, indicating no stacking fault defect formed when the system
propagated along this direction. Although the GSF energy curve comes with a saddle point along
both [4 2 1�] and [4 5 1�] directions, the one on [4 2 1�] direction shows not only higher energies of
97
γusf and γsf , but also the smaller difference between them. Those factors represent a
corresponding high energy barrier and less-stable local minimum on [4 2 1�] direction. Based on
that the most favorable path lies along the [4 5 1�] direction with a metastable stacking faults
energy of 2.73 J/m2 (GULP result), which have a displacement vector 1/2[4 5 1�].
Same as (110) results, the energy curve calculated by DFT agree with GULP results on the trend
but with a significant reduction on both γusf and γsf, as shown in Figure 6.7. We reported that the
shift of stacking faults point (from 4.63 Å to 4.87 Å) results from the different optimum
structures, lattice parameters are shown in table 6.1, which come from the two different
computational methods[85].
Figure 6.7. The generalized stacking fault energies of (104) as a function of shear displacement
along [4 5 1�], [0 1 0] directions. The triangle, round, and square dots are the calculated structures of corresponding displacement vectors, while the lines are fitted results by using the Full Width
at Half Maximum (FWHM) Gaussian function
6.7 Structure Relaxation and Electronic Structures of the Low Energy Stacking Faults
6.7.1 Idea Shear Stress on (104) and (110)
By comparing the energy curves of {110}<1 1� 0> and {104}<4 5 1�> glide systems, the
98
former shows lower values of both γusf and γsf. Nevertheless, the ratio γusf/γsf plays an important
part in determining the favorable glide system, not just the absolute values of γusf and γsf. Tadmor
and Hai[164] reported that the ratio γusf/γsf signifies the nature of slip activity. Table 6.3
summarizes the values of γusf and γsf as well as the ratio γusf/γsf, which are for {110}<1 1� 0> and
for {104}<4 5 1�>. Obviously, the lower γusf/γsf ratio range of {104}<4 5 1�> indicates that the
energy barrier that has to be overcome for sliding in this glide system is the lowest among all the
systems. It has been proposed that introduction of shear stress will lower the generalized stacking
faults energy accordingly[158]. Thus, the calculation of ISS can provide an indication on the
potentially favorable slip systems. ISS denotes the smallest stress to deform a perfect crystal.
Therefore, lower ISS value represents smaller resistance of shearing.
Table 6.3 summarizes the calculated ISS energies of {110}<1 1� 0> and {104}<4 5 1�>
glide systems derived from generalized stacking faults curves. Both DFT and empirical results
indicate the {104}<4 5 1�> is the most possible slip system. This agrees well with experimental
observations where high resolution TEM showed atomic layer deposited ZnTiO3 films have
preferential orientation of (104), and the stacking fault orientation was found to be <4 5 1�>, in
excellent agreement with our theoretical findings [85]. These analyses show that the most
favorable glide system of ilmenite ZnTiO3 is the {104}<4 5 1�> system.
Table 6.3. Values of unstable stacking fault energy (γusf), stacking fault energy (γsf), and the ratio range γsf/γusf for empirical potential and DFT calculation results
γ1usf (m/J2)
γ1sf (m/J2)
γ2usf (m/J2)
γ2sf (m/J2)
γ3usf (m/J2) γsf/γusf
ISS(GPa)
(1 1 0)/ [1 -1 0]
GULP 2.94 1.30 3.62 1.26 2.95 0.34~0.44 26.5 DFT 2.29 0.65 3.02 0.59 3.12 0.18~0.28 32.5
(1 0 4)/ [4 5 -1]
GULP 4.17 2.73 3.01 N/A 0.65~0.90 24.5 DFT 2.88 1.68 2.81 0.58~0.60 14.9
99
6.7.2 Structure Relaxation on {104}<4 5 𝟏𝟏�> System
The atomic relaxations at the saddle point of the favorable glide system obtained above
from two computational methods have been studied. In Figure 6.8, we show the schematic
atomic structure of four layers around slip plane in saddle point at 1/2[4 5 1�] along (104) surface
normal. To further study the atomic relaxation of this structure, we calculated percentage
relaxation along z direction of the two upper and lower layers around slip plane of this
metastatble structure. The results are presented in Table 6.4. The labeled atoms in first column
are shown in Figure 6.8b. The relaxation of ion from this structure are expressed as Δz,
∆𝑧𝑧 = 𝑧𝑧−𝑧𝑧𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑧𝑧𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
× 100% (6.3)
Here, z represents the z-coordinate of O, Zn or Ti after relaxation, while 𝑧𝑧𝐶𝐶𝐶𝐶𝐶𝐶𝐵𝐵 is the
unrelaxed z-coordinate determined from the as-cleaved metastable structure. And the positive
(negative) percentage means the direction upward (downward) from the corresponding layer
before relaxation.
First of all, the atoms in the -2 and -1 layers (below slip plane) move downward to the
surface normal for both methods and the displacements of empirical potential calculation are
much larger than that from DFT calculation. For the atoms in +1 and +2 layers, DFT results
show an opposite movement to the -1 and -2 layers, while GULP results indicate that most atoms
still move downward to the surface normal. Moreover, the opposite displacement predicted by
DFT calculations between -1, -2 and +1, +2 layers indicate relatively large changes in interlayer
distances separated by slip plane. Although GULP results show a consistency of displacement
directions for the for layers, the displacement of -1 and -2 layers is much larger than +1 and +2,
which also leads to an enlargement of interlayer distance between -1 and +1 layers. The structure
changes during slipping not only influence the system energies, but also have a huge impact on
100
electronic properties. We chose the initial structure, local maximum point, and saddle point along
[4 5 1�] of (104) plane as prototypes to investigate the electronic properties.
Figure 6.8. (a) Schematic structure of saddle point at 1/2[4 5 1�] at (104) surface normal; (b) atom labels for two layers with different cation sequences
Table 6.4. Structural relaxation of two upper and lower layers around slip plane at 1/2[4 5 1�] along surface normal
Layer -2 Layer -1 Layer +1 Layer +2
DFT GGA-PBE
Empirical potential
DFT GGA-PBE
Empirical potential
DFT GGA-PBE
Empirical potential
DFT GGA-PBE
Empirical potential
O1 -0.33% -1.96% -0.38% -1.53% 0.68% -0.30% 0.22% -0.53% O2 -0.39% -1.80% -0.26% -1.88% 1.33% 0.41% 0.45% -0.22% O3 -0.26% -1.66% -0.28% -1.68% 0.23% -0.88% 0.22% -0.48% O4 -0.40% -2.05% -0.67% -1.89% 0.17% -0.46% 0.28% -0.61% O5 -0.35% -1.66% 0.05% -1.33% 0.31% -0.85% 0.20% -0.58% O6 -0.45% -2.04% -0.67% -2.24% 0.31% -0.42% 0.28% -0.55% Ti1 -0.29% -2.02% -0.37% -2.06% 0.38% -0.75% 0.35% -0.59% Ti2 -0.38% -1.65% -0.50% -1.74% 0.63% 0.01% 0.19% -0.47% Zn1 -0.43% -2.09% -0.30% -1.81% 0.02% -1.18% 0.30% -0.48% Zn2 -0.32% -1.70% -0.19% -1.51% 0.05% -0.23% 0.35% -0.34%
101
6.7.3 Density of States on {104}<4 5 𝟏𝟏�> System
Figure 6.9 shows the DFT calculated partial and total density of states (DOS) of the
structures from those three points. The contribution to the DOS of O comes from the 2s and 2p
electrons, while for Zn and Ti, the contribution comes from 3d electrons. Here we only report the
DOS near Fermi level (-8eV~5eV) in order to demonstrate the DOS changes of the significant
structures along [4 5 1�] more clearly. It can be seen from Figure 6.9 that for the initial structure
the wide valence band, with an O 2p and Zn 3d electrons hybridization, lies between -5.6eV and
0 eV. This valence band is separated by an energy gap of 2.7 eV from a conduction band with
mainly Ti 3d electrons and minor O 2p electrons. These results are in agreement with previous
computational studies[105] on ilmenite ZnTiO3 structure.
For the DOS of local maximum structure, the effect of structure change around slip plane
on the DOS is significant. Although the predicted band gaps may not be reliable for compounds
with Zn and Ti (DFT underestimation), the trends in different structures with same
computational parameter sets can still be reliable. As can be seen in Figure 6.9 the broadened
valence band (from -7.8 eV to 0 eV) and conduction band (0.2 eV~5.3 eV) lead to a much
narrower band gap between them. However, the structure at saddle point has significantly larger
band gap of 1.4 eV compared to the structure at local maximum point band gap of 0.2 eV. The
DOS of the stacking faults structure is strikingly similar to the DOS of initial structure, but the
extra small states located between VBM and CBM are noticeable. The electronic properties vary
greatly according to the structure change, which may also result in the bond length change and
coordination number loss.
102
Figure 6.9. Total and partial DOS for initial, local maximum and local minimum structures of
(104)[4 5 1�] system
6.7.4 Statistical Analysis of Atom Distances on (104)[𝟒𝟒 𝟓𝟓 𝟏𝟏�] System
A statistical analysis of atom distances after DFT relaxation of the original, local
maximum point, and saddle points along [4 5 1�] of (104) plane is shown in Figure 6.10. For
these supercell models with 260 atoms, the distribution of cation~O distances (include the Zn~O
and Ti~O) and cation~cation distances (include the Zn~Zn, Zn~Ti, and Ti~Ti) are concluded, in
which the Count in y axis represents the number of pair atoms in corresponding range. First, for
103
local maximum and local minimum structures, the lower cation~O distribution in range
1.8Å~2.4Å and extra distribution above 2.4Å with respect to initial structure may result in the
Ti-O and Zn-O bonds broken and thus causing the coordination number loss. Second, compared
to the distance between cations in initial structure, the local maximum cell has extra 26 pairs of
cations in the range below 3.0 Å, while the local minimum structure has 14 pairs. As a result, the
Zn-Zn and Ti-Ti metallic bonds may be formed for those two structures and contribute to the
extra states within band gap in DOS figures.
(a) (b) Figure 6.10. Cation-cation and cation-anion distance distribution for initial, local maximum and
local minimum structures of (104)[4 5 1�] system
6.8 Conclusion
In summary, surface and stacking fault behaviors of ilmenite ZnTiO3 were studied by
combining plane wave basis set first-principles density functional theory (DFT) calculations and
partial charge empirical potential based classical calculations. Careful choice of surface
termination and application of dipole correction during DFT calculations was found necessary to
generate reliable stacking fault energies for these complex oxide surfaces. The stacking fault
energies along chosen directions from DFT calculations were used to validate the empirical
potential calculations, which were then used to map the stacking fault gamma surfaces of
ZnTiO3.
104
The results show that the (104) surfaces have a lower energy, from 0.5 to 0.8 J/m2
depending on the termination, than the (110) surfaces. The γ-surface mapping and stacking fault
energy calculations indicate that the [1 1� 0] is the preferred direction of stacking faults for (110)
planes, while [4 5 -1] is preferred direction for (104) planes. Among the gamma surfaces mapped
of the two planes, the favorable glide system of ilmenite ZnTiO3 lies in {104}<4 5 1�> due to
their significantly lower γsf/γusf values, which is in excellent agreement with experimental
observations using high resolution TEM and supports the mechanism of low friction coefficient
of these films due to the existence of large concentration of growth induced {104}<4 5 1�>
stacking faults that favor easier shear sliding in the system. Considerable structure relaxation and
change in bonding and electronic structures were observed in stacking faults as compared to the
initial bulk structure for the {104}<4 5 1�> stacking fault systems. Stacking fault formation leads
to narrowing of the band gap and creating of inter-band gap states, which can change the
electronic behaviors of the materials.
105
CHAPTER 7
SUMMARY AND FUTURE WORK
First principles density functional theory (DFT) calculations have been performed to
study the behaviors of point defects, surfaces, and stacking faults in wurtzite ZnO and ilmenite
ZnTiO3. The calculation of point defects in the case of ZnTiO3 shows a preference for oxygen
vacancy under metal-rich conditions within the whole range of band gap. While for O-rich
conditions, VO is the dominant defect within lower half of the band gap. The negative binding
energies of di-vacancies indicate that Zn/Ti and O vacancies are bound in di-vacancy clusters
under all three conditions. For wurtzite ZnO, the formation energies of the [SbZn − 2VZn]′
complex and the Zni defects are consistent with those from the literature. In addition both
formation energies and ionization energies indicate that these defect complexes are responsible
for the p-type behavior, which is observed in the annealed films conducted experimentally
The change of workfunction of low energy polar and nonpolar surfaces of ZnO as a
function of surface adsorbents coverage, including methyl, trifluoromethyl, and fluorine groups,
have been studied by using the DFT method. Dipole correction was introduced in workfunction
calculations in order to cancel the unbalanced electrostatic potential between asymmetric slabs
caused by periodic boundary conditions. In addition, as for polar (0 0 0 1)/(0 0 0 1�) surfaces,
pseudo hydrogen atoms were added to saturate the dangling bonds on the one side of the slabs,
which was found to be successful in avoiding unreasonable surface reconstruction after
relaxation. The results show that adsorption of both methyl and trifluoromethyl groups on
oxygen sites are energetically more favorable for both polar and non-polar surfaces, while
fluorine was found to stably bond only with surface Zn atoms due to its high electronegativity. In
addition, adsorption of fluorine groups leads to a significant decrease of workfunction on both
106
polar and nonpolar surfaces. The change of workfunction is found to correlate linearly with the
surface dipole moment. Increase of surface dipole moment can increase the workfunction, which
is confirmed by the charge density plots.
Furthermore, surfaces and stacking fault behaviors of ilmenite ZnTiO3 were studied by
combining DFT calculations and partial charge empirical potential based classical calculations.
The results show that the (104) surfaces have a lower energy, from 0.5 to 0.8 J/m2 depending on
the termination, than the (110) surfaces. The γ-surface mapping and stacking fault energy
calculations indicate that the [1 1� 0] is the preferred direction of stacking faults for (110) planes,
while [4 5 1�] is preferred direction for (104) planes. Among the gamma surfaces mapped of the
two planes, the favored glide system of ilmenite ZnTiO3 lies in {104}<4 5 1�> due to their
significantly lower γsf/γusf values, which is in excellent agreement with experimental
observations using high resolution TEM and supports the mechanism of low friction coefficient
of these films due to the existence of large concentration of growth induced {104}<4 5 1�>
stacking faults that favor easier shear sliding in the system. Considerable structure relaxation and
change in bonding and electronic structures were observed in stacking faults as compared to the
initial bulk structure for the {104}<4 5 1�> stacking fault systems. Stacking fault formation leads
to narrowing of the band gap and creating of inter-band gap states, which can change the
electronic behaviors of the materials.
107
APPENDIX
PUBLICATIONS RESULTING FROM THIS DISSERTATION
108
Wei Sun, Jincheng Du. Structure, energetics, and electronic properties of stacking fault defects in ilmenite-structured ZnTiO3. Modelling Simul. Mater. Sci. Eng. 24 (2016) 065015
Wei Sun, Yun Li, Jitendra Kumar Jha, Nigel D Shepherd, Jincheng Du. Effect of surface adsorption and non-stoichiometry on the workfunction of ZnO surfaces: A first principles study. J. Appl. Phys. 117, 165304 (2015)
W Sun, V Ageh, H Mohseni, TW Scharf, J Du. Experimental and computational studies on stacking faults in zinc titanate. APPLIED PHYSICS LETTERS 104, 241903 (2014)
Wei Sun, Jincheng Du. Structural stability, electronic and thermodynamic properties of VOPO4 polymorphs from DFT+U calculations. Computational Material Science. 2016. 126 (2017) 326-335
Jitendra Kumar Jha, Wei Sun, Reinaldo Santos-Ortiz, Jincheng Du, Nigel D Shepherd. Electro-optical performance of molybdenum oxide modified aluminum doped zinc oxide anodes in organic light emitting diodes: A comparison to indium tin oxide. Mater. Express, Vol. 6, No. 3, 2016
Reinaldo Santos-Ortiz, Jitendra Kumar Jha, Wei Sun, Gilbert Nyandoto, Jincheng Du, Nigel D Shepherd. Defect structure and chemical bonding of p-type ZnO:Sb thin films prepared by pulsed laser deposition. Semicond. Sci. Technol. 29 (2014) 115019
Jitendra Kumar Jha, Wei Sun, Jincheng Du and Nigel D. Shepherd. Mechanisms of AZO workfunction tuning for anode use in OLEDs: surface dipole manipulation with plasma treatments versus nanoscale WOx and VOx interfacial layers. Journal of Physics D: Applied Physics. 2016. (Under reviewing)
109
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