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1 ATOMIC STRUCTURE EFFECTS ON BULK AND SURFACE PROPERTIES OF MIXED METAL OXIDES FROM FIRST PRINCIPLES SIMULATIONS By BEVERLY BROOKS HINOJOSA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010
2 2010 Beverly Brooks Hinojosa
3 To my husband Bebo
4 ACKNOWLEDGMENTS I am sincerely grateful to my advisor Prof. Aravind Asthagiri for his years of support and advice, extreme patience, and unbelievable confidence in my abilities. His successful management of completely different work styles has been incredibly beneficial and critical to my accomplishments. I also thank my committee members Prof. Dmitry Kopelevich Prof Kirk Ziegler and Prof. Juan C. Nino. I am especially grateful to Prof. Nino who has been more like a co advisor than an external committee member. His willingness to spend hours discussing my research and p ersonal knowledge base is particularly acknowledged. I thank Prof. Simon Phillpot and Prof. Susan Sinnott for joining Prof. Asthagiri in creating the electronic materials group to provide students the opportunity to present and participate in many helpful discussions. This joint effort provided my research group a valuable bridge to coordinate with the Computational Materials Science Focus Group. Additionally, Prof. Sinnott has served as a wonderful faculty mentor who has joined me in many discussions without which I would be greatly disadvantaged. I thank the professors in the Department of Chemical and Bimolecular Engineering at the University of Houston for demanding excellence and holding every student acutely accountable. I thank Lu Cai for assistance in the drawing of Figure 3-1. I acknowledge the University of Florida High Performance Computing Center ( http://hpc.ufl.edu) for providing computational resources as well as Prof. Kopelevich Prof. Asthagiri, Prof. Jason B utler, and Prof. Tony Ladd, within the chemical engineering department, for creating and maintaining the local computing resource. I gratefully acknowledge financial support for a portion of this work provided by the National Science Foundation under NSF C HE Grant #0911553.
5 Many students have helped me enjoy my time at the University of Florida. I particularly want to acknowledge Shelley Cottrell, Can Hakanoglu, Rob Colmyer, Jun Wu, Casey LaMarche, Elena Aksel Young Min Ban, Erica Douglas, Jennifer Wohlwen d Brad and Kristen Shumbera, Heywood and Cynthia Kan, Rakesh Behera, and Chan Woo Lee. I am indebted to Jeff Hawkins for sharing an office space with me and putting up with all of my questions. On a personal note, I thank my husband for all that he does and all that he has given me. He is my reason for pushing a little harder each day, my personal comedian who makes me laugh every day, and the only person I want to annoy each night. I thank my parents for having the courage to leave all their friends and family for the hope of giving us a better life. I would not be here today without the undying faith of my mother. I thank my big brother for being my protector and confidant growing up and flying half way across the country at least once a year after I mov ed, though it might have been more to watch the gators play in the swamp than to hang out with his little sister. I am indebted to my extended family from Boston to Brazil for many words of encouragement, a strong foundation of support, and innumerable prayers. I am eternally grateful to my maternal grandmother who brought me back to the fold and took the time each day to teach me of a life I never knew and time I never saw. The loss of her taught me the importance of family over all else, a lesson that hurt more than I can say but I desperately needed to learn. I thank the Hinojosa family for always making me feel connected and for loving their Guera. I th ank God for all of my blessings, opportunities and challenges
6 TABLE OF CONTENTS ACKNOWLEDGMENTS ...................................................................................................... 4 page LIST OF TABLES .............................................................................................................. 10 LIST OF FIGURES ............................................................................................................ 12 LIST OF SYMBOLS .......................................................................................................... 16 LIST OF ABBREVIATIONS .............................................................................................. 18 ABSTRACT ........................................................................................................................ 20 C HAPTER 1 INTRODUCTION ........................................................................................................ 22 1.1 Motivation .............................................................................................................. 22 1.1.1 Motivation f or the Study of Bismuth Pyrochlores ....................................... 23 1.1.2 Dielectric Behavior in Bismuth Pyrochlores ............................................... 25 1.2 Literature Review .................................................................................................. 29 1.3 Simulation Methods .............................................................................................. 31 1.3.1 Periodic Boundary Conditions .................................................................... 32 1.3.2 Fundamentals of Density Functional Theory ............................................. 33 18.104.22.168 Schr dinger equation ........................................................................ 34 1.3. 2.2 Kohn-Sham equations ...................................................................... 35 22.214.171.124 Iterative approach to find solutions to Kohn-Sham equations ......... 36 126.96.36.199 Exchange-correlation functional ....................................................... 38 1.3.3 Calculation Details ...................................................................................... 40 1.3.4 DFT Tools and Analysis Techniques ......................................................... 43 188.8.131.52 Simulated Annealing ......................................................................... 44 184.108.40.206 Nudged Elastic Band ........................................................................ 45 220.127.116.11 Bader Charges .................................................................................. 47 18.104.22.168 Electron Localization Function .......................................................... 48 1.4 Objectives of Dissertation ..................................................................................... 48 2 CRYSTALLOGRAPHY OF PEROVSKITE AND PYROCHLORE MATERIALS ...... 50 2.1 Structure of Strontium Titanate ............................................................................ 50 2.2 Structure of Bismuth Pyrochlores ........................................................................ 51 2.2.1 Crystallography and Visualization of the Pyrochlore Structure ................. 51 22.214.171.124 Origin and cation centering in Fd 3 m space group ........................... 52 126.96.36.199 Wyckoff positions in Fd 3 m space group .......................................... 53 188.8.131.52 Sub -network interactions in ideal pyrochlores ................................. 54 2.2.2 Site Symmetry ............................................................................................. 56
7 2.2.3 Bi2O Network ............................................................................................... 59 2.2.4 Bond Val ence Model and its Application in Bismuth Pyrochlores ............. 60 3 FIRST -PRINCIPLES STUDY OF CUBIC BISMUTH PYROCHLORES ................... 63 3.1 Introduction ........................................................................................................... 63 3.2 Calculation Details ................................................................................................ 66 3.3 Resu lts and Discussion ........................................................................................ 68 3.3.1 Ideal Bismuth Pyrochlores .......................................................................... 68 3.3.2 Bismuth Pyrochlores with Atomic Displacement ....................................... 73 184.108.40.206 Atomic displacement in Bi2Ti2O7 ...................................................... 77 220.127.116.11 Atomic displacement in Bi2Ru2O7 ..................................................... 89 3.3.3 Electronic Structure .................................................................................... 91 3.4 Summary ............................................................................................................... 99 4 THE INFLUENCE OF S ULFUR SUBSTITUTION ON THE ATOMIC DISPLACEMENT IN BISMUTH TITANATE ............................................................. 101 4.1 Introduction ......................................................................................................... 101 4.2 Calculation Details .............................................................................................. 103 4.3 Resu lts and Discussion ...................................................................................... 104 4.3.1 Structural Details ....................................................................................... 104 4.3.2 Electronic Structure .................................................................................. 109 4.3.3 Bismuth Ruthenate ................................................................................... 118 4.4 Summary ............................................................................................................. 121 5 ROLE OF VACANCIES ON CATION DISPLACEMENT IN PYROCHLORES ...... 123 5.1 Introduction ......................................................................................................... 123 5.2 Calculation Details .............................................................................................. 124 5.3 Vaca ncy Configurations ..................................................................................... 125 5.4 Results and Discussion ...................................................................................... 129 5.4.1 Structural Details ....................................................................................... 129 18.104.22.168 Bi2O0.875Ru2O6 composition ........................................................... 129 22.214.171.124 Bi1.875O0.875Ru2O6 composition ...................................................... 131 5.4.2 Electronic Structure .................................................................................. 135 5.5 Summary ............................................................................................................. 136 6 BISMUTH PYROCHLORES WITH CATION SUBSTITUTES ................................. 139 6.1 Introduction ......................................................................................................... 139 6.2 Atomic Structure Ordering Predictions in BZN .................................................. 141 6.2.1 Bond Valence Analysis of BZN ................................................................ 141 6.2.2 Zn Ordering Rules on the A2O Network .................................................. 142 6.3 Calculation Details .............................................................................................. 143 6.3.1 Configuration Setup .................................................................................. 143 6.3.2 VASP details ............................................................................................... 145 6.4 Results and Discussion ...................................................................................... 146
8 6.4.1 Energetics of Zn Ordering in the A2O Network in BZN ........................... 146 6.4.2 Structural Differences between A2O Network Types in BZN ................. 152 6.4.3 Energetics and Structure in other Complex Pyrochlores ........................ 154 6.5 Summary ............................................................................................................. 160 7 ION HOPPING IN BISMUTH PYROCHLORES ...................................................... 162 7.1 Introduction ......................................................................................................... 162 7.2 Calculation Details .............................................................................................. 164 7.3 Results and Discussion ...................................................................................... 166 7.3.1 Bismuth Titanate ....................................................................................... 166 126.96.36.199 Identification of equivalent minima ................................................. 166 188.8.131.52 Ion hopping mechanisms in bismuth titanate ................................. 168 184.108.40.206 Ion hopping pathways in bismuth titanate ...................................... 171 220.127.116.11 Role of displacement correlation in bismuth titanate ..................... 173 7.3.2 Bismuth Zinc Niobate................................................................................ 174 7.3.3 Calcium Titanium Niobate ........................................................................ 177 7.4 Summary ............................................................................................................. 181 7.5 Future Efforts ...................................................................................................... 182 8 SURFACE CHEMISTRY ON LOW MILLER INDICES OF STRONTIUM TITANATE ................................................................................................................. 183 8.1 Motivation ............................................................................................................ 183 8.1.1 Project Overview ....................................................................................... 183 8.1.2 Background ............................................................................................... 183 8.1.3 DFT Component ....................................................................................... 184 8.2 A First -Principles Study of H2O Adsorption and Dissociation on the SrTiO3(100) Surface .............................................................................................. 185 8.2.1 Introduction ............................................................................................... 185 8.2.2 Calculation Details .................................................................................... 188 8.2.3 Results and Discussion ............................................................................ 190 8. 2.3.1 Half -monolayer water on SrTiO3(100) ............................................ 190 18.104.22.168 Dilute coverage of water on SrTiO3(100) ....................................... 195 22.214.171.124 Vibrational modes of water on SrTiO3(100) ................................... 206 126.96.36.199 Discussion on potential sources of DFT experiment disagreement ............................................................................................ 212 8.2.4 Summary ................................................................................................... 216 9 CONCLUSIONS ........................................................................................................ 218 9.1 Summary of Dissertation .................................................................................... 218 9.2 Outlook of Projects ............................................................................................. 222 APPENDIX A LONE PAIR FORMATION ........................................................................................ 225
9 B SUPPLEMENTAL MATERIAL FOR PYROCHLORES ........................................... 227 B.1 Dipole Moments ................................................................................................. 227 B.2 Structure Optimization ....................................................................................... 228 B.3 Charge Density Maps ........................................................................................ 229 B.4 Elec tron Localization Function Plots ................................................................. 230 B.4.1 Bi2Ru2O7 ................................................................................................... 230 B.4.2 Bi1.5ZnNb1.5O7 ........................................................................................... 231 LIST OF RE FERENCES ................................................................................................. 234 BIOGRAPHICAL SKETCH .............................................................................................. 244
10 LIST OF TABLES Table page 1 -1 Calculation details for the various simulations discussed throughout dissertation. ............................................................................................................ 42 2 -1 The percent of total lattice energy due to the A2O and B2O6 networks, and the interaction between the two subnetworks for the six simple bismuth pyrochlores. ............................................................................................................ 55 3 -1 The cubic lattice constant, oxygen positional parameter bond lengths and angle s for Bi2Rh2O7, Bi2Ti2O7, Bi2Ru2O7, Bi2Ir2O7, Bi2Pt2O7, and Bi2Os2O7.. ....... 69 3 -2 The energy change and average displacement magnitudes of simple pyrochlores after simulated annealing with respect to the ideal structure.. ......... 76 3 -3 A comparison of the crystallographic parameters of Bi2Ti2O7 obtained from two experimental studies [23,28] and the results from simulated an nealing at a lattice constant of 10.34 .. ................................................................................ 81 4 -1 The cubic lattice parameter and average bond lengths of the ideal structure i s shown for the Bi2Ti2XX6 and Bi2Ru2OX6 pyrochlore systems. ...................... 105 4 -2 The cubic lattice parameter energy change, and average displacement magnitudes are shown for the Bi2Ti2OO6, Bi2Ti2SO6, Bi2Ti2OS6, and Bi2Ru2OS6 pyrochlore systems.. ......................................................................... 107 4 -3 Bader atomic charges for the Bi, B, X, and X atoms are shown for the Bi2Ti2OO6, Bi2Ti2SO6, Bi2Ti2OS6, Bi2Ru2OO6, and Bi2Ru2OS6 pyrochlore systems. ................................................................................................................ 112 5 -1 Complete crystallographic parameters for ideal Bi2Ru2O7 from DFT for space group Fd 3 m compared with refinement for Bi1.9O0.9Ru2O6 from neutron diffraction data. ..................................................................................................... 128 5 -2 Average bond lengths in the ideal and displaced structure for the various bismuth ruth enate configurations. ....................................................................... 130 5 -3 Average bond valence sums for each atom type in the ideal and displaced structure for the various bismuth ruthenate configurations. ................................ 134 6 -1 The total lattice energy before and after atomic relaxation, as well as the average bond l engths and atomic displacements for BZN separated into the four A2O network types. ...................................................................................... 153 6 -2 The average bond lengths and atomic displacements for BZN, BZT, BMN, and BMT compounds. .......................................................................................... 158
11 7 -1 The displacement magnitude for each atom type within the CTN compound. .. 179 8 -1 Adsorption energy of water at on various high symmetry sites on the SrO and TiO2-terminated SrTiO3(100). ....................................................................... 191 8 -2 Bond lengths for the most stable water adsorbate configurations at 1/8 and 1/2 ML coverage on the SrO and TiO2-terminatated SrTiO3(100) surfa ce. ...... 192 8 -3 T he average Bader atomic charges for Sr, Ti, H, and O within the oxide and water, are shown for molecular and dissociative water on both TiO2 and SrO terminations. ......................................................................................................... 205 8 -4 DFT -derived vibrational frequencies of isolated water, molecular and dissociative water at the favored site on the 22 and 11 SrO and TiO2termination ........................................................................................................... 209 8 -5 Bulk lattice parameter of SrTiO3 and the adsorption energy for molecular and dissociative H2O at 1/8 ML on the TiO2 terminated SrTiO3(100) surface for various exchange -correlation functionals. ........................................................... 213 B-1 The polarization vectors and total polarization are summarized for the ideal bismuth titanate as well as the ten configurati ons with atomic displacement discussed in Section 188.8.131.52. ............................................................................... 228
12 LIST OF FIGURES Figure page 1 -1 Schematic of the four polarization mechanisms reproduced from Figure 2 -25 from Ref. . ......................................................................................................... 26 1 -2 Tw o dimensional schematic illustrating periodic boundary conditions. (Reproduced from Ref. .) ................................................................................. 32 1 -3 Schematic showing iterative nature of the DFT calculations. Reproduced in part based on Ref. . .......................................................................................... 38 1 -4 Figure 1 from Ref  showing the potential energy pr ofile between two minima labeled initial and final. The minimum energy path (MEP) as well as the nudged elastic band (NEB) path are shown between the two minima. ... 46 2 -1 Bulk unit cell of SrTiO3 shown near the <100> direction. ..................................... 50 2 -2 Sr centered bulk unit cell of SrTiO3 shown near the <100> direction. .................. 51 2 -3 The ideal A2B2O6O pyroc hlore structure and the two interpenetrating subnetworks A2O tetrahedra and B2O6 octahedr a .................................................... 52 2 -4 Local structure of Bi and B cations within ideal pyrochlore structure, viewed along the  and  direction. ....................................................................... 56 2 -5 Site symmetry of Bi cations within ideal pyrochlore structure, viewed along the  a nd  direction. ................................................................................. 57 2 -6 -cristobalite type displacive disorder. ................ 60 3 -1 The ideal A2B2O6O pyrochlore structure and the two interpenetrating subnetworks A2O tetrahedra and B2O6 octahedra. .................................................... 64 3 -2 The dependence of the cubic lattice constant and the oxygen positional parameter on the B4+ cation radius as determined by DFT and experiment for Rh , Ti [23, 28], Ru , Ir , Pt , and Os . ...................................... 70 3 -3 Bond lengths for Bi -O, Bi-O48 f and B -O48 f as a function of cation radius as determined by DFT and experiment for Rh , Ti [23,28], Ru , Ir , Pt  and Os . .................................................................................................... 73 3 -4 The ideal puckered ring created by the six O atoms surrounding Bi with O shown perpendicular to and along the O -Bi -O chains ........................................ 79
13 3 -5 The Bi bonding networks for the two experimental studies with the six O atoms surrounding the Bi sites for the stoichiometric case  and the deficient case  shown perpendicular and along the O Bi -O chains. ............. 80 3 -6 The 96h 96 g and 192 i displacement sites of the Bi cations observed from simulated annealing DFT of Bi2Ti2O7. .................................................................. 83 3 -7 The Bi -O bonding network determined i n all ten simulated annealings. ............. 86 3 -8 Bader atomic charges for the Bi, B, O48f and O ion presented for ea ch of the bismuth pyrochlores. ............................................................................................. 93 3 -9 Density of states for Bi2Ru2O7, Bi2Pt2O7, and Bi2Ti2O7 in the ideal cubic pyrochlores structure with no cation displacement. ............................................. 94 3 -10 Partial density of states for the Bi2Ti2O7 pyrochlore before and after displacement and volume optimization and the Bi2Ru2O7 pyrochlore ................. 96 3 -11 The electron localization of the displaced Bi2Ti2O7 for the (111) and the 1 11 plane. ...................................................................................................................... 98 4 -1 The electron localization of the Bi2Ti2OO6 with atomic displacem ent for part of the (110) plane ............................................................................................... 110 4 -2 The electron localization of the Bi2Ti2SO6 with forced atomic displacement for part of the (110) plane .................................................................................. 111 4 -3 The electron localization of the Bi2Ti2OS6 with atomic displacem ent for part of the (110) plane. ................................................................................................ 111 4 -4 Partial density of states for the Bi2Ti2O7 pyrochlore before and after atomic dis placement and lattice expansion. .................................................................... 114 4 -5 The part ial electron densities for the Bi2Ti2OO6 with atomic displacement and Bi2Ti2SO6 with forced atomic displacement for the states between 2 eV to the Fermi level for part of the (111) plane. .......................................................... 115 4 -6 Partial densit y of states for the Bi2Ti2SO6 pyrochlore before and after forced atomic displacement. ........................................................................................... 117 4 -7 Partial density of states for the Bi2Ti2OS6 pyrochlore before and after atomic displacement. ....................................................................................................... 118 4 -8 The partial electron densities for Bi2Ru2OS6 with atomic displacement for the states between -2 eV to the Fermi level for part of the 1 01 plane. .................. 120 4 -9 The electron localization function for Bi2Ru2OS6 with atomic displacement for part of the 1 01 plane. ......................................................................................... 121
14 5 -1 The Bi2O0.875 subnetwork viewed along the [ 1 1 0 ] and [ 001 ] dir ection. ............. 126 5 -2 A local O -Bi -O bonding network with the three possible Bi and two O types of displ acements reported in Table 51. .............................................................. 127 5 -3 Part of the (111 ) plane of the the VO Bi3O VBi ''' configuration showing the electron localization function and th e equivalent structure plane. ...................... 136 6 -1 The A2O network of oxygen centered tetrahedra with four types of ZnA substitution. .......................................................................................................... 144 6 -2 The relaxed energy scaled by the average of the most favored A2O network type for each of the 120 structures studied. ....................................................... 147 6 -3 The relaxed energy scaled by the average of the most favored A2O network type for each of the 120 structures studied shown for each of the four types of the A2O network. ............................................................................................. 148 6 -4 The unrelaxed energy scaled by the average of the <211> A2O network type for each of the 120 structures studied. ............................................................... 150 6 -5 The unrelaxed energy scaled by the average of the <211> A2O network type for each of the 120 structures studied shown for each of the four types of the A2O networ k. ........................................................................................................ 151 6 -6 The relaxed energy scaled by the average of the most favored A2O network type plotted versus the average ZnA to O bond length and O displacement for each of the 120 BZN configurations studied. ................................................ 152 6 -7 Energy difference between [2:2] and <211> A2O configuration sets for BZN, BZT, BMN, and BMT. ........................................................................................... 155 6 -8 Energy difference between <211> and <011> A2O configuration sets for BZN, BZT, BMN, and BMT. ................................................................................. 156 6 -9 The average displacement for the A and O site in the BZN, BMN, BZT, BMT, and BZS compounds in o rder of increasing experimental dielectric constant. .. 160 7 -1 The scaled energy for each of the six positions with the labels s1 through s6 corresponding to the structural view on the right. ............................................... 168 7 -2 T he A2O tetrahedra network for three energetically equivalent 96g sites. A single site hop and double site hop jump is represented. ................................... 169 7 -3 Energy profile for the single site hop mechanisms fo r Bi2Ti2O7. ......................... 170 7 -4 Local structure, including intermediate structures for the s2 -s3 single site hop mechanism. ................................................................................................... 172
15 7 -5 -cristobalite type displaciv e disorder in bismuth titanate. ................................................................................................... 174 7 -6 The O -Bi -O local bond angle of the pushed Bi cation is shown for a single site hop mechanism in Bi2Ti2O7 and BZN. .......................................................... 176 8 -1 The bulk unit cell of SrTiO3 highlighting the SrO termination and TiO2 termination of the SrTiO3(100) surface. ............................................................... 186 8 -2 The top layer of the SrO -terminated and TiO2-terminated SrTiO3(100) surface showing the 11 (22) surface cell outlined The high symmetry adsorption sites examined are identified. .............................................................................. 190 8 -3 Most favored half monolayer molecular and dissociative water configuration on 11-SrO and 11-TiO2 termination as viewed along the  direction and the [ 100] direction. ......................................................................................... 192 8 -4 Most favored 1/8 monolayer molecular and dissociative water configuration on 22 SrO and 22 TiO2 terminatio ns viewed along the  direction and the  direction. ................................................................................................ 196 8 -5 Eads of the most favored molecular and dissociative w ater configuration at varying coverage on the 22-TiO2 termination. .................................................. 199 8 -6 The three possible configurations of dissociatively adsorbed water at 1/4 ML coverage on the 22 TiO2 termination viewed along the  and  dire ctions ............................................................................................................. 200 8 -7 Top layer of TiO6 octahe dra for the 22 surface unit cell of TiO2-terminated with 1/2 ML of dissociated water on the Ti site viewed along the  and . ..................................................................................................................... 201 8 -8 The three possible configurations of dissociatively adsorbed water at 1/4 ML coverage on the 22 SrO termination viewed along the  and  directions ............................................................................................................. 203 B-1 The total charge density of the displaced Bi2Ti2O7 and the electron localization function for part of the 1 01 plan e. ................................................. 230 B-2 The electron localization of the displaced Bi2Ti2O7 and Bi2Ru2O7 for the (101) plane. .................................................................................................................... 231 B-3 The electron localization of the displaced BZN (111) plane and the 1 11 plane. .................................................................................................................... 232
16 LIST OF SYMBOLS displacement magnitude 0 dielectric permittivity of free space i eigenvalue for modified Hamiltonian r relative dielectric permittivity electron density i single electron wave-function n set of eigenstates o pre exponential factor for rate constants a number of cations B bond valence sum constant d(A X) bond length between cation A and anion X Eads adsorption energy EA2O energy of isolated A2O subnetwork EB2O6 energy of isolated B2O6 subnetwork EH2O gas energy of isolated water molecule EH2O slab total energy of water adsorbed on SrTiO3 slab Einteraction energy of interaction between A2O and B2O6 sub -networks En set of eigenvalues Eslab total energy of relaxed bare SrTiO3 surface or slab Etotal e nergy of total A2B2O7 lattice eV electron volts H Hamiltonian operator ki coordination number of cation i me mass of an electron
17 M mass of nucleus n number of molecules ni charge of cation i N total number of electrons P polarization r spatial position of an electron R spatial position of nucleus Rij bond length of each bond R0 bond valence sum reference bond length si bond strength of cation i Sij bond valence of each bond Tm temperature of maximum dielectric permittivity Tp desorption temper ature VBi ''' vacancy on Bi site Veff effective potential Vext external potential VHartree Hartree potential Vi valence of ion i from bond valance sums VO vacancy on O site VXC exchange -correlation potential x oxygen positional parameter zj ionic charge of anion j Z charge of nucleus
18 LIST OF ABBREVIATION S BMN (Bi1.5Mg0.5)(Mg0.5Ta1.5)O7 BMT (Bi1.5Mg0.5)(Mg0.5Ta1.5)O7 BVS bond valence sum BZN (Bi1.5Zn0.5)(Zn0.5Nb1.5)O7 BZS (Bi1.5Zn0.5)(Zn0.5Sb1.5)O7 BZT (Bi1.5Zn0.5)(Zn0.5Ta1.5)O7 CG conjugate gradient CTN (Ca1. 5Ti0.5)(Ti1.0Nb1.0)O7 DFT density functional theory ELF electron localization function EXAFS e xtended X-ray absorption fine structure GGA generalized gradient approximation GULP Generalized Utility Lattice Program HF Hartree Fock HREELS high resolution electron energy loss spectroscopy IR infrared LCAO lin ear combinations of atomic orbitals LDA local density approximation LEED l ow energy electron diffraction (LEED) LMTO linear muffin tin orbital MD molecular dynamics ML monolayer NEB nudged elastic band NVT canonical ensemble
19 Ow oxygen within water molecule Ox oxygen within oxide PAW projector augmented wave PBC periodic boundary condition PBE Perdew -Burke -Emzerhof parameterization pDOS partial density of states RPBE revised Perdew -Burke-Emzerhof parameterization RUM rigid unit mode STM scanning tunneling microscopy STO SrTiO3 XPS X -ray photoemission spectroscopy XRD X -ray diffraction TPD temperature programmed desorption UHV under ultra high vacuum UPS u ltraviolet photoemission spectroscopy VASP Vienna Ab initio Simulation Package ZPC zero point correction
20 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ATOMIC STRUCTURE EFFECTS ON BULK AND SURFACE PROPERTIES OF MIXED METAL OXIDES FROM FIRST PRINCIPLES SIMULATIONS By Beverly Brooks Hinojosa August 2010 Chair: Aravind Asthagiri Major: Chemical Engineering Oxides are a class of materials that play a critical role in many applications, includi ng both electronic and catalytic The hundreds of different oxides that vary in structure or chemical composition offer the opportunity to tune these material s for specific applications but t o r ational ly design these oxide materials requires understanding the structure-property relationships T his dis s ertation present s modeling work based on density functional theory (DFT) on two oxides to understand the structure-property relationships The predominant work of the dis s ertation fo cuses on examini ng the role of atomic displacements, oxygen vacancies, and cation substitutions in bismuth based pyrochlores. The second project involves the study of surface chemistry on SrTiO3 surfaces. Pyroc hlores have the chemical formula A2B2O7 and t he structure allows these materials to be tuned with different choices of cations on the A and B cation site. Bi based pyrochlores have received attention over the past decade for use in capacitor and high -frequ ency filter applications with extensive work focused on tuning material properties through cation substitutions. However, the myriad of effects induced by
21 cation substitutions are still not entirely understood and the relationship between cation substituti on and dielectric properties needs further elucidation. In this work DFT simulations explore the link between atomic structure and atomic displacements in Bi pyrochlores. A range of cubic Bi pyrochlores with no cation substitutions were explored and sig nificant displacement s were found only in Bi2Ti2O7. By examining the electronic structure, the formation of lone pairs on the Bi cation was found to induce cation displacement in Bi2Ti2O7. It is show n that oxygen vacancies induce cation displacement in bis muth ruthenate, resolving a discrepancy between oxygen -deficient experimental results and the DFT results of non defective bismuth ruthenate. A relationship between DFT -derived cation displacement and experimentally measured dielectric constants was found in a range of cation substituted Bi pyrochlores, Bi1.5M2+B5+ 1.5O7 (with M2+=Zn or Mg and B=Nb or Ta) Finally dielectric relaxation is determined by the ease of cation hopping between energetically degenerate positions within the pyrochlore str ucture. To explore this hopping process, a preliminary study was completed to identify transition states to cation hopping and associated energy barriers for Bi2Ti2O7, Bi1.5ZnNb1.5O7, and Ca1.5Ti1.5NbO7.
22 CHAPTER 1 INTRODUCTION 1.1 Motivation Metal oxides play an important role in modern microelectronics, catalysis, and numerous other technologies. As a class of materials metal oxides are ubiquitous partly because of their diversity. Metal oxides can be found that are highk insulators, metalli c, semiconductors, ferromagnetic, piezoelectric, ferroelectric, ferroplastic, and multiferroic. This list is a subset of the range of material properties that oxides can have depending on their composition and structure. Mixed metal oxides, such as that found in the perovskite (ABO3) and pyrochlore (A2B2O7) structures, have even a broader range of tunable properties due to the larger parameter space associated with the combination of two or more cations. Mixed metal oxides can also show both covalent and ionic nature in the distinct metal oxygen bonds. For instance in SrTiO3 the Sr -O bonds are mainly ionic while the Ti -O bonds show significant covalent bonding  The presence of this additional heterogeneity in mixed metal oxides provid es the possibility to tune the macroscopic properties through the choice of cations. For example, within the ABO3 perovskite structure family a proper choice of the A and B cation can lead to a material that is ferroelectric ( e.g. PbTiO3) versus paraelectr ic ( e.g. SrTiO3) at room temperature. This large difference in the material property between PbTiO3 and SrTiO3 can be attributed to the covalent nature of the Pb -O bond in PbTiO3 versus the purely ionic Sr O bond in SrTiO3 [1,2] As will be seen in the work presented in this dissertation the choice of cations in mixed metal oxides and the interaction between the subnetworks in these oxide structures can dramatically affect the material behavior.
23 In this dissertation, ab initio quantum based simulations are presented on two distinct projects to help elucidate the structure property relationships in two classes of mixed metal oxide structure families: perovskites and pyrochlores. This dissertation will focus on the pyrochlore system with t he details of the work on SrTi O3 surfaces discussed in Chapter 8. 1.1.1 Motivation for the S tudy of Bismuth Pyrochlores Pyrochlore oxides have received much attention the past 30 years due to their diverse physical properties, which allow for a broad range of applications. Recently pyrochlores have been considered for use in applications such as high-permittivity dielectric s  solid electrolytes in solidoxide fuel cells  potential host materials for the immobilization of plutonium and nuclear waste [5,6] and therm al barrier coatings [7 9] As will be discussed in further detail in Chapter 2, the ideal stoichiometric pyrochlores (A2B2O7) are cubic with space grou p Fd 3 m (number 227). It is often described as two interpenetrating sub networks of A4O tetrahedra and BO6 octahedra  In fact, the overall formula is typically represented as A2B2O6O to distinguish the oxygen within the two network s. This representation is one of a multitude of perspectives of the pyrochlore structure found in the literature (see Ref. [10 -14] for more extensive discussion on pyrochlore structure in addition to Chapter 2). The various representations of the structure are a reflection of the general complexity of pyrochlores, and the most helpful representation is often dependent on the particular property of interest. The complexity of the pyrochlore structure is also central to the great diversity in properties that make these materials so interesting.
24 Extensive experimental work on the synthesis and structural characterization of cubic pyrochlores have established stability fields for allowable substitution into the A and B cation site based on simple measures such as the ratio of the cation radii size and the cation electronegativity [12,15] The typical A (B) cation radii is around 1 (0.6) but variations of up to 0.2 for both cat ions have been stabilized. As the cation radii shift away from the typical values noted, the range of A/B substitutions that can stabilize into a stoichiometric cubic structure decreases sharply  Recently, it has been shown that the sta bility field of allowable cation radii rations can be extended, but these structures incorporate atomic displacements, while still retaining the cubic symmetry macroscopically. Both experimental and theoretical work has focused on finding optimal A/B combi nations for particular applications. For example, a recent extensive computational study focused on the affect of A and B cations on the thermal conductivity of the pyrochlores to develop optimal TBCs  While ideal (or simple) cubic pyrochlores are important, there has been a growing interest in more complex pyrochlores involving substitutions on the A and B cation sites. Substitutions on the cation site allow for a much broader range of stable cubic pyrochlores, and a considerable number of novel complex pyrochlores have been synthesized and studied experimentally. The Bi1.5ZnNb1.5O7 (BZN) system is one key example of these substituted pyrochlore compounds that has been s tudied extensively due to a high dielectric constant, relatively low dielectric losses, compositionally tunable temperature coefficient of capacitance and a sintering temperature of less than 950C  These properties are ideal for applications such as capacitors and high-frequency filter application  The dielectric properties of such materials are of particular
25 technological interest [3,16] Some bac kground on the behavior of dielectrics is presented below in Section 1.1.2 with further details provided in Ref. [17,18] 1.1.2 Dielectric Behavior in Bismuth Pyrochlores The dielectric behavior of an insulator is dictated by the response of the material to concentrate an applied electric field within itself by redistributing charges within the atoms or molecules of the material. This response has four contributions: atomic ( electronic ), ionic, dipolar, and space charge polarization as illustrated in Figure 11 [11,18] The total dielectric permittivity ( ) is often scaled by the dielectric permittivity of free space ( 0) to define the relative dielectric permittivit y (often simply called the dielectric constant) which is written as r or k. Each of the four polarization processes present in a material will contribute to the total dielectric permittivity. Since t he mechanism of charge rearrangement for each process v aries in both their length and time scales t hese four mechanisms will be discussed in more detail from the smallest to largest length scal es. In the atomic, or electronic, polarization process, the spherical electron cloud surrounding an atom under zero electric field conditions displaces relative to the positive nucleus creating the elongated cloud illustrated in Figure 1-1. The atomic mechanism is not sensitive to temperature variations. Since this mechanism is dependent on only the movement of electrons, atomic polarization occurs at frequencies up to ~1015 1017 Hz in all gas es, liquids, and solids. T herefore this mechanism is expected to play a role within the bismuth pyrochlores. The atomic mechanism occurs at the highest frequencies of the four types so at frequencies above ~ 1018 Hz the dielectric permittivity of any material should equal the dielectric permittivity of free space 
26 Figure 11. Schematic of the four polarization mechanisms reproduced from Figure 225 from Ref.  The ionic mechanism occurs in materials with some ionic character where the ions are bound elastically to equilibrium positions. Under an applied electric field, t he anions and cations displace towards the positive and negative electrodes respectively The displaced positions of the ions are not equilibrium positions; therefore, following the removal of the electric field, the ions will return to t heir original equilibrium positions. The ionic mechanism is also independent of temperature. Due to the larger length scale of the ions versus electrons, the ionic mechanism drops off at lower frequencies than the atomic (electronic) mechanism. T he ionic m echanism occurs up to frequencies ~ 1012 Polarization processUnpolarized state Polarized state EAtomic Ionic Dipolar Space Charge
27 1013 Hz and at frequencies above this the ionic contribution to the relative dielectric permittivity goes to 1  The ionic mechanism is expectedto occur in bismuth pyrochlores. The dipolar mechanism will only occur in materials with equivalent positions separated by an energy barrier that the ions may occupy with equal probability in the absence on an electric field. The ferroelectric t etragonal PbTiO3 is an example of such a material, where the Ti cation is shifted up or down with respect to the center position. With the Ti cation in eithe r the up or the down position, a dipole moment is created with respect to its six oxygen anions neighbors. Either the up or down position is equally occupied in the absence of an applied field, but when an electric field is applied the Ti cation will shift preferentially to have the dipole moments align with the electric field wherever possible. As illustrated in Figure 1-1, not every dipole moment will align based on the structure of the system and the location of equivalent positions with respect to the applied electric field. The dipolar mechanism occurs at low frequencies and typically drops off between ~108 109 Hz. Materials that have dipolar mechanism typically exhibit relative dielectric constants in the hundreds  Unlike either the electronic or ionic mechanism, the dipolar mechanism will depend on temperature since increasing temperature increases randomness. Increasing the temperature is expected to decrease the dipolar contribution to the static relative dielectric constant [17 ] As will be seen in Section 3.3.2 and 6.4.2, atomic displacement in bismuth based pyrochlores leads to energetically degenerate positions for both the cations and anions. The dipolar mechanism is therefore expected to contribute to the dielectric perm ittivity in bismuth pyrochlores. As will be discussed in Section 7.3.2, the temperature dependence of the
28 dipolar mechanism will allow for the approximation of the energy barrier separating the equivalent positions in BZN. The space charge mechanism shown in Figure 1-1 involves the rearrangement of charges within isolated r egions of the material. T his mechanism requires two phases within the material and will result in the accumulation of charges at the interface(s) of these regions. The space charge mechanism occurs at low frequencies, dropping off above ~ 102 Hz. Space charge is often identified in materials with dielectric constants in the thousands, such as BaTiO3 where a static relative dielectric constant is reported of 3000  Based on the dielectric data reported on bismuth pyrochlore, the space charge mechanism is not expected to contribute to the relative dielec tric permittivity. The dielectric properties are linked to the mechanism of rearrangement, which in turn depends on the atomic structure. Therefore, correlating atomic structure and dynamics to observed dielectric behavior is a critical step in realizing the promise of pyrochlores in next -generation electroceramic materials. The structure can be probed through experimental methods (see Section 1.2 for a survey of these efforts), but computational modeling can provide an inexpensive and effective form of investigating these materials at the atomic level. The goal of the work outlined here is to expand the understanding of the structure property relationships for Bi -based pyrochlores through simulations. The modeling will use first -princ ip les calculations, which may probe the role of electronic structure and allow predictive simulations of Bi -based pyrochlores. The fundamentals of these simulations are discussed in Section 1.3. The computational modeling will be an important complement to o n going experimental work in this area.
29 1.2 Literature Review A brief survey of the experimental work examining Bi pyrochlores along with efforts to model pyrochlores is present here Ideal bismuth pyrochlores structural and electronic properties have be en studied experimentally over the past three decades [15,19 -28] These pyrochlores have seen deviations from the ideal A2B2O7 structure with Bi and O deficiencies and Bi and O displacements. The known simple bismuth pyrochlores are bismuth ruthenate (Bi2 xRu2O7 y) [1922,24] bismuth rhodate (Bi1.95Rh2O6.83)  bismuth platinate (Bi2Pt2O7) [15,29] bismuth iridate (Bi1.9Ir2O6.8) [21,26] bismuth osmate (Bi2Os2O7)  and bismuth titanate (Bi2 xTi2O7 y) [23,25,28] It is known that bismuth ruthenate exists is a non-stoichiometric ideal pyrochlore Bi2 xRu2O7 y [20 -22,24] and forms stoichiometric solid solution pyrochlore with the general formula A2(Ru2 xAx)O7  or Bi2 xMxRu2O7 y [20 ] and has therefore been studied extensively. There have been computational studies of the pyrochlore system for various A and B cations mainly focused in the area of interatomic potentials fitted to experimental lattice parameters [6,9,3033] The lanthanum and yttrium family of pyrochlores were studied using DFT recently to identify the structural, elastic and bonding properties of these systems  and results compared well with available studies. A complete study of bismuth stannate, a distorted pyrochlore, included a joint experimental and theoretical approach to understanding the bonding and lone pair activity in the material  Due to the bismuth displacement and deficiency in bismuth titanate, it has been proposed as a model system to understand the properties of bismut h pyrochlores  A linear muffintin orbital (LMTO) study was performed examining the density of states and the valence electron localization function (ELF) to clarify the bonding within the Bi2O network. One
30 other LMTO study involved bismuth ruthenate pyrochlore which has also been studied extensively since the compound allows for substitution on either the A  or the B  site and sh ows potential application in solid oxide fuel cells (SOFCs)  The combined experimental and computational study in Ref.  proposes that the bismuth displacement is due to the lone pairs stereochem ical activity on the bismuth. Gaining an understanding of the simple pyrochlores in particular the role of the displacement of the bismuth cation will be important in understanding their complex multi -site substituted counterparts, but predictive computati onal work in this area has been limited. The complex bismuth based substituted pyrochlores have been extensively studied experimentally, due to their ideal combination of characteristics for capacitors and high -frequency filter applications  In the interest of time, this section will not present an exhaust ive description of all experimental work but will focus on those of pertinence to this dissertation One sy stem of particular interest is Bi1.5ZnNb1.5O7 (BZN) which has zinc substituted equally on both the A and B site in the pyrochlore structure BZN has been shown to be both zinc and oxygen deficient and to have disordered displacement on both the A and O site  Experimental studies have used both X ray and neutron powder diffraction techniques to determine the structural properties of these complex compounds [13,37] and recently, a more precise stoichiometry for BZN was determined to be Bi1.5Zn0.92Nb1.5O6.92 (BZN)  Though BZN has received much a ttention, it is certainly not the only bismuth based complex pyrochlore to be studied. Other compounds include Bi3/2ZnTa3/2O7 (BZT), Bi3/2MgNb3/2O7 (BMN), Bi3/2MgTa3/2O7 (BMT), and Bi3/2ZnSb3/2O7 (BZS) to name a few [37,38] The dielectric properties of ten of these types of compounds have been examined  and the phonon modes for BZN,
31 BZT, BMN and BMT have been determined  Based on neutron diffraction data of BZN, short range ordering of metal ions and strain induced relaxation has been proposed  The increased accuracy of structure resolution of complex pyrochlores in the last decade along with extensive spectroscopic studies have greatly increased the atomic level understanding of these pyrochlores and the combination of these probes with predictive atomistic simulations should assist in the ultimate goal of understanding and predicting structure property relationships for complex pyrochlores. T his section is concluded by noting that there is a recent trend towards exploring complex pyrochlores with multiple substitutions such as (Bi1.5Zn0.5)(Ti1.5Nb0.5)O7 (BZNT)  which reinforces the need for dependable predictive simulations of these materials to examine the large parameter space. 1.3 Simulation Methods With the initial goal of performing multi -scale modeling, the simulation techniques to be used were density functional theory (DFT) and molecular dynamics (MD). The DFT calculations were to provide an accurate basis for constructing interatomic potentials for the MD simulations. The DFT calculations proved more elaborate and provided more information about these materials than initially expected and therefore are the only calculations discussed in this work. Fu ture efforts on the development of the interatomic potentials will be discussed in Chapter 9. DFT is quantum mechanical calculations that approximately solve the Schr dinger equation without requiring any experimental input. T he commercially available program called Vienna Ab initio Simulation Package (VASP) was implemented in order to complete these calculations.
32 1.3.1 Periodic Boundary Conditions Within VASP, periodic boundary conditions (PBC) are applied in all three directions; therefore, calculations that replicate a large structure may be performed with the smallest unit of symmetry repeated infinitely in all directions. Figure 1 -2 shows a two dimensional schematic of the application of PBCs. The central box is the actual simulation cell that is inpu tted by the user. Following PBCs, the eight additional cells are created around the central cell. Each element within the central box is repeated in each additional image and these replicated elements are used when the atomic forces are calculated for each element within the original unit cell. Figure 1 2. Two dimensional schematic illustrating periodic boundary conditions. The application of PBCs leads to the circle element translating from one corner of the simulation box to another, as illustrated b y the arrows. (Reproduced from Ref.  .) As illustrated by the arrows in Figure 1-2, an element t ranslated within the original cell to reduce the forces is moved within all images. Additionally, if the translation involves the element crossing one or more of the simulation boundaries that elements Actual System Replica Replica Replica Replica Replica Replica Replica Replica
33 image will translate into the simulation box from the other side, as illustrated by the circles and arrows in Figure 12 See Ref.  for additional details regarding PBCs. The PBC approach works very well for structures with repeating units of symmetry such as bulk materials, highly ordered surfaces (assuming an appropriate amount of vacuum space is included betw een the top and bottom of the slabs in the surface normal direction), and even in straight polymers without cross -linking. The introduction of cross -linking in polymers or even polymer folding onto itself leads to a break in symmetry which would require larger base unit cells to describe the structure. As DFT is the focus of this work, a summary of the fundamentals will be discussed below in Section 1.3.2. A practical introduction to DFT that is ideal for beginners or experimentalists can be found in Ref.  and a more detailed discussion is presented in Ref.  1.3.2 Fundamentals of Density Functional Theory One of the benefits of DFT is the variation principle which states that a total energ y cannot be found that is lower than the true ground state; therefore, the true ground state energy cannot be undershot  While it should be noted, this principle does not guarantee that the current lowest energy state is th e global minimum it does provide the basis to compare configurations. Thus, t his principle allows users to compare the energies for different configurations of the same atoms to make predictions about the most stable state that is likely to be present in an experimental apparatus. This implies a key component of DFT calculations is determining the energy associated with a particular structure. The details of how these calculations are completed are detailed in the sections to follow below.
34 184.108.40.206 Schr di nger equation The basis of these calculations begins with the time-independent Schr dinger equation (Equation 1 -1) and following a few approximations the problem is computationally solvable for solid -state systems [ 41] H = E (1.1) The Schr dinger equation is an eigenvalue problem where H is the Hamiltonian operator, and is a set of eigenstates (solution) with a particular set of eigenvalues En associated with a particular solution, n. For a many -bod y problem with multiple electrons and nuclei, the Hamiltonian operator is quite complex and is defined in Equation (1.2). H = -22 me i 2+ ZIe2 ri-RI +1 2 e2 ri-rj 22 MI I 2+1 2 I i I i = 1ZIZJe2 RI-RJ (1.2) In Equation (1.2), lower (upper) case variables and subs cripts are used for electrons (nuclei), and me(M) is the mass, r (R ) is the spatial position of an electron (nucleus), and Z is the charge of a nucleus. The five terms on the right hand side of Equation (1.2) are the kinetic energy of each electron, interac tion between each electron and nucleus, the electronelectron interaction, the kinetic energy of each nucleus, and the nucleus -nucleus interaction, respectively  It is known that the mass of an electron (me) is 1800 times smaller t han the mass of a nucleus (M) and therefore an electron will respond to changes in its surroundings much faster than a nucleus. This is the basis of the Born-Oppenheimer approximation  This approximation allows for the separation of the electron and nucleus movements, which separates the mathematical solutions. The ground state of the electrons is solved for a fixed nuclei structure and the corresponding simpler Hamiltonian operator is
35 defined in Equation (1.3). The external potential, Vext, includes the effect of the nuclei on the electrons. H = -22 me i 2+ Vext( ri) +1 2 e2 ri-rj i i = 1 (1.3) The proof of two theorems by Hohenberg and Kohn lead to the formulation of DFT used today  It was first confirmed that all properties of a quantum many -body system, including the ground state energy, are unique functionals of the corresponding ground-state electron density. This lead to the replacement of the wave function, w h ich is a function of 3N variables (where N is the total number of electrons in the system), with the electron density, which is a function of only three variables (the spatial variables). This is especially important when considering the number of electrons in cluding in the calculations presented within this work. For the bismuth pyrochlores the number of electrons exceeds 700, so this theorem reduces the dimensions from 21,000 to 3. The second theorem proves total ground state energy cannot be underestimated. An electron density cannot be found that is different than the ground state electron density and results in a lower ground state energy. Therefore, i f the true functional form of the electron density were known, the exact solution to the Schr dinger equat ion could be found by varying the electron density until the ground state energy was minimized. Instead, this variational principle is combin ed with approximate forms of the functional leading to approximate solutions to the Schr dinger equation. 220.127.116.11 KohnSham equations The calculations presented within this dissertation are based on the formulation of DFT presented by Kohn and Sham shortly after the proof of the two theorems de t ail ed
36 in Section 18.104.22.168. The Kohn-Sham equations, of the form shown in Equation 1.4, express the electron density in terms of a set of equations w h ere each equation involves a single electron. -h22 me 2 + Veff( r ) i( r ) = ii( r ) (1.4) These equations break the operator into two components, the first is the kinetic energy of the electrons and the second is the effective potential defined in Equation 1.5. The effective potential combines the external potential (Vext), which was defined in Section 22.214.171.124, with the Hartree potential (VHartree) and the exchange -correlation potential (VXC)  Veff( r ) = Vext( r ) + VHartree[ ( r ) ] + VXC[ ( r ) ] (1.5) For each Kohn-Sham equation, the Hartree potential defines the Coulomb repulsion between the single electron involved in the current Kohn Sham equation and the total electron density dependent on all the electrons within the system. Since the electron being examined in the current KohnSham equation contributes to the total electron density, the Hartree potential includes the repulsion between the electron in the current Kohn-Sham equation and itself within the total electron density. This self interaction effect is unrealistic and is therefore cor rected within the exchangecorrelation potential  The exchange -correlation will be discussed in more detail in Section 126.96.36.199. 188.8.131.52 Iterative approach to find solutions to Kohn-Sham equations As may be obvious, the solution to the KohnSham equations is not as simple as a set of N independent eigenvalue problems. To solve the Kohn-Sham equations the complete operator, seen in Equation (1.4 ), must be known. However, the VHartree
37 potential depends on the electron density (defined in Equation 1 6), which is found after solving the Kohn -Sham equations for the singleelectron wave-functions ( i) To overcome the circular nature of these equations, the iterative approach outlined in Figure 13 is taken in this work  ( r ) = i( r ) 2 i (1.6) The approach begins with the definition of the initial structure and user determined parameters to set the level of accuracy for the calculation. Then an initial or tri al electron density is approximated and used to approximate the operator to solve the KohnSham equations to find the wave functions, i. With the wave functions known, the new electron density is determined from Equation (1.6). The new electron density is compared with the old electron density to determine if they are the same, with respect to a difference parameter that th e user defined If they are the same, the iteration stops, and if not then the current and previous electron densities are combined to create new electron density for the next loop. Once a self -consistent electron density is found, the structure may be evaluated to determine the total lattice energy, the forces and stresses on the material. The atomic positions may be optimized to reduce the forces and the new structure would enter the electron density optimization loop, shown in the shaded dotted inner box o f Figure 1-3 again. These two loops would be repeated until the force criteria assigned by the user was meet  Once the force criteria is reached the simulation stops and the final structure as well as its associated properties are reported. This corresponds to exiting the dashed box illustrated in Figure 13.
38 Figure 1 3 Schematic showing iterative nature of the DFT calculations. T he self consistent loop to solve the Kohn-Sham equations is shown in shaded box outlined by the dotted line. The d ashed line represents the simulation package. Reproduced in part based on Ref.  184.108.40.206 Exchange -correlation functional The exchange -correlation is a potential that corrects for two main components. First the exchange portion considers the energy change of the total system if two electrons with different coordinates are interchanged Second the correlation portion cons iders that the electrons are no longer being described as point particles, but as Self consistent? Set Parameters for Accuracy Define Structure ( r ) Initial Guess for Electronic Density [ ( r )] No Yes Output Quantities Final Structure, Energy Forces, Stresses, etc. Calculate Effective Potential V V V VXC Hart ext eff r r Solve KS Equation r r ri i i eff 2 V 2 1 Calculate Electron Density ( r ) = i ( r ) 2 i Calculate Forces Less than Force Criteria? No Yes Generate New Structure ( r ) by Moving Atoms Based on Ionic Optimization Parameter and Calculated Forces
39 electronic densities. If the electrons were treated as point particles and the position of one point particle changes a second point particle will al so change its position since the two are interacting with each other. When the point particles are replaced with electron densities, changing the position of one density does not have a direct influence onto the position of the second density. The exchange -correlation potential compensates for these two components and an exchange-correlation functional is used to approximate the potential Though the exchange-correlation functional is guaranteed by the Hohenberg -Kohn theorem, the exact form is still not kn own  In the implementation of DFT, various approximations for the exchange-correlation functional are available to allow for the determination of an approximate solution to the Schr dinger equation. The exchange correlation f unctional has been derived for uniform electron gas and is therefore the starting point of the approximations. For the calculations on the pyrochlore system presented in this dissertation the local density approximation (LDA) has been used. For this approximation, the exchange -correlation potential at each point is set equal to the known exchange -correlation potential from the uniform electron gas at that same point. The generalized gradient approximation (GGA) was implemented in this dissert ation for the SrTiO3 surface calculations GGA differs from LDA since in addition to the exchange -correlation potential depending on the uniform gas electron density it also depends on the gradient of the uniform gas electron density  While both of these approximations have been effectively applied to various materials, in general LDA is more accurate on uniform systems such as bulk metals and GGA describes systems such as molecules and surfaces  Therefore, LDA is used for all pyrochlore
40 calculations, discussed in Chapter 3 through 7 and GGA is used for the examination of SrTiO3 surfaces, discussed in Chapter 8. 1.3.3 Calculation Details All of the calculations discussed within this dissertation were performed with Vienna Ab initio Simulation Package (VASP) [44 -47] a planewave DFT code, using the projector augmented wave (PAW) pseudopotentials provided in the VASP database [48,49] There are several user -defined parameters within VASP to aid in examining the system s. While these parameters were kept relatively unchanged throughout this dissertation, there were some changes made between the various chemical systems examined. The parameters chosen for each system will be detailed here. If a parameter is not mentioned, the VASP default was used. The parameters that were varied for the results reported in this dissertation are summarized for each system in Table 11. The DFT calculations were performed within the local density approximation (LDA)  for all pyrochlores systems. The generalized gradient approximation (GGA) with the parameterization of Perdew et al  was used to examine surface chemistry on SrTiO3, detailed in Chapter 8. The Bi(5d, 6s 6p), Rh(4p, 4d, 5s) Ti(3s, 3p, 3d, 4s), Ru(4p, 4d, 5s), Os(5p, 5d, 6s), Ir(5d, 6s) Pt(5d, 6s), Nb(4p, 4d, 5s), Ta(5p, 5d, 6s), Zn(4s,3d), Mg(2p, 3s), Ca(3s, 3p, 4s), Sr (4s 4 p 5 s ), O(2s, 2p), and S(3s, 3p) orbitals were included as the valence electrons. All calculations were performed at fixed volume and shape, while the atoms were relaxed unti l the forces were less than 0.03 eV/. For some of the calculations, the force was reduced even further to less than 0.001 eV/ However, after testing it was determined that forces as low as 0.03 eV/ could reproduce the structure properties
41 within the accuracy reported here and the total energy accuracy for the calculations are equivalent, as seen in Table 1-1. Though all atoms are free to relax in these opti mized structur es the crystal lattice or shape is fixed For the simple bismuth pyrochlores, the cubic lattice constant is optimized by performing several fixed volume calculations with varying lattice constant and determining the corresponding energy versus volume profile. When compared with experimentally determined lattice constants, the DFT values are within 1 %. Once the optimized lattice constant is identified, all subsequent calculations are fixed to this value. This process was repeated after atomic displacement was identified to determine if any lattice expansion occurred. For the complex bismuth pyrochlores, the lattice constant was fixed to the experimentally determined lattice constant at the lowest temperature reported in the literature. These lattice constants were 10.55668  for Bi1.5ZnNb1.5O7 (BZN), 10.54075 for Bi1.5ZnTa1.5O7 (BZT), 10.5525 for Bi1.5MgNb1.5O7 (BMN), and 10.529 for Bi1.5MgTa1.5O7 (BMT) and 10.2301 for Ca1. 5Ti1.5Nb1.0O7 (CTN)  For the SrTiO3, the cubic lattice constant was optimized for the bulk structure then the (100) surfaces were creat ed based on the optimized bulk structure. Electronic relaxation was performed with the conjugate gradient (CG) method accelerated using Methfessel -Paxton Fermi -level smearing with a Gaussian width of 0.1 eV  for the pyrochlores. For SrTiO3, electronic relaxation was performed with a block Davidson iteration method accelerated using Fermi level smearing with a Gaussian width of 0.1 eV  A plane wave cutoff energy of 400 eV was used for all systems.
42 Table 1 1. Calculation details for the various simulations discussed throughout dissertation. System Energy Cut Off Force Criteria P seudo Potential k Mesh Total Energy Accuracy Simple pyrochlores in ideal structure (Ch. 3) 400 eV 0.001 eV/ LDA 6 6 6 0.02 eV Simple pyrochlores in displaced structure (Ch. 3, 4, 7) 400 eV 0.001 eV/ LDA 3 3 0.01 eV Simulated Annealing (Ch. 3, 4, 5, 7) 400 eV 500 steps LDA 1 1 1 0.02 eV Deficient simple pyrochlores (Ch. 5) 400 eV 0.005 eV/ LDA 3 3 0.02 eV Complex Pyrochlores (Ch. 6, 7) 400 eV 0.03 eV/ LDA 1 1 1 0.02 eV Ion Hopping (Ch. 7) 400 eV 0.05 eV/ LDA 3 3 3 (Bi 2 Ti 2 O 7 ) 1 1 1 (BZN/CTN) 0.02 eV Surface Chemistry of SrTiO 3 (Ch. 8) 400 eV 0.03 eV/ GGA PBE 8 8 1 0.01 eV
43 For the simple pyrochlores reported in Chapter 3, 4, 5 and 7 a 333 Monkhorst Pack  mesh was used. For the complex pyrochlores reported i n Chapter 6 and 7 there were a large number of configurations explored therefore only the -point (111) was used for the k -mesh. A 333 k -mesh was tested for the BZN system and there was no difference among the results within accuracy reported. These parameters were tested by performing additional calculations with higher cut off energies (450 eV, 500 eV), finer k -mesh grids (333 for BZN, 444, 666, 888 for simple pyrochlores ) and resulted in differences less than 0.02 eV in the total ener gy. Several calculations were performed to confirm that spinpolarization does not affect any of the results; therefore, all data reported are without spinpolarization. The generalized gradient approximation (GGA) with the parameterization of Perdew et al  was tested for the pyrochlore systems and provided qualitative agreement with the results reported in Chapter 37. It was confirmed that the favor a bility of atomic displacement and lone pair formation does not depend on the functionals. For the SrTiO3 system, the effects of the functional on the accuracy of the DFT calculations will be discussed in Section 220.127.116.11.1. 1.3.4 DFT Tools and Analysis Tec hniques Additional simulation technique s will be implemented in the DFT calculations to aid in investigating these materials. These methods include simulated annealing and nudged elastic band method. In order to aid in understanding the results from these calculations throughout the dissertation, details on the techniques are provided in here. Additionally, details are provided for Bader charges and electron localization functions, which are two of the analysis tools that are used to interpret the calculati ons.
44 18.104.22.168 Simulated Annealing Since there is no guarantee that DFT is capturing the global minima, simulated annealing was performed to probe for lower energy states that exhibit displacements of cations from their high symmetry sites. Simulated anneal ing is a two step process where the system is provided thermal energy for a given amount of steps and then the structure is quenched by removing all thermal energy and optimizing the structure to the final state. Both steps of t he simulated annealing calculations we re performed under fixed crystal shape and volume. The initial heating step was performed using ab initio molecular dynamics (MD)  with the relaxed ideal pyrochlore structures as the initial s tructures. The MD simulations were conducted with only the point (1 1 1 k -mesh) within the canonical (NVT) ensemble, with fixed number of atoms, volume, and temperature. The Nos thermostat was used to control temperature variation from 1000 K to 300 K o ver 500 MD steps with a time step of 0.5 femtoseconds. Though these temperatures are not exact, by providing thermal energy to the system the atoms may sample new spatial positions and potentially overcome any energy barriers separating a local minimum fro m the global minimu. Unlike any other type of calculation discussed, this step is performed for only a specific number of steps and not until a force criteria is met. After MD simulation, the system was minimized using the same approach as outlined above in Section 1.3.3 (333 mesh, CG method, etc.) and the equilibrium lattice parameter was reevaluated for the cation displaced structure While the MD simulation is over extremely short times, it was confirmed that increasing the simulation length does not affect the final fully relaxed structure.
45 22.214.171.124 Nudged Elastic Band As will be discussed in Chapter 7 and 8, it is often of interest to understand reaction pathways or minimum energy pathways between previously determined minima. DFT has been used in com bination with nudged elastic band (NEB) method  to successfully identify transition state s in proton transfer  surface diffusion  and surface chemistry, such as in the water gas shift reaction on copper  In this dissertation NEB is used to examine the minimum energy pathway between pyrochlores with different displacement patterns in Chapter 7 and reactions intermediates in water dissociation on SrTiO3 in Chapter 8 The NEB calculations involve connecting two (previously determined) minima by a discrete number of structures, or image s. These images are created by interpolating the structures between the two minima with the goal of mapping out the structural intermediates joining these stable minima along a reaction pathway. Figure 14 was reproduced from Ref  and shows a schematic of a potential energy profile between two minima, labeled initial and final. The path proposed by the interpolated structure intermediates is labeled NEB and the actual minimum energy pathway is labeled MEP. During t he NEB calculation, the two minima are fixed and each image minimizes the total energy while maintaining equal spacing along the reaction coordinate between its neighbors. This is accomplished by introducing spring forces along the band between neighboring images. The forces are detailed in the insert of Figure 1 4. The component of force perpendicular to the band is projected and minimized. This minimization should pull the initial path (labeled NEB in Figure 14) towards the actual MEP between the two min ima.
46 Figure 14. Figure 1 from Ref  showing the potential energy profile between two minima labeled initial and final. The minimum energy path (MEP) as well as the nudged elastic band (NEB) path are shown b etween the two minima. The insert details the two components that make up the NEB force FNEB: the spring force Fi S, along the tangent i, and the perpendicular force due to the potential Fi The unprojected force do to the potential Fi is also shown for completeness. One analogy to help illustrate this technique is to imagine two people by a mountain in two different valleys. A rope may be thrown from one person to the other a cross the mountain. The two people may slide the rope along the mountain to find the easiest path connecting them but neither person may more and the rope can stretch but will not break. Since the images (structure intermediates) are equal ly spac ed with respect to the reaction coordinate, the regular NEB method does not require the identification of a local saddle point. In fact, often if an even number of images is used, the saddle point will not be identified. Therefore, after the minimum energy pathway is mapped out with the regular NEB method, the climbing image a pproach is applied where the image with the highest energy is pushed towards the local saddle point. The saddle point or transition state structure minimizes its energy in all direction except along the elastic band. This type of minimization results in th e loss of equal spacing along the reaction
47 pathway between neighboring images. Due to this loss of equally spaced images, it is recommended that the climbing image approach start from a converged regular NEB calculation. 126.96.36.199 Bader Charges One of the d irect benefits of using DFT to examine oxides is the prediction of the ground state electron density. Many details may be extracted from the electron density to aid in describing the nature of bonding within the oxide and the nature of bonding will impact macroscopic observation such as insulating or metallic properties It is known that within oxides covalent bonding may occur which would result in the sharing of electrons between cations and anions. While formal charges may be assumed for each atom type, Bader proposed a method of evaluating the electron density to assign charge density at each atom  Bader identified in most molecules the charge density between atoms reaches a minimum at some distance from each atom. This results in the occurrence of a twodimensional zero-flux surface which has the ch arge density minimum perpendicular to the surface. By identifying these zero -flux surfaces the method allows one to assign a Bader volume for each atom and split the charge density between two atoms  This provides a way to approximate the total electronic charge of an atom by assigning the charges within the atoms Bader volume to each atom. T he code of Henkelman et al  was used throughout this dissertation to identify both th e Bader volumes and Bader charges in various oxides. Within DFT, atoms are assigned a specific number of electrons based on the user defined pseudopotential. The Bader determined electronic charge may be compared with the number of electro ns assigned from the pseudopotentials in order to determine the Bader charge on each atom. The deviation between the atomic Bader charges and
48 the formal ionic charge indicates the degree of covalency and can provide insight into the bonding nature within oxides. 188.8.131.52 Electron Localization Function The properties observed in bismuth pyrochlores have often been connected to the highly polarizable bismuth cation with 6s2 lone pair of electrons on the A site [13,20,3 7,52,63,64] One technique of visualizing these lone pairs is through the electron localization function (ELF) which was introduced by Becke and Edgecombe as a real space technique of mapping electrons into core, shell, bonding, and lone pair electrons based on the Pauli exclusion principle. The technique has been advanced by Savin and Silvi in an effort to provide a more quantitative description of the chemical bond [65,66] The ELF function maps the probable distribution of electrons graphically, where a fully localized (delocalized) electron has an ELF value of 1 (0). 1.4 Objectives of Dissertation As detailed above in Section 1.1 and 1.2, this dissertation objective is to help connect the atomic -level structure of mixed metal oxides with the macroscopically observable properties through ab initio simulations. In order to begin examining these oxides with DFT simulations a thorough understanding of the atom ic structure is necessary and outlin ed in Chapter 2. The pyrochlores are the main focus of this dissertation and will be detailed exclusively in Chapters 3 through 7 with the perovskite SrTiO3 discussion reserved for Chapter 8. In Chapter 3 the occurrence of atomic displacements is isolated for cubic bismuth pyrochlores without cation substitution. The electronic interactions that drive atomic displacement in Bi2Ti2O7 are further explored in Chapter 4 where sulfur is substituted on the O and O site to pert urb the Bi -O and Bi -O bonding. The role of both cation and anion vacancies on the A2O network of bismuth
49 ruthenate is presented in Chapter 5. Atomic displacement and cation substitutional ordering is explored in four complex pyrochlores in Chapter 6. Preliminary efforts to connect DFT simulations with dielectric relaxation in complex pyrochlores by identifying cation hopping mechanisms between equivalent lattice positions is presented in Chapter 7. The key observations from this work as well as expected future directions are summariz ed in Chapter 9. Background on lone pair formation is summarized in Appendix A. Additional details for the pyrochlore system not included in the body of the dissertation are discussed in Appendix B.
50 CHAPTER 2 CRYST ALLOGRAPHY OF PEROVSKITE AND PYROCHLORE MATERIALS A detailed understanding of the structure, in particular the crystallographic language used to describe these structures, is critical in understanding the results presented in this dissertation As seen in Section1.3, understanding and generating a reasonable initial structure is also the first step for the DFT calculations. Therefore, this chapter will present a detailed introduction of the two bulk crystal structures of the materials examined in this disse rtation. Given the complexity of the pyrochlore system, the simpler perovskite (SrTiO3) system will be presented first. The examination of the perovskite bulk structure here will be useful when the surface chemistry of SrTiO3(100) is discussed in Chapter 8 2.1 Structure of Strontium Titanate SrTiO3 (STO) is a perovskite material with space group Pm 3 m (number 221). The cubic bulk lattice constant is reported to be 3.90 from experiment  and 3.94 from DFT calculations  Within the cubic structure, the eight Sr2+ ions occupy the cube corners, six O2 ions are centered in each cube face, and one Ti4+ ion is in the body centered position, as shown in Figure 21. Figure 2 1. Bulk unit cell of SrTiO 3 shown near the <100> direction. Ti Sr O
51 From Figure 21, it can be seen that the Ti4+ ions are octahedrally coordinated to the six O2 on the faces. These TiO6 octahedra are aligned in the cubic structure; however, there are some reports of antiferroelectric octahedra tilting  at low temperature. Figure 2 -2 shows an alternative view with the Sr2+ centered and the twelve Sr-O bonds illustrated. Additionally, the Ti ions at the corners of the unit cell are shown in the center of each octahedron. Figure 2 2 Sr centered b ulk unit cell of SrTiO 3 shown near the <100> direction. 2.2 Structure of Bismuth Pyrochlores 2.2.1 Crystallography and Visualization of the Pyrochlore Structure Ideal stoichiometric pyrochlores (A2B2O7) are cubic with space group Fd 3 m (number 227) and can be described as two interpenetrating networ ks of corner sharing BO6 octahedra and corner sharing A4O tetrahedra  as shown in Figure 23. In fact, the overall formula is typically represented as A2B2O6O to distinguish the oxygen within the two networks. Figure 2-3 represents one of a multitude of perspectives of the pyrochlore structure found in the literature (see Ref. [10 14] for more extensive discussion on pyrochlore structure). Ti Sr O
52 (a) (b) (c) Figure 2 3. The (a) ideal A 2 B 2 O 6 O pyrochlore structure and the two interpenetrating sub-networks (b) A2O tetrahedr a and (c) B2O6 octahedr a The ideal cubic pyrochlore structure has 88 atoms in its unit cell and is highly symmetric with four distinct crystallographic sites occupied by A (site 16 d ), B (site 16 c ), O (site 48 f ), and O (site 8b ) atoms. The selection of these sites and differences between them will be discussed in detail below. Only two parameters are needed to completely describe an ideal pyrochlore, the lattice constant and the x positional parameter of the oxygen in 48f When defects and deficiencies are introduced further parame ters are needed to describe the system. 184.108.40.206 O rigin and cation centering in Fd3 m space group As mentioned above, ideal pyrochlores usually adopt the Fd 3 m space group, which has two possible origin choices at either the 8 a site (origin choice 1) or the 16c
53 site (origin choice 2)  Ori gin choice 2 is more commonly used in the literature and used exclusively within this work. Within origin choice 2, the ideal sites occupied are the 16 d site (0.5,0.5,0.5), 16c site (0,0,0), O anion at 48 f site ( x ,0.125,0.125), and O anion at either the 8a site (0.125,0.125,0.125) or 8 b site (0.375,0.375,0.375). If the larger A3+ cation is in the 16 d (0.5,0.5,0.5) site, the configuration is termed B -centered with the smaller B4+ cation occupying site 16c (0,0,0) and the O anion at 8 b (0.375,0.375,0.375). Alternatively, in the A -centered configuration, the A cation occupies the 16c site (0,0,0) with the B cation at site 16 d (0.5, 0.5, 0.5) and the O anion at 8a (0.125,0.125,0.125). Unless otherwise noted, the B -centered configuration is used in this work and, where needed, experimental works are converted to this configuration for comparison. By symmetry, only the 48f O atoms experience unbalanced forces and will therefore relax in the ideal cubic pyrochlores with the parameter x used to characterize the ox ygen position The value of x depends on the occupant of site 16c (0,0,0) and the relationship between the A and B -centered configurations is x (A) = 0.75 x (B)  There are known bounds observed in the value of x based on the oxygen polyhedral coordination at the 48f site surrounding the A and B cations. If the oxygen ions form a regular octahedron (cube), the resulting value for x (B) would be 0.3125 (0.375). 220.127.116.11 Wyckoff positions in Fd 3 m space group The spatial positions termed 16 d 16c 8 b and 48f that the four ions within the pyrochlore are located are called Wyckoff positions. While Wyckoff positions are used in all space groups to aid in describing the structure, the particular labels, i.e. 16 d are unique for each spac e group. To further explai n, the 8b site within the Fd 3 m space group does not necessarily match the 8 b site within space group Fd 3 c To understand
54 the location and particular symmetry associated with each Wyckoff position, the reader should refer to Volume A: Space group symmetry w ithin the International Tables for Crystallography, which is available online at http://it.iucr.org/A/. The Wyckoff position labeling can immediately tell the reader the multiplicity of the site based on the number compo nent. For example, the site 16d has a multiplicity of 16, which means by defining a single point in the crystal lattice, 15 additional equivalent sites are defined by symmetry. Therefore, based on the sites occupied in the ideal pyrochlore there are 16 Bi and 16 B cations in addition to 8 O and 48 O anions, thus creating the 88 atom unit cell. 18.104.22.168 Sub network interactions in ideal pyrochlores As mentioned above, often the pyrochlore structure is thought of as two separate sub-networks (A2O and B2O6); however, the two networks are not isolated from each other since the A cation is bound to oxygen ions within both networks. Within the 16 d site, the larger A3+ cation is coordinated to two O and six O ions. Alternatively, the smaller B4+ cation is coor dinated to six O ions when occupying the 16 c site. T he ratio of the total energy of the pyrochlore system for compositions without cation substitution is shown in Table 2 1. It can be seen that up to 17 percent of the total lattice energy is due to the int eraction between the two subnetworks. These percentages were determined by taking the total equilibrium structure and performing a single point calculation, i.e. no atomic movement was allowed, for the two sub-networks isolated from each other to provid e the total energy for each subnetwork. The energy due to interaction was calculated from Equation 21, where Einteraction, Et otal, EA2O ', and EB2O6 are the energy associated with the interaction between the two
55 networks, the total lattice energy, the energy of the A2O and B2O6 sub -networks, respectively. Each energy type was scaled by the systems total lattice energy in order to determine the percent contribution. Einteraction = [ ( Etotal) (EA2O + EB2O6 )] (2.1) Table 2 1. The percent of to tal lattice energy due to the A2O and B2O6 networks, and the interaction between the two subnetworks for the six simple bismuth pyrochlores. % of Total Energy Bi2Ir2O7 Bi2Os2O7 Bi2Pt2O7 Bi2Rh2O7 Bi2Ru2O7 Bi2Ti2O7 A2O' network 17% 16% 19% 18% 17% 15% B2O6 network 69% 75% 64% 65% 70% 68% Network interaction 14% 10% 17% 17% 13% 17% The local structure is depicted in Figure 2 -4 to help illustrate the structural effect leading to the Einteraction. In Figure 2 4a the isolated Bi and B cation environments are viewed along the  direction. This direction corresponds to the O-Bi -O bonding axis. By rotating normal to this direction the same local structure is shown along the  direction in Figure 24b. This direction aids in understanding how the Bi2O subnetwork extends through the open spaces created within the B2O6 sub -network. To help visualize the system, the Bi4O tetrahedra are shown as open, empty fram es instead of filled tetrahedra, as was shown in Figure 23. Additionally, the O and O ions are shown in orange and red, respectively. The O ions that are linking the corner sharing BO6 octahedra in Figure 2 4a are the six O ions coordinated to the central Bi cation. The positional parameter x discussed above for this Wyckoff site (48f ) defin es the tilting of the octahedra seen in Figure 24b. Since the Bi -O bond lengths are shorter than the Bi O bond lengths the coordination of Bi is sometimes referred to as 2+6. The longer Bi -O
56 bond lengths are necessary to satisfy the bond valence for each ion within the structure. This will be discussed further in Section 2.2.3. (a) (b) Figure 24. Local structure of Bi and B cations within ideal pyrochlore struct ure, viewed along the (a)  and (b)  direction. The key indicates each atom type and the Bi cations above and below the plane of BO6 octahedra are distinguished for visual aid. 2.2.2 Site Symmetry As discussed in Section 22.214.171.124, within the Fd 3 m space group there are specific spatial positions that may be occupied by the various constitute atoms termed Wyckoff positions. The symmetry of a particular Wyckoff site helps interpret subtle changes in the overall structure due to significant changes to the local environment. This is particularly true for the case of bismuth pyrochlores. A s will be discussed throughout this dissertation atomic displacement wi thin this material is prevalent The site symmetry of the ideally occupied positions will be discussed here. In the ideal position, the Bi cations have 3 m symmetry, which means that along the cubic primary direction, the <100> family, there is no symmetry. There is three fold inversion symmetry along the secondary direction, the <111> family, and along the Bi B O O
57 tertiary direction, the <110> family, the Bi ions have a mirror plane. To aid i n visualizing the three fold inversion the two overlapping tetrahedra in Figure 24 are isolated in Figure 25. Though a local area is isolated the three fold inversion can be visualized using the Bi ions shown in Figures 2-5a and 25b. Any n -fold inversion symmetry operation can be broken down into two components. (a) (b) Figure 2 5. Site symmetry of Bi cations within ideal pyrochlore struct ure, viewed along the (a)  and (b)  direction. The first component involves a counter -clockwise rotation about the axis of symmetry by 360 /n and the second is a reflection through the origin. In this case, if the ion labeled 1 is selected and rotated 120 about the  direction, the atom labeled 2 would be created. This component is shown as the solid (black ) arrow in Figure 2-5. Then the rotation is followed by an inversion about the origin. In Fi gure 25 the center Bi may be used as a vi sual aid of the inversion point and this inversion is illust rated further by the dotted ( green ) line in Figure 2 -5. Though the inversion point not 2 3 1 Bi O 1 2 3
58 technically correct since the Bi cations are not located at the origin, inversion through the center Bi point is equivalent (visually) to inversion through the origin. Following this inversion, the Bi ion labeled 3 would be created. The selection of the first Bi ion, i.e. the Bi ion labeled 1 in Figure 2-5, is irrelevant; each Bi ion illustrated in Figure 2-5 is reproduced following this symmetry operation. To help visualize the mirror planes along the tertiary direction (the <110> family), one plane is shown in Figure 25a as the dashed ( blue ) line. From this mirror plane the Bi ion labeled 1 would be reflected to the Bi ion labeled 2. There are two other mirror planes i n Figure 2 5a that are not indicated to minimize confusion. The two additional mirrors planes would correspond to the blue line shown in Figure 2 -5a rotated counter clockwise by 60 twice. The B4+ cations have 3 m site symmetry, which is the same site sym metry as the Bi cations. This should not be unexpected since above it is discussed that the selection of either a B -centered (chosen here) or A -centered configuration is possible. The origin was not altered between these two configurations, merely which ion would occupy the 16c Wyckoff position located at the origin. The O site symmetry is 4 3 m which means there is four fold inversion along the cubic primary direction, the <100> family, there is three fold rotation along the secondary direction, the <111> f amily, and along the tertiary direction, the <110> family, the O ions have a mirror plane. The O ions in the 48f site have 2 mm site symmetry which means that along the cubic primary direction, the <100> family, there is two fold rotation. There is no
59 sy mmetry along the secondary direction, the <111> family, and along the tertiary direction, the <110> family, the O anions have a two perpendicular mirror planes. 2.2.3 Bi2O Network Though the entire Bi2O network is not illustrated, the symmetry explain in Section 2.2.2 would extend to reproduce the full network. The Bi2O network is made of corner sharing tetrahedra and each tetrahedron is connected along the O -Bi -O bonding axis, which correspond to a direction within the <111> family. The Bi2O network is often referred to as an anti -cristobalite structure. This is because it is structurally similar to the SiO2 -cristobalite structure, which is composed of Si centered, corner sharing SiO4 tetrahedra. In contrast, the Bi2O network is constructed by O centered, corner -sharing Bi4O tetrahedra and is thus termed anti cristobalite In fact, the two compounds have the same symmetry with the anions and cations inverted, i.e. the Bi cation has the same symmetry as the O ion in the SiO2 system. One benefit of the relationship between these structures is the fact that SiO2 has been extensively studied in order to understand the dynamic disorder of the silica tetrahedra that has been reported. It has been determined that while each tetrahedron remains relatively regular, the corner atoms are displaced from the ideal high sym metry sites. This type of displacement occurs due to low frequency phonons which are referred to as rigid unit modes  Additionally, it has been observed that the displacements that occur in bismuth pyrochlores have been related to the dynamic disorder observed in silica. Namely, the displacement of the Bi and O atoms in the pyrochlore structure have shown similar concerted displacement as that observed in silica, which has been termed -cristobalite type displacive disorder  A schematic of
60 this type of correlation is illustrated in Figure 26. This type of correlation is observed for some of the pyrochlores reported in this dissertation as will be disc ussed in Chapter 3, 5, 6 and 7 Figure 26 Local structure illustrating the -cristobalite type displacive disorder. The Bi (tetrahedron corner) and O (tetrahedron center) atoms are in their ideal position with t he black arrows indicating direction of displacement 2.2.4 Bond Valence Model and its Application in Bismuth Pyrochlores Often bond valence theory is used to help examine how and why ions within pyrochlore materials exhibit atomic displacement and the concept of bond valence will be used in interpreting the DFT results in both the simple and complex pyrochlores. Bond valance model stems from Linus Paulings rules for ionic compounds. The first rule is that each cation within an ionic material will be surrounded by a polyhedron of anions and the shape of the polyhedron is based on the ratio of the cation to anion radii. For a complete list of critical ratios and the corresponding anion coordination, see Figure 2.21 in Ref.  Paulings s econd rule says that within ionic structures, the ionic charge of each anion (zj), defined in Eq. 2.2, will be (nearly) equal and opposite in charge to the sum of each cations (a) its bond strength (si)  Where a cations bond  
61 strength (si) is defined in Eq. 2.2 as its charge (ni) divided by its coordination number (ki)  zj sa i = 1 i= niki a i = 1 (2.2) Paulings third and fourth rule pertain to the linking of neighboring polyhedral. Two linked polyhedral may be corner, edge, or face sharing. The most unstable is face sharing with decreased stability for cation with large charge and low coordination numbe r. All linked polyhedral in the pyrochlore structure are corner sharing; therefore instabilities associated with face sharing are not a concern. Rule four addresses the presence of different types of cations in an ionic material. Polyhedral around cations with large charge and low coordination tend to avoid linking to other polyhedral. The extension to bond valence sum comes from Paulings second law and was identified and refined by Brown  .The bond valence (Sij) associated with each bond is related the bond length (Rij) with respect to predefined constants (R0 and B), which may be found in Appendix 1 of Ref.  The valence (Vi) of an ion is equal to the sum of all bond valences associated with that ion, as defined in Eq. (2.3). Vi= Sij= exp R0 RijB j (2.3) The bond valence sum Vi should be close to the expected valence of the ion. For exampl e, within the pyrochlore structure, the O anion in the O site are coordinated to four A site cations. Within ideal or simple pyrochlores without cation substitution, the A site is occupied by Bi cations. If the Bi cations and O anion form a perfect tetrahedr on then each O -Bi bond will contribute one quarter of the total bond valence sum for the O anion. As will be reported in Section 126.96.36.199, i n ideal Bi2Ru2O7 without atomic displacement, the Bi -O bond length is 2.21 and results in a O bond valence of 2.93
62 which is considerably larger than the expected value of 2, this could indicate the occurrence of atomic displacements. The bond valence of the Bi cations is 3.42 which again indicates an over bonding of the Bi and O atoms. However, within the Ru2O6 network, the bond valence of the O and Ru atoms are 2.01 and 4.08, respectively, which is close to their expected valence of 2 and 4.
63 CHAPTER 3 FIRST -PRINCIPLES STUDY OF CUBIC BISMUTH PYROCH LORES Reproduced with permission from Beverly Brooks Hinojosa, Juan C. Nino, and Aravind Asthagiri, Physical Review B 77, 104123 (2008). "Copyright (2008) by the American Physical Society." 3.1 Introduction As discussed in Section 2.2, i deal stoichiometric py rochlores (A2B2O7) are cubic with space group Fd 3 m and often described as two interpenetrating networks of B2O6 octahedra and A2O tetrahedra [10,15] as shown in Figure 3-1. The overall formula is typically represented as A2B2O6O to distinguish the oxygen w ithin the two networks. Figure 31 represents one of a multitude of perspectives of the pyrochlore structure found in the literature (see Ref.  for more extensive disc ussion on pyrochlore structure). From extensive experimental work on the synthesis and structural characterization of cubic pyrochlores, stability fields have been proposed for allowable substitution into the A and B cation site based on simple measures such as the ratio of the cation radii size and the cation electronegativity [12,15] The typical A (B) cation radii is around 1 (0.6) but compounds with variations of up to 0.2 for both cations are stabilized. As the cation radii shift away from the typi cal values noted, the range of A/B substitutions that stabilize into a stoichiometric cubic structure decreases sharply  Recent experimental work extended the stability field of allowable cation radii ratios with structures that incorpo rate large static atomic displacements and retain the cubic symmetry macroscopically [20,23,28] For an excellent but somewhat dated survey of known cubic pyrochlores see Ref. 
64 (a) (b) (c) Figure 3 1. The (a) ideal A 2 B 2 O 6 O pyrochlore structure and the two interpenetrating sub-networks (b) A2O tetrahedra and (c) B2O6 octahedra Recent attention is focused on bismuth based pyrochlores, where the Bi3 + cation resides in the A site, for their potential use in capacitor and high -frequency filter applications [13,14,37] Bismuth has a lone pair of electrons and the cation radius is 1.17  making it one of the larger A cations found in the pyrochlore family and results in a relatively small region of stability in the st ructure field map  In the past decade, experimental work extended this stability region, with the synthesis of
65 numerous complex Bi pyrochlores that incorporate substitutions of small cations in both the A and the B site. The most extens ively studied of these materials is Bi3/2Zn0.92Nb1.5O6.92 (BZN) [3,10,1 3,16,39,8285] Structure and spectroscopic probes reveal a large amount of displacive disorder on both the A and O ions that led Withers et al  to propose several ordering principles on the A2O network for these materials based on X -ray diffraction (XRD) studies. The large displacements of the ions in BZN and other complex bismuthbased pyrochlores, such as Bi3/2ZnTa3/2O7, Bi3/2MgNb3/2O7, and Bi3/2MgTa3/2O7, are critical to the observed high dielectric permittivity  There is still a need to better understanding the nature of the cation displacements. Interestingly, some of the more simple bismuth pyrochlores (i.e. without cation substitution) are reported to have cation displacements. In particular, large cation displacements are reported for Bi2Ru2O7  and Bi2Ti2O7 in both the stoichiometric  and defective form  The simple bismuth pyrochlores are of interest from a technological viewpoint  but also allow for examining the role of cation displacements without the influence of substitutions on the cation sites. The majority of the work on A -site bismuth pyrochlores is experimental with only a few recent density functional theory (DFT) studies of Bi2Sn2O7  Bi2Ti2O7  and Bi2Ru2O7  The DFT work on bismuth pyrochlores is restricted to its ideal cubic structure except for the work by Walsh et al [ 35] examining the low temperature, monoclinic phase of Bi2Sn2O7. DFT can be an invaluable tool in systematically exploring and providing insight into the role of structure, composition, and atomic relaxation on the properties of pyrochlores. While the e xperimental work provided details of the structure of several bismuth pyrochlores, synthesis of these materials is non-trivial and often
66 results in O deficiencies. The density and nature of these O deficiencies can vary and sometimes were not identified in early experimental work. The presence of O deficiencies introduces additional complexities, since their influence on deviations from the cubic pyrochlores is not well understood. Comparisons between experiment and DFT calculations of ideal defect -free pyr ochlores will identify deviations due to the presence of deficiencies. DFT can also help to establish connections between the electronic structure and observed structural behavior in the Bi pyrochlores. Of particular interest is the role of the Bi lone pai r in the experimentally reported large cation displacement in some bismuth pyrochlores [14,20,23,28] The lone pair of Bi is believed to create stereochemical constriction leading to more distorted structures that deviate from the ideal cubic form found in other families of pyrochlores. The reduction of this lone pair nature due to mixing of orbitals between the Bi and B -site cations is proposed to explain the lack of distorted structures in the bismuth metallic pyrochlores [26,27,89] A systematic study of the differences in electronic structure of various bismuth pyrochlores, inclu ding insulating versus metallic, will be important in clarifying the role of lone pair s in bismuth pyrochlores. T his chapter reports DFT calculations on a range of simple cubic bismuth containing pyrochlores Bi2B2O6O (B = Ti, Ru Rh, Ir, Os, Pt) that have been synthesized experimentally [15,20,23,26 -28] T he above bis muth pyrochlores were initially examined within the ideal cubic pyrochlores structure. Subsequently, the potential role of cation displacements in stabilizing the cubic bismuth pyrochlores was investigated. 3.2 Calculation Details First -principles calculations were performed with Vienna Ab initio Simulation Package (VASP)  a plane wave DFT code, using the projector augmented wave
67 (PAW) pseudopotentials provided in the VASP databas e [48,49] T he Bi(5d, 6s, 6p), O(2s, 2p), Rh(4p, 4d, 5s) Ti(3s, 3p, 3d, 4s), Ru(4p, 4d, 5s), Os(5p, 5d, 6s), Ir(5d, 6s) and Pt(5d, 6s) orbitals were included as the valence electrons. The DFT calculations were performed within the local density approximation (LDA)  The LDA is expected to underestimate lattice constants and overestimate the strength of bonding, but has been found to be more accurate than th e GGA functionals for many metal oxides [90,91] and La and Y -based pyrochlores  The generalized gradient approximation (GGA) with the parameterization of Perdew et al  was tested and provided qualitative agreement with the results reported in this chapter. Electronic relaxation was performed with the conjugate gradient (CG) method accelerated using Methfessel Paxton Fermi level smearing with a Gaussian width of 0.1 eV  The ideal Fd 3 m space group symmetry was tested and the bismuth pyrochlores examined in this work retain the ideal structure. Therefore, all calculations were performed at fixed volume and shape, while the atoms were relaxed until the forces were less than 0.001 eV/. A plane wave cutoff energy of 400 eV was used along with a 666 Monkhorst -Pack  mesh, resulting in 10 irreducible k -points. Test calculations done with a cutoff energy of 450 eV and an 888 mesh result in differences less than 0.02 eV/88atom cell in the total energy, well within the error for the results presented in this chapter. S everal test calculations were performed to confirm t hat spin-polarization does not e ffect any of the res ults; therefore, all data reported in this chapter are without spinpolarization. As discussed in Section 2.2.1 .1 pyrochlores may be defined structurally using one of two origins and with different centering options based on the selected origin. Unless o therwise noted, the B -centered configuration at origin choice 2 (see Section 188.8.131.52 for
68 more detail) is used in this chapter and, where needed, experimental work is converted to this configuration for comparison. By symmetry, only the 48 f O atoms can relax in the ideal cubic pyrochlores, and therefore the entire structure is explained by two parameters: the cubic lattice constant and the oxygen positional parameter ( x ). T he role of displacing the cations and O atoms from their high symmetry positions was examined and the details of these calculations are presented in Section 3.3.2. 3.3 Results and Discussion 3.3.1 Ideal Bismuth Pyrochlores For each of the bismuth pyrochlores (Bi2B2O6O, B= Ti, Ru, Rh, Ir, Os, Pt) the equilibrium lattice parameter was deter mine by generating energy versus volume curves. These relaxed structures retain the ideal pyrochlore structure discussed in Section 3.2. The key structural parameters (lattice parameter and oxygen positional parameter) for the relaxed structures of all the simple pyrochlores examined are reported, along with experimental data as a function of B cation radius in Figs. 32 and 3 -3 and in Table 3-1. The B cation radii, taken from Ref.  were useful in defining stability field maps for simple pyrochlores  I n several families of pyrochlores, the B cation radii were shown to correlate with structural, mechanical, and thermal properties when examined using empirical potentials [6,9,33,92] Figure 3 -2a shows that these DFT results for the lattice constants are within ~1% of the experimental values. Considering that LDA underestimates lattice constants, this level of accuracy is excellent and on par with previous DFT studies [34 ] The lattice parameter of the Bi2Ti2O7 system shows the largest deviation and is underestimated by 1.2%. The oxygen positional parameter shown in Figure 32b are within ~0.5 % of the experimental data except for Bi2Ti2O7, which shows an error of nearly 2% overestimation.
69 Table 3 1 The cubic lattice constant ( a ), oxygen positional parameter ( x ), Bi -O (d(A O)) length, Bi -O48 f (d(A O48f )) length, B-O48 f (d(B O48f )) length, and B -O -B angle in order of cation radius for Bi2Rh2O7, Bi2Ti2O7, Bi2Ru2O7, Bi2Ir2O7, Bi2Pt2O7, and Bi2Os2O7. Italics represent average values, while + and denote experimental studies conducted at room temperature and low temperature ( a () x (/ ) d (A O) () d (A O48 f ) () d (B O48 f ) () B O B () Bi 2 Rh 2 O 7 This Work 10.190 0.332 2.206 2.485 1.986 130.2 Experimental Bi 1.95 Rh 2 O 6.83 + ,a 10.2764 0.3317 2.2249 2.508 2.001 Bi 2 Ti 2 O 7 This Work 10.230 0.3244 2.213 2.547 1.96 134 Experimental Bi 2 Ti 2 O 7 *,b 10.359 0.3187 2.289 2.559 1.964 137.5 Experimental Bi 1.74 Ti 2 O 6.62 +,c 10.3569 0.3187 2.277 2.637 1.9655 Bi 2 Ru 2 O 7 This Work 10.205 0.3291 2.209 2.509 1.977 131.8 Experimental Bi 1.87 Ru 2 O 6.903 ,d 10.2699 0.3271 2.23 2.55 1.98 Bi 2 Ir 2 O 7 This Work 10.250 0.3312 2.219 2.505 1.994 130.6 Experimental Bi 1.9 Ir 2 O 6.8 +,e 10.3256 0.3299 2.236 2.534 2.003 131.4 Bi 2 Pt 2 O 7 This Work 10.345 0.3324 2.239 2.519 2.018 129.9 Experimental Bi 2 Pt 2 O 7 f 10.36 Bi 2 Os 2 O 7 This Work 10.250 0.3268 2.219 2.536 1.976 133 Experimental Bi 2 Os 2 O 7 f 10.338 aRef.  ; bRef.  ; cRef.  ; dRef.  ; eRef.  ; fRef. 
70 Bi2Ti2O7 falls outside the stability field predicted based on the ratio of cation radius, but was recently synthesized with and without O defects [23,28] Diffraction studies showed that bismuth and the O atoms displace in the Bi2Ti2O7 pyrochlore [23,28] T he differences in the structural parameters between these DFT calculations and experimental results for Bi2Ti2O7 were attributed to the Bi cation displacement. Rh (0.60) Ti (0.605) Ru (0.62) Ir (0.625) Pt (0.625) Os (0.63)10.18 10.20 10.22 10.24 10.26 10.28 10.30 10.32 10.34 10.36 (a) DFT EXPB4+ Cation Radius ()Cubic Lattice Constant () Rh (0.60) Ti (0.605) Ru (0.62) Ir (0.625) Pt (0.625) Os (0.63)0.315 0.320 0.325 0.330 0.335 (b) Oxygen Positional ParameterB4+ Cation Radius () DFT EXP Figure 3 2. The dependence of the (a) cubic lattice constant and (b) the oxygen positional parameter on the B4+ cation radius as determined by DFT (square) and experiment (triangle) for Rh  Ti [23,28] Ru  Ir  Pt  and Os  The A -centered configuration oxygen positional p arameter values reported for Bi2Ti2O7 [23,28] are converted to the B -centered configuration value (see Section 3 .2. for discussion on A versus B -centered configuration). The dashed lines drawn between the DFT values included as a visual aid. Note that Ir and Pt have the same ionic radius of 0.625 A recent study of defective Bi2Ru2O7  also reported Bi displacements with small displacement of the oxygen in the 8b site. For Bi2Ru2O7, a large deviation for either the lattice parameter or the x positional parameter was not found between the DFT results with no cation displacements and the experimentally reported structure with cation displacements  The role of Bi cation displacement in the bismuth pyrochlores is examined directly in Section 3.3.2.
71 Excluding Bi2Ti2O7, the DFT results for both lattice parameter and oxygen displacement parameter do not show any systematic correlation to B cation radii. It is important to stress the DFT results follow the same trends as the experimental data. Barker et al found that x increas es with increasing B4+ cation radius for various pyrochlores  As shown in F igure 3 -2a, this simple relationship does not hold over the entire range of bismuth pyrochlores, since the oxygen positional parameters for Rh and Os have an inverse relationship with respect to the B cation radii. A relationship between the B cation radii and cubic lattice constant has not been explicitly noted in literature, but Minervini et al observed a strong linear relationship between lattice parameter and A cation radii for several families of B cation pyrochlores  A similar trend observed for some of the bismuth pyrochlores (see Figure 32a), has some clear contradictions. Ir and Pt have the same cation radii but different lattice constants and the Os system has exactly the same lattice constant as Ir despite the larger cation radii. T he structural parameters were plotted with electronegativity and other definitions of the cation radius with no simple trends observed, indicating the bonding in the bismuth based pyrochlores may be more complex than other pyrochlore families. As will be shown in Section 3.3.3, when the electronic structure of these pyrochlores is examined there is a large degree of B cation dependent covalency in the bismuth pyrochlores. The lack of a relations hip between structure and cation radii can be attributed to the covalent interactions and a similar observation was made in a recent DFT study of lanthanum and yttrium based pyrochlores involving the polarizable Sn cation  It is worthwhile to note that the lattice constant for Bi2Ti2O7 falls in a similar region to the other bismuth pyrochlores with corresponding cation radii. Kennedy noted the larger lattice constants
72 associated with the insulator (Ti and Sn) versus metal (Rh, Ru, and Ir) bismuth pyrochlores  The ideal DFT calculations (i.e. no cation displacements) indicate that the larger lattice constant of the insulating bismuth pyrochlores (Ti and Sn) is related to the inclusion of cation displacements within these structures. As is discussed in the next section, the inclusion of atomic displacement dramatically changes the lattice constant of Bi2Ti2O7. The relevant bond lengths for the pyrochlore structure from DFT and experiment are presented in Figure 33. Overall both Bi -O and Bi -O bond lengths compare well with experimental data, but as with the lattice constant and oxygen positional parameter there is a larger difference in the Bi2Ti2O7 system. For the Bi2Ti2O7 system, the average experimental Bi -O48 f bond length compares very well, even with the exclusion of bismuth displacement. In the experimental study, the Bi -O bond length for Bi2Ti2O7 is slightly larger than that observed in other bismuth pyrochlores. Since the Bi displacement is perpendicular to the O -Bi -O chains the Bi -O bond length lengthens by this displacement. The values for the bond lengths of B -O48 f compare well with experimental data for all the systems considered. Though not shown in Figure 33, the bond angles in all six bismuth pyrochlores compare well to experimental values, but once again Bi2Ti2O7 shows the largest deviation. The B cation to O to B cation bond angles are reported in Table 31 and compared with available experimental data. Overall, the structural parameters shown in Figs. 3 2 and 33 compare well with available experimentally data indicating that DFT accurately captures the structural relaxation in the simple bismuth pyrochlore systems, at least for systems without large cation displacements observed.
73 Figure 3 3. Bond lengths for Bi O, Bi O48 f and B O48 f as a function of cation radius as determined by DFT (dark blue) and experiment ( light orange) for Rh  Ti [23,28] Ru  Ir  Pt  and Os  The l ines drawn between the values included as a visual aid. Note that Ir and Pt have the same ionic radius of 0.625. 3.3.2 Bismuth Pyrochlores with Atomic Displacement The role of cation displacements must be examined t o address the issue of the larg e deviations observed for Bi2Ti2O7 relative to the other bismuth pyrochlores. Experimentally, it is known that the bismuth ions displace from their ideal crystallographic site for some simple bismuth pyrochlores [20,23,28] Due to the symmetry of the initial structure, the Bi atoms cannot displace during relaxation; only the O48f atoms have non-zero forces at the initial structure. Therefore, it is possible that all of the relaxations are actually sitting on a saddle point with respect to displacements of the Bi atoms. Ideally, the phonon modes in the pyrochlore structure would be explored to identify any soft modes, but with the large unit cell these calculations are compu tationally expensive. Instead, a simpler approach was used to investigate the role of bismuth displacement within these pyrochlores. For each Bi pyrochlores, simulated Rh Ti Ru Ir Pt Os 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.8 2.9 Rh (0.60) Ti (0.605) Ru (0.62) Ir (0.625) Pt (0.625) Os (0.63) Bond Length ()B4+ Cation Radius ()
74 annealing was performed beginning with the relaxed structures found in Section 3.3.1, w ith the volume fixed using a b initio molecular dynamics (MD) [44 47] (see Section 184.108.40.206 for additional details). This approach forces the Bi, B cations, and O ions out of their zero -force crystallographic site s. The simulated annealing consisted of MD simulations within the canonical ( NVT ) ensemble, with fixed number of atoms, volume, and temperature. The Nos thermostat was used to control the temperature variation from 1000 K to 300 K over 500 MD steps with a time step of 0.5 femtoseconds. While the simulated annealing procedure is over extremely short times, after 100 MD steps the atoms displace from their high symmetry sites indicating that the number of steps is sufficient to accomplish the primary goal. D ue to the loss of symmetry within these displaced configurations, the ab initio MD was performed with only the point for the k mesh After simulated annealing, the system was minimized using the same approach as outlined in Section 3.2 except a 333 k -m esh (14 irreducible k -points) was used in place of the 666 k mesh Comparisons made in this section between displaced and undisplaced structures are from calculations using the 333 mesh and the decrease in k -mesh was confirmed not to a ffect any of the conclusions. Table 3 2 shows the energy change due to cation displacement and the average atomic displacements from the ideal positions of the Fd 3 m cubic pyrochlore structure for all the bismuth pyrochlores examined. Similar values for the atomic displacement reported from experimental studies of Bi2Ti2O7 and Bi2Ru2O7 are also included in Table 3 -2. In these calculations, only one system, Bi2Ti2O7, showed cation displacement improved the thermodynamic favorability by an appreciable amount of energy (0.097 eV/Bi cation). The initial simulated annealing step outlined above is performed at the
75 equilibrium volume identified for the ideal pyrochlore str ucture, but the cation displacement may be coupled to the system volume. To ensure the identification of a global minimum energy structure, the volume of the structure incorporating the cation displacement was reoptimized by examining energy versus volume curves for the displaced structure. T he volume optimization was performed for only the Bi2Ti2O7, since it is the only structure that shows any significant cation displacement. An increase in lattice parameter from 10.23 to 10.34 was found and improved th ermodynamic favorability by 0.028 eV/Bi atom. With the optimized displaced structure, the DFT lattice parameter of 10.34 for Bi2Ti2O7 compares well with experimental values of ~10.36 The volume increase corresponding with the asymmetric structure of t he displaced Bi2Ti2O7 is due to the steric effects of the lone pair. Similar increases in volume occur for the transition of rocksalt to litharge for PbO and SnO  The total energy change from the ideal configuration to the volume optimized displaced structure for Bi2Ti2O7 is 0.125 eV/Bi atom. This magnitude of energy change due to cation displacement in Bi2Ti2O7 is on par with t he energy change of 0.27 eV/Bi atom reported by Walsh et al for Bi2Sn2O7 to change from the ideal cubic pyrochlore to the monoclinic ground state structure  The bismuth pyrochlores examined in this chapter were confirmed to retain the ideal cubic structure in agreement with all the experimental evidence. The similarity between the energy associated with cation displacement in Bi2Ti2O7 versus a change in structure in Bi2Sn2O7 indicates that both are feasible mechanisms for Bi -based pyrochlores to reduce the stereochemical effects of the Bi lone pair. The more complex structural change observed in Bi2Sn2O7 is likely due to additional stereochemical effects
76 from the Sn atom in the B site. The other bismuth pyrochlores have a different mechanism in reducing the lone pair nature of the Bi cation based on the hyb ridization between the Bi 6 s and B cation d orbitals. The details of the electronic structure differences between the metal bismuth pyrochlores and Bi2Ti2O7 discussed in Section 3.3.3. Table 3 2. The e nergy change and average displacement magnitudes of s imple pyrochlores after simulated annealing with respect to the ideal structure is shown (Positive value of energy corresponds to increasing thermodynamic favorability.) Systems ordered in increasing energy with the lattice constant and oxygen positional parameter included. Bi2Ti2O7 with volume optimized represents average values from ten different configurations. Energy Change Lattice Constant x Average Displacements (eV/Bi ) () ( / ) Bi () B () O' () O () Bi2Pt2O7 9.5E -04 10.345 0.332 0.030 0.028 0.036 0.028 Bi2Rh2O7 9.5E -04 10.190 0.332 0.048 0.033 0.034 0.034 Bi2Os2O7 9.5E -04 10.250 0.327 0.05 0.0 25 0.057 0.050 Bi2Ru2O7 0.001 10.205 0.329 0.041 0.029 0.032 0.031 Bi2Ir2O7 0.002 10.250 0.331 0.036 0.027 0.025 0.026 Bi2Ti2O7 0.097 10.230 0.324 0.328 0.059 0.077 0.082 volume optimized Bi 2 Ti 2 O 7 0.146 0.009 10.340 0.324 0.38 0.02 0.07 0.02 0.11 0.01 0.10 0.02 Bi1.87Ru2O6.903 a 10.27 0.327 0.16 0 0.01 Bi2Ti2O7 b 10.359 0.431 0.43 0 0.15 Bi1.74Ti2O6.62 c 10.357 0.431 0.38 0 0 aRef.  ; bRef.  ; cRef.  Table 3 2 also shows that although there is displacement in all six of the bismuth pyrochlores, with the exception of Bi2Ti2O7, the average magnitude of these displacements are less than ~0.05. This result agrees with available experimental observations except for Bi2Ru2O7, where a Bi displacement of 0.16 has been reported
77  T he differences between DFT and ex periment for the Bi2Ru2O7 system are addressed in Section 220.127.116.11. While the general features of the displacements f oun d in Bi2Ti2O7 are similar to experiment, there are important differences which are discuss ed in detail below. 18.104.22.168 Atomic d isplacement in Bi2Ti2O7 Two experimental studies of Bi2Ti2O7 have been reported, with their main results for the structure reproduced in Table 32 [23,28] The earlier study synthesized Bi2Ti2O7 with A2O network deficiency  but subsequently Hector et al demonstrated that Bi2Ti2O7 can be synthesized in stoichiometric cubic form  T hese two studies will be referred to as deficient and stoichiometric. Both studies analyzed the structure with X ray and neutron powder diffraction and confirmed to be cubic pyrochlores in the Fd 3 m space group. The experimental work on stoichiometric Bi2Ti2O7 reports structural details from X -ray diffraction at both room temperature and 2 K. Since the DFT calculations are at 0 K, the low temperature experimental data is used for comparison. Initial attempts to fit the diffraction data using the ideal pyrochlores structure resulted in large thermal parameters in both the stoichiometric and deficient studies, which indicates static displacements in the system [23,28] Both studies saw the (422) reflection within the X ray diffraction data collected, which is not possible in the ideal pyrochlore structure with isotropic atoms occupying the ideal a b c d or f sites of the space group and only becomes possible if the atoms are anisotropic or occupy the e g h or i positions  Incorporation of static displacements of Bi dramatically improved the fit to the diffraction data and reduced the thermal parameters to plausible values [23,28] Hector et al still found high O thermal parameters in the stoichiometric system, which were reduced by
78 the incorporation of static displacements in a fraction of the O atoms. Both studies reported no significant static displacements in the B2O6 network. Below the three main features of the Bi2Ti2O7 structure (Bi cation displacement, O displacement, and the B2O6 network) will be discussed in detail and compared with the two experimental studies. While both experimental studies find large Bi cation displacement, they differ in the details of the resulting displaced structure. Moreover, the stoichiometric system requires the additional O displacement discussed above. It is important to stress that establishing th e details of the static displacements experimentally is non-trivial and multiple possibilities may be indistinguishable. DFT results can assist in determining the significance of the particular displacements reported in the experiments. A comparison betwee n the DFT results and the experiment indicates the same general features of large Bi cation displacement and smaller displacements in the other atoms. Before the details of the Bi cation displacement are discussed it is helpful to examine the ideal local e nvironment around the Bi cation shown in Figure 3 -4. Two views of the bonding network are shown, one perpendicular to and one along the O -Bi -O chains. The Bi cation is in a scalenohedral cage with two O atoms and six O48f atoms at the edges. The six O48f atoms form a puckered ring around the Bi cation outlined in black in Figure 3 -4, with three slightly above and three slightly below the Bi atom. Both experimental studies examined several different possible Bi cation displacements. The best fit to the diffraction data for the stoichiometric and deficient Bi2Ti2O7 was Bi in the 96 g and 96h crystallographic site, respectively [23,28] There are six equivalent positions around the Bi cation for both the 96 g (x x z ) and 96h (0, y -y )
79 sites as illustrated in Figure 3-4. The assumption is that the Bi cation displaces randomly to one of these six equivalent sites, which when averaged over the bulk material results in an average ideal bulk pyrochlore structure. (a) (b) Figure 3 4. The ideal puckered ring (outlined in black) created by the six O48 f ( labeled O ) atoms surrounding Bi (green) at site 16c with O ( tan) at 8a shown: (a) perpendicular to and (b) along the O -Bi -O chains. Dark red is used to distinguish the O48 f atoms slightly below Bi. Small ions represent the six 96 g (x, x, z) sites (light gray) and the six 96 h (y, y, 0) sites (dark blue) for Bi atom displacement. The 96g (x x z ) site brings the Bi closer to one of the six O48 f atoms within the pucker ring and the di splacement occurs along the direction of the Bi O48f bond (see Figure 34b), while the 96h (0, y -y ) site involves a displacement between two of the six Bi -O48 f bonds. Hector et al note that the fitting quality between the 96g and 96 h sites results in only a small difference and these two sites may be indistinguishable within the diffraction data.
80 (a) (b) Figure 3 5. The Bi bonding networks for the two experimental studies with the six O48 f (red) atoms surrounding the Bi sites f or the stoichiometric case at 96 g (x x z ) (light gray)  and the de ficient case at 96 h (y -y 0) (dark blue)  and O (tan) at 8 b shown: (a) perpendicular to and (b) along the O -Bi -O chains. Only one of the six equivalent sites shown for both 96 g and 96 h along with the bond lengths in (light gray for 96g and dark blue for 96h ) between these two sites and the six O48f and the two O. Dark red used to distinguish the O48f atoms slightly below Bi. The main differences between Bi occup ying 96 g and 96h Wyckoff sites can be seen within the altered Bi -O48 f and Bi -O bonding networks shown in Figure 3-5. The ideal Bi -O48 f bonding network, seen in Figure 34, has six equal bond lengths. For the ideal Bi2Ti2O7 structure determined with DFT (s ee Section 3.3.1), a lattice constant of 10.23 and an oxygen positional parameter of 0.324 was found, which corresponds to a Bi -O48 f bond length of 2.55 There will be four or three unique Bi -O48f bond lengths if the Bi occupies the 96 g or 96 h site respectively, as seen in Figure 3-5b. Here, Figure 3 5b shows the bonding network along the O Bi -O chains for both the stoichiometric and deficient experimental structures [23,28] Additionally, the site 96 h displaces the Bi perpendicular to the O -Bi -O chains result ing in equal Bi -O bond lengths, while the 96g site is not directly perpendicular resulting in two different bond
81 lengths as illustrated in Figure 3-5a. T he details of the refined crystallographic parameters for Bi2Ti2O7 from the two experimental studies along with these DFT results are reported in Table 3 3. Table 3 3 A comparison of the crystallographic parameters of Bi2Ti2O7 obtained from two experimental studies [23,28] and the results from simulated annealing at a lattice constant of 10.34 The experimental temperatures differed with + and denoting room temperature and low temperature ( Experimental Bi2Ti2O7 Ref.  10.359 Atom Site x / a y / b z / c Bi 96 g 0.0167 0.0167 -0.0357 Ti 16 d 1/2 1/2 O 48 f 1/8 1/8 0.43128 O'(1) 8 a 1/8 1/8 1/8 O'(2) 32 e 0.215 0.215 0.215 Experimental Bi1.74Ti2O6.62 + Ref.  10.357 Atom Site x / a y / b z / c Bi 96 h 0 0.02745 0.97255 Ti 16 d 1/2 1/2 1/2 O 48f 1/8 1/8 0.43126 O' 8 a 1/8 1/8 1/8 This Work Bi2Ti2O7 10.34 Atom Site x / a y / b z / c Bi 96 h 0 0.02418 0.97582 Bi 96 g 0.01451 0.01451 0.96615 Bi 192 i 0.99033 0.01934 0.02901 Ti 192 i 0.49275 0.5 0.5 O 48f 1/8 1/8 0.4256 O' 192 i 1/8 0.12984 0.13467 Comparisons between experiment and the DFT results for displaced Bi2Ti2O7 must be qualified due to the restricted system size in the DFT calculations. The DFT calculations are performed on one unit cell with periodic boundary conditions and therefore will not include the full set of all possible displacements sampled in a real crystal. To ensure the conclusions about Bi cation displacement from DFT are valid, ten
82 distinct simulated annealing and/or relaxation runs were performed at the optimized Bi2Ti2O7 lattice parameter of 10.34 In addition to annealing from the ideal structure with Bi at site 16 c similar runs were performed on structures with small random initial displacements ( 0.005 to 0.005 ) on the Bi cations. Obviously, ten runs will not result in the appearance of all possible displacements, but as discussed below the observed displacements in the DFT calculations can be grouped into specific site types. T he relaxed Bi2Ti2O7 structures were modified by shifting Bi atoms into other equivalent displacement sites and reoptimized the structure. Stability of such struc tures gives confidence the site types identified in these DFT calculations are fully occupied. For example, if Bi relaxing to a 96 g site is observed then occupation of all six possible displacements into a 96 g site in the real crystal is expected. Differe nces between the energy of the 10 optimized structures are negligible (less than 0.003 eV/Bi cation). The similarity in energy between the relaxed structures and the ability to identify additional equivalent displacements indicates that if a large number o f simulations were performed or at a larger system size, all the equivalent sites would appear with equal probability. C orrelation among the atomic displacements in Bi2Ti2O7 is discussed at the end of this section. From the ten DFT simulations an average B i cation displacement of ~0.38 0.02 is found, which again is in good agreement with the experimental values of 0.43 (0.38) in the stoichiometric (deficient) system (see Table 32) [23,28] Within the DFT study it is found that the bismuth cation displaces to both the 96 g and the 96 h sites in addition to the 192 i (x y z ) site (see Table 3-3). The displacements were determined by comparing the final relaxed positions for bismuth in each simulation with the initial ideal
83 configuration and binned the displacement magnitudes in 0.05 intervals. Figure 36 illustrates each of the three sites observed in these DFT calculations for the Bi displacement, includ ing the six equivalent 96 h sites in Figure 3 -6a, the six 96 g sites in Figure 3 6b, and the twelve 192 i sites in Figure 3 -6c. (a) (b) (c) Figure 3 6. The (a) 96 h (b) 96 g and (c) 192 i displacement sites of the Bi cations observed from simulated annealing DFT of Bi2Ti2O7. The two O atoms included for orientation and all equivalent Bi displacement sites shown with one labeled. The DFT results present a different scenario for Bi displacement than both the stoichiometric and deficient experimental results. Both experimental studies note that the 96 g and 96 h sites may be indistinguishable since they overlap to a great extent. The fitting quality to the experimental diffraction data in switching between the 96g and 96h 2 of 4.83 and 4.86 for the 96 g and 96h sites, respectively  DFT was used to probe whether there is a detectable diffe rence between the two experimental models. Beginning with the experimentally determined crystallographic structures, the energetics of the two experimental structures were examining at the experimental lattice constants using the structure optimization met hod detailed in Section 3.2. There was no discernible difference in the energetics of the two
84 starting structures, which further indicates that these two crystallographic sites are likely both occupied in the real Bi2Ti2O7 crystal. From the literature nei ther experimental study examined the possibility of multiple displacement sites. It would be worthwhile to revisit the experimental data and evaluate the fit to the diffraction data for a model incorporating the 96g 96 h and 192 i displacement sites as predicted by DFT. It is suspect ed that the quality of the fit between the two experimental models and the DFT prediction will be indistinguishable with the diffraction data, and other more precise probes of the Bi2Ti2O7 system will be needed to resolve this i ssue. Specifically, the use of infrared spectroscopy could identify any mode splitting of the Bi -O mode, as seen in complex bismuth based pyrochlore  T he issue of O displacement is now addressed where there is also a substantial diff erence between the two experimental studies. The deficient structure study fixed the O occupation and therefore the O anions do not undergo any displacement. Although the isotropic thermal parameter for O was larger than that of the other atoms, it rema ined reasonable at 0.02 2  In the stoichiometric case, Hector et al present a structural model with less than 100% occupation at the ideal site. They confirmed there were no O deficiencies and the initial model with no O displacement resulted in a relatively large isotropic thermal parameter at ~0.05 2 indicating disorder on this site  Instead they found the best fit to the diffraction data when the O occupied one of two sites; the ideal 8 a site at 90% site occupancy and the 32 e (x, x, x ) at 10% site occupancy. Displacement to the 32e (x, x, x ) site would cause the O to move from its center position between four Bi cations toward two and simultaneously away from two of the Bi cations, resulting in two different bond lengths. The magnitude of the O
85 displacement, for the 10% of O atoms that displace, determined was quite large at 0.15 (see Table 3-2 for details of the structure). While the displacement of only 10% of the O atoms in the stoichiometric system was selected based on ideal site occupation, and it is difficult to find a plausible driving force for such a selective mechanism. Though it is assume d that they examined models with 100% displacement of O, they did not report the quality of such models  Similar to the stoichiometric system, O ion displacement was observed in the DFT cal culations, though the average magnitude of displacement was smaller at 0.11 0.01 as compared to 0.15  and the displacement occurred at every O site unlike the 10% reported in the experimental study. In a single unit cell there are eight O atoms and within each of the ten simulations, the initial position was at th e ideal 8b site. Within this chapter every O ion displaces by approximately the same magnitude and in the same direction within one simulation. In the DFT relaxed structures, the displaced O do not occupy the 32e site seen within the stoichiometric stud y. Instead, all of the displacement vectors of the O atoms belong to the < 0.1, 0.05, 0.0 > family, resulting in O occupying the 192i site (see Table 3 3). To illustrate the single displacement vector of all eight O ions within the unit cell of one si mu lation, the relaxed O ions are shown in Figure 3 7 with Bi displacement sites identified along with the 16c and 8b ideal sites. With the  direction indicated in Figure 37, it can be seen that the O displaced ions follow the same vector within one u nit cell and there appears to be a relationship between the O and Bi displacements further discussed below. This O directional displacement is consistent with other complex bismuth based pyrochlores refinement, but in those systems the O displacement is along the  direction 
86 The occurrence of O anion displacement coincides with Bi cation displacement in Bi2Sn2O7  Bi2Ru2O7  and Bi2Ti2O7  The difference in nature and magnitude of O displacement between these structures depends on the B2O6 network. Even though the O displacement within this work differs from the experimental works for simple bismuth pyrochlores, the identification follows the coupling of O displacement and Bi cation displacement. Figure 3 7. The Bi O bonding network with the 8 a site (tan), the displaced O (dark red), the 16 c site (light blue) and the 96 g 96 h and 192i Bi displacement sites (dark green) determined in all ten simulated annealings. The bonding network direction provided indicates the displaced sites appear correlated along the  direction, with an alternating of the orientation with respect to the ideal site, either in front of or behind. T he discussion on the structure of Bi2Ti2O7 is concluded with the B2O6 network. In both experimental studies the oxygen at the 48f si te was unaffected and both determined an oxygen positional parameter of 0.31 87 Since there wa s no change in the oxygen positional parameter and Ti atoms were at site 16 d in the experimental  
87 model s, the Ti -O48 f bonding remained completely unaffected by the Bi cation displacement. By contrast, within this chapter the Ti atoms show ed displacement with an average of 0.07 0.02 but did no t have a significant effect on the average Ti -O48 f bond length found at 1.96 5 matching b oth experimental studies. Although the average bond length found in this study was unaffected by displacement, the distribution of Ti -O48 f bond lengths observed within each TiO6 octahedron ranged from 1.88 to 2.07 with the average in each octahedron equal to 1.965 The small average magnitude of the Ti displacement seen in this study may indicate that these d isplacements could be absorbed by the thermal parameters experimentally and without direct probing may be overlooked. Based on the observations made within this study, reexamination of the experimental structural parameters with emphasis paid on the site( s) of Bi and any potential displacement of the Ti cations could provide further insight into this system. Since this study identified multiple Wyckoff sites occupied by displaced Bi cations, it would be beneficial to determine if this is also observed within the experimental crystal and if there is any short range ordering of displacement sites. The DFT results indicate that the multiple displacement positions around the ideal Bi site are energetically equivalent, but the barriers to hop among these displac ement sites have not been addressed in this chapter (see Chapter 7 for details) This hopping mechanism has been proposed in bismuth-based pyrochlores by Nino et al  Levin et al  Vanderah et al  and Somphon et al  for other pyrochlore compounds exhibiting catio n displacement. Future DFT studies probing the potential energy surface of
88 bismuth cation hopping between these displacement sites will assist in understanding of the dielectric properties of these materials. Up to this point the structure of Bi2Ti2O7 has concentrated on terms of an average structure where it is assume d that the displacements discussed occur randomly but with equal probability. The experimental studies could not probe any correlation within the atom displacements in Bi2Ti2O7, since they on ly have information on the average structure. Bismuthbased pyrochlores have recently been studied as spinice py rochlores, where the tetrahedron bonding of O ions around the central Bi cation are compared to the ice equivalent of oxygen centered tetr ahedrons with two hydrogen ions covalently bonded at a smaller length and two more distantly bonded by hydrogen bonding  Structural analysis of these complex pyrochlores has shown short -range order  Such characteristics are mirrored within the observed displacements of bismuth in Bi2Ti2O7. Both the Bi displacement and O displacement seen within this study follow the two -long, two-short bonds, resulting in each tetrahedron of BiO4 comprised of, on average, two bonds at 2.2 and two bonds at 2.3 A relationship between the Bi displacement sites can be seen in Figure 3 -7 with the displacement site alternating between in front of or behind the ideal 16c site along the  direction. It is f ou nd that within any given simulation these two alternating sites belong to two of the three Wyckoff sites identified for Bi displacement (i.e. 96 g 96 h 192 i ), occurring in pairs of either (96 g and 96h ) or (96 g and 192 i ). It is stress ed that for no individual simulation of a unit cell are all three Bi disp lacements observed indicating there is a strong correlation between the Bi cation displacements.
89 Previous studies have investigated the displacement within bismuthbased pyrochlores to identify any correlation with results depending on the cation(s) occupying the B site [14 ] In the case of Bi2InNbO7, the observation of diffuse streaking at particular Bragg reflections in electron diffraction patterns indicates cristobalite type displacive disorder in the Bi2O sub -structure  In this structure along with Bi1.89Fe1.16Nb0.95O6.95, displacement within the BiO4 tetrahedron network is highly correlated and believed to be related to the dielectric properties [10,63,96,97] Similar correlation between Bi displacements are observed within this study of Bi2Ti2O7 (Figure 3 -7) and withi n the electronic structure detailed in Section 3.3.3. A lthough electron diffraction patterns from experimental studies of this system are not known, it would be of interest to investigate the presence of similar indicators of displacive disorder within suc h patterns. 22.214.171.124 Atomic d isplacement in Bi2Ru2O7 Experimentally, Bi2Ti2O7 is not the only simple bismuth pyrochlore that exhibits Bi cation displacement. Avdeev et al report a bismuth displacement of 0.16  i n Bi2Ru2O7 In their investigation, it is determined that Bi occupies the 96 h site and O occupies the 32 e site. The displacement of the O ions is relatively small at 0.01 However despite multiple attempts with simulated annealing including at various volumes no substantial bismuth displacements is observed for the Bi2Ru2O7 system. The energetics and magnitude of cation displacement are both negligible in this investigation, with only 0.006 eV/Bi cation associated with an average Bi displacement of only 0.05 Similar to Bi2Ti2O7, calculations w e re performed on the experimentally reported displaced structure, with the experimental structure found to be less stable by
90 1.8 eV/Bi cat ion when compared to the ideal undisplaced structure that is report ed in Section 3.3.1. When compared to the energy change in the titante system, the magnitude indicates that the experimental structure is highly unfavorable. Since the lattice constant may be coupled to the Bi displacement, the volume was varied within the experimental structure, but it remains less favorable than the ideal undisplaced system by 1.7 eV/Bi. Recently, Goodwin et al have examined the defective Bi2Ru2O7 using diffuse electron diffraction  They find a structural model involving beta cristobalite type displacement for Bi cations best fits the diffraction features. A lso DFT calculations wer e performed using the structural model of Goodwin et al  and again c ation displacement is not favored. Cation displacement was again not favorable with the consideration of spin polarization for both the simulated annealing of the ideal configuration and in the atomic relaxation of the experimental structure. Between the simulated annealing runs performed on both the ideal starting structure and the experimentally determined displaced structure, the DFT results suggest that cation displacement is not critical to stabilize the Bi2Ru2O7 system. As show n in the next section, based on the electronic structure of the metallic bismuth pyrochlores, there is no indication that Bi2Ru2O7 should behave dramatically different than the other metallic bismuth pyrochlores. The source of the discrepancies between DFT and the recent experim ental studies is not currently clear. The Bi2Ru2O7 in the experiments is defective in O and Bi, possible triggering the observed bismuth cation displacement, but the role of O vacancies on the structure of the bismuth pyrochlores was not probed in this chapter Both experimental studies assumed that the vacancies do not play a significant role in the observed cation displacements. Goodwin et al find a
91 good fit to the diffraction patterns without the O vacancies, which gives support to this assumption  From the literature, there is no experimental study of stoichiometric Bi2Ru2O7. Future DFT and experimental studies are needed to clarify the role of O vacancies and may assist in resolving the conflict between DFT and experimental results. The role of O vacancies is directly examined using DFT in Chapter 5. 3.3.3 Electronic Structure To shed light on the observation of large cation displacements solely in the Bi2Ti2O7 system, the electronic structure was examined f or the simple bismuth pyrochlores and the displaced Bi2Ti2O7. Structure distortions in Bi and Pb based oxides are quite common [35,94] and are usually attributed to the sterochemical activity of the lone pair on the cation. Recently, electronic structure calculations of SnO, PbO and Bi2O3 oxides have shown that the classical picture of lone pair formation due to the hybridization of s and p orbitals of the metal cation is not correct [94,99,100] Instead, they found that the asymmetric electronic structure in these materials is due to interactions between the metal cation s and p and O 2p states, and it is these interactions that stabilize the distorted structure [94,99,100] The Bi pyrochlores are more complex since there are two cations and the above understanding in the oxides does not shed light dire ctly on why only the Bi2Ti2O7 shows appreciable distortions. Below details the electronic structure obtained from the DFT results using Bader atomic charges, total and partial electronic density of states (DOS and pDOS), and the electron localization funct ion (ELF) for each of the bismuth-based pyrochlores. It is found that the atomic displacements in Bi2Ti2O7 are favored due to the Bi s and p orbitals interactions with the O p orbital in a manner similar to that reported by Walsh et al for distorted PbO, SnO, and Bi2Sn2O7 [94,99,101] The metallic bismuth pyrochlores do not show
92 displacement due to weaker Bi -O interactions combined with stronger Bi -O interactions at the Fermi level. These differences in bonding between Bi -O are sufficient to suppress the formation of the lone pair upon cation displacement and therefore large cation displacement is not favored in the metallic bismuth pyrochlores. An examination of the ELF of the distorted Bi2Ti2O7 with Bi cation displacement clearly displays the asymmetric electronic distribution around the Bi cation indicative of a lone pair and alters the bonding network. The distribution of charges within the system of the ions aids in characterizing the bonding between atoms. T he atomic charges were determined based on the Bader criteria for decomposing the electron density  using the code of Henkelm a n et al  Figure 38 presents the Bader charges for all the Bi pyrochlores in this study plotted against the B cation radius. The values for the various ions are comparable to values seen in a recent DFT investigation of Laand Y based pyrochlores  T he trends of the Bader charges versus the cation radius were examined, shown in Figure 3 -8 but similar to the bond lengths (see Section 3.3.1) a simple trend was not found. The deviation between the atomic Bader charges and the formal ionic charge indicates the degree of covalency. All ions deviate from their formal charge values, indicating the importance of covalency in these systems. T he smallest difference between the DFT atomic charge and the formal charge is found in the Bi2Ti2O7 system, indicating that the Ti -O bond is less covalent than the B -O bond in the metallic pyrochlores. The degree of covalency of the cation-O bonds in bism uth pyrochlores was previously explored using electronegativity values of the ions  The larger the
93 differe nce in electronegativity between the two bonded atoms, the more ionic the bond. Based on this relationship, Wang et al determined that the B -O bond should be least covalent in Bi2Ti2O7, which matches the conclusion from the DFT atomic charges. Therefore, in systems that do not show large cation displacements the B -O bonds are more covalent. Figure 38 also indicates the change in atomic charges in Bi2Ti2O7 due to atomic displacement. Though dramatic changes were not seen in the charge values, but Bi and O atoms undergo the greatest change due to increased Bi -O interaction after displacement. Figure 3 8. Bader atomic charges in units of e for the Bi, B, O48 f and O ion presented for each of the bismuth pyrochlores. The charges within the Bi2O network are darker while the charges associated with the B2O6 network are lighter colored. Lines are drawn as a visual aid, solid for the cations and dashed for the oxygen ions. Averaged Bader charges for the Bi2Ti2O7 displaced system at 10.34 are indicated by the stars. The DOS of each system was examined to identify electronic structure characteristics. Figure 3 -9 presents the DOS for some of the bismuth pyrochlores without cation displacement. The known metallic bismuth pyrochlores of Bi2Ru2O7, Rh (0.60) Ti (0.605) Ru (0.62) Ir (0.625) Pt (0.625) Os (0.63)-1.2 -1.0 Bader ChargeB4+ Cation Radius () qO qO'1.4 1.6 1.8 2.0 2.2 qBi qB
94 Bi2Ir2O7, Bi2Rh2O7, and Bi2Os2O7 all have qualitatively similar DOS that indicate metallic behavior; therefore, Figure 3-9a shows only the Bi2Ru2O7 DOS. As seen in Figure 39b, the DOS for Bi2Pt2O7 is slightly different from the other transition metal bismuth pyrochlo res and shows a small band gap of 0.5 eV. Experimental work identified Bi2Pt2O7 as a semi metal  but did not provide information on the band gap. It is known experimentally that Bi2Ti2O7 behaves as an insulator and Figure 3 -9c shows the DFT results predict a band gap of 1.8 eV compared with the value of 2.95 estimated from UV Vis adsorption spectrum by Yao et al  In general DFT is known to underestimate band gaps, and the value d etermined here is comparable to a value of 2 eV determined by Seshadri  from linear muffin-tin orbital calculations. Figure 3 9. Density of states in arbitrary units for (a) Bi 2 Ru 2 O 7 (b) Bi 2 Pt 2 O 7 and (c) Bi2Ti2O7 in the ideal cubic pyrochlores structure with no cation displacement. The zero energy value represents the Fermi level. -12 -10 -8 -6 -4 -2 0 2 4 6 8 (c) Energy (eV)-12 -10 -8 -6 -4 -2 0 2 4 6 8 (b) -12 -10 -8 -6 -4 -2 0 2 4 6 8 (a)
95 The pDOS is shown for undistorted and distorted Bi2Ti2O7 and Bi2Ru2O7 in Figure 3 -10. Again only the Bi2Ru2O7 pDOS is shown but the other metallic bismuth pyrochlores have essentially the same features. S imilar mi xing of states qualitatively matched prior DFT studies of Bi2Ru2O7 and undistorted Bi2Ti2O7 [14,20] The Bi 6s and 6 p B cation d and the O 2p states appear in the conduction and v alence bands. The discussion is initially focused on the main features that impact the observed cation displacement in Bi2Ti2O7. A comparison of the pDOS of undistorted Bi2Ti2O7 with recently reported studies of PbO, SnO, and Bi2Sn2O7 [35,94,99] shows several sharp similarities. Bi s states show mixing with O p states in the lower part of the valence band ( 10 to 8 eV) and near the Fermi level at the top of the valence band. The Bi s O p mixing near the Fermi level observed in undistorted Bi2Ti2O7 (Figure 3-10a) has been reported in PbO, SnO, and Bi2Sn2O7, and is associated with antibonding interactions [35,94,99] There is mixing between Bi p states with O p states in the top region of the valence band, which is further enhanced in the Bi2Ti2O7 after cation displacement and volume optimization (see Figure 310a and 3-10b). The Bi s O p mixing near the Fermi level is also enhanced by cation displacement. A recent study by Walsh and Watson on the distorted -phase and undistorted cubic -phase of Bi2Sn2O7  shows similar di fferences in the Bi -O interactions as the undistorted and distorted Bi2Ti2O7 pDOS. The Bi p (Bi s O p ) interactions are associated with the asymmetric electron density of the lone pair and is the same mechanism observed in several other oxides that d istort due to the presence of a lone pair [35,94,99,101]
96 Figure 3 10 Partial density of states in arbitrary units of Bi (6 s ), Bi (6 p ) and B ( d ) for the Bi2Ti2O7 pyrochlore (a) before and (b) after displacement and volume optimization and the Bi2Ru2O7 pyrochlore (c) undistorted and (d) with forced displacement. Other oxides such as PbO and SnO do show a larger change in the degree of the mixing of Bi cation O upon a change from high to low symmetry structures. Therefore, the formation of the lone pair is not due to direct Bi s Bi p hybridization, but instead i s a result of Bi s and p hybridization with O p states. There are interactions between Bi and -12 -10 -8 -6 -4 -2 0 2 4 6 0 1 2 Ti (d) O' (p) O (p)Energy (eV)0 1 2 (a) Bi (s) Bi (p) O' (p) O (p) -12 -10 -8 -6 -4 -2 0 2 4 6 0 1 2 Ti (d) O' (p) O (p)Energy (eV)0 1 2 (b) Bi (s) Bi (p) O' (p) O (p) -12 -10 -8 -6 -4 -2 0 2 4 6 0 1 2 Ru (d) O' (p) O (p)Energy (eV)0 1 2 (c) Bi (s) Bi (p) O' (p) O (p) -12 -10 -8 -6 -4 -2 0 2 4 6 0 1 2 Ru (d) O' (p) O (p)Energy (eV)0 1 2 (d) Bi (s) Bi (p) O' (p) O (p)
97 O p states in Bi2Ti2O7, but the Bi s O p interactions are less pronounced than the Bi s O p interactions and occur at an energy further down from the Fermi level. As seen in a comparison of Figure 3-10a and 3 10c, there are many differences between the pDOS of the undistorted Bi2Ti2O7 and the metallic bismuth pyrochlores (Bi2Ru2O7). Bi2Ru2O7 shows more overlap between Ru d and O p states over a broad range than found between Ti d and O p in Bi2Ti2O7, which reinforces the conclusion from the Bader atomic charges that the B -O bond is more covalent in the metallic pyrochlores. The Bi s O p mixing in Bi2Ru2O7 is similar to Bi2Ti2O7 for the lower energy region in the valence band, but the Bi s and p mixing with O p states at the top of the valence band shows several differences. Firstly, the Bi s O p interactions in Bi2Ru2O7, similar to the antibonding interactions found in Bi2Ti2O7, occur 2 eV below the Fermi level. There is also mixing between the Bi s and p states with O48f across the Fermi level along with some mixing of the Bi s and O p states. In general, there is more mixing between the Bi states and O48 f p states in Bi2Ti2O7 and while additional mixing of the Bi s and O p states is observed at the Fermi level it is much less pronounced than the Bi O48 f interactions. The net effect of the greater mixing of Ru d and O48f p states on the top of the valence band is a reducti on of the Bi -O interactions. This reduction is sufficient to eliminate the interactions that favor lone pair formation upon cation displacement in Bi2Ru2O7. T he pDOS for a displaced Bi2Ru2O7 is shown in Figure310d and as large an enhancement of Bi -O mixing that occurs in displaced Bi2Ti2O7 is not found. The ELF for the Bi2Ti2O7 displaced system on the ( 111 ) and 1 11 plane is shown in Figure 3 -11a and 3 -11b, respectively. Figure 3 -11a provides details of the Bi -O48 f
98 environment, while Figure 311b highlights the interaction between the Bi -O network and the Ti -O48 f network. Within Figure 311a, the Bi cation bonds strongly to three of the six O48 f anions, leaving a small space between the remaining three O48 f anions a nd the Bi cation. (a) (b) Figure 3 11. The electron localization of the displaced Bi 2 Ti 2 O 7 at 10.34 for the (a) (111) plane with Bi in the center and the (b) 1 11 plane with the Bi, Ti, O, and O48f ions indicated. The ELF values range from 0 (light red) to 1 (dark purple) as shown by the scale A more strongly localized lobe of electron density, corresponding to the lone pair, partially occupies the space. Figure 311a indicates that the Bi cation displacement allows the lone pair on the Bi cation to occupy space that would be otherwise unavailable within the ideal configuration. A similar ELF structure has been observed for the bismuth based SrBi2Ta2O9 by Seshadri  where the electrons preferentially localize to one half of the ion. Figure 311b illustrates the correlation with the Bi cation displacements explained in Section 126.96.36.199, with the Bi displacement simultaneously 0.0 0.25 0.50 0.75 1.0 Bi Ti O O48f
99 alternating towards and away from the Ti -O48f networks. As mentioned in Section 188.8.131.52, Somphon et al observed this correlated Bi displacement within the more complex Bi2InNbO7 and Bi2FeNbO7 compounds and indicated that it would affect the displacive disorder, and therefore the dielectric properties  Though not shown here (included in Section B.4.1) the ELF for Bi2Ru2O7 provides no indication that atomic displacement is needed to stabilize the st ructure. 3.4 Summary Using Density Functional Theory (DFT) calculations, a range of cubic Bi2B2O6O (B = Ti, Ru, Rh, Ir, Os, Pt) pyrochlores was examined and the structural parameters along with the electronic structure of the bismuth pyrochlores in their ideal cubic, defect -free structure are reported The role of cation displacements within the cubic structure was investigated, with the identification of atomic displacement in the Bi2Ti2O7 compound. The atomic displacement stabilizes this structure with an energy change of similar magnitude to that of other bismuth-based pyrochlores. In Bi2Ti2O7, an average displacement of 0.38 for the Bi cation, 0.07 for the Ti cation, 0.11 for the O anion was found, and an energy change of ~0.15 eV/Bi atom. The Bi cation displacements appear to be correlated as seen in other more complex bismuth pyrochlores. Examination of the electronic structure shows the main d ifference between Bi2Ti2O7 and the metallic bismuth pyrochlores is the extent of Bi -O interactions. In Bi2Ti2O7, more overlap of Bi s and p states with O 2 p states was observed similar to the reported electronic structure of Bi2Sn2O7, PbO and SnO [94,99] which leads to the asymmetric electronic structure around the Bi cation and the displacement of cations in Bi2Ti2O7. These DFT results match the general understanding from experimental studies, but underestimate the displacement in Bi2Ru2O7 compared with experimental results of
100 defective Bi2Ru2O7  W ork directly examining the role of vacancies may assist in resolving the differences between these DFT results and experimental work on Bi2Ru2O7 and is discussed in Chapter 5.
101 CHAPTER 4 THE INFLUENCE OF SUL FUR SUBSTITUTION ON THE ATOMIC DISPLACEMENT IN BISMUTH TITANATE Reproduced in part with permission from Beverly Brooks Hinojosa, Paul M. Lang, and Aravind Asthagiri, Journal of Solid State Chemistry 183 262-269 (2010). "Copyright (2009) by Elsevier Inc." 4. 1 Introduction A general feature observed in all the Bi pyrochlores that show high dielectric permittivity is large ion displacement from the ideal Fd 3 m sites of the cubic pyrochlore structure. This atomic displacement results in multiple energetically equivalent sites that facilitate cation hopping. Because compounds such as BZN incorporate cation substitutions onto both the A2O and B2O6 networks it is difficult to fully separate the myriad of influences on cation displacement. In order to overcome this difficulty in C hapter 3 the investigation of a series of cubic bismuth pyrochlores without cation substitution (Bi2B2O6O, B= Ti, Ru, Rh, Os, Ir, Pt)  was discussed Only Bi2Ti2O7 showed a preference for atomic displacement and the magnitude from DFT matched the experimentally reported values [23,28] Through the analysis of the electronic structure of Bi2Ti2OO6 (see Section 3.3.3) the atomic displacement was attributed to the development of a lone pair on the Bi cations. The lone pair does not form for the other bismuth pyrochlores examined confirming that the B2O6 network has some influence in the lone pair formation. In this c hapter the ro le of the B2O6 versus A2O network in the formation of the lone pair in Bi2Ti2OO6 is further developed by examining the effect of sulfur substitution on the two networks. Walsh and co workers showed that substitution of sulfur for oxygen suppressed the fo rmation of the lone pair observed in PbO [94,105,106] This suppression of the
102 lone pair explains the difference in structure between PbO (litharge) and PbS (rocksalt). More importantly, the work on PbO an d PbS shows that the lone pair formation is dependent on the cationanion hybridization and not just due to hybridization of 6s and 6 p orbitals of Pb  Section 3.3.3 showed that the B2O6 network influences lone pair formation in Bi pyrochlores through a secondary effect. For the semi metallic and metallic systems the Bi cations interact more strongly with the O anions within the B2O6 network  This stronger Bi -O interactions weakens the Bi -O interactions, which in turn suppresses the formation of the lone pair. This model for lone pair formation essentially matches the conclusion from DFT studies of lead oxides and bismuth oxides by Walsh et al [94,100,105,106] Specifically, in lead oxides and bismuth oxides t he nature of the cationanion direct bonding is critical to induce lone pair formation, but in pyrochlores the B2O6 network can influence the primary Bi -O interactions. In this chapter, a similar approach to Walsh and c o workers is employed to further elucidate the lone pair formation in Bi2Ti2OO6 through sulfur substitution. Though some pyrochlore compounds have been synthesized with S on the O site [15,107] the goal of this study is not to detail stable pyrochlore compounds with sulfur substitutions. Instead a purely theoretical approach is taken to provide further insight into the lone pair formation in Bi2Ti2OO6. Sulfur may substitute onto either of the two sites occupied by oxygen, which are the 8 b ( X) site within the A2O network and 48 f (X) site within the B2O6 network. T hree configurations were created with sulfur substituted (i) only on the 8 b (X) site, (ii) only on the 48 f (X) site and (iii) on both the 8 b (X) and 48f (X) sites with the molecu lar formula e of Bi2Ti2SO6, Bi2Ti2OS6 and Bi2Ti2SS6, respectively. It is f ound that S substitution only
103 on the O site can suppress cation displacement, confirming the primary importance of the Bi -O interactions. Comparisons of the electronic structure in Section 4.3.2 shows that the S substitution on the O site modifies the Bi anion hybridization which suppresses the formation of the lone pair. Section 4.3.3 show s through S substitution on the B2O6 network of Bi2Ru2O6O cation displacement and lone pair formation can be induced in the metallic Bi pyrochlores by disrupting the Bi -O interactions. 4.2 Calculation D etails First -principles calculations were performed with Vienna Ab initio Simulation Package (VASP)  and details nearly match that reported in Section 3.2. T he Bi(5d, 6s, 6p), Ti(3s, 3p, 3d, 4s), O(2s, 2p), and/or S(3s, 3p) orbitals were included as the valence electrons for the projector augmented wave (PAW) pseudopotentials. The DFT calculat ions were performed within the local density approximation (LDA)  It was confirmed that the favorability of atomic displacement and lone pair formation does not depend on the functionals. All calculations were performed at fixed volume and shape, while the atoms were relaxed until the forces were less than 0.03 eV/. Though all atoms are free to relax in these optimized structures the crystal lattice is fixed to the cubic pyrochlore structure. A plane wave cutoff energy of 400 eV was used along with a 333 Monkhorst Pack  mesh, resulting in 14 irreduc ible k points. Test calculations done with 450 eV cutoff and a 444 mesh resulted in differences less than 0.01 eV/88atom unit cell in the total energy, well within the error for the results presented in this chapter. To probe minima associated with dis placed cations, simulated annealing (see Section 184.108.40.206) calculations we re performed under fixed crystal shape and volume using ab initio molecular dynamics (MD) [44 -47] with the relaxed ideal pyrochlore
104 stru ctures (from the above procedure) as the initial structures. The MD simulations were conducted with only the point within the NVT ensemble starting at 1000 K and cooling to 300 K over 500 MD steps with a time step of 0.5 femtoseconds. After MD simulation, the system was minimized using the same approach as outlined above (333 mesh, CG method, etc.) and the equilibrium lattice parameter was reevaluated for the cation displaced structure. While the MD simulation is over extremely short times, it was confi rmed that increasing the simulation length does not affect the final fully relaxed structure. 4.3 Results and D iscussion 4. 3.1 Structural D etails For each of the bismuth pyrochlores (Bi2Ti2XX6, X= O or S) the equilibrium lattice parameter was determined by generating energy versus volume curves. Table 4 1 contains the equilibrium lattice parameters for the Bi2Ti2OO6, Bi2Ti2SO6, Bi2Ti2OS6 and Bi2Ti2SS6 pyrochlore system. Though there is no experimental data available for comparison in the sulfur containing systems, in the Bi2Ti2OO6 system there is excellent agreement between the DFT value of 10.34 and the experimental values of 10.357 for Bi1.74Ti2O6.62  an d 10.35907 for Bi2Ti2O7  In the earlier study, DFT predicted lat tice parameters for other Bi -based pyrochlores were within 1% of experimental values providing confidence in DFTs ability to reproduce these structures  Bi2Ti2SO6 has only 8 sulfur atoms per unit cell and shows the smallest lattice expansion from 10.23 to 10.65 When the number of sulfur atoms is increased to 48 per unit cell in the Bi2Ti2OS6, a m uch larger lattice expansion to 12.0 is observed. In
105 the Bi2Ti2SS6 with sulfur on both the X (8 b ) and X (48f ) sites, the lattice expands slightly more to an equilibrium lattice parameter of 12.3 Table 4 1. The cubic lattice parameter in and average bond lengths in of the ideal structure (before annealing) is shown for the Bi2Ti2XX6 and Bi2Ru2OX6 pyrochlore systems. Lattice Parameter () Expansion Percentage (%) Bond Length () Bond Expansion Percentage (%) Bi X Bi X BX Bi X Bi X BX Bi2Ti2OO6 10.23 2.21 2.55 1.96 Bi2Ti2SO6 10.65 4.1 2.31 2.69 2.02 4.5 5.5 3.1 Bi2Ti2OS6 12.00 17.3 2.59 2.91 2.35 17.2 14.1 19.9 Bi2Ti2SS6 12.30 20.2 2.66 3.03 2.38 20.4 18.8 21.4 Bi2Ru2OO6 10.205 2.21 2.51 1.98 Bi2Ru2OS6 11.75 15.1 2.54 2.88 2.28 15.1 14.8 15.2 It is well known that S anions are larger than O anions so the lattice expansion is not unexpected. In order to understand if the lattice expansion depended only on the incorporation of the larger sulfur anion on various lattice sites, the real space volum e occupied by each atom w as identified within the lattice. Utilizing the code of Henkelman et al  the Bader volumes we re determined based on the Bader criteria for decomposing the electron density  Since the Bader volumes are calculated using the zero-flux surfa ces in the electron density gradient, the sum of Bader volumes in a crystal will exactly equal the total cell volume. Therefore, each Bader volume will depend on the equilibrium lattice volume. To compare the atomic volume of an atom on a specific site bet ween the various crystal systems examined, each Bader volume w as scaled by its corresponding total unit cell volume. As expected, it was found that the scaled Bader volume of the Ti cation and O anion remain the same between the Bi2Ti2OO6 and the Bi2Ti2S O6 system. However, the scaled Bader volume of the Bi cation was reduced by 8% while the S increased the X (8 b ) site volume by 23%. In the
106 Bi2Ti2OS6 system the S anion increased the X (48 f ) site volume by 15% in comparison to the Bi2Ti2OO6 system. Between the Bi2Ti2OO6 and the Bi2Ti2OS6 systems, the Bi, Ti, and O atoms showed volume reductions of 27%, 24%, and 27%, respectively. Based on the variations in the Bader volumes, the equilibrium lattice parameter may not be determined a priori with only the atomic volumes of the constituent atoms. Table 4 1 also shows the Bi X, Bi X, and B X bond lengths, where X can represent either O or S atom. As expected, the lattice expansion increases all the bond lengths significantly. In the ideal py rochlore structure only the O atom experiences nonzero forces and will undergo atomic relaxation to optimize both the Bi -O and Ti -O bond lengths. Therefore, the Bi X bond length scales directly with the lattice parameter Some variations are observed when the bond lengths corresponding to the O site are examined. In the Bi2Ti2SO6 system, there is a 4.1% lattice expansion but the Ti -O bond only increases by 3.1%. This smaller Ti -O bond length indicates the Ti atom bonds more strongly to the O atoms and th erefore pulls the O atoms in closer, which then elongates the Bi -O bond by 5.5%. In both the Bi2Ti2OS6 and Bi2Ti2SS6 systems, the Bi S bond is reduced with respect to the lattice expansion and the Ti -S bond is elongated, indicating the Bi S interactions are stronger than the Ti -S interactions. Atomic displacement was probed for in the three materials by performing simulated annealing as discussed in Section 4.2, with the results summarized in Table 4 -2. Since the lattice parameters vary considerably between the systems, both the average displacement magnitudes and these magnitudes scaled by the equilibrium lattice parameter are included in Table 4-2. In Bi2Ti2SO6, atomic displacement is not favored since no significant energy chang e or atomic displacement was observed
107 Table 4 2. The cubic lattice parameter in energy change in eV/Bi atom, and average displacement magnitudes in after simulated annealing with respect to the ideal structure with positive energy indicating an increase in thermodynamic favorability are shown for the Bi2Ti2OO6, Bi2Ti2SO6, Bi2Ti2OS6, and Bi2Ru2OS6 pyrochlore systems. The average bond lengths are provided in Lattice Parameter () Energy Change (eV/Bi) Displacement Magnitude () [Scaled by Lattice Parameter (/ )] Bond Length () Bi B X Bi X Bi X BX Bi2Ti2OO6 10.34 0.146 0.001 0.38 0.02 [0.037 0.002] 0.07 0.02 [0.007 0.002] 0.11 0.01 [0.011 0.001] 2.26 0.04 2.34 0.11 1.96 0.05 Bi2Ti2SO6 10.65 0.000 0.00 0.00 [0.000 0.000] 0.01 0.00 [0.001 0.000] 0.00 0.00 [0.000 0.000] 2.31 0.00 2.69 0.00 2.02 0.00 Bi2Ti2OS6 12.00 0.533 0.51 0.14 [0.043 0.012] 0.40 0.14 [0.034 0.012] 0.71 0.18 [0.059 0.015] 2.33 0.27 2.85 0.14 2.39 0.08 Bi2Ru2OS6 11.75 0.064 0.23 0.03 [0.019 0.002] 0.01 0.00 [0.001 0.000] 0.20 0.00 [0.017 0.000] 2.56 0.35 2.89 0.04 2.28 0.01
10 8 Additionally, atomic displacement was attempted by forcibly pushing the Bi cations out of the ideal 16 d site and after atomic relaxation it was observed that Bi2Ti2SO6 does not retain the pyrochlore structure. The resulting structure is not energetically favorable with respect to the un -displaced pyrochlore structure; therefore, the ideal pyrochlore structure without cation displacement is taken as the preferred structure for the Bi2Ti2SO6 system. Alternatively atomic displacement is favored in the Bi2Ti2OS6 system with an energy change of 0.53 eV per Bi cation. When comparing this value with that reported in the Bi2Ti2OO6 system of 0.15 eV per Bi cation  the atomic displacement here is 3.5 times more favorable. This larger preference for the displaced structure is attributed to the disruption of the Bi -Ti2O6 interactions. The Bi cation interacts less with the Ti2S6 network than the Ti2O6 network, which in turn allows for stronger Bi -O hybridization. As will be discussed in more detail in Section 4.3.2, the Bi -O in teractions leads to lone pair formation and corresponding atomic displacement. For Bi2Ti2OS6, these stronger Bi -O interactions make the asymmetric atomic displaced structure much more favorable over the ideal structure with no cation displacement. In the Bi2Ti2OO6 system the Bi cation displaces on average 0.38 and in Bi2Ti2OS6 the Bi cations are displacing on average 0.51 Additionally, in Bi2Ti2OS6 the 0.7 atomic displacement of the Ti atoms is more than four times that seen in the Bi2Ti2OO6 system. The average bond lengths are included in Table 4 -2. The Bi displacement in the Bi2Ti2OS6 system reduces the Bi -O bond length from 2.59 to 2.33 and is comparable to the 2.26 bond length of Bi2Ti2OO6. Table 42 shows the averaged Ti -S bond length is not significantly altered by the large displacement of Ti.
109 However, within each of the Ti centered octahedron there are variations in the six local Ti -S bonds with up to a 0.2 variance of the Ti -S bond lengths within some of the octahedrons. T h e simulated annealing of Bi2Ti2SS6 shows similar results to Bi2Ti2SO6 with minimal atomic displacement observed initially and lattice distortion upon forcible atomic displacement. Since atomic displacement in Bi2Ti2OO6 can be suppressed by S substitution on the O site, the Bi2Ti2SS6 system is redundant and convolutes the role of the A2O and B2O6 networks and therefore will be excluded from the following discussion. 4. 3.2 Electronic S tructure The results presented in Section 4.3.1 clearly indicate that S substitution on the A2O network is needed to suppress cation displacement. This section will focus on understanding the differences in the electronic structures between the Bi2Ti2SO6 and Bi2Ti2OS6 systems and how these systems relate to the Bi2Ti2OO6 pyrochlore both before and after atomic displacement. The electron localization function (ELF) maps the probable distribution of electrons graphically, where a fully localized or delocalized elec tron has an ELF value of 1 or 0, respectively (see Sect ion 220.127.116.11 for additional details). T he ELF function is use d below to identify active lone pair formation on the Bi cations as a function of S substitution on the X (48f ) and X (8 b ) sites. The ELF is plotted in Figure 41 for the Bi2Ti2OO6 pyrochlore af ter atomic displacement in the (110) plane intersecting the Bi -O and Ti -O bonding environment. After simulated annealing, the Bi cation displaces to the right with an asymmetric lobe of increased electron localization forming to the left of the Bi cation i n the vacant space. It is through the development of this asymmetric lone pair that atomic displacement is
110 favored in the Bi2Ti2OO6 system. To understand why atomic displacement is not observed in the Bi2Ti2SO6 system the electronic structure was examined for this system with the same atomic displacement as that seen in the Bi2Ti2OO6 pyrochlore without allowing atomic relaxation. Figure 4 1 The electron localization of the Bi 2 Ti 2 OO 6 with atomic displacement for part of the (110) plane passing through one Bi atom (center) and four O ions (in the corners). The ELF values range from 0.0 (red) to 0.8 (purple) as indicated by the scale. The ELF for the Bi2Ti2SO6 pyrochlore with forced displacement is plotted in Figure 4 -2 in the same (110) plane. Though the Bi cation in the center of Figure 42 is displaced in the same fashion as that in Figure 41 the electron localization observed in Figure 42 surrounding the Bi cation shows negligible asymmetry character, indicating a p -type lone pair does not form in this system In fact, th e ELF for the Bi2Ti2SO6 system more closely resembles t he symmetric localization previously reported for Bi2Ti2OO6 pyrochlore without atomic displacement on the A2O network  In the earlier study t he observed symmetric lobes are due to site symmetry with the Bi and O in their ideal 16 d and 8 b crystallographic sites and are termed a s -type lone pair In the systems, it is interesting to note within the same atomic structure (i.e. displacement pattern) th at Bi -O interactions lead to atomic displacement and an asymmetric p -type electron localization for Bi2Ti2OO6 (Figure 4 -1 ) while Bi S suppresses p -type electron localization for Bi2Ti2SO6 (Figure 4-2 ) 0.0 0.2 0.4 0.6 0.8
111 Figure 42 The electron localization of the Bi2Ti2SO6 with forced atomic displacement for part of the (110) plane passing through one Bi atom (center) and four O ions (in the corners). The ELF values range from 0.0 (red) to 0.8 (purple) as indicated by the scale. Figure 4-3 illustrates the ELF plotted in the same (110) plane for the Bi2Ti2OS6 pyrochlore after simulated annealing. It is clear the atomic displacement in this system does not exactly match that observed in the Bi2Ti2OO6 pyrochlore but surrounding the Bi cation is an asymmetric lone pair. Therefore the development of a p-type lone pair on the Bi cation is observed in the two pyrochlore with favorable atomic displacement and not observed in the pyrochlore with forced atomi c displacement. The effect of S substitution onto the X site on the p-type lone pair formation is explored in more detail in the discussion of pDOS below. Figure 4 3. The electron localization of the Bi 2 Ti 2 OS 6 with atomic displacement for part of the (110) plane passing through one Bi atom (center) and four O ions (in the corners). The ELF values range from 0.0 (red) to 0.8 (purple) as indicated by the scale. To quantify charge transfer in these systems the Bader atomic charges  we re determined using the code of Henkelman et al  The Bader charges for the most 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8
112 favored structure, i.e. either with or without displacement, of each of the pyrochlore systems examined are included in Table 4 -3. The importance of covalency can be seen in these systems since all ions deviate from their formal charge values. Wang and coworkers  addressed covalent versus ionic bonding in bismuth pyrochlore using differences in the ion electronegativity. Ionic (covalent) bonding would dominate with the larger (smaller) difference in electronegativity between the two bonded atoms. In the earlier study, the Bi -O bonds were identified to be more ionic in nature than the B -O bonds in metallic pyrochlores (B=Ru, Os, Ir, and Pt). Inversely, for Bi2Ti2O7, the Bi -O bonds are more covalent in nature than the Ti -O bonds. These observations suggest that to observe cation displacement and p-type lone pair formation requires either to increase the strength of the Bi -O interactions or alternatively weaken the B -O interactions. As is show n below this simple rule is not suffi cient to describe the behavior of Bi pyrochlores upon S substitution. Table 4 3. Bader atomic charges in units of e for the Bi, B, X, and X atoms are shown for the Bi2Ti2OO6, Bi2Ti2SO6, Bi2Ti2OS6, Bi2Ru2OO6, and Bi2Ru2OS6 pyrochlore systems. The indicates systems that prefer atomic displacements. Lattice Parameter () Bader Charge ( e ) Bi B X X Bi2Ti2OO6* 10.34 1.88 2.18 -1.22 -1.15 Bi2Ti2SO6 10.65 1.63 2.21 -0.81 -1.15 Bi2Ti2OS6* 12.00 1.47 1.62 -1.14 -0.84 Bi2Ru2OO6 10.205 1.90 1.61 -1.27 -0.96 Bi2Ru2OS6* 11.75 1.37 0.66 -1.13 -0.49 T he Bader charges for both the ideal and S substituted Bi2Ti2O7 pyrochlores are reported in Table 4 -3 These Bader values may be interpreted in conjunction with the changes to the electronegativity differences upon sulfur substitution. With increasing
113 sulfur content, the Bi Bader charge decrease from 1.88 e for Bi2Ti2OO6 to 1.63 e for Bi2Ti2SO6 to 1.47 e for Bi2Ti2OS6, in dicating the degree of covalent bonding is increasing with increasing sulfur content. Using Paulings  electronegativity (EN) for Bi, Ti, and O, in Bi2Ti2OO6 the difference in EN between Bi and O is 1.42 while the difference between Ti and O is 1.9. In Bi2Ti2SO6, the difference in EN between Bi and S is 0.56 significantly lower than the Ti -O; therefore, the Bi -S bond should be more covalent in comparison to the Ti -O bond Table 4 -3 shows the Bader charges of Bi and S are significantly less than their respective formal charges and that the Ti and O charges do not change between Bi2Ti2OO6 and Bi2Ti2SO6. With sulfur occupying the X site, the difference in EN between Ti and S is 1.04, between Bi and O is 1.42 and between Bi and S is 0.56. It can be predict ed that the Bi -O bond should be more ionic than the Bi S bond which should be more ionic than the Ti -S bond. The calculated Bader charges again agree with this prediction with reduction of the Bi, Ti, O and S charges. While the Bader charges reflect the changes in covalency expected from using the differences in EN, they fail to predict the systems were cation displacement occurs. For example, th e Bi cation becomes more covalent going from Bi2Ti2OO6, Bi2Ti2SO6, and Bi2Ti2OS6 but only the two end systems show atomic displacement. While Bi2Ti2SO6 shows larger charge transfer between the Bi and X than in the Bi2Ti2O O6 system, it is only the lat ter that produces a strong p -type lone pair and atomic displacement. As is discuss ed below, cation displacement and lone pair formation are dictated by the nature of the Bi X cation hybridization, which is distinctly different in the Bi2Ti2O O6 and Bi2Ti2SO6 systems.
114 Figure 44 compares the partial density of states (pDOS) determined for the previously reported Bi2Ti2OO6  (a -d) before and (eh) after atomic displacement, with the states separated for each atom type. Figures 44 a and Figure 44 e compare the O2s and 2p states before and after displacement, respectively. At the low energy band corresponding to the Bi 6s states (see Figure 4 -4 b and 44 f), there is considerable hybridization between the O 2s and 2p states to overlap with the Bi 6 s orbital. Figure 4 4 Partial density of states of O 2 s O 2 p Bi 6 s Bi 6 p O 2 p and Ti 3 d in states per electron volt for the Bi2Ti2O7 pyrochlore at 10.23 (a-d) before and (e -h) after atomic displacement and lattice expansion to 10.34 Additionally, displacement significantly alters the valence band near the Fermi level with increased hybridization of the Bi 6s and 6p states and improves the overlap with the O 2p orbital. The central features of the pDOS responsible for lone pair formation can be identified by examining the partial electron density at specific energy windows. T he partial electron density was examined for the low energy band ( 11 to -8
115 eV), the intermediate energy band ( 6 to 2 eV) and th e top of the valence band ( 2 eV to the Fermi level). The energy range at the top of the valence band shows the most clear asymmetric electron density and can be attributed with the lone pair with the Bi 6s and 6 p orbitals strongly overlapping with the O 2 p orbital in the Bi2Ti2OO6 displaced system (see Figure 44 f). Figure 45 shows the partial electron density for Bi2Ti2OO6 (Figure 45a) with atomic displacement and Bi2Ti2SO6 (Figure 4-5b) with forced atomic displacement. The planes illustrated in F igure 4-5 correspond to a Bi cation, in the center of the plane, displaced upward and surrounded by its six O anions. (a) (b) Figure 4 5 The partial electron densities for the (a) Bi 2 Ti 2 OO 6 with atomic displacement and (b) Bi2Ti2SO6 with forced atomic displacement for the states between -2 eV to the Fermi level for part of the (111) plane passing through one Bi atom (center) and six O ions (edges) with the scale indicating the contour levels between 0.0 (blue) and 0.1 e /3 (red). Similar to Figure 4 -1, the lone pair on the Bi cation can be observed in the opposite direction of the displacement. In both cases, an asymmetric lobe is formed but the size and magnitude of density are much smaller in Bi2Ti2SO6 (Figure 4 5b) than in Bi2Ti2OO6 (Figure 45a), agreeing with the observations made from the ELF that lone pair formation is not favored for Bi2Ti2SO6 (Figure 42). 0.000 0.025 0.050 0.075 0.100
116 Figure 46 shows the pDOS for the Bi2Ti2SO6 pyrochlore (a d ) at equilibrium and ( e -h ) with forced atomic displ acement. The forced displacement system is the same structure mentioned above with regards to Figure 4 2 Although Bi2Ti2SO6 is insulating like Bi2Ti2OO6 ( Figure 4-4 ), the magnitude of the band gap has been reduced from 1.8 eV to 1.5 eV. The pDOS associated with the S site, shown in Figure 4-6 a, shows the 3s and 3 p states directly overlap in the low energy band corresponding to the Bi 6s states (see Figure 46 b) which differs from that observed for O in Bi2Ti2OO6 (Figure 4 4a and 4 -4 e). In the upper valence band, the S 3 p states are considerably lower in energy than equivalent O states resulting in a peak split of the Bi 6 p at the low energy range of this band. This split results in a small energy window where only the Bi 6p orbital interacts with the S 3 p orbital, which is not observed in any other pyrochlore. After forced displacement (Figure 4-6e -f), this interaction is shifted in energy towards the other interactions within the upper valence band. Additionally, the direct overlap of the S 3 s and 3 p states seen in the low energy valence band is reduced and mirrors that observed for Bi2Ti2OO6 (Figure 4-4a and 4 -4e). While there is an increase in hybridization of the Bi 6s and 6 p (Figure 46f) near the Fermi level after forced displacement its magnitude is much smaller than that seen in Bi2Ti2OO6 (Figure 44f). This corresponds to the previous observation of the symmetric electron localization (Figure 42) surrounding the Bi cations within this structure. Figure 47 compares the pDOS for Bi2Ti2OS6 (a -d ) before and (e-h) after atomic displacement. In the Bi2Ti2OS6 system, the band gap between the conducting and valence band is not observed but the energy gap between the two valence bands is retained
117 Figure 4 6 Partial density of states of S 3 s S 3 p Bi 6 s Bi 6 p O 2 p and Ti 3 d in states per electron volt for the Bi2Ti2SO6 pyrochlore at 10.65 (a d) before and (e-h) after forced atomic displacement. The O 2s and 2 p states (Figure 4 -7e) in the lower valence band overlap wit h the Bi 6 s states (Figure 4 7f) at the higher and lower energy respectively. This hybridization matches that seen for Bi2Ti2OO6 (Figure 44e -f). In the upper valence band, atomic displacement significantly increases the overlap of the Bi 6s and 6p orbitals near the Fermi level (see Figure 4-7f) with the O 2 p states (see Figure 47e), corresponding to the formation of the lone pair observed in Figure 43. The essential interactions between Bi 6s and O states at the low energy band and subsequent hybridi zation between the Bi 6 p near the Fermi level that mediate the forming of the lone pair in Bi2Ti2OO6 is similar to the findings of Walsh and co workers for a range of oxides [94,100,105,106]
118 Fig ure 4 7 Partial density of states of O 2 s O 2 p Bi 6 s Bi 6 p S 3 p and Ti 3 d in states per electron volt for the Bi2Ti2OS6 pyrochlore at 12.0 (a-d) before and (e-h) after atomic displacement. 4.3.3 Bismuth R uthenate This study is concluded by returning to a representative metallic bismuth pyrochlore, Bi2Ru2O6O, that in the earlier DFT study did not show any cation displacement  (see Section 18.104.22.168) Experimental studies of Bi2Ru2O6O showed atomic displacement  but the samples had significant oxygen deficiency on the A2O network. This oxygen deficiency might force the experimentally observed displacement and disagreement with DFT  but the stud y of O vacancies on cation displacement is left for discussion in Chapter 5. Section 3.3.3 on metallic Bi pyrochlores concluded atomic displacement is not favored due to the stronger interactions between Bi and the O atoms on the B2O6 network in these syst ems. For Bi2Ti2O6O the cation displacement is enhanced and the p -type lone pair formation more pronounced when S is substituted in the B2O6 network. This result suggests that substituting sulfur in the B2O6 network for
119 the metallic pyrochlores may induce cation displacement. Below this hypothesis is tested by examining sulfur substitution on Bi2Ru2O6O. T he lattice optimization procedure discussed in Section 4.3.1 is repeated for the Bi2Ru2OS6 structure to determine the equilibrium lattice parameter and favorability of atomic displacement. The structural details of the ideal pyrochlore both with and without sulfur are included in Table 41. Following s imulated annealing, it is found that atomic displacement is favored in the Bi2Ru2OS6 pyrochlore, with the details summarized in Table 4 2. The Bi cations in the Bi2Ru2OS6 pyrochlore displace towards one of the two O anions thereby creating two long and two short Bi -O bonds within each A4O tetrahedron (see Figure 23 b). In other bismut h based pyrochlores, this type of Bi cation displacement has been dis cussed as spin ice tetrahedron bonding  The pyrochlore O anions are equated to oxygen within ice with two hydrogen ions covalently bonded and two hydrogen ions more weakly bound at significantly l onger distances. While the spinice arrangement does not affect the average Bi -O bond in the pyrochlore, structurally a single long and short Bi -O bond can be seen by looking at the atoms included in Figure 48 In addition to showing some structural details, Figure 4-8 illustrates the partial electron density for a portion of the 1 01 plane, which slices through the S -Bi -O bonding network. The energy range of interest, 2 eV to the Fermi energy, is the same as that examined in the Bi2Ti2O7 pyrochlore (F igure 4-5). A similar asymmetric electron density to that seen in the Bi2Ti2O7 is observed for the Bi2Ru2OS6 pyrochlore after atomic displacement, but the electron density maximum is larger ( 0.1 e /3 versus 0.05 e /3) in Bi2Ti2O7.
120 Fig ure 4 8 The partial electron densities for Bi 2 Ru 2 OS 6 with atomic displacement for the states between 2 eV to the Fermi level for part of the 1 01 plane passing through one Bi atom (center), two O (atoms included) and two S ions with the scale indicating the contour levels between 0.0 (blue) and 0.1 e /3 (red). Figure 4 9 shows the ELF for Bi2Ru2OS6 with atomic displacement in the 1 01 plane and illustrates an asymmetric lobe form s along the long Bi -O bond direction created behind the Bi cation displacement towards the opposite O. Though the magnitude of electron localization in Bi2Ru2OS6 ( Figure 4 -9 ) is lower than that seen in Bi2Ti2OO6 (Figure 4 1 ), it can see n in Bi2Ru2OS6 t he lone pair formation is favored and triggers the atomic displacement. The Bader charges for the Bi2Ru2OO6 and Bi2Ru2OS6 systems are reported at the bottom of Table 4 -3. Similar to the bismuth titanate systems, sulfur incorporation increases the degree of covalent bonding in the system, as indicated by the reduction of the Bi, Ru, O and O/S site Bader charges. These results suggest that modifying the B2O6 network to weaken Bi O interactions is a plausible strategy to increase cation displacement and in turn induce larger dielectric permittivity in these materials. It will be interesting to compare the lone pair formation and role in cation displacement in the more complex Bi pyrochlores such as BZN to the observations of Bi2Ti2O7. 0.000 0.025 0.050 0.075 0.100
121 Fig ure 4 9 The electron localization function for Bi 2 Ru 2 OS 6 with atomic displacement for part of the 1 01 plane passing through one Bi atom (center), two O (atoms included) and two S ions. The ELF values range from 0.0 (red) to 0.8 (purple) as indicated by the scal e. 4. 4 Summary In this chapter, DFT calculations were employed to further clarify the lone pair formation in Bi2Ti2OO6 through sulfur substitution on the O and O sites. Atomic displacement is not favored in the case of S substituted on the O site, conf irming the primary importance of the Bi -O interactions. Analysis of the partial density of states, ELF and partial electron density shows that the S substitution on the O site suppresses the formation of the lone pair by modifying the Bi anion hybridizat ion. The electronic structure was examined f or the Bi2Ti2SO6 pyrochlore fixed with the same displacement pattern as that seen in Bi2Ti2OO6. In Bi2Ti2SO6 with forced displacement the ELF displayed symmetric lobe around the bismuth cations with a signifi cant reduction of the asymmetric electron density in the energy range of -2 eV to the Fermi level. In the case of sulfur on the O site, atomic displacement is significantly favored energetically in Bi2Ti2OS6 and showed the largest atomic displacement. Spe cifically, the Ti and O atoms displaced considerably more in Bi2Ti2OS6 than in Bi2Ti2OO6 which is attributed to weaker Bi S versus Bi -O interactions that leads to stronger hybridization between the 0.0 0.2 0.4 0.6 0.8
122 Bi and O states that favors the lone pair formation. Finally, it is show n that by substituting S on the B2O6 network of Bi2Ru2OO6 cation displacement and lone pair formation can be induced in the metallic Bi pyrochlores by disrupting the Bi -O interactions. This result suggests that modifying the B2O6 network to reduce the interactions with the A2O network will assist in promoting cation displacement.
123 CHAPTER 5 ROLE OF VACANCIES ON CATION DISPLACEMENT IN PYROCHLORES 5.1 Introduction As shown in Chapter 3, even without cation substitutions, various bismuth pyrochlores exhibit atomic displacements [20,23,28,35] Sec tion 3.3.2 examined cubic bismuth pyrochlores without cation substitutions, with molecular formula Bi2B2O6O (B4+=Ti, Ru, Rh, Ir, Os, and Pt), and directly probed for atomic displacements  It was found that atomic displacement was energetically favorable only in Bi2Ti2O7 and the observed displacement magnitudes and patterns  agreed very well with experimental reports [23,28] These simulations provided critical insight into the main driving force behind the atomic displacement through electronic structure features. In Bi2Ti2O7, it was s hown increased overlap between Bi s Bi p and O 2 p states not observed in any of the other compounds considered and these interactions lead to the formation of an asymmetric p -type lone pair on the Bi cations. These interactions were suppressed in the metallic pyrochlores (e.g. Bi2Ru2O6O) due to strong O -B4+ and O -Bi interactions. In Chapter 4, sulfur was substituted onto the oxygen site in both subnetworks to determine the necessary interactions t hat lead to lone pair formation. Sulfur substitution on the bismuth ruthenate, in Section 4.3.3, showed lone pair formation could be induced by modifying the O -B4+ and O -Bi interactions. Bismuth ruthenate pyrochlores have been synthesized with several st oichiometries including oxygen and bismuth vacancies [20,22,109,110] and in general can be considered a non-stoichiometric compound. Further, based on neutron diffraction data atomic displacement occurs on both the bismuth and oxygen sites  As discussed in Section 22.214.171.124, the first principles calculations  of stoichiometric bismuth ruthenate
124 directly disagreed with the experimental reports on the favorability of atomic displacements  Based on the discussion in Section 4.3.3, the observed atomic displacement was attributed to the O vacancies. However, Goodwin et al question the likelihood of the oxygen vacancies in triggering the observed displacements since their reverse Monte Carlo models of the stoichiometric system showed atomic displacement and the simulated electron diffraction patterns were qualitatively com parable to experimental results  Section 4.3.3 for stoichiometric bismuth ruthenate (i.e. Bi2Ru2O7) showed atomic displacement may be induced by strengthening the Bi -O interactions as a secondary effect of weakening the Bi -O interactions by substituting S onto the Ru2O6 sub network  In the case of O and Bi vacancies the local Bi -O bonds should be strengthened due to the under -coordination, and likely trigger atomic displacements within the deficient bismuth ruthenate. This chapter used DFT calculations to (1) provide additional confirm ation that oxygen and bismuth vacancies trigger atomic displacements within bismuth ruthenate pyrochlore and (2) illustrate the displacement patterns and electronic structure differ for vacancy triggered displacement from the displacement observed in Bi2Ti2O7 attributed to the formation of a p type lone pair  5.2 Calculation Details As discussed in Section 2.2.1, t he pyrochlore structure is often viewed as two interpenetrating networks of Bi2O and Ru2O6  (see Figure 2-3) T his chapter focus es on the Bi2O network since the vacancies and atomic displacements are isolated to this subnetwork  For completeness, the crystallographic details of the structure reported in Section 3.3.1 for stoichiometric Bi2Ru2O7  and that determined based on neutron diffraction data taken at 12 K by Avdeev et al  for Bi1.9O0.9Ru2O6
125 are both reported in Table 51. This ideal structure previously d etermined for Bi2Ru2O7 served as the starting point for all calculations which were performed with Vienna Ab initio Simulation Package (VASP) [44 -47] with details matching that reported in Section 3.2 except the force criteria was slightly lowered to 0.005 eV/ T he Bi2Ti2O7 system before (from Section 3.3.1) and after atomic displacement (from Section 126.96.36.199) was reexamined with the force criteria used here and confirmed the energy of displacement and displacem ent pattern are equivalent to that previously reported in Section 188.8.131.52. 5.3 Vacancy Configurations The four pyrochlore bismuth ruthenate compositions reported in the literature are Bi2O0.9 5Ru2O6  Bi2O0.9Ru2O6  Bi2O0.7Ru2O6  and Bi1.9O0.9Ru2O6  with the oxygen vacancy attributed to the O site. In the ideal pyrochlore there are eight symmetrically equivalent O anions within the unit cell. Therefore, the experimental composition (B i2O0.9Ru2O6)  is nearly matched by removing one O atom from the unit cell to give Bi2O0. 875Ru2O6 as the simulated composition. Since the choice of which O anion is removed is irrelevant due to symmetry, one O anion in the middle of the ideal unit cel l was selected to serve as the O vacancy site. Figure 51 illustrates the Bi2O subnetwork with two unit cells connected along the  direction of this configuration viewed along the 1 1 0 (in Figure 51a) and  (in Figure 51b) direction. T he two empty unit cells along the 1 1 0 direction are outlined to aid in visualizing the perspective between the two filled cells. This type of structure is termed VO with the O vacancy site outlined by an empty frame. The experimentally reported composi tion of Bi1.9O0.9Ru2O6  was simulated by removing one of the 16 Bi cations to give a nearly equivalent simulated composition of Bi1.875O0.875Ru2O6.
126 (a) (b) (c) Figure 51. The Bi2O0.875 subnetwork viewed along the (a) [ 1 1 0 ] and (b) [ 001 ] direction. For visual aid, the tetrahedra at the O vacancy site is distinguished by an empty frame, the unit cell is outlined by a dashed line, and the (empty) unit cell along [ 1 00] and  are included. (c) The Bi1.875O0.875 local structure is isolat ed with two types of Bi vacancy sites distinguished by half and quarter filled atoms. Following the selection of the O vacancy the symmetry of the structure reduces the number of distinguishable Bi vacancy sites to two types. The first type has the bismut h vacancy within the tetrahedron outlined as the empty frame and one possibility is illustrated as a half filled Bi cation site in Figure 51c. This type of structure places the Bi O
127 Bi vacancy as the first nearest neighbor to the O vacancy and is refer r ed t o as VO VBi '''. The second type of vacancy configuration places the Bi vacancy at the second nearest neighbor site or one tetrahedron away from the oxygen vacancy. T his configuration is named VO Bi3O VBi ''' and one of the possible Bi sites meeting this symmetry is highlighted in Figure 51c as a quarter filled Bi cation. The crystallographic details for all three deficient structures after simulating annealing are reported for the Bi and O atoms in Table 51. Figure 52a shows the O Bi -O structure with the three different Bi displacement types 96h 96 g 192 i from light to dark isolated in Figure 5-2b and the two types of O displacements isolated in Figure 25c along with the experimentally reported displ acement parameters as the gray ellipsoids. (a) (b) (c) Figure 5 2 A local (a) O Bi O bonding network with the three possible Bi (from light to dark 96 h 96 g 192 i ) and two O types of displacements reported in Table 5 -1. An alternative view along the  direction isolating the (b) Bi and (c) O local displacement pattern. The displacement reported by Avdeev et al  is included as the translucent gray ellipsoids. i h g
128 Table 5 1. Complete crystallographic parameters for B -centered ideal Bi2Ru2O7 from DFT for space group Fd 3 m with a = 10.205 compared with refinement for Bi1.9O0.9Ru2O6 from neutron diffraction data tak en at 12 K with a = 10.270 Crystallographic parameters for Bi and O following simulated annealing for Bi2O0.875Ru2O6, and both Bi1.875O0.875Ru2O6. Atom Site x y z Occupancy Hinojosa et al Bi2ORu2O6  Bi 16 d 0.5 0.5 0.5 1 Ru 16c 0 0 0 1 O 48f 0. 329 0.125 0.125 1 O 8 b 0.375 0.375 0.375 1 Avdeev et al Bi1.9O0.9Ru2O6  Bi 96 h 0 0. 240 0. 76 0 0.157 Ru 16c 0 0 0 1 O 48f 0.3 27 0.125 0.125 1 O (model 2) 8 b 0. 37 5 0. 37 5 0. 37 5 0.906 O (model 3) 32 e 0. 3 82 0. 3 82 0. 3 82 0.226 Bi 2 O 0.875 Ru 2 O 6 (configuration V O ) Bi 96 h 0 0. 240 0. 76 0 0.1 6 7 O 32 e 0.388 0.388 0.388 0. 125 O 32 e 0.380 0. 380 0. 380 0.094 Bi 1.875 O 0.875 Ru 2 O 6 (configuration V O V Bi ''') Bi at V O 192 i 0.5 0 6 0. 5 1 2 0. 520 0.016 Bi 96 h 0 0.238 0.762 0.125 O at V Bi ''' 32 e 0.391 0. 391 0.391 0. 094 O 32 e 0.380 0. 380 0. 380 0.125 Bi 1.875 O 0.875 Ru 2 O 6 (configuration V O Bi3O V Bi ''') Bi at V O 96 h 0 0. 238 0. 762 0.0 42 Bi 96 g 0.5 0 9 0.5 0 9 0. 487 0. 115 O at V Bi ''' 32 e 0.389 0.389 0.389 0. 125 O 32 e 0.380 0.380 0.380 0.094
129 5.4 Results and Discussion 5.4.1 Structural Details 184.108.40.206 Bi2O0. 875Ru2O6 composition Within the VO configuration, the Bi cations directly surrounding the O vacancy site experience unbalanced forces when occupying the high symmetry sites; therefore, following minimization, the local Bi cations displace off their ideal site towards the missing O. Following simulated annealing, the other Bi cations within the structure also displace but the difference in the total system energy is negligible between the two states. The Bi cation displaces to a 96 h ( 0, y -y ) Wyckoff site with y =0.010 As seen in Figure 52b, this type of Wyckoff site has six equally occupied positions with a displace ment magnitude of 0.14 from the ideal high symmetry site. The Bi cations near the vacancy site before and after annealing showed the Bi displaced preferentially towards the open space created by the vacancy. This is further illustrated by the change in b ond lengths following displacement, summarized in Table 52. In the ideal structure, the Bi -O bond length was 2.21 but the Bi -O bond length is shorten to 2.11 after displacement for the Bi cations located at the O vacancy, i.e. the Bi cations illus trated as half filled in Figure 51c. For the Bi cations illustrated as quarter filled in Figure 51c, the Bi -O bond lengths are elongated to 2.28 However, the Bi cations linking two complete (OBi4) tetrahedra follow the equal probability occupation o f the equivalent Wyckoff positions. The bond lengths for these Bi cations are 2.23 and match that seen for the ideal structure. The Bi displacement splits the six Bi -O bonds into three types with bond lengths of 2.41 2.51 and 2.65 which agrees v ery well with experimentally determined bond lengths [20,110]
130 Table 5 2. Average b ond le ngths (in ) in the ideal and displaced structure for the various configurations all standard deviations are less than 0.005. T he Bi -O bond lengths near and away from th e vacancy site are separated and any short/long bonds at the vacancies are distinguis hed. B ond Lengths Bi -O Bi -O at vacancy Bi -O Ru -O Ideal Bi 2 Ru 2 O 7 2.21 2.51 1.98 Bi 2 O 0.95 Ru 2 O 6 a 2.23 2.55 1.98 Bi 2 O 0.9 Ru 2 O 6 b 2.23 2.55 1.98 Displaced Bi 2 O 0.875 Ru 2 O 6 ( V O ) 2.23 2.11,2.28 2.41,2.51,2.65 1.98 Bi 2 O 0.95 Ru 2 O 6 a 2.23 2.42,2.56,2.69 1.98 Bi 1.875 O 0.875 Ru 2 O 6 ( V O V Bi ''' ) 2.21 2.09,2.31 2.39,2.51,2.69 1.97 Bi 1.875 O 0.875 Ru 2 O 6 ( V O Bi 3 O V Bi ''' ) 2.23 2.09,2.31 2.39,2.51,2.70 1.97 Bi 1.9 O 0.9 Ru 2 O 6 c 2.23 2.42,2.55,2.67 1.98 aRef.  bBased on structure in Ref.  cBased on structure in Ref  Additionally, following annealing the O anions displace off the high symmetry site to retain appropriate bonding distances to the displaced Bi cations. However, the degree of displacement depended on the proximity to the vacancy site. For the O anions th at are directly bound to the Bi cations forming the empty frame illustrated in Figure 51a or the vacancy sites nearest neighboring Bi cations, the O anions displace into a 32e (x x x ) type site with an x parameter of 0.388 and 4 equally probable positi ons 0.23 away from the ideal site. The O anions away from the first nearest Bi cations displace with a magnitude of 0.09 also into a 32e site. The O anions bound to the half filled Bi cations in Figure 5-1c displace preferentially towards these three fold coordinated Bi cations and the O anions away from the vacancy equally occupy the four equivalent 32 e sites. The preferential displacem ent of the O anions towards the half filled in Figure 5 1c leads to the splitting of the Bi -O bond lengths detailed above. As will be show n later through bond valance sums, this difference in displacement
131 magnitude is due to the degree of under -coordinat ion of the Bi at the vacancy site. The resulting displacement pattern for both the Bi and O determined here match well to those reported by Avdeev et al  One notable discrepancy is the assignment for the O anions, Avdeev et al also reports a 32e (x x x ) site however the magnitude of displacement is significantly lower at 0.12 The displacement pattern of both types of O reported here are within the ellipsoids predicted by the thermal parameter reported by Av deev et al  as seen in Figure 52c. In Bi2Ti2O7, it was observed that following atomic displacement the cubic lattice expands from 10.23 to 10.34 to accommodate the Bi and O displacements and Bi lone pair fo rmation  A similar expansion is anticipated in bismuth ruthenate, especially since the lattice constant is underestimated by DFT at 10.205 versus the experimental values from 10.270  to 10.299  Following simulated annealing, lattice expansion was investigated by reevaluating the lattice constant versus total lattice energy profile with the annealed structure. N o lattice expansion was found in any of the three configurations. Though this seems to contradi ct previous observations, when considering the displacement patterns and lone pair formation discussed in Section 5.4.2, the bismuth ruthenate lattice is less perturbed by atomic displacement than the bismuth titanate lattice. The displacements are smaller in magnitude and the atoms tend to displace towards the open space created by the vacancy near the vacancy. 220.127.116.11 Bi1.875O0.875Ru2O6 composition For the VO VBi ''' configuration, the Bi and O atoms in the ideal sites immediately displace due to unbalanced forces and the displacement magnitudes depend on the proximity to the vacancy sites, as seen in Table 52. Near the O vacancy, the Bi cations
132 displace into a 192 i (x y z ) type site with x =0.006, y =0.012, and z =0.020 and a displacement magnitude of 0.24 This displacement pattern would result in 12 equivalent positions, shown in Figure 5 2b, but again the Bi cations preferentially displace towards the vacancy si te. Away from the O vacancy site, the Bi cations displace into a 96 h (0, y -y ) type with y =0.012 and a displacement magnitude of 0.17 As with the VO configuration, the Bi cations away from the vacancy site occupy the six equivalent positions equally The O anions near the Bi vacancy displace off the high symmetry site by a much larger magnitude of 0.29 into a 32e site with x =0.391. The O anions away from the Bi vacancy follow the same pattern as those O anions in the VO configuration with a d isplacement magnitude of 0.09 The larger Bi and O displacements shortened the Bi -O bond length to 2.09 for the half filled Bi cations in Figure 51c and slightly elongate the Bi -O bond length to 2.31 for the quarter filled Bi cations in Figure 51c. The Bi -O bonds are slightly altered from the VO configuration, as summarized in Table 52. Similarly, the VO Bi3O VBi ''' configuration showed the same type of displacement pattern with regards to proximity to the vacancy sites. The Bi cati ons near and away from the O vacancy displace into a 96 h and 96 g site with a magnitude of 0.17 and 0.18 respectively Again all O anions displace into a 32e site but the 0.09 or 0.24 magnitude of the displacement depends on the proximity to the Bi v acancy. Despite the differences in the displacement patterns seen in Figure 52, the bond lengths in Table 52 for the VO Bi3O VBi ''' configuration essentially match the VO VBi ''' configuration. In both configurations with O and Bi vacancies the atomic displacements are favored by 1.38 eV/unit cell over the ideal high symmetry sites and that annealing does
133 not significantly affect the final displacement pattern or total energy. In comparison, displacement and lattice expansion was favored on average by 2.3 eV/unit cell in Bi2Ti2O7  The clustering of th e vacancies into one tetrahedron (the VO VBi ''') is slightly more favorable by 0.25 eV/unit cell than the VO Bi3O VBi ''' configuration. Unlike Goodwin et al  a long range correlation is not fo und in the displacements of the Bi cations. In Bi2Ti2O7, displacement correlation reduced the local lattice strain due to stereochemical restriction of the bismuth lone pair  However, in the Bi1.875O0.875Ru2O6 system the local network is not strained due to the open space created by the vacancy Therefore, correlation of the displacement may not be necessary to stabilize the atomic displacement to reduce the local strain. Goodwin et al  used Bi2ORu2O6 and upon displacement the corner sharing tetrahedra rotate in a -cristobalite type displacive disorder with minimal tetrahedron distortion as described by rigid unit modes  Though the concentration of oxygen vacancies is low it essentially matches the experimental concentrations. At these concentr ations, within a single unit cell with periodic boundary conditions, a single oxygen vacancy disrupts the long range network every third tetrahedron along the <110> direction. This frequent disruption is sufficient to overcome the rigid unit mode type of d isplacive disorder and leads to tetrahedra distortion and rotation. Bond valance sums (BVS) is often used to measure the degree of bonding strain at the local bonding level by combining the local bond lengths and geometry to determine if the coordinatio n satisfies each atoms oxidation state [80,112] see Section 2.2.3 for details Us ing the appropriate parameters [80,113] the BVS was determined for the ideal stoichiometric and deficient structures and compared them to that of the
134 fully relaxed deficient structures, which are summarized in Table 53. In the ideal structure, the O vacancy introduces a significant decrease in the large BVS of the Bi cations neighboring the vacancy site from 3.4 v.u. to 2.7 v.u. With the inclusion of a Bi vacancy, the local O BVS reduces from 2.9 v.u. to 2.2 v.u. It is clear to see with these significant changes in the local bonding environm ent that the O and Bi cations would alter their local bonding networks through displacement to reduce the local strain. Table 5 3 Average b ond valence sums (BVS) in valence units (v.u.) for each atom type in the ideal and displaced structure for the various configurations all standard deviations are less than 0.05 For the Bi and O types the BVS near (away from) the vacancy site are distinguished. BVS Bi Ru O O Ideal Bi 2 Ru 2 O 7 3.4 4.1 2.9 2.0 Bi 2 O 0.875 Ru 2 O 6 2.7 (3.4) 4.1 2.9 2.0 Bi 2 O 0.9 Ru 2 O 6 a 3.0 4.0 2.8 1.9 Bi 1.875 O 0.875 Ru 2 O 6 2.7 (3.4) 4.1 2.2 (2.9) 2.0 Displaced Bi 2 Ru 2 O 7 b 3.0 4.1 2.2 2.0 Bi 2 O 0.875 Ru 2 O 6 3.0 (3.3) 4.1 2.8 2.0 Bi 1.875 O 0.875 Ru 2 O 6 3.1 (3.3) 4.2 2.2 (2.7) 2.0 Bi 1.9 O 0.9 Ru 2 O 6 c 3.2 4.0 2.8 1.9 aBased on structure in Ref.  bRef.  cBased on structure in Ref.  Within (Bi1.5Zn0. 5)(Zn0.5Nb1.5)O7 (BZN), Withers et al  sho wed BVS play a key role in pushing the under -coordinated Bi (2.5 v.u.) cations towards the under bonded O (1.7 v.u.) within the in B2O6 network to reduce both the Bi and O under -bonding instead of displacing towards the O anions with a BVS of 2.0 v.u. wit hin the A2O network. Within bismuth ruthenate, after displacement it is observed that the BVS (in Table 5 3) of Bi near the O vacancy is 3.1 v.u. and 3.3 v.u. away from the vacancy, with corresponding Bi -O bond lengths (in Table 52) of 2.09 and 2.23 respectively. Also, the BVS of O near the Bi vacancy is 2.2 v.u. while away it is 2.7 v.u. This reduction in the total BVS across the Bi2O network leads to a slight increase in the BVS
135 of Ru from 4.1 v.u. in the ideal structure to 4.2 v.u. in the displaced system, with only a slight decrease in the Ru-O bond length from 1.98 to 1.97 The average BVS values were determined based on the structures reported for the experimental compounds [20,22] and find good agreement with the DFT values. For Bi2O0.9Ru2O6,  within the ideal pyrochlore structure, the BVS values are 3.0 v.u., 4.0 v.u., and 2.8 v.u. for Bi, Ru, and O, respectively and for Bi1.9O0.9Ru2O6  the average BVS values only differed for Bi w ith 3.2 v.u. When the values are averaged, the Bi values agree very well but the experimental based Ru values are slightly lower and the O values are slightly higher than what was determined here 5.4.2 Electronic Structure To help identify changes in t he bonding within the displaced structure the electron localization function (ELF) was examined for various planes. The ELF provides a real space mapping of the probable distribution of electrons based on the Pauli exclusion principle where a fully local ized (delocalized) electron has an ELF value of 1 (0) [65,66,114] The ELF is plotted in Fig ure 5 3a for the fully relaxed VO Bi3O VBi ''' configuration in the 1 11 plane isolating the Bi cations near the O vacancy and the O anions at the Bi vacancy with the scale shown. Figure 5 -3b outlines the equivalent structure plane in a black box. The three Bi cations displaces toward the center of the O vacancy site in Figure 53a. An asymmetric lobe of increased electron localization forms at each of the Bi cations and each of t hese lobes point towards the vacant space. In other bismuth pyrochlore systems that exhibit displacement, the lone pair forms in the opposite direction to the observed displacement; however, here the lone pair and displacement are in the same direction. T his difference is due to the open space
136 created by the vacancies within the structure. In fact, unlike stoichiometric bismuth pyrochlores, within the ideal high symmetry sites, the Bi cations still exhibit an asymmetric lobe pointed into the open space created by the vacancy. Within the other vacancy configurations the Bi cations also illustrated an asymmetric lobe both before and after displacement off the high symmetry site. (a) (b) Figure 5 3 Part of the (1 11) plane of the VO Bi3O VBi ''' configuration showing the (a) electron localization function with scale and the (b) solid black box outlining the equivalent structure plane including the three Bi cations near the O vacancy (center of empty tetrahedron frame) and the two O anions at the Bi vacancy (quarter filled atom). For visual aid two RuO6 octahedra are included and Bi cations above and below the plane are lighten. 5.5 Summary Through DFT calculations, it is shown that vacancies on the Bi2O network trigger atomic displacements of both the Bi and O atoms due to alterations of the local Bi -O interactions. F or the three vacancy configurations considered, it is found that the O anions displace off the high symmetry site into a 32e site but the magnitude of 0.0 0.2 0.4 0.6 0.8
137 displace ment depends both on the type of configuration as well as the proximity to the vacancies present in the structure. In all cases, the O displacement predicted by DFT is well within the displacement pattern predicted by neutron diffraction data  The magnitude and type of Bi displacement varied both by system and proximity to vacancies. No correlation of the atomic displacements is found, indicating even a small concentration of O vacancy is sufficient to reduce the local stretic strain on the A2O network due to Bi lone pair formation from atomic displacement. This work suggests the intentional inclusion of oxygen vacancies could be used to induce atomic displacements in bismuth pyrochlores and disrupt long range c orrelation which should improve the observed dielectric properties. Du et al  examined various compositions of BZN and found the dielectric constant dec reased with increasing oxygen vacancy concentration; however, this reduction was partially attributed to the substitution of the less polarizable Zn cation for the Bi cations due to valance compensation for the increasing oxygen vacancy concentration on th e O site, in addition to the formation of secondary phases. Therefore, it would be interesting to see a similar study of vacancy concentration influence on dielectric properties in a bismuth pyrochlore with only Bi on the A site. Additionally, it was previously shown that displacement may be triggered by weakening the Bi -O interactions and thereby strengthen the Bi -O interactions  T his chapter showed altering the bond strain at a local level by the inclusion of Bi and O vacancies also induces atomic displacement. Though the role of vacancies is sufficiently important since many synthesized bismuth pyrochlores are slightly deficient in the A2O network, this strain affect should not be limited to the presence of vacancies
138 alone. Intentionally doping the system with cation substitutes on the A2O or B2O6 network would create varying local bond strain affects as well as significant alterations of the Bi -O and Bi -O interactions and either may trigger atomic displacements.
139 CHAPTER 6 BISMUTH PYROCHLORES WITH CATION SUBSTITU TES 6.1 Introduction Chapter s 3, 4, and 5 focused on isolating and understanding the triggers behind atomic displacement in bismuth pyrochlores without cation substitution. Section 3.3.3 showed the interactions between Bi -O and Bi -O are critical to atomic displacement ind uced by lone pair formation. Section 5.4 showed atomic displacement may also be induced by altering the local interactions, through the introduction of a vacancy. Though these simple pyrochlores are also important, t he majority of work investigating the bismuth pyrochlore family has focused on compounds that include Bi1.5ZnNb1.5O7 (BZN)  Bi1.5ZnTa1.5O7 (BZT), Bi1.5MgNb1.5O7 (BM N), Bi1.5MgTa1.5O7 (BMT), and Bi1.5ZnSb1.5O7 (BZS) due to their dielectric properties [37,38] in addition to compounds incorporating multiple substitutions such as (Bi1.5Zn0.5)(Ti1.5Nb0.5)O7 (BZNT)  These complex bismuth based pyrochlores are often studied due to their combination of ideal characteristics for capacitors and high-frequency filter applications  The phonon modes for BZN, BZT, BMN and BMT have been indentified  and the di electric properties of ten bismuth pyrochlore compounds have been examined  .Though each of the compounds detailed above have interesting properties, one system of particular interest is BZN, with Bi1.5Zn0.92Nb1.5O6.92 as the molecular formula has been shown to be both zinc and oxygen deficient and to have disordered displacement on both the A and O site  As discussed in Section 18.104.22.168, bond valence sum analysis o f BZN has shown the atomic displacement observed is necessary to alleviate the under -coordination of O and Bi  Based on the structural analysis in Section 5.4, it is expected in BZN the O
140 and Zn deficiencies would also contribute to the atomic displacements locally with the displacements magnitudes decreasing with increasing distance from the vacancy sites. Experimentally the atomic displacements that have been reported through the neutron and X ray diffraction data have been lin ked to the dielectric permittivity  It is proposed that the atomic displacement of Bi cations is due to the formation of an active lone pair. T he lone pair formation has been linked to the atomic displacement in Section 3.3.3 and 4.3.2 in addition to 5.4.2. It is therefore expe cted that the Bi cations within these complex pyrochlores would also exhibit lone pairs. The active lone pair and local dipole moments should both contribute to the overall dielectric permittivity as discussed in Section 1.1 2 As discussed by Melot et al the magnitude of the A and O atom ic displacement is correlated to the dielectric permittivity due to the dipole moments created between these atoms  Due to all of these structural details that affect the dielectric permittivity, the bond lengths and atomic displacement magnit udes will be discussed in Section 6.4.2. Based on neutron diffraction data of BZN, short range ordering of metal ions and under -coordination of cations induced relaxation has been proposed  As will be discussed in Section 6.2, there are various ways the cation substitutes could be arranged within the ideal pyrochlore structure. DFT may examine various atomic structure changes in the complex bismuth pyrochlores to identify energy differences between the atomic structures. To help foc us the investigation S ection 6.4.1 and 6.4.2 will examine the energetic s of Zn ordering and predicted structural details including atomic displacements in BZN Then Section 6.4.3 will compare both the energetics and structures of BZN, BZT, BMN, and BMT
141 6.2 Atomic Structure Ordering Predictions in BZN 6.2.1 Bond Valence Analysis of BZN Using the local chemistry of the pyrochlore structure, Withers et al  predicted favorable configurations and identified four potential rules f or the Zn substitution into the A2B2O7 compound. Withers et al completed a bond valence analysis for both the A2O and B2O6 network for Bi, Zn, and Nb cations and O and O anions to determined the magnitude of atomic displacement needed to satisfy each at oms bond valence (see Section 2.2.3)  Using an assumed stoichiometric system (Bi1.5ZnNb1.5O7), the bond valence sums calculations predicted a Nb5+-O bond length of 1.98 and a Zn2+-O bond length of 2.11 for the (Zn0.5Nb1.5)O6 n etwork. These bond lengths were compared to the average cation to oxygen bond length on the B site from experiment of 2.01 indicating the Nb would pull O in by about 0.03 and the ZnB would push O away by about 0.09 In the (Bi1.5Zn0.5)O network, the predicted Bi3+-O bond length was 2.35 and the Zn2+-O bond length was predicted to be 1.96 Based on the structural parameters determined experimentally, the average A -O bond length is 2.287 [11,18] This would cause the Bi cations to pull the O in by only 0.06 but the Zn will push the O out by 0.33 Based on these displacement magnitudes, i t can be seen that the four Zn cations that substitute onto the A2O network cannot occupy the same tetrahedron, indicating there are favorable arrangements of Zn substitutions in the A2O network. In contrast, t he displacement magnitudes predicted in the B2O6 network indicate this network is more r eceptive to the Zn substitution and t herefore, Withers et al predicted that the B2O6 network does not require specific ordering.
142 6.2.2 Zn Ordering Rules on the A2O Network Second, the A2O is quite sensitive to Zn substitution so preferred configurations are identified to alleviate the strain created by Zn substitution. In each tetrahedron formed by four A cations (see Figure 2-3b ), only one Zn will be favorably accepted resulting in a three to one ratio of Bi to Zn cations. The structural differences between the [2:2] ratio and the [3:1] ratio can be seen in Figure 6-1a and Figure 61b, respectively. Thirdly, within the A2O network there are two directional ordering possible, while maintaining the [3:1] ratio of B i:Zn, one along <011> and the second along <211>. Figure 61b shows a [3:1] ratio of Bi:Zn with all the Zn cations ordering along the <211> direction and Figure 61c shows the [3:1] ratio with all the Zn cations ordering along the <011> direction. Withers et al  determined the <211> direction, illustrated in Figure 6 -1b, will be almost exclusively favored for the Zn substitution. These four rules were confirmed through comparing diffraction patterns from Monte Carlo models with those determined experimentally. Fourthly, even in with an optimal ordering of the Zn substitutes, the atoms within BZN will exhibit displacement to satisfy their respective bond valence. The first three rules provide a framework to reduce the number of possible configurations that incorp orate the Zn substitution to a subset that may be probed using density functional theory. Section 6.3.1 outlines the various arrangements of Zn substitution on the A2O network tested within this chapter and illustrated in Figure 61
143 6.3 Calculation Det ails 6.3.1 Configuration Setup As discussed in Section 2.2, th e pyrochlore structure single unit cell contains sixteen A and sixteen B cation sites. In the BZN crystal, the Zn cations equally distribute into both the A and B cation site, resulting in four of the sixteen sites in both networks occupied by Zn cations. Even with the restriction on the relative distribution of the Zn cations between the A2O and B2O6 networks, there exists a wide range of possible short -range and long-range cation ordering conf igurations. A large system containing multiple unit cells would be required to completely probe the role of short -range ordering of the Zn substitutions but such calculations using DFT are not computationally practical. Instead, a single unit cell was used and the effect of the Zn distribution was sampled by examining 120 different configurations. Using the results of Withers et al  discussed in Section 6.2, ordering of the Zn in the A2O network, termed ZnA, was applied and a random distribution of Zn in the B2O6 network, termed ZnB. Based on this observation, all of the configurations examined in this chapter fall into four types distinguished by the ZnA cation ordering in the A2O network. Figure 61 illustrates the differences be tween these four types of configurations. The first, labeled [2:2], has two of the four Zn cations sharing the center tetrahedron, as seen in Figure 6 1a. The remaining three types restrict the Bi to Zn ratio to [3:1] within each tetrahedron. As mentioned in Section 6.2, with the first ZnA cation substitution site selected there are two directions to orient the remaining Zn cations in the unit cell  either along the <211> or the <011> direction. The second and third types strictly follow the <211> and <011> directional ordering and can be seen in Figure 6-1b and Figure 61c, respectively.
144 Figure 6 1. The A 2 O networ k of oxygen centered tetrahedra with Zn A (black) substitution in a (a) [2:2] ratio of Bi:Zn within the center tetrahedral and in a [3:1] ratio of Bi:Zn within all tetrahedra strictly ordering (b) along <211> or (c) <011> and (d) combining <211> with 1/3 <011> of the ZnA substitutes. The arrows provided indicate the <211> (solid) and <011> (dashed) direction betwe en the ZnA cations. The last configuration, illustrated in Figure 6-1d, combines the <211> and <011> ordering of the ZnA-ZnA, with 2/3 of the ZnA-ZnA cations oriented along <211> and 1/3 of the cations along <011> It should be noted that the [2:2] configurations has 2/3 of the ZnA-ZnA oriented along <211> and 1/3 along <011>. For simplicity, these four types of (a) <211> <011> (b) <211> (c) <011> (d)<211> <011>
145 ZnA ordering within the A2O network will be referred to as [2:2], <211>, <011>, and 1/3 <011> throughout the chapter. While the A2O network shows short -range ordering, to date there is no experimental evidence of ordering of the Zn substitutions within the B2O6 network for BZN. The 120 configurations were created by selecting 30 different random distribut ions of Zn substitutions within the B2O6 network and pairing the B2O6 network with each of the four A2O networks detailed above. From an analysis of the energy distribution among the configurations, preferred ordering can be identified along with informat ion on the average structure. This approach used here would not identify any long -range or superlattice ordering within BZN, but currently no experimental evidence indicating such long -range ordering in BZN has been reported. 6.3.2 VASP details First -princ iples calculations were performed with Vienna Ab initio Simulation Package (VASP),  and details match those discussed in Section 3.2. T he Bi(5d, 6s, 6p), O(2s, 2p), Nb(4p, 4d, 5s ), Ta( 5p, 5d, 6s), Zn(4s,3d), and Mg(2p, 3s) orbitals were included as the valence electrons for the projector augmented wave (PAW) pseudopotentials provided in the VASP database [48,49] The DFT calculations were performed within the local density approximation (LDA)  and a plane wave energy cutoff of 400 eV. All calculations were performed for a cubic cell with the lattice constant fixed to the low temperature experimental valu e of 10.55668  for BZN, 10.54075 for BZT, 10.5525 for BMN, and 10.529 for BMT The force criteria was re duced to 0.03 eV/ with no significant changes observed between structures optimized to 0.03 eV/ and those optimized to 0.0 0 1 eV/ Due to the number of configurations explored only the point (1 1) was used for the k -mesh T his level of accuracy was confirmed
146 sufficient to capture the preferences in chemical ordering and atomic displacements by examining a few configurations with a cutoff energy of 500 eV and a k -mesh of 3 3 3 with no difference to the level of accuracy reported within this chapter seen in structures 6.4 Results and Discussion 6.4.1 Energetics of Zn O rdering i n the A2O N etwork in BZN Figure 62 shows the energy after relaxation for all 120 configurations examined. Based on the average energy of each configuration type, the most s table Zn ordering is found in the <211> type A2O network, illustrated in Figure 61b. The <211> configuration type is more favored by 0.47, 1.63, and 0.49 eV/unit cell than the [2:2], <011>, and 1/3 <011>, respectively. The average total lattice energy of all 30 configurations with the <211> A2O network provides the reference zero energy state for all energies values reported. Figure 62 shows the scaled energies of all 120 configurations with respect to this reference energy state. Figure 62 also includ es the average for each A2O type (horizontal lines) and error bars (vertical lines on left and right of the data set) associated with one standard deviation from the average. Most of the individual configurations follow the stability order suggested by th e average values but there are some outliers. This suggests that the ZnB distribution on the B2O6 network can have some influence on the stability of BZN. There is one B2O6 configuration that combines with the <011> A2O network to give a structure whose energy is well outside three standard deviations and is therefore removed from Figure 62 and the resulting discussion. Nevertheless, the ordering of the ZnA in the A2O network is the dominant factor influencing the stability of the BZN system.
147 Figure 6 2. The relaxed energy in eV scaled by the average of the most favored A 2 O network type for each of the 120 structures studied. The 30 B2O6 configurations (numbered 1-30 along the x axis) are shown for each of the four types of the A2O network separated into the [2:2] ratio (open diamond), the [3:1] ratio with only <211> ordering (square most favored), the [3:1] ratio with only <011> ordering (triangle) and the [3:1] ratio with 1/3 <011> ordering (open circle). The observed preference of the <211> ordering within the A2O network over the other three A2O networks agrees with Withers et al  However, the magnitude of energy by which the <211> type A2O network is favored over the [2:2], <011>, and 1/3 <011> type networks indi cates the energy cost of a higher local ZnA concentration is not as great as the energy needed order ZnA along the <011> direction. It is important to note there is a configuration where the [2:2] A2O network is more favored in the relaxed structure by a degree of about 0.1 eV. Though the 3 to 1 ratio is more often preferred, the B2O6 configuration of Zn cation substitution sites can mediate the unfavored cation ratio within the A2O network by acquiring an appropriate 0 5 10 15 20 25 30 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 Energy (eV) [2:2] <211> <011> 1/3 <011>
148 arrangement. However, since this is observed in only one of 30 B2O6 configurations it suggests that <211> would be preferred within the real crystal but the [2:2] is not forbidden. Figure 62 contains a lot of information but the degree of scatter for some of the configurations can lead to c onfusion. To help simplify the information, the energies shown in Figure 62 are compressed into just the differences between the A2O network and the B2O6 network is no longer distinguished. The compressed view is shown in Figure 63. Figure 6 3. The relaxed energy in eV scaled by the average of the most favored A 2 O network type for each of the 120 structures studied. The 30 B2O6 configurations are shown for each of the four types of the A2O network separated into the [2:2] ratio, the [3:1] ratio wi th only <211> ordering (most favored), the [3:1] ratio with only <011> ordering and the [3:1] ratio with 1/3 <011> ordering. [2:2] <211> <011> 1/3 <011> -0.5 0.0 0.5 1.0 1.5 2.0 2.5 Energy (eV)
149 T he degree of scatter of these energies is seen by examining the energy difference between the most and least favored configuration for each A2O type in Figure 63 The energy range for the configurations with the [2:2], <211>, <011>, 1/3 <011> A2O networks are 1.25 eV, 1.0 eV, 1.75 eV, and 2.0 eV, respectively. The range for the <211> set is the smallest with the [2:2 ] set following as the next smallest. This ordering matching the order of favorability based on the average scaled energies. However, the 1/3 <011> average energy is nearly the same as the [2:2] set but the energy range for the 1/3 <011> set is the largest at 2.0 eV. To understand the effect of atomic displacement with the BZN compound, the energy distribution was examined among the 120 configurations before atomic relaxation occurred. On average for the BZN system the <211> is (0.250.59) eV more favorab le than a [2:2] in the unrelaxed ideal crystal structure. After atomic relaxation, the <211> is more favorable by (0.47 0.28) eV. The total unrelaxed energy scaled to the average of the <211> configurations for each of the 30 configurations with the [2:2] A2O network is shown as open diamonds in Figure 64. The [2:2], <211>, <011>, and 1/3 <011> A2O networks are shown as open diamonds, filled squares, filled triangles, and open circles, respectively. As with Figure 6 -2, the average of each A2O confi guration is shown in solid (dashed) lines for the filled (open) symbols with error bars for one standard deviation included on the far left and right side of the graph. It is interesting to note that the <011> and 1/3 <011> set averages are more favorable (lower thermodynamic energy) than the <211> set unlike in the relaxed structures. This indicates the <211> configurations significantly improve the
150 thermodynamic stability through preferred atomic displacements to reduce forces for all atoms. Additionally by comparing the averages for each A2O set in both Figure 64 and 6 -2 the order of the unrelaxed and fully relaxed scaled energies change. Before optimization, the order of thermodynamic stability is the [2:2], <211>, <011>, and 1/3 <011> but after the atomic forces are minimized, the order is <011>, 1/3 <011>, [2:2], and <211>, with the 1/3 <011> and [2:2] average energies being nearly equivalent (as seen in Figure 6 2). Figure 6 4. The unrelaxed energy in eV scaled by the average of the <211> A 2 O network type for each of the 120 structures studied. The 30 B2O6 configurations (numbered 1-30 along the x axis) are shown for each of the four types of the A2O network separated into the [2:2] ratio (open diamond), the [3:1] ratio with only <211> orderi ng (square most favored), the [3:1] ratio with only <011> ordering (triangle) and the [3:1] ratio with 1/3 <011> ordering (open circle). As before in Figure 62, the degree of scatter for the scaled energy with each configuration is very large and the re is a lot of information included in Figure 6-4 making 0 5 10 15 20 25 30 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Energy (eV) [2:2] <211> <011> 1/3 <011>
151 it difficult to compare these A2O sets. To help visual the differences between these configurations Figure 65 illustrates the same energies but the configurations are only distinguished by the A2O network. As seen above for Figure 6 -3, the energy range of the A2O networks before optimization can be teased out and compared for each type of ZnA distribution. Before optimization, the energy range for the [2:2], <211>, <011>, 1/3 <011> sets are 2.25 e V, 2.0 eV, 2.0 eV, and 1.75 eV, respectively. In contrast, following atomic displacement the energy ranges are 1.25 eV, 1.0 eV, 1.75 eV, and 2.0 eV. It can be seen reducing the atomic forces significantly improves the spread of the [2:2] and <211> sets and only slightly improves the spread of the <011> set. However, the energy range of the 1/3 <011> set increases following relaxation from 1.75 eV to 2.0 eV. Figure 65 The unrelaxed energy in eV scaled by the average of the <211> A2O network type for each of the 120 structures studied. The 30 B2O6 configurations are shown for each of the four types of the A2O network. [2:2] <211> <011> 1/3 <011> -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Energy (eV)
152 6.4.2 Structural Differences between A2O N etwork Types in BZN To determine the primary atomic interactions that affect BZN stability the correlation of the energy to the average bond lengths in the system for the Bi -O, Bi-O, ZnA-O, ZnA-O, Nb -O, and ZnB-O bonds were examined. T he correlation plot for ZnA-O showed interesting results and is included in Figure 6-6 a. There is a much narrower distribution of ZnA-O bond lengths for the configurations associated with the <211> type ordering. The average and standard deviation of the bond lengths for each A2O network type are included in Table 6 -1. Figure 66 The relaxed energy in eV scaled by the average of the most favored A2O network type plotted versus (a) the average ZnA to O bond length () and (b) O displacement () for each of the 120 BZN configurations studied. Each of the four types of the A2O ne twork are separated into the [2:2] ratio (open diamond), the [3:1] ratio with only <211> ordering (square most favored), the [3:1] ratio with only <011> ordering (triangle) and the [3:1] ratio with 1/3 <011> ordering (open circle). 1.90 1.92 1.94 1.96 1.98 2.00 2.02 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 (b) (a) Energy (eV)d(ZnA-O') ()0.40 0.45 0.50 0.55 0.60 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 [2:2] <211> <011> 1/3<011> O' ()
153 Table 6 1. The total lattice energy in eV before and after atomic relaxation, as well as the average bond lengths and atomic displacement s for each of the configurations studied is averaged and separated into the four A2O network types. [2:2] <211> <011> 1/3<011> Unrelaxed Energy (eV) 0.25 0.59 0 0.53 -0.17 0.44 -0.22 0.46 Relaxed Energy (eV) 0.47 0.28 0 0.23 1.63 0.39 0.49 0.44 d(Bi -O) () 2.341 0.024 2.340 0.020 2.326 0.023 2.352 0.025 d(Bi -O) () 2.347 0.011 2.343 0.016 2.356 0.036 2.323 0.022 d(ZnA-O) () 1.946 0.030 1.933 0.021 1.963 0.036 1.937 0.027 d(ZnA-O) () 2.129 0.046 2.138 0.051 2.137 0.052 2.175 0.050 d(Nb-O) () 2.001 0.002 2.000 0.002 2.002 0.002 2.001 0.003 d(ZnB-O) () 2.089 0.008 2.090 0.007 2.094 0.012 2.088 0.008 ) 0.433 0.099 0.435 0.080 0.399 0.110 0.432 0.086 ) 0.154 0.068 0.155 0.065 0.140 0.063 0.142 0.056 A () 0.685 0.086 0.673 0.070 0.683 0.104 0.630 0.064 B () 0.116 0.048 0.125 0.052 0.107 0.052 0.088 0.033 ) 0.496 0.020 0.516 0.024 0.452 0.049 0.493 0.020 C orrelation of the energy to the average of each atom types displacement was examined the in the relaxed crystal. The results for the correlation wn in Figure 6-6 b. Since a connection between the lattice energy and the ZnA-O bond length was observed, a link between the energy and the atomic displacement of either the ZnA or the O atoms is not unexpected. The observation of the O atomic displacement correlating to the overall lattice energy is attributed to the type of displacement ZnA atoms undergo. Within the configuration with the <211>
154 ordering, the displacement pattern of every ZnA site was determined and 95 percent of the time the ZnA cation would not displace towards the O anions. Therefore, any variation in the ZnA-O bond length must be attributed to the movement of the O anion. T his information helps explain why the energy costs are so high for the <011> ordering with respect to the [2:2] network. If each O anion displaces towards the ZnA cation within its tetrahedron, the <011> type ordering could lead to a splitting of the unit cell. The network illustrated in Figure 6-1c can provide further explanation. In this configuration, the O anion located at the top and bottom of the unit cell would displace upwards along the z axis, while the O anion in the center of the cell would displace downwards. This could resul ts in a spatial separation of the previously evenly distributed corner sharing tetrahedra. Additionally, this type of displacement would lead to a reduction of bonding with the two Bi cations forming t he top of the center tetrahedron In the case of the [2 :2] type ordering, most of the ZnA cations are ordered along the <211> direction with only one set having the <011> directional ordering. As seen in Figure 6 1a, the O anions can displace towards the ZnA cations evenly in all directions. Also, the O anion in the center tetrahedron has multiple displacement pathways towards either one of the two ZnA c ations with in the tetrahedron or towards both ZnA cations simultaneously. 6.4.3 Energetics and Structure in other Complex Pyrochlores As seen in Section 6.4. 1, the <211> A2O set is the most favored for BZN. For the other complex bismuth pyrochlores examined (BZT, BMN, and BMT), it was of interest to compar e the <211> A2O set with the [2:2] and <011> A2O sets Instead of showing the energy associated with each configuration, as shown for BZN in Figure 62, the difference in energy between the <211> A2O and either the [2:2] or the <011> A2O type
155 for each B2O6 configuration was determined The average and standard deviation for the difference between the [2:2] and <211> A2O set s are shown in Figure 6-7 for the four complex compounds Figure 68 shows a similar plot for the difference between the A site substitutes (either ZnA or MgA) ordering along the <211> versus the <011> directions. The positive value s indicate the <211> A2O type is more thermodynamically favorable. Since the total lattice energies varied considerably between the four different pyrochlores, the energies reported are scaled by A site cations, or 16, in order to make comparisons between t he different pyrochlores. Figure 6 7 Energy difference (eV/A site) between [2:2] and <211> A 2 O configuration sets for BZN, BZT, BMN, and BMT. As seen in Figure 67 the [3:1] ratio is preferred in the BZN, BZT, BMN and BMT compounds. Upon inspection of the energy difference between the [2:2] and <211> for the individual configurations for BZT, BMN, and BMT, it was observe d that some structure s show the [2:2] A2O type to be favored over the <211> A2O type as was BZN BZT BMN BMT-0.02 0.00 0.02 0.04 0.06 Energy (eV/A site)
156 s een for one structur e in the BZN compound. O ne configuration was found in the BZT and BMT compound where the [2:2] A2O set is more favorable than the [3:1] with <211> ordering. The ZnB/MgB distribution in these two configurations did not match each other or the previously di scussed BZN configuration with favorable [2:2] ZnA distribution. For t he B MN compound, three structures were found where the [2:2] A2O is more favorable than the [3:1] with <211> ordering. Of these three structures, one of the MgB distributions matched th e ZnB distribution from the BZN compound with favorable [2:2] A2O type. Figure 6 8 Energy difference (eV/A site) between <211> and <011> A 2 O configuration sets for BZN, BZT, BMN, and BMT. When comparing the <211> and <011> A2O sets, seen in Figure 68 the <211> configuration is favored by a larger degree over the <011> set than the <211> A2O was favored over the [2:2] set for all four compounds. Unlike for the [2:2] A2O set, there are BZN BZT BMN BMT0.02 0.04 0.06 0.08 0.10 0.12 0.14 Energy (eV/A site)
157 no ZnB/MgB distributions in any of the four compounds w here the <011> A2O ZnA/MgA is more thermodynamically favorable over the <211> A2O ZnA/MgA distribution. There are a few ZnB/MgB distributions with energy differences between the <211> and <011> A2O types above 0.2 eV/A site, resulting in larger standard deviations for the BZN, BZT, and BMN compounds. The BMT does not show this effect and therefore the standard deviation is only about 0.01 eV/A site versus 0.03 eV/A site for the other four compounds. Based on the differences in energies, the <211> direct ional ordering of ZnA/MgA cations is predicted to be preferred in the BZN, BZT, BMN, and BMT complex pyrochlores. While the [3:1] ratio of ZnA/MgA to Bi cations within each tetrahedron is preferred with the majority of ZnB/MgB distributions tested, the [2: 2] ratio is not forbidden. In addition to examining the energetics of Zn/Mg distributions, the average bond lengths and displacement magnitudes were also identified in each of the complex pyrochlores, which are summarized in Table 6-2. All four compounds show similar Bi -O and Bi -O bond lengths between 2.32 2.34 and 2.34 2.35 respectively. The ZnAO bond lengths for BZN and BZT between 1.941.95 are slightly lower than the MgAO bond lengths in BMN and BMT between 2.02.03 In all four syst ems the ZnA/MgAO bond lengths are between 2.13 2.16 On the B2O6 network, the Nb-O bond lengths in BZN and BMN at about 2.0 are only slightly longer than the Ta -O bond lengths in BZT and BMT at about 1.98 However, there are larger standard deviations of the B -O bond lengths in the BZT,
158 BMN, and BMT systems than that observed in BZN. The ZnB/MgB-O bond lengths in all four compounds are between 2.09 2.10 Table 6 -2. The average bond lengths and atomic displacement s for BZN, BZT, BMN, and BMT compounds BZN BZT BMN BMT d(Bi -O) () 2.340 0.023 2.341 0.170 2.321 0.171 2.331 0.161 d(Bi -O) () 2.342 0.021 2.3 45 0.16 7 2.353 0.164 2.350 0.162 d( ZnA/ MgA-O) () 1.945 0.029 1.942 0.029 2.030 0.024 2.001 0.020 d(ZnA/MgA-O) () 2.145 0.050 2.129 0.15 6 2.159 0.142 2.155 0.171 d(Nb/Ta -O) () 2.001 0.002 1.984 0.06 5 2.001 0.080 1.983 0.057 d(ZnB/MgB-O) () 2.090 0.009 2.100 0.060 2.088 0.043 2.091 0.042 ) 0.425 0.094 0.414 0. 087 0.422 0.106 0.492 0.059 /Ta () 0.148 0.063 0.123 0.053 0.163 0.084 0.089 0.037 A/MgA () 0.668 0.081 0.686 0. 118 0.654 0.088 0.574 0.058 B/MgB () 0.109 0.046 0.110 0.045 0.145 0.033 0.069 0.028 ) 0.489 0.028 0.476 0. 097 0.433 0.125 0.453 0.092 The average atomic displacement magnitudes for each atom type are also included in Table 6-2. On average, the Bi cations displace by a larger degree in the BMT at 0.49 than in the BZN, BZT, and BMN compounds. In contrast, the Ta cations in BMT and BZT displace by smaller about of 0.09 and 0.12 respectively, versus the Nb cations in BZN and BMN with displacement magnitudes between 0.15 0.16 The substitute cations on the B site displace by neraly the same magnitude as the Nb/Ta cation for the BZT and BMN compounds. For the BZN and BMT compounds, the ZnB/MgB displacements are only slightly smaller than the Nb/Ta. However, on the A site, the substitute cations displaced by a larger degree in all four compounds. For the
159 compounds with Zn on the A site, the displacement magnitudes are between 0.67 0.69 versus the 0.41 0.43 for the Bi cations However, the Mg at the A site displaces by a much larger degree in B MN at 0.65 versus 0.57 in BMT. For the O anions, the displacement magnitude is larger in the zinc substituted compounds between 0.48 0.49 versus in the magnesium substituted compounds between 0.43 0.45 Recently, Melot et al  has shown a connection between the average A site displacement magnitudes and the experimentally determined dielectric constant in zinc substituted bismuth pyrochlores. The average A site and O site atomic displacement magnitudes from DFT are plotted in Figure 6-9 with the experimentally determ ined relative dielectric permittivity For BZN, BMN, BZT, and BZS the relative permittivity is reported at 1 MHz. However, there is no data in the literature for BMT at 1 MHz therefore, the value shown in Figure 6 -9 corresponds to BMT at 900 GHz. The corre lation observed between these structures should not be affected by using the different frequency result for BMT, since the ordering of the BZN, BMN, BZT, and BMN are the same for 900 GHz as that seen at 1 MHz. Since the displacement magnitudes and dielect ric constants on BZN, BZT, BMN, and BMT were so similar, Bi1.5ZnSb1.5O7 (BZS) was also examined, where a much lower of average A site (~ 0.32 ) and O site (~0.38 ) displacement magnitude is reported from neutron powder diffraction data  The details of the structure determined by DFT for the BZS systems will not be detailed but both the A site (0.36 ) and O site (0.42 ) average displacement magnitudes in Figure 6-9. It can be seen from Figure 69 that the correlation between the average A site displacement
160 magnitudes and the dielectric constants is seen for the DFT predicted atomic displacements in BZS, BMT, BZT, BMN, and BZN. The O site average displacement magnitudes correlate for the zinc substituted systems but are lower for the magnesium substituted compounds. Also, in F igure 69, it can be seen that the large jumps in dielectric permittivity occur with changing the occupation on the B site, with BZN and BMN being grouped with the highest r, then BZT and BMT, and then BZS. As was seen with the energetics of the various configurations, this observation indicates the importance of the B2O6 network in these systems which should be examined further. Figure 6 9 The average displacement for the A and O site in the BZN, BMN, BZT, BMT and BZS compounds in order of increasing experimental dielectric constant 6.5 Summary DFT calculations were performed on a range of cation substituted Bi pyrochlores, Bi1.5M2+B5+ 1.5O7 (with M2+=Zn or Mg and B=Nb or Ta). The distribution of the cation BZN BZT 0.35 0.40 0.45 0.50 0.55 20 40 60 80 100 120 140 160 Displacement () A site O' siter' BZS BMN BMT
161 substitutes was var ied with specific rules on the A2O network and a random arrangement on the B2O6 network By combining 4 ordered A2O distributions with 30 random B2O6 distributions, 120 configurations were tested to probe the affects of cation ordering on the A2O networ k. The cation distributions on the A2O varied the local concentration (difference between [2:2] and [3:1] bismuth to substitute ratio) and the substitute to substitute directional ordering (either along the <211> or the <011> directions). In all four comp lex pyrochlores, on average a more distributed [3:1] ratio was observed for each tetrahedron, for some B2O6 configurations the [2:2] A2O configuration was favored. The ordering of both Zn and Mg cations along the <211> direction was found in all four comp ounds confirming the predictions from Monte Carlo models previously reported  Though previous reports have suggested the importance of the A2O network over the B2O6 network, large deviations in the energies of the configurations with randomly distributed cation substitutes on the B2O6 network indicate some ordering on the B2O6 network may be preferred. A relationship between DFT determined A site cation displacement magnitudes and the experimentally reported dielec tric constants.
162 CHAPTER 7 ION HOPPING IN BISMU TH PYROCHLORES 7.1 Introduction The dielectric properties of pyrochlores are often attributed to ion hopping of A and O atoms [13,52] and a better atomic level understanding of the hopping mechanisms and energetics will be invaluable in tailoring substitutions. A general feature observed in all the Bi pyrochlores that show high dielectric permittiv ity is large ion displacement from the ideal Fd 3 m sites of the cubic pyrochlore structure. This atomic displacement results in multiple energetically equivalent sites as discussed in Section 22.214.171.124 which facilitate cation hopping. Nevertheless, o pen que stions remain on the relative influence of substitutional disorder  on the A2O versus B2O6 network (Figure 2-3b ,c ) on the ion hopping in these systems  Within Bi1.5Zn0.92Nb1.5O6.92 (BZN)  the infrared (IR) active phonon modes were indentified and the contribution of each mode to the dielectric permittivity was assigned  Later, Nino et al explored how the IR modes that contributed the most to the dielectric permittivity may affect the dielectric relaxation observed in BZN  Nearly 80% of the ionic dielectric permittivity is accounted for by the phonon modes associated with O -A-O and O -A-O bend modes  In contrast, in Bi1.5ZnTa1.5O7 (BZT) and Bi1.5MgTa1.5O7 (BMT) these modes are associated with 66 % and 56 % of the permittivity, respectively  While in Bi1.5MgNb1.5O7 (BMN), these modes also account for 81 % of the ionic dielectric permittivity  Based on the IR results Nino and co workers proposed that the relaxation observed in BZN is due to ion hopping mechanisms associated with the A and O atoms between equivalent positions [11,64]
163 Investigating this proposed mechanism would be difficult experimental ly due to the rapid movement of the atoms within a crystal even at low temperatures. Sensitiv e structure techniques such as neutron or X -ray diffraction used to examine changes at the atomic scale average over the entire bulk structure and require a finite measuring time. The identification of pair distribution function to examine the interatomic distances between atom types can be used to approximate changes in the local bonding environment  Extended X -ray absorption fine structure (EXAFS) has been used in BZN to refine displacement patterns of both Bi and ZnA based on the local bonding environment of the nearest O atoms  However, these measurements are taken over the course of some measurement time and if applied instantaneously following the application of an electric field to examine hopping mechanisms the collect ed information would contain a larger component of final state structure information tha t may dilute any transitional state structure. Additionally, the frequency of these hopping mechanisms events may be faster than the measurement time hindering the capt ure of intermediate structures within the atom hopping mechanisms. This limitation may be overcome through simulations using density functional theory (DFT). The DFT calculations discussed up to this point in the dissertation have been restricted to identi fication of local minima. While this information provides insight on cation hopping behavior, for instance the identification of multiple local minima for Bi cations displaced from the ideal Fd 3 m site, they do not directly probe the cation hopping process. One approach to probe cation hopping would be to directly simulate the pyrochlore system at relevant temperatures over the time scale of the cation hopping using molecular dynamics (MD). While ab initio MD can be performed on the pyrochlore system the appropriate time
164 scales to observe cation hopping are computationally expensive. Performing MD simulations using classical potentials would allow cation hopping to be observed on experimentally time and length scales, but sufficient accurate potentials for th e Bi pyrochlores are not yet available. Instead of MD simulations DFT was used in combination with nudged elastic band (NEB) method [55 -57] to identify transition states between the displaced local minima. NEB calculations have been used to identify transition states in proton transfer  surface diffusion  and surf ace chemistry, such as in the water gas shift reaction on copper  The background of how NEB calculations are implemented into DFT is detailed in Section 126.96.36.199. In this work, NEB calculations are applied on various bismuth pyrochlores to identify pathways to cation hopping and the associated energy barriers. While there is no guarantee that the lowest energy pathway has been identified since the NEB method depends on providing two relevant connected minima, several different possible p athways were examined to determine the relevant hopping mechanisms. The various minima that serve as input to the NEB calculations are associated with the specific spatial location of the cations with respect to the ideal position. As mentioned before, the characterization of these spatial positions is done by using Wyckoff positions which have been explained in Section 2.2 and Section 188.8.131.52. 7.2 Calculation Details First -principles calculations were performed with Vienna Ab initio Simulation Package (VASP)  with details matching those reported in Section 3.2. T he Bi(5d, 6s, 6p), Ti(3s, 3p, 3d, 4s), O(2s, 2p), Nb(4p, 5s, 4d), Zn(4s, 3d), Ca(3s, 3p, 4s) orbitals were included as the valence electrons. The DFT calculations were performed within
165 the local density approximation (LDA)  All calculations were performed at fixed volume and shape, while the atoms were relaxed until the forces were less than 0.03 eV/. The cubic lattice constants for the BZN and Ca1. 5Ti1.5Nb1.0O7 ( CTN) systems are fixed to their experimentally determined values of 10.55668  and 10.2301  Though all atoms are free to relax in these optimized structures the crystal lattice is fixed to the cubic pyrochlore structure. A plane wave cutoff energy of 400 eV was used along with a 333 Monkhorst -Pack  mesh, resulting in 14 irreducible k -points for the Bi2Ti2O7 system an d only the point for the BZN and CTN systems. To probe for equivalent minima associated with displaced cations, one bismuth cation was selected and moved it into an equivalent site within the same Wyckoff position where it was held fixed. Simulated annealing calculations were then performed under fixed crystal shape and volume using ab initio molecular dynamics (MD)  The MD simulations were conducted with only the point within the NVT ensemble starting at 1000 K and cooling to 300 K over 500 MD steps with a time step of 0.5 femtoseconds. After MD simulation, the system was minimized using the sam e approach as outlined above. Following all of these steps, the fixed cation was released and the forces were relaxed. The climbing -image nudged elastic band (cNEB) method w as used to identify the minimum energy paths between equivalent minima [55 -57] The process was completed in two steps. The first step used one image, or intermediate structure along the pathway, to confirm the presence of an energy barrier between two minima and the second step provided more detail to the resulting pathway by adding two additional images resulting in a total of three intermediate structures. The calculation details of the
166 compounds minima were repeated within the NEB calculations, e.g. cutoff energy, k mesh, etc. For the NEB calculations a force criteria of 0.05 eV/ was used and confirmed the equivalency of both the energy profile and structure intermediates are well within the values reported. Though the force criteria is lowered for the intermediate structures, test calculations were performed to confirm the energy barriers determined are accurate to within 0.02 eV. As shown in Chapter 6, the distribution of cation substitute in BZN and CTN may vary significantly. For this work the distribution was limited to a single configuration for the BZN and CTN systems. One other distribution was tested for each system to confirm the energy barriers determined were not dependent on the distribution. 7.3 Results and Discussion 7.3.1 Bi smuth Titanate 184.108.40.206 Identification of equivalent minima The first step in studying the hopping mechanism within Bi2Ti2O7 is to identify possible linked minima. Within bismuth titanate, a stable minimum was attempted for each of the six equivalent Wycko ff positions for two different bismuth cations in the unit cell. The two bismuth cations were selected based on the occupancies of the 96g and 96 h positions and their positions with respect to the correlation observed in bismuth titanate (see Section 220.127.116.11 for details on the Bi2O correlation). Though two cations were forcibly pushed into each of the five equivalent Wyckoff positions, these minima were created independently by moving one cation for each minimum thereby creating ten different structures. F or one of the bismuth cation in the 96h site, following the simulated annealing and releasing the cation to reduce its forces (see Section 7.2 for fully details of procedure), the cation shifted out of two of the equivalent positions back
167 into one of the other four positions. This indicates that though the six equivalent positions could be stabilized when starting from the ideal pyrochlore structure, once the displacements are initiated the probability of each position being occupied is not equal. However, for the second bismuth cation that was pushed, each of the equivalent 96g positions could be stabilized. Since six equivalent positions were stabilized with this Bi cation, these structures will be focused on for the ion hopping pathways d iscussed below. Figure 71 plots the total energies for each of the six minima at the 96g sites scaled by the lowest energy structure (s2). For visual aid, the energy of the sixth position, or s6, is shown to the left of s1 and to the right of s5. This i s to help reflect the spatial arrangement of these points and visualize the difference in energy between s1 and s6. Additionally, the pushed bismuth cations final position for each of the structures is also shown in the inset of Figure 7-1. Though there is some shift in the cations, the characteristic ring of the 96g Wyckoff site is maintained including the displacement component along the O -Bi -O bonding axis, which is perpendicular to the view shown. The difference in the total lattice energy between the structures is about 0.2 eV/88 atoms, which is about 0.002 % of the total energy. In comparison to the energy difference observed following atomic displacement and volume expansion from the ideal pyrochlore structure discussed in Section 18.104.22.168, the total energy difference between these structures is 0.0125 eV/Bi cation which is only 8% of that reported for atomic displacement and volume expansion. This degree of difference is minimal and should be expected among the various structures.
168 Figure 7 1. The scaled energy for each of the six positions with the labels s1 through s6 corresponding to the structural view on the right. 22.214.171.124 Ion hopping mechanisms in bismuth titanate Once these six minima are established, possible hopping mechanisms may be examined between the minima. T hree different types of mechanisms were considered : single site doub le site or triple site hop mechanism. Figure 7 2 illustrates the difference between the initial structure, shown in red, and the final structure for the single site hop, in orange, and the double site hop mechanism, in yellow. Note in Figure 72, the numb ers centered in the schematic to the right of the tetrahedra network correspond to the hopping mechanism and not the site assignment, as was used in Figure 71. From Figure 72, it can be seen that though a single bismuth cation is pushed into equivalent positions, multiple bismuth respond to this perturbation. Based on previous work in pyrochlores and other materials with connected tetrahedra, the concerted motion is not unexpected and this type of response has been termed rigid unit modes  and discussed in Section 2.2.3. s6 s1 s2 s3 s4 s5 s6 0.00 0.05 0.10 0.15 0.20 E96g-site
169 Figure 7 2 The A 2 O tetrahedra network for three energetically equivalent 96g sites. A single site hop and double site hop is represented by the red to oran ge and red to yellow tetrahedra network, respectively. Several attempts to examine a three site hop mechanism yielded only large barrier mechanism (Eb of 1.4 eV). Upon inspection of the electronic localization function (ELF), it is found that along the triple site hop mechanism, the transition state places multiple lone pair towards each other, reducing the lone p air to lone pair spacing, producing large sterical strain, and the resulting high energy barrier. The triple site hop mechanism will be discussed further in Section 126.96.36.199. T he single site hop mechanism was examined for transition between each of the s ites in Figure 71. The energy pathways are illustrated in Figure 73 with respect to the spatial pathway of the pushed bismuth cation. The pathways are broken down by color and symbol for each of the transitions between each position. In addition to the p athways, the scaled energy for each minimum was included in the Figure 73 as the black stars. Upon inspection of the pathways, the barriers between the minima fall into two categories, no barrier or a barrier between 0.11 and 0.16 eV. Note, the accuracy of
170 these calculations was tested and the total energy is accurate to within 0.02 eV, therefore the range of this barrier is 0.11 0.02 eV to 0.16 0.02 eV. Figure 73. Energy profile for the single site hop mechanism s for Bi2Ti2O7. Each original minimum is included as the black star with each component of the pathway shown as a separate series There is no barrier from s1 to s2, from s1 to s6, or between s3 and s4, in either direction. This indicates that s1 is not a true minimum since there is no barrier between s1 and either of its single site hop neighbors. Given this observation, the phonon modes for each minimum should be examined using the finite differences method within VASP. This would determine all real and imaginary phonon modes for the bismuth titanate in each minimum. In addition to providing the frequency for each mode, the structure movement at that frequency would be reported by VASP. The occurrence of any imaginary p honon modes within any structure would indicate there is some instability within the structure. In that case, the atoms involved in the unstable mode would need 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 EBi Pathway () s6-s1 s3-s4 s1-s2 s4-s5 s2-s3 s5-s6
171 to be pushed in the direction identified. Following relaxation, the structure could result in reproducing one of the other Wyckoff positions or even an entirely different configuration. These calculations are currently underway to confirm this hypothesis. As mentioned above, there are two different reasonable hopping mechanisms within bismuth tit anate, a single site hopping and double site hopping mechanisms. T he double site hop mechanisms was also examined and showed energy barriers between 0.20 and 0.32 eV. For double site hop mechanisms that started or ended with the s1 minimum, the barrier out of the s1 site was 0.03 eV or less and the barrier into the s1 site was 0.25 eV or higher. For two hop site pathway from s6 to s2, the s1 structure nearly matches the intermediate structure along the pathway. These observations regarding the barriers aroun d the s1 minimum further support that the structure determined for the s1 site is not a stable minimum. 188.8.131.52 Ion hopping pathways in bismuth titanate In addition to examining the energy profile along the various single site hop pathways, the local str ucture of the pushed Bi cation provides some insight into these hopping mechanisms. Figure 7-4 shows the two Bi4O tetrahedra connected by the pushed Bi cation for each of the end points and the intermediate structures along the s2 to s3 pathway. The d arkness of the atoms illustrated is increased along the pathway from light (s2) to dark (s3) for both the Bi (purple) and the O (red). The outline for the tetrahedra of only the stable minima is included for visual aid. Though it may be difficult to view the O -Bi -O bond angle for the Bi cation that was pushed (center atom) is relatively unchanged along the pathway. This may be easier to see by looking at the front corner of the upper tetrahedron since the push of one Bi cation results in the subsequen t transition of additional bismuth cations to maintain
172 appropriate bonding with the O anions. The front corner Bi cation in the upper tetrahedron illustrates from light to dark the pathway from the s2 to s3 structures involves movement of the bismuth cati on away from the ideal high symmetry site. This type of bismuth displacement was observed for each of the single site and double site hop mechanisms examined. The detailed variation of the O -Bi -O bond angle of the pushed bismuth cation (center cation i n Figure 7-4) along the s2 to s3 single site hop mechanism is shown in Figure 7 6 and will be discussed below in Section 7.3.2. Figure 7 4. Local structure including intermediate structures for the s2 s3 single site hop mechanism. The atom colors (purple for Bi and red for O) are graded from light (s2) to dark (s3) along the pathway T he double site hop mechanism structure intermediates were also examined and the show similar structures along the pathway as the single site hop mechanism. The intermediate structures have the bismuth cation following a pathway away from the high symmetry site and the transition is not isolated to the pushed cations. The tetrahedra within the intermediate structures do not exhibit any distortions or signi ficant changes to the Bi -O bond lengths.
173 184.108.40.206 Role of displacement correlation in bismuth titanate As mentioned above, the triple site hop mechanism was examined and the large energy barrier of 1.4 eV indicates this is not a reasonable pathway to ion hopping in bismuth titanate. Upon inspection of the determined structural intermediates, it is found in the three hop mechanism the bismuth cations transition near the high symmetry center site. The previously observed correlation of Bi and O atoms in sta ble bismuth titanate minima  may provide further understanding of the intermediate structures observed in the triple site hop mechanism The -cristobalite type displacive disorder discussed in Section 2.2.3 is repeated in Figure 7 5 to show the concerted displacement along the  directio n and corn er sharing tetrahedra in bismuth titanate. It is expected the atomic correlation in bismuth titanate will affect the cation hopping mechanisms. I on hopping was probed for two different bismuth cations located at different places within the pyrochlore struc ture. These two cations experience different types of displacement based on the correlation observed. One cation shows downward displacement along the  direction while one exhibits the rotation illustrated in Figure 7-5. The hopping mechanisms associa ted with these two cations were examined and both showed the same energy barriers and equivalent structure intermediates. Upon examination of each intermediate structure for any single site or double site hop mechanism and at any point along the pathway, t -cristobalite type displacive disorder is maintained. With regards to the triple site hop mechanism, the two stable minima exhibit completely opposite correlation. To further explain, the bismuth cation that displaces downward along the  directio n in Figure 7-5 for one minimum will displace upwards in the other minimum. The NEB calculations try to map out a reasonable pathway
174 connecting these two minima and the initial intermediate structures placed the bismuth cation transitioning through the high symmetry sites. This type of transition would result in the loss of the correlation. Following relaxation, the bismuth cations shift out of the high symmetry site as was seen in the single site and double site hop mechanisms but the ELF plots for the int ermediate structure show a reduction in the lone pair -lone pair spacing. The reduced lone pair spacing results in the loss of the correlation within the intermediate structures. It is predicted that the correlation is needed to stabilize the bismuth displa cement within the bismuth titanate compound and therefore destabilizes the triple site hop mechanism. This observation will be further discussed in Section 7.3.2 where the hopping mechanisms in BZN are examined, since correlation of the atomic displacement was not observed in BZN. Figure 75. Local structure illustrating the -cristobalite type displacive disorder in bismuth titanate. The Bi (purple) and O (red) atoms are in their ideal position with t he black arrows indicating direction of displace ment 7.3.2 Bismuth Zinc Niobate As discussed in Section 7.1, the dielectric properties of BZN have been linked to O -A-O and O -A -O bending phonon modes  Additionally, the dielectric relaxation observed in BZN is proposed to be due to the A and O atoms hopping between  
175 equivalent positions [11,64] T he same minimum finding procedure was used for the BZN compound as discussed for the Bi2Ti2O7 c ompound. Similar to the bismuth titanate system, it was found that each of the six equivalent Wyckoff positions for the pushed Bi cation are not equally favorable after atomic displacement has occurred. F our of the six minima were stabilized f or two different Bi cations. T he single site hop and double site hop mechanisms were examined between these stable minima. It was found for the single site hop mechanism the energy barriers are between 0.12 and 0.20 eV. For the double site hop mechanism, similar energy barriers are observed with nearly the same energy range of 0.12 to 0.31 eV Unlike in bismuth titanate, a triple site hop mechanism was stabilized in BZN with energy barriers within a range of 0.12 to 0.35 eV. It is proposed that the triple site hop mechanism may occur in BZN since there is no correlation of atomic displacement in the A2O network. Based on these energy barriers, the bismuth cation in various positions may respond to an applied electric field through all three mechanisms. M inima a ssociated with pushing a Zn cation at the A site were also indentified. The energy barriers associated with both the single and double site hop mechanism range from 0.12 to 0.32 eV. Smooth energy profiles were observed along the hopping pathways for both t he single site and double site hop transitions Though a triple site hop mechanism was not examine d for the Zn cation based on the observation for the Bi cation the triple site hop mechanism should be possible for the Zn cation as well. Experimental reports have shown dielectric relaxation in BZN and with changing applied electric field frequency the maximum permittivity occurs at shifting temperature (Tm). An Arrhenius function may fitted to the frequency versus Tm and yields an
176 activation energy of ~0.1 4 eV for the BZN system [3,16,64] Since the relaxation observed in BZN is attributed to these ion hopping events, the activation energy determined experimentally is predicted to be for the jump mechanism of ions sampling multiple equivalent minina. While there is no guarantee that DFT is capturing the same activated hopping mechanisms, the activation energy determined by experiment of 0.14 eV compares well to the DFT predicted energy range of 0.12 to 0.35 eV Figure 7 6 The O Bi O local bond angle of the pushed Bi cation is shown for a single site hop mechanism in Bi2Ti2O7 (s2 -s3 circles) and BZN ( s1 s2 squares). Each original minimum is included as the black star Further, examination of the intermediate structure for all three hopping mechanisms showed the Bi cations transition nearly through the ideal high symmetry s ite in contrast to the bismuth titanate system. Therefore, this type of transition of Bi cations within BZN may lead to the similarity among the energy barriers associated with 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 155 160 165 170 175 180 O'-Bi-O' Bond Angle ()Bi Pathway () BZN Bi2Ti2O7
177 all three hopping mechanisms. The DFT predicted bismuth cation pathway in BZN a grees well with experimental predictions since there are significant variations in the local O -Bi -O bond angle along the pathway but minimal Bi -O bond length variations. Figure 76 details the variation of the O -Bi -O bond angle of the pushed bismuth cation along the s1 to s2 single site hop mechanism for the BZN system and the s2 to s3 single site hop mechanism for the bismuth titanate system. Within BZN, the bond angle varies significantly between the s1 (166.3 ) and s2 (177.5) minima. In compariso n, the local O -Bi -O bond angle in bismuth titanate varied only by couple degrees between the s2 (158.2) and the s3 (156.2 ) minima Along the pathway, for both systems, the bond angles vary within the range created by their respectively minima with the exception of one intermediate structure in bismuth titanate with a bond angle of 155.7. 7.3.3 Calcium Titanium Niobate It has been thought that bismuth pyrochlores exhibit interesting dielectric properties such as high permittivity and dielectric relaxation, due to the highly polarizable Bi cation occupancy of the A site. However, recently a pyrochlore compound within the CaO:TiO2:Nb2O5 (CTN) system has been synthesized and it shows high permittivity and dielectric relaxation without the incorporation bismuth cations  Though the dielectric constant is lower for the CTN pyrochlore versus the BZN system, the Arrheni us function fitted to the dielectric relaxation data yields a much larger activation energy of 0.32 eV  versus the 0.14 eV reported for the BZN system [3,16,64] This activation energy is higher than any values previously reported for the pyrochlore systems but it is reasonable when the compared with other fluorite relat ed structures such as the weberite Gd2(Gd,Nb)2O7 where an activation energy of ~0.45 are
178 reported  The CTN forms a pyrochlore solid solution with compositions ranging from Ca1.48Ti1.48Nb1.02O7 to Ca1.41Ti1.37Nb1.14O7  In order to examine ion hopping mechanisms in CTN, the CTN system was simulated based on the Ca1 .48Ti1.48Nb1.02O7 and the composition was approximated to Ca1. 5Ti1.5Nb1.0O7. Unlike the complex pyrochlores discussed so far, the A site and the B site have a stoichiometry of (Ca1.5Ti0.5) and (Nb1.0Ti1.0), respectively. For the purpose of these simulations, the stoichiometry means there will be 12 A sites occupied by Ca, 4 A sites occupied by TiA, 8 B sites occupied by Nb, and 8 B sites occupied by TiB. T he CTN configuration was simulated assuming the A2O network would prefer a [3:1] ratio of C a to TiA cations within each tetrahedron and that the TiA cations would order along the <211> direction. Variations of the A2O network have been discussed for other complex pyrochlore in Chapter 6 and in all complex pyrochlores previously discussed this A2O configuration was favored. These assumptions were made in order to examine ion hopping mechanisms more rapidly in the CTN system. As a follow up additional A2O configurations should be tested to confirm the preference of the [3:1] ratio and <211> ordering. The dielectric permittivity and relaxation observed in the CTN system are due to the atomic displacement from the ideal high symmetry sites therefore accurately capturing these displacements is important. Based on the X -ray diffraction refinement, t he Ca, TiB and Nb ions are located at their respective high symmetry sites  The TiA ion displace by 0.7 into the 96 g position and the O ions displace by 0.48 into the 32 e position 
179 Structural minima for the CTN were identified fol lowing the procedure discussed in Section 7.2. Following atomic relaxation, the atomic displacements observed are reported in Table 7-1.The experimental values included in Table 7 -1 are for the Ca1.46Ti1.38Nb1.11O7 pyrochlore using X -ray diffraction at roo m temperature  Though the experimental composition of this pyrochlore does not match the simulated composition, the authors indicate the displacement parameters are the same for the Ca1.48Ti1.48Nb1.02O7 compound. Though DFT determines additional displacement of Ca, Nb, and TiB cations, the magnitude of the TiA and O atomic displacement agree very well with the experimentally reported values. Additionally, the displacement patterns agree between DFT and experiment determining the TiA and O atoms displacing to a 96 g and 32e Wyckoff position, respectively. Table 7 1 The displacement magnitude for each atom type within the CTN compound. Atomic Displacement DFT Experiment a site) 0.23 0.00 site) 0.75 0.70 site) 0.50 0.48 site) 0.16 0.00 site) 0.16 0.00 x O (B site) 0.3234 0.3237 aFrom Ref.  Following the determination of a stable minimum that compar es well with the experimental reports, focused was placed on identifying additional minima with the TiA and Ca cations occupying equivalent Wyckoff positions, using the procedure discussed in Section 7.2. B oth A site cations were examined to discern if it is the TiA cation that is participating in activated cation hopping mechanisms. Similar to both bismuth titanate and BZN systems, it was found that all six equivalent Wyckoff positions are not equally
180 probably when starting with a previously displaced structure. Starting with the ideal high symmetry Wyckoff site both the TiA and Ca cations it was confirmed that each cation will equally occupy the equivalent positions. F our different minima were stabilized for both the TiA and Ca cations and the single site and double site hop mechanisms were examined among these minima. For the Ca cations, energy barriers of 0.05 eV or less were identified indicating these hopping mechanisms are not those being observed experimentally. The TiA cations hopping between equivalent minima exhibit energy barriers between 0.18 to 0.42 eV for both single site hop and double site hop mechanisms. The TiA cations in the intermediate structures transition away from the high symmetry site similar to the Bi cations in the bismuth ti tanate system. In contrast to the transition in BZN, the minimum changes to the O -TiA-O bond angle would indicate if IR phonon mode information were available for CTN the O -A-O phonon mode would contribute to the dielectric permittivity to a lower degr ee than that observed in BZN. Often for the double site hop mechanisms, when three images are used, the middle structure along the pathway matches the stable minimum between the two minima. For example, the hopping mechanism from s1 to s3 for the TiA cati on within the CTN system transitions through the s2 site combining the two single site hop mechanisms on either side. The energy barriers and transition state structures match for both pathway components (the s1 to s2 and s2 to s3 mechanisms). When the TiA cations are forced through the high symmetry site the energy barrier increases to ~1 eV. As with BZN, there is no guarantee that DFT is capturing the exact hopping mechanisms that are approximated experimentally. T he larger energy barriers present
181 in CTN versus those observed in both bismuth titanate and BZN indicate s DFT is able to capture differences among the hopping mechanism between these three systems. 7.4 Summary Using DFT, three different ion hopping mechanisms (single, double or triple site) were examined in bismuth titanate, bismuth zinc niobate, and calcium titanium niobate. In bismuth titanate, the energy barriers ranged from 0.11-0.16 eV, 0.200.32 eV, and ~ 1.4 eV for the single, double, and triple site hop mechanisms, respectively. The large energy barrier associated with the triple site hop mechanism is attributed to the breaking of the correlation of atomic displacement observed in the A2O network. In BZN, the energy barriers for the single, double, and triple site hop mechanisms ranged fr om 0.120.20 eV, 0.120.31 eV, and 0.12-0.35 eV, respectively, and agree well with the experimentally predicted activation energy of ~0.14 eV [3,16,64] In CTN, Ca ions hopping between equivalent minima has an energy barrier of 0.05 eV or less indicating these mechanisms are not being captured in the experimental study. However, DFT predicts energy barriers associated with both single and double site hop mechanisms of TiA between 0.18 and 0.42 eV. Experimentally, an activation energy of 0.32 eV is approximated from the dielectric measurements  The transition state struct ures were examined in all mechanism and for the bismuth titanate and CTN system the A cation transitions away from the high symmetry site, while in BZN both the Bi and Zn cations transition through the high symmetry site. This transition pathway leads to v ariations of the O -A-O bond angle by up to 10 degrees in BZN. This predicted pathway agrees well with experimental IR phonon mode assignment which show ~80% of the dielectric permittivity in BZN is attributed to O -A-O and O -A-O bending modes 
182 7.5 Future Efforts The minima determined by moving cations from one Wyckoff site to an equivalent Wyckoff site, within the same Wyckoff position must be thoroughly tested for each site. It was observed that some positions, though ideally equally likely in nature, may not be obtained by forcible moving select cations. For CTN, Section 7.3.3 showed the number of images used to capture the ion hopping pathways is important since, as seen with the s1 to s3 pathway, the double site hop mechanism often resulted in the combination of two single site hop mechanisms (s1 to s2 and s2 to s3) with the center image structure matching the middle minima structure (s2). Therefore, the number of images should be increased for all mechanism (single, double, and triple) for each system studied. The calculations reported in this chapter were using a single substitutional configuration for the BZN and CTN systems. Though a few single and double site hop mechanisms were tested to confirm the energy barriers and transition pathways did not significantly vary, an exhaustive investigation was not completed. Addition al single, double, and triple site hop mechanisms should be investigated and both the A2O and B2O6 configurations should be varied to confirm these results.
183 CHAPTER 8 SURFACE CHEMISTRY ON LOW MILLER INDICES O F STRONTIUM TITANATE 8.1 Motivation 8.1.1 Project Overview Surface chemistry is a board term to define a large component of the chemical engineering industry with focus in chemical separation  and improved catalytic prop erties leading to efficient chemical reactions. In addition to environmental considerations, there is increasing interest in improving the efficiency of various chemical reactions in order to fully utilize the current resources. The chapter will focus on p art of a larger effort at the University of Florida to design methods to deliver metal clusters to specific sites on oxide surfaces through the use of organometallic ligands with hydroxyl and/or amine terminal groups. 8.1.2 Background Transition metal surfaces have provided a rich source of reactivity in a variety of areas, but with a large price tag. Attempts have been made to merge the reactivity of expensive transition metals with inexpensive substrates, such as metal oxides. Often these heterogeneous catalytic surfaces are created through simple techniques, such as co precipitation or wet impregnation, which results in nonuniform metal particle distributions. In fact, the poor control of where and how the metal particles will arrange leads to a large portion of the expensive metal particles occupying inactive surface sites. By exploiting the selective binding nature of specific functional groups to low coordination surface sites, this work hopes to accomplish atomic level control of the metal particle layer on the metal oxide support. This would be accomplished by
184 depositing the metal particle in two steps. In the first step, the metal particle is delivered to the surface as a part of an organometallic complex with NH2 and OH functional groups. The second step involves the selective removal of the organic ligands using oxygen or hydrogen plasmas as low temperatures, as to not affect the metal adsorption sites. While this ultimate goal will require multiple levels of investigation, DFT may provide reactiv ity and adsorbate configuration information for various small molecules that would replicate the functional groups. 8.1.3 DFT Component The initial investigation is focused on examining if NH3 and H2O selectively bind to specific sites on various SrTiO3 (STO) surfaces. These probe molecules are used to gauge how the organometallic will bind to the oxide surface. The STO surfaces that are of interest include the low Miller index surfaces, (100), (110), and (111) in addition to higher Miller index surfaces. The (621) surface is selected as the initial high Miller index surface based on its steps, kinks, and terrace size. The DFT results will complement experimental work being conducting in the Jason Weaver research group. They will perform various surface sc ience techniques under ultra high vacuum (UHV) on the order of 1x1010 torr. Temperature programmed desorption (TPD) will utilized to determine the desorption temperatures and masses of adsorbates from the surface. T PD is a technique that can be used to approximate adsorption energy  Conversely, desorption temperatures can be approximated f r om the calculated binding energies to compare effectively Additionally, if there are multiple peaks present in the TPD spectra, indicating the possibility of multiple adsorbate phases, DFT can predict the adsorbate configurations corresponding to each TP D peak to provide further insight. The ability of the experimentalists to probe the desorbed molecules mass allows for the
185 use of isotopes to investigate the occurrence of dissociation or even atom exchange with the surface. Additional comparisons may be made between experimental work in UHV and DFT calculations, such as determination of the vibrational modes. As part of this work, a manuscript was completed regarding the adsorption of water of STO(100) included in S ection 8.2. Within this section, the det ails of these results are discussed and comparisons are made to available experimental data. 8.2 A F irst P rinciples Study of H2O A dsorption and D issociation on the SrTiO3(100) Surface Reproduced with permission from Beverly Brooks Hinojosa, Tim Van Cleve, and Aravind Asthagiri, Molecular Simulation (2010). "Copyright (2010) by Taylor & Francis 8.2. 1 Introduction The adsorption and reactivity of water on low Miller index oxide surfaces serves as a model system to understand the structure-reactivity relationship of oxide surfaces. Water oxide interfaces are also important in a range of applications in catalysis, environmental remediation, and biomineralization. Even under ultrahigh vacuu m (UHV) conditions it is often difficult to avoid contamination of low coverages of water, which can affect the reactive properties of the surface. An important first step in understanding the reactive behavior of H2O/oxide interfaces is to resolve if water adsorbs associatively or dissociatively on the oxide surface. While one may expect that the characterization of the structure of water on low Miller index oxide surfaces under UHV conditions should be readily resolved with a combination of experimental and theoretical studies, the literature, for example on H2O on rutile TiO2(110)  and MgO(100) [121,124-126] shows that conflicting models can exist and are non-trivial to resolve. This challenge in resolving the structure of water on the oxide is due in part to a
186 combination of the difficulty of controlling oxide surface reconstructions, c ontaminants in UHV, coverage effects, the potential presence of complex water clusters, and approximations that must often be made in first -principles studies (e.g. size of system, accuracy of exchange -correlation functionals). Water adsorption on SrTiO3(100), including the role of defects such as oxygen vacancies and step edges, has been the focus of several experimental studies  Like many oxides, SrTiO3 can have multiple terminations for a given surface plane and SrTiO3(100) can be terminated by a SrO or a TiO2 plane (see illustration in Figure 8 -1). The presence of the TiO2-termination of SrTiO3(100) has motivated some of the studies of H2O on this surface, since it offers an opportunity to compare with the extensive studies of TiO2 surfaces to shed light on the role of structure on the chemistry of the oxide  SrTiO3 also garners interest for potential use as a catalyst in photoelectric splitting of water [134,135] Figure 8 1. The bulk unit cell of SrTiO 3 highlighting the SrO termination (top) and TiO 2 termination (middle) of the SrTiO3(100) surface. U ltraviolet and X-ray photoemission spectroscopy (UPS, XPS) studies of water on SrTiO3(100) under UHV conditions suggests that water adsorbs molecularly on the Ti Sr O
187 pristine surface, but upon the inducement of oxygen vacancies via ion bombardment  or step edges by vacuum fracturing [128,129] the presence of dissociated water is detected. High resolution electron energy loss spectroscopy (HREELS) has also been used to probe the phonon modes of H2O and D2O on stoichiometric and reduced SrTiO3(100)  While the identification of phonons attributed to the adsorbate is complicated by the overlap with surface phonons of the oxide, the HREELS study suggests that water adsorbs dissociatively on SrTiO3(100) only in the presence of oxygen vacancies [130,131] Lopez et al report in another HREELS study that the presence of Na on the SrTiO3(100) surface can also result in dissociative adsorption of water  Wang a nd co workers have performed to date the only temperature programmed desorption (TPD) experiments in UHV of water adsorption on SrTiO3(100)  They found that water adsorbs on the pristine SrTiO3(100) with a maximum peak desorption temperature of 260 K which shifts to lower temperatures with increasing water coverage. The TPD study also concludes non -dissociative adsorption on the SrTiO3(100) surface with the primary evidence that the observed peak temperature in the TPD is too low to be assigned to dissociated H2O. They approximate the surface to consist of ~80% TiO2-termination and have confirmed via low energy electron diffraction (LEED) that the bare surface is initially the (11) unreconstructed SrTiO3(100) surface. After sputtering the surface the TPD spectra shows a long tail starting at ~ 300 K and decaying at around 480 K. This tail in the TPD spectra is assigned to dissociated water on O vacancies and matches the general conclusions from earlier XPS and UPS studies of reduced SrTiO3(100) surfaces. Wang et al suggest that the lack of bridging O atoms on the pristine TiO2-terminated SrTiO3(100)
188 surface prevents strong H -bonding with the surface O atoms that would make it favorable to strip the H atoms from the water molecule  In comparison, water binds dissociatively (associatively) on the defect -free TiO2(100) (TiO2(110)) surfaces  DFT calculations suggest that on the TiO2(100) the adsorbed water molecule has acces s to low -coordinated bridging O atoms, but on TiO2(110) the distance between the H atoms on the water molecule and the bridging O atoms is longer and prevents strong formation of H bonds with the oxide surface O atoms [127,140] In summary, the experimental results to date point to a model of associative adsorption of H2O on pristine SrTiO3(100) surface, with dissociation only observed at defects such as O vacancies and step edges. T his chapter report s on DFT cal culations of H2O adsorption and dissociation on both terminations of SrTiO3(100). There has been a recent hybrid HF -DFT study examining water on both terminations of SrTiO3(100) and they find that molecular water is favored over dissociated water on both t erminations with this difference more pronounced on the TiO2termination  This earlier DFT study examined coverages down to 1/2 monolayer (ML) on a 11 surface cell (see Section 8.2.2 for definition of 1 ML for this system). M ore dilute coverages are explored down to 1/8 ML using a 2 2 surface cell and it is f ou nd that both the adsorption energy and structure of water is sensitive to coverage. At coverages below 1/2 ML, DFT predicts dissociated H2O is favored on both terminations of SrTiO3(100). P ossible sources of the differences betw een the DFT results and the existing experimental studies are discussed 8.2. 2 Calculation Details All the DFT calculations in this chapter were performed with the Vienna ab initio simulation package (VASP) [45 -47,142] us ing the projector augmented wave (PAW)
189 [48,49] pseudopotentials provided in the VASP database T he Sr (4 s 4 p 5 s ), Ti (3 s 3 p 3 d 4 s ) and O (2s 2 p ) orbitals were include d as valence states and a plane wave cutoff energy of 400 eV was used. Calculations have been done using the generalized gradient approximation P erdew Burke-Emzerhof (GGA-PBE) exchange correlation functional  T he effects of the functional on the accuracy of the DFT calculations will be discus sed in Section 220.127.116.11.1. Electronic relaxation was performed with a block Davidson iteration method accelerated using Fermi level smearing with a Gaussian width of 0.1 eV  All calculations used a 6 -layer slab with the bottom two layers fixed. The system is relaxed until all the forces are less than 0.03 eV/. A n 8-layer and 10layer slab was tested to confirm the binding energy and adsorbate structure match that found o n the 6layer slab. The water adsorbates were examined in increments of 1/2 and 1/8 ML for the 11 or 22 surface unit cells, respectively. A monolayer ( ML ) is defined as 2 water molecules per surface cation, which matches the definition used by the earlie r HF -DFT study  A vacuum region of about 24 is included to prevent the slab from interacting with its periodic image in the surface normal direction. All calculations were completed using dipole corrections in the surface-normal direction  The effects of spin polarization corrections were checked and found not to affect the adsorption energy for any of the high symmetry sites. An 881 and 441 Monkhorst -Pack mesh  was used for the 11 and 22 surface unit cells, respectively. A bulk lattice constant of 3.95 for SrTiO3 was found which matches fairly well to that previously reported from experiment (3.90 )  and DFT (3.94 )  Minimum energy pathways and the barrier to water dis sociation are calculated using the nudged elastic band (NEB) method 
190 8.2. 3 Results and D iscussion 8.2. 3.1 Half monolayer water on SrTiO3(100) On the (11) -SrTiO3(100) surface four high symmetry adsorption sites were examined for both the SrO and TiO2terminated surfaces, with each shown in Figure 82. Figure 82a, 8-2b, 8-2c, and 82d highli ght the fourfold hollow site, the bridge site between the surface O and Sr, on top the Sr and O atoms, respectively, for the SrO terminated surface. Figure 8-2e, 82f, 82g, and 82h illustrates the surface sites for on top O, the bridge site between surfa ce Ti and O, on top surface Ti, and the fourfold hollow site, respectively, for the TiO2terminated surface. For simplicity these sites will be referred to as fourfold, bridge, Sr, Ti, and O. Figure 8 2. The top layer of the (a) (d) SrO terminated and (e) (h) TiO 2 terminated SrTiO3(100) surface showing the 11 (22) surface cell outlined by dashed (solid) lines. The h igh symmetry adsorption sites examined are identified on the SrO -termination as (a) fourfold, (b) bridge, (c) Sr, (d) O and on the TiO2t ermination as (e) O, (f) bridge, (g) Ti, and (h) fourfold. The adsorption of one water molecule on a single high symmetry site for this surface unit cell corresponds to a surface coverage of 1/2 ML, with ML defined in Section 8.2.2. I nitially molecular water was placed with the oxygen atom in each of the high symmetry surface sites with various orientations for the hydrogen atoms ranging from pointing away from, perpendicular to, and downward towards the surface atoms.
191 After relaxation, the final adsorbate configurations depend only on the surface adsorption site (i.e., the initial oxygen position) and not on the initial hydrogen orientation. Table 8 1. Adsorption energy (Eads) of water at 1/8 and 1/2 ML on various high sym metry sites, defined in Figure 8-2 on the SrO and TiO2terminated SrTiO3(100). T he values in parenthesis correspond to a dissociative state. Adsorption Site E ads (eV/H 2 O) SrO termination TiO 2 termination 1/8 ML on 22 1/2 ML on 11 1/8 ML on 22 1/2 ML on 11 Fourfold 0.90 (1.31) 0.58 (0.85) 0.23 0.05 Bridge unstable (1.14) 0.84 (unstable) 0.89 (1.11) 0.79 (0.59) Sr / Ti converges to fourfold converges to fourfold 0.76 0.61 O 0.11 0.06 0.72 converges to top Ti Table 8 1 reports the adsorption energy for each of the surface adsorption sites at both 1/2 and 1/8 ML coverage. The adsorption energy (Eads) is defined in Equation 8 1 where Eslab is the total energy of the relaxed SrTiO3 slab in the absence of any adsorbed H2O, EH2O gas is the energy of an i solated water molecule, EH2O slab is the total energy of the H2O/SrTiO3 slab, and n is the total number of water molecules in the H2O/SrTiO3 slab. Eads= Eslab+ n EH 2 O gas EH 2 O slab n (8 -1) Figure 83a to 83d illustrate the most favored associative and dissociative H2O configuration on the SrO and TiO2-terminations of SrTiO3(100) at 1/2 ML. Table 82 presents all the relevant bond lengths associated with the structures illustrated in Figure 8 -3a to 8-3d, along with similar values at the 1/8 ML configuration. This section focus es on the results at 1/2 ML and low coverage results are detailed in Section 18.104.22.168. To distinguish between hydroxyl groups consisting of O atom coming from the water versus the oxides surface the O atom from water (oxide surface) will be denoted by Ow (Ox).
192 (a) (b) (c) (d) Figure 8 3. Most favored half monolayer molecular [(a) and (b)] and dissociative [(c) and (d)] water configuration on 11-SrO [(a) and (c)] and 11-TiO2 [(b) and (d)] termination as viewed along the  direction (above) and the  direction (below). The 11 surface unit cell is outlined by the dashed box. Table 8 2. Bond lengths for the most stable water adsorbate configurations at 1/8 and 1/2 ML coverage on the SrO and TiO2-terminatated SrTiO3(100) surface. Bond Length SrO termination TiO 2 termination 1/8 ML (22) 1/2 ML (11) 1/8 ML (22) 1/2 ML (11) H 2 O H OH H 2 O H OH H 2 O H OH H 2 O H OH O w H 1 1.03 0.97 0.97 0.97 0.98 0.97 0.98 0.97 O w H 2 1.03 1.59 1.07 1.30 1.00 2.84 1.00 2.23 (Sr,Ti) O w 2.71 2.59 2.58 2.55 2.21 1.84 2.27 1.90 O x H 1 1.62 3.39 3.04 2.97 2.18 2.77 2.42 2.91 O x H 2 1.62 1.01 1.45 1.13 1.88 0.98 1.82 0.99 Note: Molecular and dissociative states are identified as H2O and H OH respectively. The most favored state for molecular H2O is the bridge site for both terminations. On the SrO -termination the bridge site is more stable by 0.26 eV over the fourfold site and similarly on the TiO2-termination the bridge site is favored over water adsorption on the Ti atom by 0.18 eV. A previous investigation using hybrid HF -DFT with a linear combinations of atomic orbitals (LCAO) calculations studied water on SrTiO3(100) and found both stable molecular and dissociative states on both terminations  To test
193 the favorability of dissociative binding one hydrogen was separated from the converged molecular H2O/SrTiO3(100) and placed near the neighboring surface oxygen. On both terminations, it was found that after minimization that the dissociated H atom recombin es with the hydroxyl group to form molecular water. Evarestov et al also reported difficulties stabilizing a dissociative state at 1/2 ML coverage on the 11 surface cell of SrTiO3(100)  To overcome these difficulties, they used the stabilized dissociated water on SrZrO3(100) surface s and scaled the resulting configuration to the SrTiO3(100) surface. Upon relaxation of this structure they were able to obtain a stable dissociated configuration on the SrTiO3(100) surfaces. A dissociative state was finally stabilized by using an initial configuration based on the structure reported by Evarestov et al  The primary feature of the dissociated structure of Evarestov et al is the distance between the hydroxyl group and the surface cation. If this distance is appropriate in the initial dissociated structure, a local mini mum of the dissociative H2O can be stabilized on both terminations. However, the dissociative state on TiO2 is found to be less stable by 0.2 eV/H2O and there is a negligible difference between associative and dissociative H2O (0.01 eV/H2O) on the SrO term ination (see Table 8 1). These DFT values for the relative stability of associative versus dissociative H2O on both terminations are very similar to the earlier HF -DFT study where they report molecular H2O is favored by 0.1 eV/H2O on the TiO2termination a nd 0.003 eV/H2O difference on the SrO -termination  As can be seen in Figure 8 -3c, water rotates out of the bridge site and into a fourfold site on SrO following dissociation. The values for the bond lengths of the m inima structures determined in this study also agree well with those r eported by Evarestov et al with
194 slight under estimation of the OwSr at 2.55 vs. 2.61  and over estimation of the Ow-Ti at 1.90 vs. 1.88  The primary conclusion from the earlier DFT study and this work at 1/2 ML is that H2O should be expected to adsorb associatively on the defect -free SrTiO3(100) surface. This conclusion agrees with the general findings of various experimental studies [1271 33] especially if the expectation of predominance of TiO2-termination in these experiments is taken into consideration  While adsorption energies were not extracted in the TPD study of H2O adsorption on SrTiO3(100)  these values may be approximated from the peak temperature (Tp) associated with dilute coverages (~260 K) and assuming first order desorption with no adsorbate interaction and a prefactor ( o) of 11013. With these ass umptions and a heating rate of 1 K/s  a value for the adsorption energy of 0.92 eV can be ex tracted for Tp = 260 K. This adsorption energy is larger than the DFT value of 0.79 eV and 0.84 eV on the TiO2and SrO -termination respectively, which is not unexpected since the 260 K peak temperature is associated with dilute coverages but suggests that the adsorption energies from DFT are qualitatively similar to that derived from experiment. Ignoring the adsorbateadsorbate interactions a value of 0.63 eV for the adsorption energy (Tp ~ 180 K) at high coverages can be obtained from the TPD spectra. While these values are not very accurate due to the assumptions, they do illustrate a strong coverage dependence on the adsorption energy that can also be observed from the strong shift in the TPD spectra with coverage 
195 8.2. 3.2 Dilute coverage of water on SrTiO3(100) To probe coverage affects on the adsorption energy and structure of water on the SrTiO3(100) surface a 22 surface unit cell was used, which reduces the possible coverage increment to 1/8 ML. W ater was examined in the same adsorption sites as above and the Eads for each site is summarized in Table 8 1. Figure 8 -4 shows the most stable minima for associative and dissociative H2O found on both terminations at 1/8 ML. For both terminations there is a general increase in adsorption energy for both the dissociated and molecular water states, but more importantly a crossover to a preference f or the dissociative state occurs when the coverage is decreased from 1/2 to 1/8 ML. For the SrO -termination a stable dissociated state was observed at the fourfold site with an adsorption energy of 1.31 eV/H2O and shown in Figure 8 -4c. As discussed in Sec tion 22.214.171.124 dissociated water on the bridge site could not be stabilized in the 11 surface cell, and the dissociative state could only be stabilized by shifting to a fourfold site (see Figure 8 -3c). The lower coverage further stabilizes the dissociative state as evident by the more substantial separation of the Ow-H and free H as seen in Figure 84c versus Figure 83c. At a coverage of 1/8 ML, the bridge site does have a stable state associated with dissociated water (not shown), but this site is less favorable over the fourfold by 0.17 eV/H2O. For molecular H2O the effect of lower coverage acts in an inverse way and destabilizes the water molecule at the bridge site. At 1/2 ML on the 11 SrO -termination the molecular water is prevented from dissociating due to repulsive neighbor interactions with an adjacent hydroxyl group in the  direction due to periodic
196 boundary conditions, but at lower coverages this obstacle is removed and the water molecule dissociates as it would prefer in the absence of this repulsive interaction. (a) (b) (c) (d) Figure 84. Most favored 1/8 monolayer molecular [(a) and (b)] and dissociative [(c) and (d)] water configuration on 22 SrO [(a) and (c)] and 22 TiO2 [(b) and (d)] terminations as viewed along the  direction (above) and the  direction (below). The 22 surface unit cell is outlined by the solid box for visual aid of the top view along the  direction.
197 The most stable molecular water state at 1/8 ML is then found on the fourfold site (see Figure 84a), and while this state gains 0.32 eV in stability due to the decrease in coverage, the adsorption energy of 0.90 eV/H2O is weaker by 0.41 eV/H2O than the dissociated state. Therefore, thes e DFT results suggest that at lower coverages dissociated water should be favored on the SrO termination. On the TiO2 termination of the 22 surface cell, an increase in molecular water stability was found at each site with respect to the 1/2 ML coverage on the 11 TiO2 terminated surface. The dissociated water structure at the bridge site on the TiO2 termination shown in Figure 8 4d has an adsorption energy of 1.11 eV/H2O, which is 0.22 eV/H2O more stable than the most stable molecularly bound water shown in Figure 8 -4b. Therefore, similar to the SrO -termination there is a crossover to the dissociated state at lower coverages on the TiO2-termination. Evarestov et al describe the dissociated water configuration on the 11 TiO2 termination as having two hyd roxyls groups in a trans -position to each other (see Figure 8-3d). On the 22 TiO2 termination, a cis -type configuration of the two hydroxyl groups was identified, as illustrated in Figure 8 -4d. T he stability of a trans -type configuration of the two hydroxyl groups was tested at the coverage of 1/8 ML and a nearly energy equivalent state was found with an Eads of 1.13 eV/H2O. The structural information for the most stable water configurations at 1/8 ML are reported in Table 8-2 and can be compared with the values at 1/2 ML. As noted above for the SrO -termination there is a shift in the site of the preferred molecular water, therefore some of the changes in the structural parameters is due to this change. For dissociated H2O on the SrO -termination, t he primary change is a decrease in the Ox-H
198 distance from 1.13 to 1.01 when the coverage decreases to 1/8 ML. The distance of 1.01 is very close to the value of the hydroxyl group from the water molecule (0.97 ), which implies that the hydroxyl group associated with Ox increases in strength upon the decrease in coverage. On the TiO2termination there are less dramatic changes in the structure of water (associated or dissociated) with coverage, but instead the changes occur in the noninteracting bond distances. For molecular water there is a decrease of 0.24 in the bond distance between one of the H atoms on H2O and the adjacent Ox atom (Ox-H1) and the Ow-Ti bond distance shrinks by 0.06 These changes again reflect the increase in the strength of the water -surface interactions for the molecular configuration with decreasing coverage. For the dissociated water on TiO2-termination there is a similar decrease in the Ow-Ti bond. But the more important effect of the lower coverage is made clear by compa ring Figure 84d and Figure 8-3d, which represent the dissociated minima at 1/8 and 1/2 ML, respectively. At 1/2 ML coverage the two hydroxyl groups (Ox-H and Ow-H) are restricted in the ability to relax to optimize the interaction due to the presence of h ydroxyl groups on both sides. This restriction is lifted at 1/8 ML and as can be seen in Figure 84d the two hydroxyl groups relax and rotate away from each other. The net effect of this additional relaxation is an increase in the stability of the dissociated H2O molecule at the lower coverage. To probe the observed crossover in the preferred structure of water 1/8, 1/4, 3/8, and 1/2 ML coverages were examined on the 22 surface cell. The use of th e 22 surface cell also allows for the study of any size e ffects due to periodic boundary
199 conditions at 1/2 ML. Figure 85 plots the adsorption energy of the most favored configuration identified as a function of coverage (ML) on the 22 TiO2-termination along with the earlier values at 1/2 ML found on the 11 Ti O2-termination. Figure 8 5. E ads of the most favored molecular (triangle) and dissociative (circle) water configuration at varying coverage on the 22-TiO2 termination. The adsorption energy for the molecular and dissociated H2O on the 11 TiO2 termin ation are included as open symbols at 1/2 ML. Figure 86a to 86c illustrates the relaxed dissociative water at 1/4 ML coverage (2 water molecules/22 surface unit cell) in three different possible adsorbate configurations. The configurations vary in the distance between water molecules (Figure 8 -6a versus Figure 8-6b and Figure 86c) or the relative orientation (Figure 86b versus Figure 86c). W ater binding molecularly was also examined for each of these three configurations. For molecular binding the adsorbed water interacts similarly for all of the configurations with Eads of 0.82, 0.88, and 0.88 eV/H2O, for molecular equivalents of the 0.000 0.125 0.250 0.375 0.500 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Molecular Dissociative Eads (eV/H2O)Coverage (ML)
200 dissociative configurations in Figure 8-6a, 86b, and 86c, respectively. However, for dissociative binding there ar e significant variations in the adsorption energy with surface configuration with an Eads of 1.02, 1.06, and 0.56 eV/H2O for Figure 8 -6a, 8 6b, and 86c, respectively. The reason for this significant reduction in Eads for the configuration shown in Figure 8 6c is due to repulsive interactions between the OxH and the Ow-H groups. (a) (b) (c) Figure 8 6. The three possible configurations of dissociatively adsorbed water at 1/4 ML coverage on the 22 Ti O2 termination viewed along the  and  directions with Eads values of (a) 1.02 (b) 1.06, and (c) 0.56 eV/H2O. The 22 surface unit cell is outlined by the solid box for visual aid of the top view along the  direction. The unfavorable configur ation associated with Figure 8-6c is similar to the configuration f ou nd for 1/2 ML on the 11 surface cell (see Figure 83c). At a coverage of 1/4 ML the water adsorbates can avoid this unfavorable interaction by taking on configurations as shown in Figure 8 6a and 86b, which allow for rotations of the two adjacent hydroxyl groups. However, the addition of one more water to the surface in any surface site will create this unfavorable row of hydroxyl groups that are restricted by having 2 neighbors on either side. Therefore, at 3/8 ML coverage the molecular water
201 state is favored over dissociative adsorption, as illustrated by the Eads in Figure 8-5 the crossover to favorability of dissociative water occurs at coverages below 3/8 ML. Using the 22 surface unit cell the favored configurations found on 11-TiO2 termination were revisited. For the dissociative state at 1/2 ML on the 22-TiO2, the cis type hydroxyl group configuration binds with slightly higher adsorption energy of 0.63 eV/H2O than the trans -type hydroxyl group configuration found on the 11-TiO2 termination with an adsorption energy of 0.59 eV/H2O (see Figure 85 at 1/2 ML). The adsorbate configuration on the 22 TiO2 termination is mediated by surface TiO6 octahedra rotation. Periodic bound ary conditions restrict this tilting in the 11 -TiO2 system thus reflecting the lower adsorption energy. Figure 8-7 isolates the top layer of TiO6 octahedra in order to aid in visualization of this tilting. Figure 8 7. Top layer of TiO 6 octahedra for the 22 surface unit cell of TiO 2 terminated with 1/2 ML of dissociated water on the Ti site viewed along the  (top) and  (bottom). The 22 surface unit cell is outlined by the solid box for visual aid of the top view along the  directi on.
202 This tilting occurs because at this coverage oxygen within each hydroxyl group binds to a surface Ti atom and thereby completes the octahedral coordination. The hydrogen freed from each water then binds to the neighboring surface oxygen and triggers a buckling up or down of the surface oxygen and results in each octahedron rotation. Specifically, the Ti atom buckles up by about 0.2 while the oxygen atoms buckle up by 0.31 or down by 0.23 In addition to the buckling normal to the surface, in the t op view shown in Figure 87 there is movement of the oxygen perpendicular to the surface normal direction. T he role of the surface response was tested by fixing the surface atoms to the ideal (optimized bare surface) position and both molecular and dissoci ated water at 1/8 ML coverage were examined on the 22 TiO2 surface. T he dissociated water recombines to the molecular state. With the molecularly bound water, the Eads on the fixed surface is reduced by 0.30 eV/H2O with respect to the 1/8 ML molecular state. Similar to the 22 TiO2-termination different configurations at 1/4 ML were examined on the 22 SrO -terminated surface. The configurations for dissociated water following relaxation are shown in Figure 88a, 8 -8b, and 8-8c with an adsorption ener gies of 1.01, 1.22, and 1.20 eV/H2O, respectively. Molecular water in similar configurations shown in Figure 8-8a, 88b, and 88c binds with adsorption energies of 0.85, 0.70, and 0.76 eV/H2O and therefore, as at 1/8 ML, the dissociated water configurations will be favored at 1/4 ML on the SrO -termination. Unlike on TiO2, the adsorption energy for dissociated H2O only slightly decreases (0.1 to 0.3 eV/H2O) with respect to the favored dissociated configuration at 1/8 ML (Eads = 1.31 eV/H2O). Additionally, t he configuration that was found to be least stable on the
203 TiO2-termination (Figure 8-6c) is found to be the most stable on the Sr O -termination (Figure 88c). Figure 8-8c shows the dissociated hydroxyl groups cluster instead of lining up in a row along the surface cations as was seen in Figure 86c. This clustering is due to the favored adsorption site of the hydroxyl group at 1/4 ML on the SrO termination, namely a transition from a bridge to a fourfold type site as illustrated in Figure 82a. Within the pr imitive 11 unit cell of the SrO termination there are two equivalent fourfold sites. The close proximity of the second fourfold site allows one of the dissociated water groups to rotate the hydroxyl portion to the neighboring fourfold site and away from t he initial fourfold site, which was aligned along the  direction. (a) (b) (c) Figure 8 8. The three possible configurations of dissociatively adsorbed water at 1/4 ML coverage on the 22 SrO termination viewed along the  and  directions with Eads values of (a) 1.01, (b) 1.22, and (c) 1.20 eV/H2O. The 22 surface unit cell is outlined by the solid box for visual aid of the top view along the  direction. At 3/8 ML, an adsorption energy of 1.12 eV/H2O was found for the dissociated water, which is less stable by 0.19 eV to that determined at 1/8 ML. The molecular water configuration has an adsorption energy of 0.71 eV/H2O and therefore at 3/8 ML as at 1/4 and 1/8 ML the dissociated water configuration is favored on the SrO
204 termination. Finally at 1/2 ML coverage some repulsive interactions are seen between the adsorbates with an adsorption energy of 0.97 eV/H2O and one of the water molecules recombines resulting in a mixed adsorption configuration. A pure molecular configuration was identified with the H2O molecules sitting on the fourfold sites. This configuration had an adsorption energy of 0.64 eV/H2O, compared to a val ue of 0.58 eV/H2O on the 11 surface cell. This slight increase can be attributed to the ability of the water molecules to take on alternative up and down orientation (not shown), which slightly stabilizes the molecular water layer. Nevertheless, by compar ing the adsorption energy of the mixed adsorption configuration (0.97 eV/H2O) with molecular water on the bridge site of the 11 SrO (0.84 eV/H2O) or at the fourfould sites of the 22 SrO (0.64 eV/H2O), it is found that the mixed configuration is favored. The mixed adsorbate configuration on the 22 SrO surface, the inability to stabilize water molecules on the bridge sites on the 22 SrO surface, and the observed octahedra tilting of the TiO6 octahedra of the 22 TiO2 surface indicate that for even high coverages a minimum 22 surface unit cell is needed to fully probe the structure of water due to these subtle adsorbate adsorbate interactions. If the DFT -predicted configurations at 1/2 ML are accurate they suggest that LEED probing the water layers on SrTi O3(100) should show doubling of the unit cell. Utilizing the code of Henkelman et al [62,144] the Bader charges we re determined based on the Bader criteria for decomposi ng the electron density  Table 8 -3 summarizes th e Bader charges for the 11 and 22 surface unit cell of both the SrO and TiO2 termination. In each case, the hydrogen atom is assigned zero electrons according to the Bader criteria due to the lack of zero -flux surfaces in the electron
205 density gradient be tween the hydrogen and oxygen atoms. Though not shown, the electron charge density was examined to confirm the lack of a gradient between hydrogen and oxygen. Therefore, the hydrogen atoms electron is assigned to the bonding oxygen. This is evident by the significant change in charge associated with the terminal oxygen before and after dissociation. After dissociation, the water oxygen losses 0.44 (0.31) electrons on the TiO2 (SrO) termination and the surface oxygen bound to the free hydrogen gains 0.43 (0.54) electrons. Table 8 3. The average Bader atomic charges in units of e for Sr, Ti, H, and O within the oxide and water, are shown for molecular and dissociative water on both TiO2 and SrO terminations. TiO 2 termination Molecular (1/8, 1/4, 1/2) Dissociative (1/8, 1/4, 1/2) Sr 1.58 1.58 Ti 2.14 2.14 O surf 1.24 1.24 ( 1.67) O water 1.97 1.53 H 1.00 1.00 SrO termination Molecular (1/8, 1/4, 1/2) Dissociative (1/8, 1/4, 1/2) Sr 1.60 1.60 Ti 2.16 2.14 O surf 1.24 1.23 ( 1.77) O water 2.09 1.78 H 1.00 1.00 Note: For the dissociative adsorption, the surface oxygen bound to the free hydrogen is distinguished in parenthesis. Though the Bader charges determined in this study underestimate the ionic character of both the Sr and Ti atoms with respect to that determined by Evarestov et al  the general features agree well. Similar to Evarestov et al ., it was determine d that the Sr, Ti, and O (not bound to the free hydrogen) charges are unaffected by th e dissociation of water  On b oth terminations, the charges are unaffected by changes in coverage or surface unit cell, e.g 1/2 ML molecular water on 11 TiO2 has
206 the same charges as 1/2 ML and 1/8 ML molecular water on 22 TiO2. This observation suggests that the observed changes in stability of coverage is not linked with changes in bonding, but instead are dependent on electrostatic interactions. Calculations with larger unit cells (44) would also be useful in determining the role of long-range substrate mediated versus electrostat ic interactions, but such analysis are left to future studies. The larger charge transfer on the SrO termination can help examine the clustering affect observed on the 22 SrO surface with 1/4 ML coverage, illustrated in Figure 8-8c. The initial configuration, with the two dissociated water groups aligning along the surface Sr cations, places each hydroxyl group between two surface oxygen bound to two free hydrogen atoms. Since each of these surface oxygen atoms gain 0.54 electrons following adsorption of free hydrogen atoms, each of the two surface oxygen atoms would have a charge of 1.7 e The two hydroxyl groups from the two adsorbed water would also have a charge of 1.7 e The close proximity of these charges would lead to significant repulsion between the surface based and water based hydroxyl groups. This repulsion in combination with the close proximity of another free favorable surface adsorption site on the 22 surface leads to the rotation previously mentioned. 126.96.36.199 Vibrational modes of water on SrTiO3(100) A useful experimental probe in attempting to resolve between molecular versus dissociated water is HREELS or other vibrational spectroscopy tools [137,138] Lopez et al reported vibrational spectra of water on both clean and sodium modified SrTiO3(100) at room temperature  The absence of the OH band at 3600 cm1 indicated water d id not adsorb on clean SrTiO3 but this is likely due to the experiments being performed at room temperature. Wang and co workers in their TPD study have demonstrated
207 lower temperatures are needed to have uptake of H2O on pristine SrTiO3(100)  However, after sodium atoms are coadsorbed with water on the SrTiO3(100) a feature was observed at 3660 cm1, attributed to the OH band  This feature was independent of the water exposure indicating surface saturation at low water exposure. Though not discussed, the absence of a feature near 1595 cm1 would indicate that water was adsorbed dissociatively, as this feature is attributed to H -O -H bond angle scissoring. Cox. et a l completed an HREELS study of water adsorption on SrTiO3(100) at 100 K and found two main peaks associated with the a moderate dosage (4 Langmuir) of water  a sharp peak at 3662 cm1 and another peak at 1686 cm1. Cox et al attributed the high frequency mode to symmetric stretching of the OH bond and the low frequency mode with the bond angle bend ing mode, thereby concluding water binds molecularly on pristine SrTiO3. However, due to a poor noise to signal ratio, the asymmetric OH stretching mode was not resolved. Overall, while there have been vibrational studies of H2O/SrTiO3(100) the coverage in these studies is not known and full mode assignment, including resolving shifts of frequencies upon adsorption, has not been made. DFT was used to determine the vibrational modes of the most stable dissociative and molecular water structure at 1/8 ML on the SrO and TiO2-termination. To allow for a computationally tractable but accurate evaluation of the vibrational modes associated with the water adsorbate and its interaction with the surface, the oxide atoms not participating directly in the water adsor ption were fixed. A finite central difference method with a displacement of 0.05 is used to calculate the Hessian matrix. Neglecting the surface modes the zero point corrections (ZPC) to the adsorption energy
208 are 0.056 and 0.092 eV/H2O molecule for the 0.125 ML coverage dissociative and molecular state on the TiO2 termination, respectively. These corrections are not sufficient to change the favorability of the dissociative state over the molecular state and therefore do not affect the primary conclusions and ZPC adsorption energies are not reported in this chapter. The frequencies and mode assignments are summarized in Table8 4. The mode associated with stretching of the H -O bond is split for molecular water on the TiO2 termination. The hydrogen orientat ed away from the surface oxide (i.e. H1 in Table 8 2 and 8 -4) has the characteristic frequency of 3609 cm1 (3666 cm1), while the hydrogen oriented towards the oxide (i.e., H2) is assigned a frequency nearly 300 cm1 lower at 3337 cm1 (3254 cm1) at 1/8 ML (1/2 ML) coverage. The bond angle scissoring mode has a frequency of 1585 and 1571 cm1, for the 1/8 and 1/2 ML coverage, respectively, are very near the expected 1595 cm1. The dissociatively bound water at 1/8 ML, has vibrational modes of 3762 cm1 a nd 3644 cm1 associated with O -H stretching of the water hydroxyl group and surface oxide, respectively. The same modes for the 1/2 ML coverage are slightly down shifted at 3734 cm1 and 3513 cm1. At either coverage, the low frequency mode near 1595 cm1 was not observed due to the loss of the H -O -H bond angle scissoring. Onal et al computationally investigated water adsorption on small clusters representative of (101) and (100) surfaces of anatase TiO2 structure  Vibrational frequencies determined compare well to available experimental data. Similar to the DFT findings reported here there were marked differences between the frequencies associated with molecular and dissociative adsorption.
209 Table 8 4. DFT -derived v ibrational frequencies in cm1 of isolat ed water, m olecular and dissociative water at the favored site on the 22 and 11 SrO and TiO2termination and the corresponding mode assignment. For isolated H2O the experimental values  are given in parentheses. Frequency Mode Assignment Isolated H 2 O H O w 3838 ( 3756 ) cm 1 Asymmetric stretching H O w 3723 ( 3657 ) cm 1 Symmetric stretching H O w H 1560 ( 1595 ) cm 1 Bending (scissoring) 1/8 ML H 2 O on 2 2 TiO 2 H 1 O w 3609 cm 1 Asymmetric stretching H 2 O w 3337 cm 1 Asymmetric stretching H 1 O w H 2 1585 cm 1 Bending (scissoring) 1/8 ML H OH on 2 2 TiO 2 H 1 O w 3762 cm 1 S tretching H 2 O x 3644 cm 1 S tretching 1/2 ML H 2 O on 11 TiO 2 H 1 O w 3666 cm 1 Asymmetric stretching H 2 O w 3254 cm 1 Asymmetric stretching H 1 O w H 2 1571 cm 1 Bending (scissoring) 1/2 ML H OH on 1 TiO 2 H 1 O w 3734 cm 1 S tretching H 2 O x 3513 cm 1 S tretching 1/8 ML H 2 O on 22 Sr O H 1 O w H 2 2593 cm 1 S ymmetric stretching H 1 O w H 2 2542 cm 1 Asymmetric stretching H 1 O w H 2 1546 cm 1 Bending (scissoring) H 1 O w H 2 1248 cm 1 Pseudo b ending (scissoring) 1/8 ML H OH on 22 Sr O H 1 O w 3788 cm 1 Stretching H 2 O x 2864 cm 1 Stretching H 1 O w H 2 1111 cm 1 Pseudo bending (scissoring) H 1 O w H 2 1047 cm 1 Pseudo bending (scissoring) 1/2 ML H 2 O on 11 Sr O H 1 O w 3780 cm 1 Asymmetric stretching H 2 O w 2118 cm 1 Asymmetric stretching H 1 O w H 2 1579 cm 1 Bending (scissoring) H 1 O w H 2 1113 cm 1 Pseudo b ending (scissoring) 1/2 ML H OH on 11 Sr O H 1 O w 3 785 cm 1 Stretching H 1 O w H 2 1543 cm 1 Pseudo bending (scissoring) H 1 O w H 2 1345 cm 1 Pseudo bending (scissoring) H 1 O w H 2 1251 cm 1 Pseudo bending (scissoring)
210 On (101) TiO2, molecular adsorption exhibited three features at 3005, 3661, and 1605 cm1 and dissociative adsorption only two at 3666 and 3713 cm1. The dissociated water on TiO2(101) frequencies are within 50 cm1 to those determined on the TiO2terminated SrTiO3(100) surface. For the molecular state on TiO2(101), there is also a do wnward shift in the frequency of the O -H stretching mode by 600 cm1 similar to that observed on SrTiO3(100) but with larger magnitude. The vibrational modes of molecular and dissociative water at 1/8 ML on the 22 SrO and 1/2 ML on the 11-SrO surface are included in Table 84. For dissociative water at 1/8 ML, the hydrogen oxygen stretching within the dissociated hydroxyl group has an upward shift in frequency to 3788 cm1 versus the same mode on the 22-TiO2 surface at a frequency of 3762 cm1. The disso ciated hydrogen to surface oxygen stretching is significantly altered on the 22SrO surface (2864 cm1) versus 22-TiO2 surface (3644 cm1). The change in this mode between the two surfaces is related to the adsorbate structures, and adsorption sites. On the TiO2 termination, the dissociated hydrogen stretches into the free space above the surface. In the fourfold adsorption site on the SrO termination, the dissociated hydrogen stretches between the surface oxide and the oxygen from the dissociated water. Additional modes not observed on the TiO2 termination are identified on both the 11and 22SrO surface due to the open nature of the SrO surface plane. The dissociated hydrogen stretches into the open fourfold high symmetry site [see Figure 82a]. This stretching results in a pseudo bond angle bending mode with two frequencies of 1111 cm1 and 1047 cm1 at 1/8 ML on 22-SrO and three frequencies of 1543 cm1, 1345 cm1, and 1251 cm1 at 1/2 ML on 11-SrO. These low frequency pseudo bending
211 modes are als o observed for the molecular states on both 11and 22-SrO. For 1/2 ML, the expected Ow-H stretching has a frequency of 3780 cm1 and 2118 cm1, the bond angle bending mode at 1579 cm1 and one pseudo bond angle bending modes at 1113 cm1. At 1/8 ML, the molecular water adsorbed in the fourfold site leads to a downshift in the frequencies. Symmetric stretching of the water hydrogen atoms has a frequency of 2593 cm1 versus the isolated water mode at 3838 cm1. The asymmetric stretching of the hydrogen ato ms is 2542 cm1, nearly 1200 cm1 shift in the mode. The bond angle bending mode has frequency of 1546 cm1 similar to the isolated molecule at 1560 cm1. The pseudo bond angle bending occurs at 1/8 ML as well but at a frequency of 1248 cm1 versus 1113 c m1. Despite this pseudo bond angle bending, the ~400 cm1 downward shift in the frequency should be sufficiently large to distinguish between a dissociatively or molecularly bond water at a given coverage on the SrO termination of SrTiO3(100). However, ca re must be taken in identifying the coverage of water on the surface since the pseudo bending mode at 1/8 ML molecular water on SrO (1248 cm1) is nearly the same as that of 1/2 ML dissociative water on SrO (1251 cm1). As mentioned in the introduction, w ater adsorption on MgO (100) has been extensively studied with varying results. Chizallet et al examined IR features of OH groups on MgO using both cluster and periodic DFT calculations  The vibrational modes determined were compared to experimental spectra showing a broad low frequency feature between 3200 and 3650 cm1 and a sharp high frequency feature between 3650 and 3800 cm1. The theoretical investigation identified various features based on the water adsorbate configuration providing some insight into the broad low frequency feature. Lower frequency modes arou nd 1800 cm1 are associated with H -O -
212 H bending modes for the slightly dissociated water molecule, similar to observations made here. 188.8.131.52 Discussion on potential sources of DFT -experiment disagreement T he introduction (Section 8.2.1) discussed the experimental studies that conclude the presence of only molecular water on pristine SrTiO3(100). The only study of water at low coverages is the TPD study of Wang and coworkers  and while they see peak temperatures close to 260 K, which is associated with molecular H2O, there is no direct evidence, for example from HREELS vibrational spectra or STM images of the state of H2O at low coverages versus at high coverages. As noted earlier conflicting models between DFT and experiment in the resolution of structure of water on oxide surfaces is not new, as in both TiO2(110) and MgO (100) there have been initial disagreement. Below some possible sources of the difference between the DFT results and experimental studies are addressed. 184.108.40.206.1 Evaluation of the exchange -correlation functional One often raised concern with DFT studies is the impact of the functional on the accuracy of the calculations. Could the favorability of dissociated H2O found in these calculations be due to error associated with the PBE functional? There are numerous DFT studies of water adsorption on various ox ide surfaces [125,145,147,148] including the discussed HF -DFT study of H2O/SrTiO3(100)  which suggest in general GGA -type functional will more accurately describe H2O adsorption than the LDA functional. Nevertheless, within GGA there are several different implementations and hybrid functionals, such as PBE0 [149,150] and HSE06 [151,152] are expected to be more accurate than PBE. At least one drawback to the hybrid functionals is the add itional computational expense in their current implementation in planewave codes such as VASP [153,154]
213 Table 8 5. Bulk lattice parameter () of SrTiO3 and the adsorption energy (eV/H2O) for molecular and dissociative H2O at 1/8 ML on the TiO2 terminated SrTiO3(100) surface for various exchange-correlation functionals. Functional Lattice parameter E ads (molecular) E ads (dissociated) PBE 3.95 0.89 1.11 RPBE 3.98 0.84 1.13 PBEsol 3.90 2.67 2.83 PBE0 3.90 HSE06 3.90 experiment 3.90 To characterize the error in the GGA -PBE calculations reported in this chapter, select calculations with other GGA functional are repeated Table 8 -5 shows the bulk SrTiO3 lattice parameter and the adsorption energy at a coverage of 1/8 ML on the TiO2-term inated SrTiO3(100) surface for both associated and dissociated H2O. T he testing of the functional was focused on TiO2-termination since the experimental SrTiO3(100) surface is expected to be dominated by this termination  PBEsol and the two hybrid functionals reproduce the experimental lattice parameter value of 3.90 precisely, while the P BE and RPBE functionals are less accurate with values of 3.95 and 3.98 respectively. These results match earlier studies that have shown that PBEsol is very effective in capturing the bulk structure without the expense of the hybrid functionals [68,154] Unfortunately, applying the PBEsol to calculate the adsorption energy gives values that are nearly double the PBE results, a trend that matches earlier tests of this functional  but even within PBEsol the dissociated structure is favored. The RPBE functional gives very similar results as PBE and most importantly confirms the preference for dissociat ed water at low coverages. Unfortunately, similar tests of the adsorption energies using the hybrid functional were not possible due to the computational expense. Future testing with a more computationally tractable system (i.e. point and thinner slabs) will help to clarify any
214 quantitative changes to the adsorption energy, but a reversal of the favorability of the water structure is not expected using the hybrid functionals due to the large energy difference (~0.20 eV) between the two structures in the PBE -DFT calculations. 220.127.116.11.2 Clustering effects favored dimer configurations. Clustering of water molecules could dramatically affect the stable structures on the surface. For example, co adsorbed HO -H2O complexes have been identified on PdO(101)  TiO2(011)  TiO2(110)  MgO(001) [124,157] While a thorough study of cluster effects on the structure and stability of water on SrTiO3(100) is beyond the scope of this chapter, stable dimer configurations on the 22 TiO2 termination have been attempt ed The goal is to probe on 22 surface unit cell if any dimer complexes (H2O H2O, OH -OH, or HO H2O) are more favored over the isolated dissociated water structure identified at 1/4 and 1/8 ML. T he stabilized molecular and dissociative water structure at 1/8 ML from Section 18.104.22.168 were used with molecular water added to neighboring high symmetry sites from Figure 8 2. It was found when molecular water is added near the dissociated water either th e dissociative water recombines or the molecular water shifts over to the next bridge site, thereby increasing the interatomic distance. For most of the configurations with two molecular water adsorbates, the two molecules separate in order to increase the interatomic distance between molecules to a state similar to the molecular equivalent of configuration in Figure 8-6c. In one case the second water adsorbs to the neighboring fourfold site, see Figure 82h, and orients itself perpendicular to the original state with one hydrogen pointed towards the surface and one pointed away. For each of these configurations, the total adsorption energy is found to be between 0.11 and 0.16 eV/H2O less favored than the molecular equivalent shown
215 in Figure 8 -6c. Therefore, these configurations do not stabilize dissociative adsorption and on the 22 TiO2 termination the isolated dissociated H2O remains the most favored configuration identified. However, the cluster configuration of molecular water adsorbed to the bridge and fourfold site provides some insight into how the TiO2 termination may favorably accommodate surface saturation at high coverages of 1 ML. While this initial study of dimer configurations has not found any favored clustering on the TiO2termination, future studies using larger surface unit cells and probing more configurations for stable clusters of water molecules would assist in more firmly resolving the role of intermolecular interactions of water on the SrTiO3(100) surface. 22.214.171.124.3 Kinetic barrier to dissociation One potential scenario that would resolve the difference between DFT and experiment is the presence of an energy barrier to dissociation of H2O on the surface that is larger than the desorption energy. Under these conditions the molecular state would be kinetically trapped. T he NEB method was first used to identify the barrier on the 11 SrO and TiO2-termination. For the 11 SrO surface, the energy difference between the molecular and dissociative state was 0.01 eV (see Table 8 -1). Both the b arrier to dissociation and association on the 11 SrO -termination are negligible at 0.00 and 0.05 eV respectively. On the 11 TiO2 termination surface, where the molecular water is favored, the dissociation (association) barrier is 0.29 (0.13) eV. Both these barriers are relatively small in comparison to the adsorption energies and would suggest that kinetic barriers will not play a large role in the observed species on the surface. Nevertheless, the crossover to dissociated water occurs at lower coverages and therefore the energy barrier on the 22 TiO2 was evaluated at a coverage of 1/8 ML to examine both the role of system
216 size and adsorbate coverage. The cis -type dissociative structure was used in order to compare with the results for the 11 TiO2 system Additionally, to reduce the computational expense the force criterion was slightly reduced for the NEB calculations from 0.03 to 0.05 eV/. The resulting dissociation barrier is 0.08 eV, while recombination has a barrier of 0.34 eV. Therefore, the dissoc iation barrier in fact becomes smaller and is negligible at lower coverages. While not explicitly calculated for the 22 SrO -termination similar results are expected for the dissociation barrier. These calculated energy barriers suggest the discrepancy bet ween DFT and experimental observations cannot be resolved by invoking large kinetic barriers to dissociations. 8.2. 4 Summary The stability of molecular versus dissociated water as a function of coverage on both terminations of SrTiO3( 100) was studied using DFT. At a coverage of 1/2 ML the general results of an earlier hybrid HF -DFT study  were reproduce which predicts that associative H2O will be the favored structure on TiO2-termination and negligible differences in stability between associative and dissociative H2O on the SrO termination. Artificial size effects at coverages of 1/2 ML due to periodic boundary conditions were identified by comparison of 11 and 22 surface unit cells. As the coverage is decreased below 3/8 ML a crossover to dissociative water on t he TiO2termination was found. On the 22 SrO -termination a mixed adsorbate structure can be stabilized at 1/2 ML and further decrease in coverage stabilizes pure dissociative H2O. The DFT results reported in this chapter at lower coverages conflicts with the general conclusions of experiments examining H2O/SrTiO3(100), but these experiments have not yet presented a fully detailed picture of water on SrTiO3(100). It remains an open question of the source of the differences between these DFT results and the
217 experimental studies. P ossible stabilization mechanisms for molecular water were probed, but clustering or kinetic barriers to dissociation were not found to stabiliz e molecular over dissociated water. A possibility that would dramatically affect the compa rison between experiment and DFT would be adsorbate induced surface reconstructions. As noted in the introduction the experimental studies note a unreconstructed 11 SrTiO3 surface, but several studies have shown that there are numerous reconstructions for SrTiO3(100) and the surface structure is sensitive to both the environment and preparation [158,159] The TPD study of H2O on SrTiO3(100) does report LEED patterns before the water exposure indicate a 11 unreconstructed surface  but they did not probe the surface structure during and after exposure to H2O. The SrTiO3 surface could reconstruct upon adsorption which would lead to different surfaceadsorbate interactions and under that scenario this DFT study would not be examining the relevant configurations. To probe possible reconstructions upon H2O adsorption would be nontrivial using DFT without any input from experiment. Future experimental studies that probe the surface structure during water exposure, for example with STM, would be critical to resolve the role of clustering and/or ordering of water and any possible adsorbate induced reconstructions. DFT studies to elucid ate the role of O vacancies and step edges on water structure, in particular vibrational modes specific to these defects, would assist in the interpretation of experimental data. Such combined DFT -STM studies have been critical for example to resolve the w ater structure on TiO2(011)  and it is expected that the results reported in this chapter will motivate similar studies on SrTiO3(100).
218 CHAPTER 9 CONCLUSIONS 9.1 Summary of Dissertation To rationally design oxide materials for specific applications requires understanding the structure-property relationship which ideally would allow one to link structure on an atomic -level up to macroscopic properties such as dielectric constants or cataly tic activity. In this dissertation modeling work based on density functional theory (DFT), an ab initio quantum based method, was presented for two different oxide families with the goal of understanding the structure-property relationships. The majority of the dissertation focused on isolating and understanding the role of atomic displacements, oxygen vacancies, and cation substitutions on the dielectric properties of bismuth based pyrochlores. DFT results for the adsorption and dissociation of H2O on SrT iO3(100) was also presented. These calculations provide a portion of a collaborate effort at the University of Florida on SrTiO3 surfaces to design methods of delivering metal clusters to specific sites on oxide surfaces. In Chapter 3, DFT was used to exp lore a range of cubic Bi2B2O6O (B = Ti, Ru, Rh, Ir, Os, Pt) pyrochlores with no cation substitutions and found significant cation displacement in only Bi2Ti2O7. The atomic displacement stabilizes this structure with an energy change of similar magnitude t o that of other bismuth based pyrochlores. In Bi2Ti2O7, an average displacement of 0.38 for the Bi cation, 0.07 for the Ti cation, 0.11 for the O anion, and an energy change of ~0.15 eV/Bi atom is found. The Bi cation and O displacements appear to follow a -cristobalite type displacive disorder as seen in other more complex bismuth pyrochlores.
219 T he electronic structure shows the main difference between Bi2Ti2O7 and the metallic bismuth pyrochlores is the extent of Bi -O interactions. In Bi2Ti2O7 th ere is more overlap of Bi s and p states with O 2 p states similar to the reported electronic structure of Bi2Sn2O7, PbO and SnO [94,99] which leads to the asymmetric electronic structure around the Bi cation and the displacement of cations in Bi2Ti2O7. These DFT results match the general understanding from experiment al studies, but underestimate the displacement in Bi2Ru2O7 compared with experimental results of defective Bi2Ru2O7  The lone pair formation in Bi2Ti2OO6 is further examined in Chapter 4 through sulfur substit ution on the O and O sites in an effort to isolate the Bi -O and Bi -O interactions. The primary importance of the Bi -O interactions was confirmed since atomic displacement is not favored in the case of S substituted on the O site. The electronic structu re analysis, including the partial density of states, electron localization function and partial electron density, showed that the S substitution on the O site suppresses the formation of the lone pair by modifying the Bi anion hybridization. The electron ic structure of the Bi2Ti2SO6 pyrochlore fixed with the same displacement pattern as that seen in Bi2Ti2OO6 was also examined. In Bi2Ti2SO6 with forced displacement the ELF displayed a symmetric lobe around the bismuth cations with a significant reduction of the asymmetric electron density in the energy range of -2 eV to the Fermi level. Alternatively, atomic displacement is significantly fav ored energetically in Bi2Ti2OS6. The Ti and O atoms displaced considerably more in Bi2Ti2OS6 than in Bi2Ti2OO6 which was attribute to stronger hybridization between the Bi and O states that favors the lone pair formation due to weaker Bi -S versus Bi -O interactions.
220 Substituting S on the B2O6 network of Bi2Ru2OO6 resulting in favorable cation displacement and induced lone pair formation in the metallic Bi pyrochlores by disrupting the Bi -O interactions. This result suggests that modifying the B2O6 netw ork to reduce the interactions with the A2O network will assist in promoting cation displacement. Since vacancies on the A2O network could alter the local interactions of Bi, t he bismuth ruthenate compound was examined in Chapter 5 with multiple stoichi ometries. As expected the inclusion of vacancies on the Bi2O network triggered spontaneous atomic displacements of both the Bi and O atoms due to alterations of the local Bi -O interactions. This finding resolves a discrepancy between experimental resul ts using oxygen -deficient and the earlier DFT results of ideal nondefective bismuth ruthenate. T he O anions displace off the high symmetry site into a 32 e site but the magnitude of displacement depends on the proximity to the vacancies present in the str ucture. T he O displacement predicted by DFT is well within the displacement pattern predicted by neutron diffraction data  The magnitude and type of Bi displacement varied both by stoichiometry and proximity to v acancies. Unlike Bi2Ti2O7, there is no correlation of the atomic displacements in Bi2Ru2O7 indicating even a small concentration of O vacancy is sufficient to reduce the steric strain on the A2O network due to Bi lone pair formation from atomic displacement. In Chapter 6, a r ange of cation substituted Bi pyrochlores, Bi1.5M2+B5+ 1.5O7 (with M2+=Zn or Mg and B=Nb or Ta) was examined. Preferred substitute ordering was found on the A2O sub -networ k and the occurrence of locally higher substitute concentrations was found for some B2O6 configurations. The ordering of both Zn and Mg cation along
221 the <211> direction was found in all four compounds confirming the predictions from Monte Carlo models prev iously reported  Though previous reports have suggested the importance of the A2O network over the B2O6 network, large deviations in the energies of the configurations with randomly distributed cation substitutes on the B2O6 networ k indicates some ordering on the B2O6 network may be preferred. A dditionally, a relationship between DFT determined A site cation displacement magnitudes and the experimentally reported dielectric constants was identified. Finally, while cation displacement seems to be correlated to higher dielectric constants in Bi pyrochlores, the dielectric properties have been linked to the ease of cations in hopping between energetically degenerate positions within the pyrochlore struct ure. To explore this hopping process, a preliminary study was discussed in Chapter 7 to identify transition states to cation hopping and associated energy barriers for Bi2Ti2O7, Bi1.5ZnNb1.5O7, and Ca1.5Ti1.5NbO7. This study shows that the hopping process can be quite distinct within the pyrochlore family and future studies will explore these differences in more detail. For the SrTiO3 oxide, Chapter 8 detailed the stability of molecular versus dissociated water as a function of coverage on both terminations of SrTiO3(100). At higher coverages of 1/2 ML DFT predicted molecular H2O will be the favored structure on TiO2termination and negligible differences in stability between molecular and dissociative H2O on the SrO termination which confirmed with the gene ral results of an earlier hybrid HF -DFT study  Comparison of 11 and 22 surface unit cells identified artificial size effects at coverages of 1/2 ML due to periodic boundary conditions. At coverages below 3/8 ML a crossover between dissociative and molecular
222 water was found on the Ti O2-termination. On the 22 SrO -termination a mixed adsorbate structure was stabilized at 1/2 ML and pure dissociative H2O was stabilized lower coverage. The DFT results reported at lower coverages conflicted with the general conclusions of experiments exam ining H2O/SrTiO3(100). The source of the differences between the DFT results and the experimental studies remains an open question. Future experimental studies that probe the surface s tructure during water exposure would be critical to resolve the role of clustering and/or ordering of water and any possible adsorbate induced reconstructions. DFT studies to elucidate the role of O vacancies and step edges on water structure, in particular vibrational modes specific t o these defects, would assist in the interpretation of experimental data. Such combined DFT -STM studies have been critical for example to resolve the water structure on TiO2(011)  9.2 Outlook of Projects Both the bismuth pyrochlores and surface chemistry on SrTiO3 surfaces will require additional examination to aid in the optimal design of future applications using these materials. For the SrTiO3 project, part of the preliminary effort of DFT calculations have been completed to examine small molecule adsorption. However, only the water adsorption on SrTiO3(100) has been examined Since it is expected that step edges and surface polarity will play a role in the adsorption of water, the examination should be extended to other low Miller index surfaces such as the (110) and the (111) in addition to high Miller indicies surf aces that exhibit under -coordinated kink sites along step edges. Further, the study should be expanded to NH3 on similar surfaces on SrTiO3 in an effort to identify favorable adsorption and dissociation sites to aid in the oxide support design.
223 There are several potential pathways to extend the investigation of bismuth pyrochlores. One area of interest would be to examine the role of the small system size s on the atomic displacements observed. In Section 126.96.36.199, a correlation between the cation and anion displacements was observed on the A2O network. Within DFT it has been confirmed that a 2x2x2 super -cell structure cannot be successfully modeled due to the large number of atoms. It would be interesting to examine a 2x2x1 super -cell to determin e if perio dic boundary conditions affect the observed displacement patterns. As was seen in Chapter 6, ordering is likely to occur with the inclusion of cation substitutes on bismuth pyrochlores. The B2O6 subnetworks influence was identified on the overall stabil ity of the substitutional ordering on the A2O network but i t would be of interest to re examine the BZN compound to determine if any rules exist f or substitutional ordering on the B2O6 network. As discussed in Section 7.5, the preliminary work examining ion hopping in pyrochlores needs to be expanded and further analyzed. Specifically, the phonon modes should be determined for the minima created by moving cations from one Wyckoff site to an equivalent Wyckoff site to confirm structure stability. The numb er of images along each path should be increased to at least three or more depending on the intermediate structures. Additional cation substitution configurations should be tested for the BZN and CTN systems. The hopping mechanisms in bismuth ruthenate, wh ere vacanc ies triggered atomic displacement, should be examined to determine if vacancies significantly affect hopping pathways or energy barriers. The initial aim of this work was to perform DFT calculations as part of a multi -scale modeling approach to understand the structure property relationships in bismuth
224 pyrochlores. The DFT calculations would serve as the fitting data to develop core -shell i n teratomic potential s. Fitting interatomic potentials was attempted without success using the fitting potential algorithm within the Generalized Utility Lattice Program (GULP)  The bismuth titanate system was the focus of the f itting efforts with the interatomic potential for the Aurivillius phase Bi4Ti3O12 as the starting point  Future attempts to adjust this potential for the pyrochlore phase should implement the restrained fitting option within GULP where the fitted parameters are bound between the user -defined range of values.
225 APPENDIX A LONE PAIR FORMATION The atomic displacement observed in bismuth titanate was attributed to the lone pair formation in Section 3.3.3. Both lone pair formation and a p -type lone pair (seen in Section 3.3.3) versus a s -type lone pa ir (discussed in Section 4.3.2) are detailed here for further understanding. To simplify the structure, here the PbO and PbS systems will be focused on with details from Walsh et al [94,100,105,106] used. It had been thought that the lone pair on the Pb atom in PbO was due to hybridization of the 6s and 6 p orbitals of Pb but Walsh et al showed it is actually the catio n anion hybridization that leads to an asymmetric electron density indicative o f a lone pair [94,100,105,106] Walsh et al further showed it is due to the asymmetric nature of this active lone pair that the layered litharge structure is favored for PbO over the cubic rocksalt structure in direct contradiction to the PbS system. For PbS the lone pair does not display as stron g a hybridization between Pb the 6s and 6p state with the S 3 p states resulting in a s type lone pair This was confirmed by partial density of states, partial electron density plots, and an analysis of the crystal orbi tal overlap populations (COOP). This p -type of lone pair was observed for Bi2Ti2O7 following atomic displacement, as discussed in Section 3.3.3. The lone pairs observed in Bi2Ti2O7 with atomic displacement are termed p -type since they result from the hybridization of the 6s and 6p states of the Bi cations with the O anions. The hybridization results in asymmetric nature of these lone pairs as seen in Figure 4-1. Alternatively, when the energy states of the Bi 6s and 6 p orbitals do not constructively interact with the O site, as was seen in Section 4.3.2 for the Bi2Ti2SO6 material, when sulfur replaced oxygen on the O or X (as it was termed in Chapter 4 to avoid confusion), a p -type lone pair is not observed.
226 Instead the electron density surrounding the Bi cation remains relatively symmetr ic as seen in Figure 4 2, even though the bismuth cation was forcibly pushed out of its high symmetry site. This symmetric lobe had been observed previously for bismuth titanate without atomic displacement and is termed a s -type lone pair  These lone pairs, both s -type and p -type, visually look similar to the orbitals they are named after. For the s -type lone pair, the electron density is spherical and symmetrical similar to the s orbital For the p -type lone pair, the electron density appears asymmetrical similar to the orbitals which combine the shapes of the spherical s and dumb bell p orbitals following sp hybridization. These p -type lone pairs show increased electron density in the opposite direction of cation displacement which corresponds to the direction wi th reduced cation and anion bonding, as was seen in PbO [94,100,105,106]
227 APPENDIX B SUPPLEMENTAL MATERIA L FOR PYROCHLORES Additional details regarding the bismuth pyrochlores that were not included in the body of the dissertation are presented here. For the simple pyrochlores, focus will be placed on the Bi2Ti2O7 and Bi2Ru2O7 systems since these are the only simple pyrochlores that show ed atomic displacement. Additionally, the electron localization function is examined for the Bi1.5ZnNb1.5O7 system. B.1 Dipole Moments As discussed in Section 1.1.2, the dielectric permittivity is expected to depend on the dipole moments created within the material. In addition to the local dipole moments contributing to the total permittivity, the summation of all the dipole moments over the structure can result in a net polarization in a specific direction which will affect the dielectric permittivity The total polarization was calculated for the ideal Bi2Ti2O7 system as well as all ten configurations with atomic displacement to determine if a net dipole moment occurs in this system. These polarization calculations assumed a formal charge for each of the atoms. The p olarization vectors in C/cm2, for these 11 systems are detailed in Table B-1. The Px, Py, and Pz components correspond to the polarization along the x -, y -, and z axis, respectively. The total polarization (P) is also included in Table B 1, in addition to the vector components. As expecte d, the cubic system with the cations in their ideal positions shows no net dipole in any direction as indicated by the zero values for each component of the polarization vector. However, after atomic displacement and volume expansion, all ten systems show some nonzero polarization The total polarization was found to be
228 between 6.44 and 11.76 C/cm2 and three different groupings were observed, ~6.5, ~8.5, ~11.0 C/cm2. Table B -1. The polarization vectors and total polarization in C/cm2 summarized for the ideal bismuth titanate as well as the ten configurations with atomic displacement discussed in Section 188.8.131.52. Structure Px Py Pz P Ideal 0 .00 0.00 0.00 0.00 Displaced 1 2.00 8.11 0.31 8.36 Displaced 2 0.26 9.85 4.27 10.74 Displaced 3 6.24 0.60 1.49 6.44 Displaced 4 6.65 0.71 1.50 6.85 Displaced 5 6.56 0.60 1.09 6.67 Displaced 6 6.69 9.65 0.56 11.76 Displaced 7 9.57 5.45 0.88 11.05 Displaced 8 3.61 1.44 10.32 11.03 Displaced 9 1.86 2.86 7.90 8.60 Displaced 10 3.60 10.55 1.05 11.20 Fannin et al  examined bismuth titanate using density functional theory and identified a soft phonon mode that lead to a crystal system transition from the cu bic (Fd 3 m ) to a monoclinic (Cm) structure. This monoclinic structure exhibited a net polarization of 20 C/cm2 along the  direction As can be seen from Table B -1, the polarization vector and magnitude determined here is lower than that found by Fannin et al  B.2 Structure Optimization The occurrence of a nonzero polarization vector indicates the cubic structure is not stable for the bismuth titanate system with atomic displacement. Since the DFT
229 calculations are performed at fixed volume and shape, the stability of the cubic structure was initially tested for the ideal structure and was discussed in S ection 3.2. However, this test was not repeated following atomic displacement. Therefore, after the polarization results, the ideal Fd 3 m (number 227) space group was again tested for the bi smuth titanate with atomic displacement. It was found that the cubic structure is not the lowest energy structure for bismuth titanate with atomic displacement. A monoclinic structure was determined with lattice constants of a=10.35 b=10.30 c=10.38 Though Fannin et al  do report a monoclinic structure is the most energetically favorable, no structural details are reported for compariso n. Fannin et al  state that the monoclinic structure is relatively more stable than the ideal Fd 3 m structure by 240 meV or 0.24 eV In fairly good agreement, the monoclinic structure determined here is found to be more energetically favorable than the cubic structure by 0.18 eV. The difference between these relative energies may be attributed to differences between the parameters used in both studies. B.3 Charge Density Map s In Section 4.3.2, the electronic structure was examined for Bi2Ti2OO6 and bismuth titanate with sulfur substitutions on the O or the O sites. Figure 45 examined the partial electron density for both the displaced Bi2Ti2OO6 and the Bi2Ti2SO6 with for ced atomic displacement. It was shown that an asymmetric lobe of increased electron density occurs in the wake of the bismuth cation displacement path for both compounds. However, the degree of asymmetry for the Bi2Ti2SO6 system was considerably smaller than that of the Bi2Ti2OO6 system. This electron density profile included only the energy states between -2 eV and the Fermi level. Figure B 1 a shows the total charge density
230 for all energy states for part of the 1 01 plane. It can be seen that the total charge density does not show any asymmetry after atomic displacement. To compare, the electron localization function (ELF) is plotted for the equivalent plane in Figure B 1 b and the signature asymmetric lobe in observed, cor responding to the lone pair. (a) (b) Figure B 1 The (a) total charge density of the displaced Bi 2 Ti 2 O 7 and the (b) electron localization function for part of the 1 01 plane with Bi in the center wit h the Bi and O ions shown in (a) The contour lines are incremented by 0.1 eV/ 3 and t he ELF values range from 0 (light red) to 1 (dark purple) as shown by the scale. B.4 Electron Localization Function Plots B.4.1 Bi2Ru2O7 As discussed in Section 184.108.40.206 atomic displacement was examined in Bi2Ru2O7. Atomic displacement could not be stabilized by either simulated annealing or by using the experimentally reported displaced atomic positions. Since the electronic structure of Bi2Ti2O7 was key to understanding why atomic displacement occurs (Section 3.3.3) it was desired to examine the electronic structure of Bi2Ru2O7 in similar detail. To examine the electronic structure of the Bi2Ru2O7 system both with and without atomic displacement, a structure was created with the same atomic dis placement pattern as one of the Bi2Ti2O7 configurations. The density of states was discuss ed in Section 3.3.3 0.0 0.25 0.50 0.75 1.0
231 for the Bi2Ru2O7 system with forced atomic displacement but the electron localization functi on (ELF) was not included in Chapter 3. Figure B2 sh ows the ELF for the (101) plane for the (a) Bi2Ti2O7 and the (b) Bi2Ru2O7 with the same displacement pattern. As indicated by the included key, there is an increase in electron localization behind the Bi cation displacement in the Bi2Ti2O7 system creating an asymmetric lobe surrounding the bismuth cation. However, in Figure B-2 b the electron density surrounding the bismuth cation is more symmetric in nature and in fact this symmetry makes the Bi cation seem visually less displaced than in t he Bi2Ti2O7 system. (a) (b) Figure B 2 The electron localization of the displaced (a) Bi 2 Ti 2 O 7 and (b) Bi 2 Ru 2 O 7 for the (1 0 1) plane with Bi in the center wit h the Bi, Ti/ Ru and O48f ions indicated. The ELF values range from 0 (light red) to 1 (dark purple) as shown by the scale B.4.2 Bi1.5ZnNb1.5O7 Figure 311 compared the ELF plots for the (111) plane and the 1 11 plane for the Bi2Ti2O7 pyrochlore following atomic displacement. An atomic level correlation was found for this system and discussed in Section 2.2.3, as well as in Chapter 3, 4, 5, 6, and 7. In Section 3.3.3, i t was proposed that the lone pair on the displaced Bi cations in Bi2Ti2O7 leads to the correlation as a method of reducing the stereochemical strain of 0.0 0.25 0.50 0.75 1.0 Bi Ti/ Ru O48f
232 incorporating large lone pairs in a relatively small space. It was stated that the cristobalite type displacive disorder was not found in Bi1.5ZnNb1.5O7 ( BZN ) due to the presence of ZnA cations along the O -A-O bonding network. Figure B -3 shows the ELF plots for one of the BZN structures examined for the (a) (111) plane with Bi in the center and the (b) 1 11 plane with the Bi, ZnA, Nb, ZnB, O, and O48f ions indicated. Figure B -3 a looks nearly identical to Figure 3-11a with the exception of the delocalized electron profile near the ZnB cations. The increased localization of electrons just below the center Bi cation shows the lone pair also forms in the BZN configuration, as was seen in Figure 3-11a for the Bi2Ti2O7 case. (a) (b) Figure B 3 The electron localization of the displaced B ZN at 10.34 for the (a) (111) plane with Bi in the center and the (b) 1 11 plane with the Bi, ZnA, Nb, ZnB, O, and O48f ions indicated. The ELF values range from 0 (light red) to 1 (dark purple) as shown by the scale Figure B -3 b shows the electron density for both the ZnA and ZnB sites are very similar, with only a slight increase in electron density around the ZnA cations. Though it 0.0 0.25 0.50 0.75 1.0
233 may be difficult to see, the ZnA cations shown in Figure B 3 b all displace towards the O 48f anions surrounding the ZnB cation. Due to the layout of ZnA sites, if all the Bi cations displaced away from these O anions then the -cristobal ite type displacive disorder observed in Bi2Ti2O7 could still be maintained. Instead, two of the Bi cations displace towards the same O 48 f anions surrounding the ZnB cation resulting in five of the six surrounding A cations displaced towards this BO6 octah edra. Therefore, the correlation observed for Bi2Ti2O7 was not observed in BZN and the ELF plot shows the Bi lone pairs are not orienting themselves to reduce the interactions with the BO6 octahedra.
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244 BIOGRAPHICAL SKETCH Beverly L. Brooks was born in Houston, Texas, to William L. Brooks and Karen E. Brooks. She graduated from Spring High School in May 2000 where she focused on math, science, and photography. She continued her education at the University of Houston. In May 2005, she received her Bachelors of Science in chemical engineering. During part of her undergraduate studies, she worked at the Freeport site for the chemical company BASF Corporation through an internship program. Beverly and Jose Hinojosa were married in July 2005 before joining the chemical engineering department at the University of Florida. In January 2006, she began working for Prof. Aravind Asthagiri, performing computational based research on bulk met al oxides. In May 2008, she began examining small molecule interactions with mixed metal oxide surfaces.