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First Principles Calculations of Intrinsic Defects and Extrinsic Impurities in Rutile Titanium Oxide

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1 FIRST PRINCIPLES CALCULATIONS OF INTRINSIC DEFECTS AND EXTRINSIC IMPURITIES IN RUTILE TITANIUM DIOXIDE By JUN HE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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2 Copyright 2006 by Jun He

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3 To my parents and my wife

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4 ACKNOWLEDGMENTS I would like firstly to express my appreciation and respect to my advisor, Dr. Susan B. Sinnott, who has been supporting, guiding and believi ng in me for over five years. I feel very fortunate to receive her guidan ce and appreciate her openness and confidence in my ideas during this research. I would also like to thank my current committee members (Dr. David Norton, Dr. Eric Wachsman, Dr. Simon Phillpot and Dr. Ha i-Ping Cheng) and former committee members (Dr. Darryl Butt and Dr. Jeffrey Krause) for their assistance and participation on my supervisory committee. Special thanks go to Dr. Micheal W. Finnis (Imperial College London, UK) and Dr. Elizabeth C. Dickey (Pennsylvania State Univ ersity) for their numerous guidance and helpful suggestions. I would also like to thank Dr. Sinnotts group and Dr. Phillpots group (former and current members are acknowledged) for supporting and pr oviding a pleasant working environment. My research has been benefited from the discus sions with Dr. Douglas Irving, Dr. Yanhong Hu, Rakesh Behera, Wendung Hsu and Dr. Jianguo Yu. I would also like to thank my friends, Lewei Bu, Hailong Meng and Qiyong Xu. Their friendship has made these 5 years in UF wonderful. Finally, I thank my family, without whom this thesis would not have been possible. I thank my parents; their encouragement and love have accompanied me through this journey. I thank them for their understanding and their belief in me. I am also blessed to have my wife with me through this process.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............12 CHAPTER 1 BACKGROUND....................................................................................................................14 Defect Chemistry in Rutile TiO2............................................................................................16 Oxygen Vacancies are the Only Pred ominant Defect in the System..............................19 Titanium Interstitials are the Only Predominant Defect in the System...........................20 Simultaneous Presence of Oxygen Vacancies and Titanium Inte rstitials in the System......................................................................................................................... .22 Presence of Dopants and Impurities in the System.........................................................24 Experimental and Theoretical Studies of Defect Formation..................................................26 Experimental Studies of Def ect Formation in Rutile TiO2.............................................26 Theoretical Studies of Defect Formation in Rutile TiO2.................................................30 Theoretical Studies of Electronic Structure............................................................................34 Summary........................................................................................................................ .........36 2 INTRODUCTION OF DENSITY FUNCTIONAL THEORY AND ITS APPLICATION IN DEFECT STUDY..................................................................................49 Overview of Density Functional Theory................................................................................50 Kohn-Sham Theory.........................................................................................................50 Exchange-Correlation Functional....................................................................................52 Pseudopotential Approximation......................................................................................55 Implementation and Benchmark Test..............................................................................57 Application of Density Functi onal Theory in Defect Study...................................................59 Supercell Approximation.................................................................................................59 Band Gap and Defect Levels...........................................................................................60 Charge State and Compensation......................................................................................62 3 DFT CALCULATIONS OF INTRINSIC DEFECT COMPLEX IN STOICHIOMETRIC TIO2......................................................................................................66 Introduction................................................................................................................... ..........66 Computational Details.......................................................................................................... ..68 Model Development.............................................................................................................. .70 Results and Discussion......................................................................................................... ..71

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6 Summary........................................................................................................................ .........75 4 CHARGE COMPENSATION IN TIO2 USING SUPERCELL APPROXIMATION...........81 Introduction................................................................................................................... ..........81 Computational Details.......................................................................................................... ..83 Results and Discussion......................................................................................................... ..85 Summary........................................................................................................................ .........88 5 PREDICTION OF HIGH-TEMPERATURE POINT DEFECTS AND IMPURITIES FORMATION IN TIO2 FROM COMBINED AB INITIO AND THERMODYNAMIC CALCULATIONS..................................................................................................................94 Introduction................................................................................................................... ..........94 Computational Methodology..................................................................................................96 Electronic Structure Calculations....................................................................................96 Defect Formation Energies of Intrinsic Defects..............................................................97 Thermodynamic Component...........................................................................................98 Charge Compensation.....................................................................................................99 Results and Discussion of Intrinsic Defects...........................................................................99 Electronic Structure of Defects in TiO2...........................................................................99 Structural Relaxation.....................................................................................................101 Defect Formation Enthalpies.........................................................................................103 Extrinsic Impurities in Nonstoichiometric TiO2...................................................................109 Background....................................................................................................................109 Computational Details...................................................................................................110 Results and Discussion of Alumin um Ambipolar Doping Effects................................113 Summary........................................................................................................................ .......115 6 ELECTRONIC STRUCTURE OF CHARGED INTRINSIC N-TYPE DEFECTS IN RUTILE TIO2.......................................................................................................................131 Introduction................................................................................................................... ........131 Computational Details..........................................................................................................132 Results and Discussion.........................................................................................................133 Analysis of the Density of States...................................................................................133 Charge Density Difference Analysis.............................................................................136 Summary........................................................................................................................ .......138 7 CONCLUSIONS..................................................................................................................145 LIST OF REFERENCES.............................................................................................................149 BIOGRAPHICAL SKETCH.......................................................................................................156

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7 LIST OF TABLES Table page 1-1 Selected bulk properties of rutile TiO2..............................................................................38 2-1 Calculated lattice constants, CPU times convergence data, and total energies of rutile TiO2..........................................................................................................................63 3-1 Comparison between the calculated structur al parameters and experimental results for rutile TiO2.....................................................................................................................76 3-2 Positions of the Ti interst itial site in the Frenkel defect models shown in Figure 2-2......76 3-3 Calculated Schottky DFEs for rutile TiO2.........................................................................76 3-4 Calculated Frenkel DFEs for rutile TiO2...........................................................................76 3-5 Comparison of DFT calculated Frenkel and Schottky DFEs to published experimental and theoretical values for rutile TiO2...........................................................77 5-1 Calculated lattice parameters a nd Ti-O bond lengths for rutile TiO2 compared to the theoretical values and experimental values......................................................................117 5-2 Structural relaxation around defects. The relative change s from original average distances from perfect bul k are listed in percent.............................................................117 5-3 Calculated defect formation enthalpies for their most stable charge states of defects under three typical conditions: standard condition (T=300 K, F= 1.5eV, pO2= 1 atm), reduced condition (T=1700 K, F= 2.5eV, pO2= 10-10 atm), and oxidized condition (T=1200 K, F= 0.5eV, pO2= 105 atm).............................................................................118 6-1 Calculated band gap and ba nd width for perfect rutile TiO2 structure and defective structure with a fully charged titanium interstitial and with a fully charged oxygen vacancy........................................................................................................................ ....139

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8 LIST OF FIGURES Figure page 1-1 Bulk structures of rutile and anatase TiO2.........................................................................39 1-2 Diagram of the TiO6 octahedral struct ure in rutile TiO2....................................................39 1-3 Phase diagram of the Ti-O system.....................................................................................40 1-4 Calculated defect concentration in undoped TiO2 (A=0) at different temperature ranges as a function of pO2 using reported equilibrium constants......................................40 1-5 The logarithm of weight change of rutile as function of logarithm of oxygen partial pressure....................................................................................................................... .......41 1-6 Thermogravimetric measurement of x in TiO2-x as function of oxygen partial pressure....................................................................................................................... .......41 1-7 Electrical conductivity measurement of TiO2-x as function of oxygen partial pressure....42 1-8 Electrical conductivity measurement of TiO2-x as function of oxygen partial pressure....42 1-9 Defect formation energies of (a) CoTi (b) Coint (c) CoTiVO, (d) VO defects as a function of Fermi level in the oxygen-rich limit................................................................43 1-10 Defect formation energies as a function of the Fermi level, un der the Ti-rich (left panel) and oxygen-rich (right pane l) growth conditions, respectively..............................43 1-11 Calculated total DOS for TiO2 per unit cell compared to experimental UPS and XAS spectra for TiO2 (110) surface............................................................................................44 1-12 Calculated valence density difference maps for (a) (110) and (b) (-110) lattice planes. (c) shows the experimental electron dens ity map in (-110) plane in rutile TiO2...............44 1-13 Calculate density of states for the 8 8 10 rutile model with 0, 1, 5, and 10% oxygen vacancies on a large energy scale showing the development of a tail of donor states below the conduction band minimum................................................................................45 1-14 Comparison of density of states (DOSs) between defective and perfect rutile TiO2.........45 1-15 Calculated total and partial density of states (DOSs) of anatase TiO2..............................46 1-16 Calculated density of states (DOSs) of rutile-structured RuxTi1-xO2 with different Ru concentrations compared with experimentally determined spectra...................................46 1-17 Calculated spin polarized density of states (DOSs) of the Co-doped anatase (left) and rutile (right) Ti1 xCoxO2.....................................................................................................47

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9 1-18 Calculated DOS of the (a) re laxed defective structure with Cd0 (b) relaxed defective structure with Cd, and (c) unrelaxed defective structure with Cd0...................................47 1-19 Total ( A ) and partial ( B ) density of states (DOS) for doped anatase TiO2 calculated by FLAPW....................................................................................................................... ..48 2-1 Flow-chart describi ng Kohn-Sham calculation.................................................................64 2-2 Illustration of difference between all-electron scheme (solid lines) and pseudopotential scheme (dashed lines) a nd their corresponding wave functions..............64 2-3 The influence of supercell size on the defect formation energy of a neutral oxygen vacancy as a function of supercell size, as calculated with DFT (both using single point energy, geometry optimization includi ng electronic relaxation and full atomic relaxation) and an empirical Buckingham potential..........................................................65 3-1 The Schottky defect models considered in this study........................................................78 3-2 The Frenkel defect models considered in this study..........................................................79 3-3 The densities of states of perfect and defective TiO2. The valance-band maximum is set at 0 eV.................................................................................................................... .......80 3-4 Possible octahedral Ti inte rstitial sites in rutile TiO2........................................................80 4-1 Schematic illustration of the use of PBCs to compute defect formation energies for an isolated charged defect in a supercell approximation...................................................90 4-2 Calculated defect formation energies and defect transition levels in different supercells..................................................................................................................... .......91 4-3 Calculated defect formation energies for various charge states of the titanium interstitial in a 72-atom supercell in TiO2 as a function of the Fermi level (electron chemical potential) with and without a pplication of the Makov-Payne correction...........92 4-4 Calculated defect format ion energies for intrinsic de fects at 300 K and 1400 K with and without the Makov-Payne corr ection under reduced conditions (pO2=10-20)..............93 5-1 Electrical conductivity of rutile TiO2 single crystals as func tion of the oxygen partial pressure in the temperature range 1273-1773K...............................................................119 5-2 The influence of system size on the de fect formation energy of a single oxygen vacancy calculated by atomic-level simu lations using the empirical Buckingham potential...................................................................................................................... ......120

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10 5-3 Calculated band structure (a) and defect transition levels (defectq1/q2) (b) after the band gap lineup correction for TiO2. All the thermodynamic transition levels are calculated with respect to the valence band maximum regardless of their donor or acceptor character............................................................................................................1 21 5-4 Ball-and-stick models showing rela xation around a titanium interstitial (a) an oxygen vacancy (b) and a titanium vacancy (c) in a TiO2 supercell...............................122 5-5 Cross-sectional cont our maps of structure (a) and charge density difference around a titanium interstitial of differing charges [ Tii in (b) and Tii in (c) ]..............................123 5-6 Calculated defect formation enth alpies (DFEs) of point defects ( VO, Tii, Oi and VTi) as a function of Fermi level, oxygen partia l pressure, and temperature [(a)-(f)].............124 5-7 Calculated defect formation enthal pies (DFEs) of defect complexs [ (a) Frenkel defect; (b) anion-Frenkel defect; (c) Schottky defect] as a function of Fermi level at 1900 K when pO2=10-10....................................................................................................125 5-8 Calculated defect formation enth alpies (DFEs) of point defects ( VO, Tii, Oi and VTi) as a function of Fermi level, oxygen partial pressure, and temperature..........................126 5-9 Two-dimensional defect fo rmation scheme as a function of oxygen partial pressure and temperature calculated at three different Fermi levels [ F= 0.5 eV in (a) 1.5 eV in (b) and 2.5 eV in (c) ]......................................................................................................128 5-10 Contribution of vibrational energy and en tropy to the defect fo rmation energy of the indicated defects relative to the defect-free structur e as calculated with the Buckingham potential......................................................................................................129 5-11 Calculated defect formation enthalpies (DFEs) of aluminum impurities doped as interstitials ( Ali) and substitutionals on the Ti site ( AlTi) as a function of Fermi level and temperature [(a)-(d)] in the reduced state (log(pO2)= -20)........................................129 5-12 Calculated defect formation enthalpies (DFEs) of aluminum impurities doped as interstitials ( Ali) and substitutionals on the Ti site ( AlTi) as a function of Fermi level and temperature [(a)-(d)] in the reduced state (log(pO2)= -20)........................................130 6-1 A 2 1 1 supercell model for rutile TiO2 structure. X shows the center of an O6 octahedral structure..........................................................................................................1 40 6-2 Total and partial DOS of pristine rutile TiO2...................................................................140 6-3 Total and partial DOS comparison between pristine and defective TiO2 with a +4 charged Ti interstitial.......................................................................................................1 41 6-4 Total and partial DOS comparison between pristine and defective TiO2 with a +2 charged oxygen vacancy..................................................................................................142

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11 6-5 Valence density difference maps for: (a ) (110) and (b) (-110) lattice planes of pristine TiO2 structure......................................................................................................143 6-6 Valence density difference maps before and after atomic relaxation for: (a) Ti interstitial along apical bond direction; (b) Ti interstiti al along four equatorial bond direction; and (c) oxygen vacancy in (1-10) lattice plane...............................................144

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12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FIRST PRINCIPLES CALCULATIONS OF INTRINSIC DEFECTS AND EXTRINSIC IMPURITIES IN RUTILE TITANIUM DIOXIDE By Jun He December 2006 Chair: Susan B. Sinnott Major Department: Materials Science and Engineering Titanium dioxide has been intensively studied as a wide band-gap transition metal oxide due to its n -type semiconducting propert y. In this dissertation, fi rst the defect formation enthalpies of Frenkel and Sc hottky defects in rutile TiO2 are calculated. The results predict that Frenkel defects are more energetically favorable th an Schottky defects and bot h of them prefer to cluster together in TiO2. The possible diffusion routes for interstitial Ti atoms are also investigated. Secondly, the dependence of defect formation energies on supercell size is investigated. The results indicate that the el ectrostatic Makov-Payne correction improves the convergence of defect formation energies as a function of supe rcell size for charged titanium interstitials and vacancies. However this correction gives the wr ong sign for defect formation energy correction for charged oxygen vacancies. This is attributed to the shallow nature of the transition levels for oxygen vacancies in TiO2. Next, a new computational a pproach that integrates ab initio electronic-structure and thermodynamic calculations is given and applied to determine point defect st ability in rutile TiO2 over a range of temperatures, oxygen partial pressu res, and stoichiometries. The favored point defects are shown to be controlled by the relative ion size of the defects at low temperatures, and

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13 by charge effects at high temperatures. The orderi ng of the most stable po int defects is predicted and found to be almost the same as temperature increases and oxygen partia l pressure decreases: titanium vacancy oxygen vacancy titanium interstiti al. Also it is found that the formation energies of Schottky, Frenkel, and anti-Frenkel defect complexes do not change with the Fermi level. At high temperatures the formation of these complexes will restrict the further formation of single point defects, such as oxygen vacancies In the study of ambipolar doping behavior of aluminum in TiO2, the concept of pseudo-state is proposed to describe thermodynamic equilibrium procedure between impurities and host i ons. It is predicted that at high temperatures aluminum substitutional defects become the predominant dopant in TiO2 while n -type doping of aluminum interstitials is limited by high concen trations of titanium interstitials and oxygen vacancies. Finally, the origin of shallow level n -type conductivity in rutile TiO2 is discussed. The calculated densities of states fo r defective structures with fully charged titanium interstitials show a broader lower conduction band, which ma y enhance short-range cation-cation orbital overlap and thus lead to the fo rmation of shallow donor levels.

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14 CHAPTER 1 BACKGROUND Transition metal oxides remain one of the mo st difficult classes of solids on which to perform theoretical predictions us ing first-principles calculations. This is due to their complex crystal structures and the fact that they usually exhibit a wide range of properties, including acting as insulating, semiconducting, superconducti ng, ferroelectric, and magnetic materials. As a prototypical semiconducting trans ition metal oxide, titanium dioxide (TiO2) has been the focus of extensive experimental and theore tical studies for over f our decades due to its numerous technological applicatio ns [1, 2]. For example, TiO2 is widely used in heterogeneous catalysis, as gas sensor, as a phot ocatalyst, as an optical coat ing, as a protective coating, as biomaterial implants and as varist ors in electric devices. Many of these applications are tightly connected to the point defects and impurities introduced in the structure. These defects and impurities can be found in the bulk, on the surf ace and at the grain boundary. Consequently, there is great interest in tryi ng to understand defect structures and the mechanisms responsible for their creation. Since the discovery of photol ysis applications on TiO2 surfaces by Fujishima and Honda [3], it has been well established that surface defe ct states play an important role in surface chemistry phenomena (such as mass transport and waste decomposition). For one thing, these surface defects strongly affect the chemical and el ectronic properties of oxide surfaces. Recently there has been considerable interest in usi ng powerful instruments and techniques such as scanning tunneling microscopy (STM) and tr ansmission electron mi croscopy (TEM) to investigate surface structures and defect diffusion on the TiO2 surface. For instance, Diebold et al. reported a series STM studies combined with theoretical calcu lations to determine the image contrast in STM analysis of the oxygen-deficient rutile TiO2 (110) surface [4-6]. Based on their

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15 observations, the local electronic properties indu ced by intrinsic and extrinsic defects on this surface are discussed [7]. For example, they f ound in scanning tunneling spectroscopy (STS) that oxygen-deficient defects do give rise to defect states within the band gap. Also their observations indicated that those impurity atoms, whic h have a positive charge state and cause n -type doping, may cause a localized downward band bending. In addition, Schaub et al. studied the oxygenmediated diffusion of O vacancies on the TiO2 (110) surface using quant itative analysis of many consecutive STM images [8]. All these findings called for a reinterpretation of the defect chemistry of oxygen vacancies on the TiO2 (110) surface and opened the lead for further experimental investigations as well as theore tical calculations for su rface defect models. In contrast, our fundamental understanding of defect formation and diffusion mechanisms in the bulk and at grain boundari es is still unclear. One reason fo r this is that there are few experimental techniques that can be used to ex plore the nature of defect formation in bulk materials. Experimental techniques, such as thermogravimetry and electrical conductivity measurements, have been used to study de viations from stoichiometry in bulk TiO2 as a function of temperature and oxygen partial pressure us ing reasonable assumptions since the early 1960s [9-14]. Individual defects and impurities at grain boundaries in oxides can be analyzed by high resolution transmission electr on microscopy (HRTEM) [15], Z -contrast imaging, and electron energy loss spectroscopy (EELS) in scanning transmission elect ron microscope (STEM) [16]. For example, Bryan et al. studied Co2+ and Cr3+ doped nanocrystalline TiO2 by HRTEM and electronic absorption spectroscopy and found that the most important factor for activating ferromagnetism in nanocrystalline Co-doped TiO2 is the creation of grain boundary defects, which is identified as oxygen vacancies [17]. It has also been demonstrated that EELS is

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16 sensitive to the changes in the oxygen concentrati on and could be used to probe the effect of individual defects on the local electronic structure. Even thou gh all these techniques are proven to be most sensitive to the heavier elements in the crystal structure, the oxygen atom, which is the element that in many cases pl ays the largest role in determining the electronic properties in oxides, is the least well charac terized in these experiments. Although these experimental t echniques have been successfully used for the study of defect concentrations, further understanding of defect formation mechanisms is still limited due to the extreme sensitivity of the electronic and physical properties of metal oxides to minute concentrations of defects and impurities in the bulk material. It is therefore important to understand the de fect structure and formation mechanisms in bulk metal oxides, such as TiO2, especially when they influen ce the materials conductivity in different ways, depending on gas adsorption and temperature fluctuation. By controlling the nature and concentration of point defects and impurities, one can image a new means of tailoring the conductivity of semiconducting transition meta l oxides. In this di ssertation, the defect structures of intrinsic defect complexes, such as Schottkey and Frenkel def ect pairs in bulk rutile TiO2, are studied using the density functional theo ry (DFT) method. In addition, the influence of temperature and oxygen partial pressure on the st ability of intrinsic defects (including oxygen vacancies and interstitials, and titanium vacancies and intersti tials) and extrinsic impurities (including aluminum, niobium) are also investig ated using a new approach that integrates ab initio DFT and thermodynamic calculations. Finally, the electronic structures of these defects are calculated and compared with the electronic structure of the pristine structure. Defect Chemistry in Rutile TiO2 Before giving a literature overview of experi mental studies of def ect formation in bulk TiO2, an introduction to defect chemistr y in this material is necessary.

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17 Titanium dioxide has three major different crystal structures: rutile (tetragonal, mnm P Dh/ 42 14 4 a=b=4.594 c=2.959 ), anatase (tetragonal, amd I Dh/ 41 19 4 a=b=3.733 c=9.370 ), and broo kite (rhombohedral, Pbca Dh15 2, a=5.436 b=9.166 c=5.135 ) [2]. Among these three structures, only rutile a nd anatase play important roles in applications that make use of TiO2 and are, consequently, of great interest to researchers. Their unit cells are shown in Figure 1-1. The structure studied in this dissertation is rutile TiO2. Table 1-1 shows selected bulk properties of this material. In the rutile structure, the basic building block consists of a titanium atom surrounded by six oxygen atoms, TiO6, in a distorted octahedral configuration. In this structure, the bond length between the titanium and the oxygen atoms for two apical bonds along the linear (twofold) coordination is slightly longer than that of four bonds along the rectangular (fourfold) coordination (s ee Figure 1-2). Also, the neighboring octahedral structures share one corner along the <110> directions and are stacked with their long axis alternating by 90. In the <001> direction, there are edge-sharing octahedral TiO6 structures connecte d by their edges with the edges of neighboring oc tahedral structures. As shown in early studies by Wahlbeck and Gill es [18], titanium oxide can occur in a wide range of nonstoichiometric structures determ ined by temperature and oxygen pressure (see Figure 1-3). For example, order-disorder transf ormations take place over the entire composition range of the Ti-O solid solution depending on the temperature. As with most of the nonstoichiometric oxides, the thermochemical and electronic prope rties of rutile TiO2 are directly influenced by the type a nd concentration of point defects. The defect structure of rutile TiO2 has been studied since the early 1950s. For an oxygendeficient oxide like TiO2-x, the deviation from stoichiometry, x, has been studied by

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18 thermogravimetry and electronic conductivity measurements. Usually, based on the assumption that there is only one predominant defect residing in the oxide, th e deviation x is described as a function of oxygen partial pressure, x PO2 -1/n. In addition, it was determined that n is very sensitive to temperature [19, 20]. Thus we can sa y the deviation x can be described as a function of both temperature and oxygen partia l pressure in homogeneous TiO2-x. In order to describe the point defects in te rms of equations, it is important to have a notation system. The Krger-Vink notation is empl oyed in this dissertatio n [19, 20]. In this notation system the type of defects is indica ted by the combination of a major symbol, a subscript and a superscript. The major symbol de scribes the type of de fect and the subscript shows its occupation site. The supers cript is used to describe the ch arge state of this defect. The charge can be described as the actual charge s of defects, for example, +1, +2. However, considering the contribution of def ect charge to the whole perfect structure, it is more convenient to assign an effective charge to the defect. Generally the zero eff ective charge is symbolized as a cross (X) in the superscript, the positive effective charge is indicated by a dot () and the negative effective charge is shown by a prime (). For example, the normal titanium and oxygen atoms on the regular lattice sites have zero eff ective charge, and so are written as X TiTi and X OO. The intrinsic single point defects in TiO2 include vacancies (VO and VTi), self-interstitials (Oi and Tii), and antisites (OTi and TiO). The oxygen vacancies with two, one or no electrons localized around the vacancy site are written as X OV, OV and, OV respectively. The titanium vacancies with four, three, two, one or no holes localized around the vacancy site are written as X TiV, 'TiV, 'TiV, ' 'TiV, ' 'TiV, respectively. There are also numerous opportunities for dopa nts and impurities to be present in the system. For example, aluminum, gallium, iron, magnesium, niobium, zinc, and zirconium have

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19 been reported to appear in rutile TiO2 samples [21, 22]. These impurities may occupy the regular titanium site or an empty interstitial site. When an impurity ion M occupies a regular titanium site, it is written as MTi. On the other hand, if it occupies the em pty interstitial site, it is written as Mi. The effective charge state will be di scussed in the following chapter. It is commonly suggested that the intrinsic n-type defects, such as oxygen vacancies and titanium interstitials, are responsible for oxygen-deficient rutile TiO2. So, first let us only consider the defect chemistry of oxygen vacancies in this system under low pO2. Oxygen Vacancies are the Only Pred ominant Defect in the System Assume a neutral oxygen vacancy (VO X) is formed in the structure with two electrons trapped around the vacancy site. Then, depending on the temperature, these two trapped electrons may get excited one by one from the v acancy site. At the same time, the neutral oxygen vacancy acts as an electron donor and becomes singly charged (VO ) or doubly charged (VO ). The defect reactions are then de scribed by the following equations. ) ( 2 / 12g O V OX O X O (1-1) 'e V VO X O (1-2) 'e V VO O (1-3) The corresponding defect equilibrium equations are also written as a O X OK p V 2 1 2] [ (1-4) ] [ ] [X O b OV K n V (1-5) ] [ ] [ O c OV K n V (1-6) where n = [e ] defines the electron concentration.

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20 Because the oxygen vacancies and the comple mentary electrons are the predominant defects in the oxygen deficien t oxide, the charge neutra lity principle requires ] [ 2 ] [ O OV V n (1-7) The concentration of electrons and the neutral, singly and doubly charged oxygen vacancies are related through the above four equations [equations (1-4)-(1-7)]. Then, by combining these equations, expressions for each of the defects may be obtained. However we need some assumption to solve these equations The concentrations of oxygen vacancies are given by the following limiting assumptions: if [VO X] [VO ]+[VO ], [VO]total= KapO2 -1/2 if [VO ] [VO X]+[VO ], [VO]total= (KaKb)1/2pO2 -1/4 if [VO ] [VO X]+[VO ], [VO]total= (1/4KaKbKc)1/3pO2 -1/6 Thus the concentration of oxygen vacancies in an oxygen-deficient oxide may have an oxygen pressure dependence that ranges from pO2 -1/2 to pO2 -1/6. Titanium Interstitials are the Only Predominant Defect in the System Assume the titanium interstitials are formed in the structure with simultaneous formation of titanium vacancies by Frenkel defect reactio n. The titanium vacancies may also help the formation of oxygen vacancies through Schottky de fect reactions. The entire defect reaction scenario can be described as i Ti X TiTi V Ti' ' (1-8) 22 ' ' 2TiO V V O TiO Ti X O X Ti (1-9) 2 ) ( 2 12e g O V OO X O (1-10) The defect reaction equilibrium and charge neutrality can then be described as

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21 1] ][ ' ' [ K Ti Vi Ti (1-11) 2 2] ][ ' ' [ K V VO Ti (1-12) 3 2 1 22] [ K p n VO O (1-13) ] [ 4 ] [ 2 ] ' ' [ 4 i O TiTi V V n (1-14) To finally obtain the relationship of these defects to the oxygen partial pressure, detailed information of the equilibrium constants K1, K2, and K3 are necessary. However as Bak et al. [14] pointed out, the equilibrium constants of Schottky and Frenkel defect reactions, K1 and K2, are not yet available. The lack of data could be attributed to the fact that these two defect reactions assume equilibrium at substantially higher temperatures than that of other defect reactions reported in the literatur e. Therefore to overcome this, reasonable assumptions are made here. On the other hand, this case can be simplif ied from the analysis done by experimental researchers as shown in the follo wing condition. Assume that one neutral titanium interstitial is formed in the structure with four trapped electron s, and then the charged in terstitial forms step by step. ) ( 22g O Ti O TiX i X O X Ti (1-15) e Ti Tii X i (1-16) e Ti Tii i (1-17) e Ti Tii i (1-18) e Ti Tii i (1-19) Then we can obtain the defect reac tion equilibrium equations as

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22 A O X iK p Ti 2] [ (1-20) ] [ ] [X i B iTi K n Ti (1-21) ] [ ] [ i C iTi K n Ti (1-22) ] [ ] [ i D iTi K n Ti (1-23) ] [ ] [ i E iTi K n Ti (1-24) Also the charge neutra lity principle requires ] [ 4 ] [ 3 ] [ 2 ] [ i i i iTi Ti Ti Ti n (1-25) Therefore, the concentration of titanium interstitials is given by the following limiting assumptions (according to experimental data, only tit anium interstitials with +3 or +4 charge are considered to be viable defects): if ] [ ] [ ] [ ] [ i i i iTi Ti Ti Ti 4 1 2 4 1) 3 ( 33 0 3 / ] [ O ip K n Ti if ] [ ] [ ] [ ] [ i i i iTi Ti Ti Ti 5 1 2 5 1) 4 ( 25 0 4 / ] [ O ip K n Ti where K = KAKBKCKD, K = KAKBKCKDKE. Thus the concentration of titani um interstitials in an oxygen-deficient oxide may have an oxygen pressure dependence that is related to pO2 -1/(m+1), where m is the charge of the titanium interstitial. Simultaneous Presence of Oxygen Vacancies and Titanium Interstitials in the System Sometimes it is difficult to predict whether titanium interstitials or oxygen vacancies are the predominant point defect in an n -type system. In fact, both def ects may be important, at least in certain temperature and oxygen pa rtial pressure ranges. If this is the case, it is not easy to predict the relationship of defect concentration to oxygen partial pressure. Here, let us assume

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23 the oxygen vacancies with +2 charge and titani um interstitials with +3 charge are the predominant defects in the system. Let us further assume that the titanium interstitials with +3 charge remain in equilibrium with titanium interstitials with +4 ch arge. The resulting total defect reactions are therefore ) ( 2 1 22g O e V OO X O (1-26) ) ( 3 22g O e Ti O Tii X O X Ti (1-27) e Ti Tii Ti (1-28) It therefore follows that we get the follo wing defect reaction equilibrium equations: K p n VO O 2 1 22] [ (1-29) K p n TiO i 23] [ (1-30) ] [ ] [ Ti iTi K n Ti. (1-31) Also the charge neutra lity principle requires ] [ 4 ] [ 3 ] [ 2 i i OTi Ti V n (1-32) The combination of the above equations [equa tions (1-29) to (132)] results in the following expression for pO2: 2 1 2 2 3 ) 2 / 1 () 12 3 (2 K nK K n K K n pO (1-33) Equation (1-33) allows for the determina tion of the relationship between n and pO2. Consequently, the concentration of all possible ma jor defects may be determined as a function of pO2 using Equations (1-29) to (1-33). However knowledge of th ese equilibrium constants is required to solve these equations.

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24 In considering this problem, Bak et al. presented their theoretical work on the derivation of defect concentrations and diagrams for TiO2 based on experimental data of equilibrium constants reported in the literature [14]. The fully charged oxygen vacancies and titanium inte rstitials with +3 or +4 charge are all c onsidered as possible major n -type defects in the system. Additionally, the temperature considered in the defect di agram ranges from 1073 K to 1473 K. The resulting defect diagram for undoped TiO2 is shown in Figure 1-4. They found that at low pO2, the slope of -1/4 represents titanium interstiti als with +3 charge. And at high pO2, the slope of -1/6 is consistent with doubly charged oxygen vacancies. Also, the pO2 at which the concentration of oxygen vacancies surpasses the concen tration of titanium in terstitials increases with temperature. Finally, the concentration of titanium interstitials with +3 charge prevails over the titanium interstitials with +4 charge at lower pO2, while at higher pO2 the titanium interstitials with +4 charge become the minority defects (the slope is assumed to be -1/4). These derived defect diagrams therefore indicate that undoped TiO2 exhibits n -type conductivity over the entire range of pO2 and does not exhibits an n p transition in the temperatur e range of 1073 K 1473 K. This is an interesting result. However the accuracy of their predicted defect concentrations highly depends on the experimental equilibrium constants reported in the literatures, which actually vary over a significant range duri ng measurement. This is due to the practical problems during the measurement of equilibrium defect concentra tions. Since the determination of these intrinsic electronic equilibrium constants is not the main objective of this dissertation, the reader is referred to [14, 23] for additional details. Presence of Dopants and Impurities in the System In the presence of dopants and impurities in th e system, the defect reactions rely on a few additional complicating factors. These factors incl ude the valences of the impurities relative to

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25 the host compound, the lattice positions that the impurity at oms occupy, and the current predominant native defects in the structure. If a trivalent cation is dissol ved in the structure, it may o ccupy the intersti tial site or substitute on the normal titanium site, dependi ng on its ionic radius. Taking aluminum as an example, the aluminum ion may o ccupy the interstitial si te, and the resulting defect reaction is ) ( 2 3 6 22 3 2g O e Al O Ali (1-34) With the presence of the aluminum interstitial, the electron concentration will increase, and the concentration of oxygen vacancies or tita nium interstitials will decrease according to equations (1-5), (1-6) and (1-21) to (1-24). Since aluminum has a smaller ionic radius th an titanium, it may substitute on the titanium site. Assuming the fully charged oxygen vacancies are the predominant nature defects in the system, then the defect reaction and the charge neutrality will be X O O Ti TiOO V Al O Al 3 222 3 2 (1-35) ] [ 2 ] [ O TiV Al n (1-36) As a result, the electrical conductivity will be proportional to pO2 -1/4, not pO2 -1/6. Thus it is possible that the relation ship of defects to pO2 -1/4 may reflect the presence of single charged oxygen vacancies, or tri-valent Ti interstitials, or even tri-valent impurities that occupy normal titanium lattice sites. If the fully charged titanium interstitials are the predominant defects in the system, then the reaction and charge neutrality equation are 2 3 23 4 4 2 TiO Ti Al Ti O Ali Ti X Ti (1-37) ] [ 4 ] [ i TiTi Al n (1-38)

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26 On the other hand, self-compensation may also need to be considered in the case of aluminum. Aluminum can be considered to disso lve both substitutionally and interstitially due to its small ionic radius. Therefore, the reaction can be written as 2 3 23 3 3 2 TiO Al Al Ti O Ali Ti X Ti (1-39) In the case of pentavalent impurities, th ey are always treated as donors in TiO2. Taking niobium as an example, the reactions can be written as ) ( 2 5 2 22 5 2g O e Nb O NbTi (1-40) ) ( 2 5 10 22 5 2g O e Nb O Nbi (1-41) In this case the electron con centration increases and the concentration of oxygen vacancies or titanium interstitials decreases. Experimental and Theoretical Studies of Defect Formation Experimental Studies of Def ect Formation in Rutile TiO2 Based on the above reasonable assumptions, the nonstoichiometry of rutile TiO2-x as a function of temperature and oxygen partial pressu re has been studied us ing thermogravimetry measurement techniques since the ea rly 1960s. Kofstad first gave a detailed derivation of defect formation in oxygen-deficient rutile TiO2 [9]. Afterwards, different de fect models that assume different defects are dominant were proposed for reduced rutile TiO2. However, there has been no conclusive experimental eviden ce to indicate which of the above defect models is, in fact, the best description of experimental systems. In fact, when the oxygen pressure dependence of x is expressed as PO2 -1/n, it is found that plots of log x vs. 1/T do not yield a straight-line relationship. This suggests that the defect stru cture cannot be interpreted in te rms of a simple model in which either oxygen vacancies or tita nium interstitials predomin ate. As a matter of fact,

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27 nonstoichiometric rutile TiO2-x is thermodynamically stable ove r a wide range of oxygen partial pressures and temperatures. The first prestigious thermogravim etric study on nonstoichiometric TiO2 was reported in 1962 by Kofstad, showing that at low oxygen pres sure and high temperature (1350 K-1500 K), the weight change of TiO2 is proportional to pO2 -1/6 (see Figure 1-5) [9]. This shows the existence of the fully charged oxygen vacancies. At low temperature (1250 K-1300 K) the results show a transition in the oxygen pre ssure dependence from pO2 -1/6 to pO2 -1/2. Neutral oxygen vacancies were considered in the rutile, and their formation energy was measured to be 5.6 eV (129 kcal/mol). However, soon it was realized that neither the oxygen pre ssure dependence of pO2 -1/2 nor pO2 -1/6 is a stable observation in this temp erature range. Frland also worked on thermogravimetric studies of rutile [10] (see Figu re 1-6). He measured the weight change of the TiO2 as a function of temperature (1133 K-1323 K) and oxygen pressure (13-520 torr) and found the weight change to be proportional to pO2 -1/6, which was interpreted as being due to fully charged oxygen vacancies. He also measured a formation energy of 3.91 eV for oxygen vacancy reaction shown in equation (1-10). In contrast to these studies, Assayag et al. had a different explanat ion for the predominant point defects in rutile TiO2 (cited in [9]). They measured the equilibrium oxygen pressure and corresponding weight loss of a ru tile sample while it was heated in oxygen (the oxygen pressure was between 10-2 and 10-4 atm) over a temperature range of 1318 K-1531 K. They found the weight loss to be approximately proportional to pO2 -1/5, which was attributed to the existence of fully charged titanium interstitials. In 1967 Kofstad [24] proposed a new appro ach to study point de fects in rutile TiO2 and suggested that they simultaneously comprise both fully charged oxygen vacancies and titanium

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28 interstitials with +3 and +4 charges, where the former is dominant at high pO2, and the latter are predominant at low pO2 (see Figure 1-6 and Figure 1-7). In the 1960s electrical conduc tivity measurement was also shown to be an important experimental technique to explore defect structures in TiO2. For example, Blumenthal et al. presented the results of their electr ical conductivity measurements in the c direction over temperature range 1273 K 1773 K and oxyge n partial pressure range 1 10-15 atm [12] (see Figure 1-7). They found that the value of n in pO2 -1/n that were calculated from the slopes were not integers, but varied from 4.2 to 4.8 at 1773 K at low oxygen partial pressures, and were around 5.6 at high oxygen partial pressures at 1273 K. Thus, the nonsto ichiometric defect structure could not be described in terms of either a single pre dominant defect model or a single predominant charge state for the defects. They also suggested that the conduc tivity of rutile in air below 1223 K appeared to be impurity controlled due to the presence of aluminum rather than due to intrinsic defects. Recently, a conductivity experiment performed by Knauth et al. showed that titanium interstitials were the dominant defect in TiO2. And more interestingly, the conductivity in nanoand micro-crystalline TiO2 was reported to be independent of oxygen partial pressure [13]. The experimental study of Garcia-Belmonte et al. also indicated that at hi gh defect concentrations the point defects may not be randomly distributed in the material but were instead clustered or associated as a consequence of the interac tions between the defect s and incipient phase separation [25]. This was proved by the existenc e of the crystallographic shear planes (CSP) structure in reduced TiO2. The defect formation energy of a titanium interstitial was reported to be 9.6 eV, and the formation of a titanium vacan cy was reported to be 2.2 eV. They also found that TiO2 exhibited a p -type regime when the oxygen partial pressure is high (105 > pO2/Pa > 10)

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29 and the temperature is below 1200 K. In contrast when the oxygen partial pressure is low (10-9 > pO2/Pa >10-19), TiO2 exhibits n -type behavior, as expected. At atmospheric oxygen pressures, high-purity rutile is believed to ac t as an excellent insulator. Bak et al. also reported on charge transpor t in undoped polycrystalline TiO2 using electrical conductivity and thermopower measurements. The pO2 range was between 10 Pa and 70 kPa and the temperature range was 1173 K-1273 K [26]. They found that the slope of log vs. log pO2 at low partial pressures were -1/8.3, -1/6 and -1/4.7 at 1173, 1223 and 1273 K, respectively. At high partial pressures the sl opes were 1/7.6 and 1/11.9 at 1173 and 1223 K, respectively (see Figure 1-8). These values are obviously different from the ideal n values derived based on the above assumptions that fully charged oxygen vacancies (-1/6) or titanium interstitials with +3 and +4 ch arges (-1/4 and -1/5) are the pr edominant ionic defects. This departure indicates that there is more than one kind of major point defect taking part in conduction as charge carriers in the studied ranges of pO2 and temperatures in TiO2. For impurities studies, Slepetys and Va ughan measured the solubility of Al2O3 in rutile TiO2 at 1 atm of oxygen over the temperature range 1473-1700 K [27]. They found that the solubility increased from 0.62 wt% Al2O3 at 1473 K to 1.97 wt% Al2O3 at 1700 K. This means that high temperature helps aluminum impurities to remain dissolved in the structure. Also at 1700 K the density of the sample with 1.60 wt% Al2O3 did not significantly change compared with undoped TiO2. As a result, they suggested that alum inum dissolves both substitutionally and interstitially. However, it should be noticed th at their conclusions were only valid for the atmospheric environment. Considering the influence of the oxygen part ial pressure, Frland showed that the solubility of Al2O3 increased with decreasing oxygen partia l pressure in the temperature range

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30 11731473 K in the low partial pressure range (1 to 10-15 atm) [10]. His results also suggested that aluminum dissolve in the interstitial sites. In the contrast, Yahia m easured the electrical conductivity of Al2O3-doped rutile TiO2 as a function of oxygen partial pressure over the temperature range 950 1213 K [28]. The conductivity was found to change from n -type to p type. This can only be explained by assuming that a large fraction of aluminum was dissolved substitutionally in addition to existing as interstitial defects. In summary, two possible temperature regime s have been considered in experimental studies of defect formation mechanisms in ru tile structures. The intrinsic defects are more thermodynamically predominant at high temperat ures, and the concentration of defects can change over a range of oxygen pa rtial pressures. At low temper atures, the doping of triand pentavalent ions should lead to the n -type conductivity. However the impurity studies discussed above were all performed at high te mperatures, so behavior at lower temperatur es is not as well understood. Additionally, by using electrical co nductivity measurement, only charged single point defects were able to be considered. There is therefore no reported electrical conductivity study of neutral defects or even intrinsic defect comple xes such as Schottky and Frenkel defects. Thus, a detailed computational examination of a ll these various findings is necessary to fully understand the doping mechanisms responsible for all possible defects and impurities in nonstoichiometric rutile TiO2. Theoretical Studies of Defect Formation in Rutile TiO2 Although these experimental st udies provide important info rmation about the preferred defect structures in bulk TiO2 and on TiO2 surfaces, a full understanding of the various defect structures and formation mechanisms for TiO2, even in the bulk, is still elusive. For this reason theoretical calculations are employe d for defect structures in TiO2 and play an important role that is complementary to the experimental studies.

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31 Most calculations use one of three classes of theoretical approaches : empirical and semiempirical methods such as tight binding and Mott-Littleton me thods, molecular dynamics (MD) simulations, and more theoretically rigorous firs t principles approaches. The major methods in the third class include Hartree-Fo ck (HF) methods and density f unctional theory (DFT) methods. All of these approaches have been applied to the study of de fects in titanium dioxide. For example, Catlow et al. performed an extensive series of Mott-Littleton calculations on TiO2 and found that the Schottky defect was en ergetically more stable than th e Frenkel defect in rutile [2931]. They also concluded that vacancy disorder will predominate in TiO2. In another important study, Yu and Halley calculated th e electronic structure of point de fects in reduced rutile using a semi-empirical self-consistent method [32]. They worked with titanium intersti tials and oxygen vacancies and found donor levels in the range of 0.7-0.8 eV for isolated defects in each case. They also predicted the presence of defect clustering in nearly stoichiometric rutile with multiple defects. Although empirical studies are instructive and provide good in sight into point defect behavior in TiO2, these calculations cannot provide e nough predictive and accurate information about defect formation. In the worst case, they may lead to wrong conclusions. For example, the potential parameters for atomistic simulations are generally determined from perfect crystal properties such as cohesive en ergy, equilibrium lattice consta nts and bulk modulus. This may cause discrepancies due to the complex nature of metal oxides such as TiO2 and the different chemical environments in the perfect and defect ive lattices. Another problem is related to the charge state of the defects in th e system. In the empirical potentia l calculations, it is difficult to deal with variable charges on defects and impur ities. Thus the defects are usually treated as either neutral or fully charged, which obvi ously does not include all possible states.

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32 In last few years, DFT calculations using re asonably sized supercells have become more popular and practical for the study of defect s in transition metal oxides such as TiO2. For instance, the active surface site responsible for the dissociation of water molecules on rutile TiO2 (110) was explored by Schaub et al. by using DFT calculations combined with STM experiments [33]. Their results showed that the dissociation of water is ener getically possible only at oxygen vacancies sites. Dawson et al. also used the DFT method to study point defects and impurities in bulk rutile [34]. Their results show that isolat ed Schottky and Frenkel defects are equivalent energetically. The Schottky defect formation en ergy (DFE) was calculate d using two different approaches. In the first approach, the defect formation calculation involved calculating the formation energy of the Coulombically bound Schottky trio VTi+2VO in a 72 atoms supercell (2 unit cell). The Schottky form ation energy of this model was calculated to be 4.66 eV. In the second approach, the Schottky formation energy was expressed as the sum of the energies of one isolated titanium vacancy and twice the value of one is olated oxygen vacancy minus the cohesive energy per unit cell. The isolated defect calculations were carried out on a 12 atom cell. The Schottky formation energy was calculated to be 17.57 eV using this approach. The Frenkel formation energy was also calculated through the combination of one tit anium interstitial and titanium vacancy in an unrelaxed 72 atom cell. The value of the Frenkel DFE was found to be 17.72 eV. While this study consider ed variations in the structur es of the Schottky and Frenkel defect complexes, the system size (of a 12 at om supercell) was quite small and may have introduced self-interaction errors into the forma tion energy. An additional problem is that the calculation of DFE of the Frenkel defect was carried out using an unrelaxed supercell because of the high computational cost associated with allowing atomic relaxation. This is problematic, as it is well known that atomic relaxati on influences system energies.

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33 Recently, a few DFT studies have been repor ted on the topic of intrinsic defect and impurity formation in bulk TiO2. For example, Sullivan and Erwin reported their first-principles calculations of the formation ener gy and electrical activity of Co dopants and a variety of native defects in anatase TiO2 [35]. They found that under oxygenrich growth conditions the Co dopants would be formed primarily in neutral substitutional form, which conflicts with the experimentally observed behavior of Co-doped samples (see Figur e 1-9). Thus, they concluded that the growth conditions were most likely oxygen poor. When they considered oxygen-poor conditions they predicted roughly equal concentra tions of substitutional and interstitial Co. Na-Phattalung et al. also investigated intrinsic defect formation energies in anatase TiO2 at 0 K without considering the temperature influenc e and electrostatic inte raction correction [36]. They found that the tita nium interstitials ( Tii) has very low formation energy in both n -type and p -type samples. Thus they believed that titani um interstitials are th e strongest candidates responsible for the native n -type conductivity observed in TiO2. (see Figure 1-10) They also predicted that VO has a higher formation energy than Tii. However, after considering the lower kinetic barrier needed to create VO relative to the barrier to create Tii from perfect TiO2, they suggested that the post -growth formation of VO is also possible, especially after the sample has been heated for a prolonged time. Cho et al. reported DFT calculations of neutral ox ygen vacancies and titanium interstitials in rutile TiO2 [37]. They calculated the DFEs for these two defects and found them to be 4.44 eV and 7.09 eV for the oxygen vacancy and Ti interstit ial, respectively. This indicates that the formation of oxygen vacancy is energetically favo red. However neither th e charge state nor the temperature/oxygen partial pressure wa s considered in this study. Weng et al. also performed

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34 DFT calculations for Co-dope d rutile and anatase TiO2. They found that the Co prefers to substitute on the Ti site [38]. Theoretical Studies of Electronic Structure A detailed understandin g of Ti-O bonding is essential for explaining n -type semiconducting behavior in rutile TiO2, or even more generally, the electronic properties of transition metal oxides. Detailed information a bout the band structure and density of states (DOS) for perfect rutile has been reported by Glassford and Chelikowsky [39], and Mo and Ching [40]. The calculated total DOS is in good agreement with experiment [39]. (see Figure 111) Importantly, these calculations found a signifi cant degree of covalent bonding in the charge density contour map (see Figure 1-12). Since th e focus of this dissertation centers on the computational and theoretical study of defective st ructures, we will not discuss these studies in detail. However, it should be i ndicated that the electronic struct ure of pristine rutile is a good benchmark reference for the defective structure. A lot of theoretical works have been devoted to the study of defective TiO2 structures, beginning with an early ti ght-binding study by Halley et al. [41], a linear muffin-tin orbital (LMTO) study by Poumellec et al. [42], a DFT study by Glassford and Chelikowsky [43], and a full-potential linearized augmented plane wave (FLAPW) formalism by Asahi et al. [44]. For example, Halley et al. presented a tight-binding calculation to describe the electronic structure of a defective TiO2 with oxygen vacancies [41]. In Fi gure 1-13 they show the effects of different concentrations of oxygen vacancies at randomly selected sites on the DOS. They found donor states tailing into the band gap below the conduction band that increases with vacancy concentration. In addition, Poumellec et al. reported electronic stru cture, LMTO method calculations [42]. Their results show that there is a significant O2pTi3d mixing in the valence band and a weaker O2pTi3d mixing in the conduction band. This contradicts the general

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35 assumption that TiO2 is an ionic compound. They also me ntioned that the non-cubic environment may allow Ti4p-Ti3d hybridization, which woul d explain the observed preedge and edge features in the Ti x-ray absorption spectrum. Cho et al. reported DFT calculations on neutral oxyge n vacancies and titanium interstitials in rutile TiO2 [37]. They found that the oxygen vacancy does not give rise to a defect level within the band gap while titanium interstitial cr eate a localized defect level 0.2 eV below the conduction band minimum (see Fi gure 1-14). Na-Phattalung et al. also investigated the native point defects in anatase TiO2 using DFT calculations and found that the defect states for Tii and VO were predicted to be the Ti d states above the conduction band minimum (see Figure 1-15). Impurity doping is always important for the study of defects in semiconducting metal oxides. In order to achieve a high free-carrier concentration at low temperature, a high concentration of dopant impurity is obviously required under conditions of thermodynamic equilibrium. Generally the dopant concentr ation depends on temper ature, oxygen partial pressure, and the abundance of th e impurity as well as the host constituents in the growth environment. For cation impurity doping, frequently studied impurities include ruthenium, cobalt, and cadmium. For instance, Glassford and Che likowsky studied Ru doping in rutile TiO2 using DFT [43]. They found that the Ru-induced de fect states occur within the TiO2 band gap about 1 eV above the O 2p band, which is in good agreement w ith absorption and photoelectrochemical experiments (see Figure 1-16). A dditionally, these states were f ound to be localized on the Ru with t2g-like symmetry. In their DFT DOS calculations and absorption analysis for Co-doped rutile and anatase TiO2, Weng et al. suggested that the p d exchange interaction between the O 2 p and Co 3 d

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36 electrons should be ferromagnetic, which means that intrinsic ferromagnetism should occur in the Co-doped TiO2 systems (see Figure 1-17). This is prove n by their optical magnetic circular dichroism (MCD) spectra measurement. Errico et al. presented details about the electr onic structure of the Cd-doped TiO2 using FLAPW methods and the results show that th e presence of Cd impurity leads to the Cds levels at the bottom of the valence band, and impurity states at the t op of the valence band (see Figure 1-18). In addition, the Cd was found to introduc e fairly anisotropic atomic relaxation in its nearest oxygen neighbors [45]. While considering that cation metal impurities often give quite localized d states deep in the band gap of TiO2 and result is a recombination center of carriers, Asahi et al. suggested using anion dopants, instead of cation dopants, in orde r to ensure photoreduc tion activity [44]. They calculated the DOS of the substitutional doping of C, N, F, P, or S for O in the anatase TiO2 crystal using the FLAPW formalism. What they found is that the subst itutional doping of N was the most effective doping because its p states contribute to th e band-gap narrowing by mixing with O 2 p states. Furthermore, due to its large ioni c radius and much higher formation energy, S doping in the structure was believed to be difficult (see Figure 1-19). Lastly, Umebayashi et al. reported the band structure of S-doped anatase TiO2 using ab initio DFT calculations and found that the band gap ge ts narrower due to the S doping into the substitutional site. This obviously or iginates from the mixing of the S 3p states with the O 2p states in the valence band [46]. Summary This chapter gives a brief introduction to th e experimental and th eoretical studies of defects and impurities in TiO2. The defect formation behavior and electronic property changes are also discussed. It should be noted, however, that the influen ce of crystal structure on various

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37 point defects is still poorly understood due to the experimental and theoretical complicacy involved in such determinat ions. Additionally, many computational studies of TiO2 do not consider the influence of temper ature and oxygen partial pressure. Th erefore it is truly necessary to perform DFT calculations combined with thermodynamic calculations to obtain a more thorough understanding of the def ect chemistry in rutile TiO2.

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38 Table 1-1. Selected bulk properties of rutile TiO2. Values Atomic radius (nm) O 0.066 (covalent); Ti 0.146 (metallic) Ionic radius (nm) O(-2 ) 0.14; Ti(+4) 0.064 Melting point (Kelvin) 2143 Standard heat capacity at 298K (J/mol C) 55.06 Linear coefficient of thermal expansion at 0500K ( 10-6, C-1) 8.19 Anisotropy of linear co efficient of thermal expansion at 30-650K ( 10-6, C-1) Parallel to c -axis, =8.816 10-6+3.653 10-9 T +6.329 10-12 T2; Perpendicular to c -axis, =7.249 10-6 +2.198 10-9 T+1.198 10-12 T2; Dielectric constant Perpendicular to c -axis, 160; Along c -axis, 100; Band gap (eV) 3.0

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39 Figure 1-1. Bulk structures of rutile and anatase TiO2 [2]. Figure 1-2. Diagram of the TiO6 octahedral struct ure in rutile TiO2.

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40 Figure 1-3. Phase diagram of the Ti-O system [18]. Figure 1-4. Calculated defect concentration in undoped TiO2 (A=0) at different temperature ranges as a function of pO2 using reported equilibrium constants [14].

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41 Figure 1-5. The logarithm of we ight change of rutile as functi on of logarithm of oxygen partial pressure [9]. Figure 1-6. Thermogravimetric measurement of x in TiO2-x as function of oxygen partial pressure [23].

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42 Figure 1-7. Electrical conduc tivity measurement of TiO2-x as function of oxygen partial pressure [12]. Figure 1-8. Electrical conduc tivity measurement of TiO2-x as function of oxygen partial pressure [26].

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43 Figure 1-9. Defect form ation energies of (a) CoTi (b) Coint (c) CoTiVO, (d) VO defects as a function of Fermi level in the oxygen-rich limit [35]. Figure 1-10. Defect formation energies as a fu nction of the Fermi level, under the Ti-rich (left panel) and oxygen-rich (right panel) growth conditions, respectively. The slope of the line is an indication of the ch arge state of the defect. Th e band gap is set to be the experimental band gap. The vertical dotted line is the calculated band gap [36].

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44 Figure 1-11. Calculat ed total DOS for TiO2 per unit cell compared to experimental UPS and XAS spectra for TiO2 (110) surface [39]. Figure 1-12. Calculated va lence density difference maps for (a) (110) and (b) ( 10 1) lattice planes. (c) shows the experiment al electron density map in ( 10 1) plane in rutile TiO2 [39].

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45 Figure 1-13. Calculate density of states for the 8810 rutile model with 0, 1, 5, and 10% oxygen vacancies on a large energy scale show ing the development of a tail of donor states below the conduction band minimum [41]. Figure 1-14. Comparison of dens ity of states (DOSs) between defective and perfect rutile TiO2. (a) Total DOS for TiO2 with the oxygen vacancy compar ed with that of the perfect crystal. The Fermi levels EF are shown as vertical lines. (b) Partial DOS for the supercell containing the oxygen vacancy. Tinear indicates one of three Ti atoms neighboring the vacancy site and Tifar is the Ti atom furthest from the vacancy site. (c) Total DOS for TiO2 with Ti interstitial compared w ith that of the perfect crystal. (d) Partial DOS for the interstitial Ti atom Tiint and the Ti atom furthest from the interstitial site [37].

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46 Figure 1-15. Calculated tota l and partial density of stat es (DOSs) of anatase TiO2 [36]. Figure 1-16. Calculated density of states (DOSs) of rutile-structured RuxTi1-xO2 with different Ru concentrations compared with expe rimentally determined spectra [43].

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47 Figure 1-17. Calculated spin pol arized density of states (DOSs) of the Co-doped anatase (left) and rutile (right) Ti1 xCoxO2 [38]. Figure 1-18. Calculated DOS of the (a) relaxed defective structure with Cd0 (b) relaxed defective structure with Cd, and (c) unrelaxed defective structure with Cd0 [45].

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48 Figure 1-19. Total (A) and partial (B) density of states (DOS) for doped anatase TiO2 calculated by FLAPW [44].

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49 CHAPTER 2 INTRODUCTION OF DENSITY FUNCTIONAL THEORY AND ITS APPLICATION IN DEFECT STUDY The success of density functiona l theory (DFT) is clearly de monstrated by the numerous books, reviews and research articles that have b een published in the last two decades reviewing development of the theory and presenting result s obtained with DFT [47]. Many researchers in this field can be divided into three classes: those who develope d the fundamentals of the theory and/or new extensions and functi onals, for example, W. Kohn, L.J. Sham, J.P. Perdew, and D. Vanderbilt; those theoretical scientists who are concerned with numerical implementation, for example, R. Car, M. Parrinello, M.C. Payne, and J. Hafner; and those application scientists the vast majority who use the codes to study material s and processes that are important to different research areas. It is important that application scientists should have a sound knowledge of both the theory and its applications, and understand its limitation s and numerical implementation. Consequently, in this chapter, a brief overview of density function al theory is provided in the first section. Then in the second section, which is the main focus of this chapter, the application of DFT methods in materials science is reviewed, w ith a particular emphasis on the st udy of point defects. Particular attention is paid to the computationally technica l aspects that are unique to defect calculation. For example, questions about how to apply th e supercell approximation to describe real materials, how to deal with the band gap problem and interpret defect tr ansition levels, and how to consider charge states and ch arge compensation. It should be pointed out that there are some additional issues that influence the results, such as overbinding, self-interaction, dipole interaction, and strong correlati on effects. However, these probl ems are minimal in solid-state, total energy calculations of bulk TiO2, so these issues are not discussed in this dissertation.

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50 Overview of Density Functional Theory Kohn-Sham Theory Solid state physics explicitly describes solids as a combination of positively charged nuclei and negatively charged electrons. If there are onl y time-independent interactions in the system, and the nuclei are much heavier th an the electrons, the nuclei can be considered to be static relative to the electrons. This greatly simplifies th e system, which can be treated as an isolated system with only N interacting electrons moving in the (now ex ternal) potential of the nuclei while maintaining instantaneous equilibrium w ith them. This is the idea behind the BornOppenheimer nonrelativistic approximation. Usi ng this approximation, the classic Schrdinger equation can be simplified and described as [48] E H (2-1) where E is the electronic energy, is the wave function, E is the total ground-state energy, and H is the Hamiltonian operator. These latter two terms are described as ee neV V T H (2-2) ij N j i i N i i N i ee ner r V V T E 1 ) ( ) 2 1 (1 2 1 (2-3) On the right side of equation (2 -3) there are only three terms that need to be evaluated. The first term, T, is the kinetic energy of the electron gas; the second one, neV, is the potential energy of the electron -nucleus attraction; and the third term, eeV, is the potential energy due to electron-electron interactions. Th is many-body problem, while much eas ier than it was before the Born-Oppenheimer approximation, is still far too difficult to solve. Several methods have been developed to reduce equation (2-2) to an approximate but more readily solvable form. Among these approaches, the Thomas-Fermi-Dirac method, the Hartree-

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51 Fock (HF) method, and the density functional theory (DFT) method are the most prevalent. Only the DFT method will be discusse d in this dissertation. The DFT method was formally established in 1964 by the theoretical formulation proposed by Hohenberg and Kohn (Hohenberg-Kohn theorem), a nd has been widely used in practice since the publication of the Kohn-Sham equations in 1965 [49]. In the K ohn-Sham formula, the ground-state energy of a many-body system is a unique functional of the electron density, ] [0 E E In addition, the electron density is parame terized in terms of a set of one-electron orbitals representing a non-inte racting reference system as 2| ) ( | ) ( i ir r The Hamiltonian and total-energy functional can therefore be described as xc ee ne KSV V V T H 0 (2-4) ] [ ] [ ] [ ] [ xc ee ne KSV V V T E (2-5) In equation (2-5), the ground-state energy is in the Kohn-Sham form, EKS[], which is defined as the sum of the kinetic energy T[], the external potential energy Vne, the electronelectron potential energy Vne[], and the exchange-correlation (xc) potential energy Vxc[]. Although the exact form of T[] for a fully interacting set of electrons is unknown, the kinetic energy of a set of non-in teracting electrons, T, is known ex actly. Kohn and Sham proposed an indirect approach to use this well-defin ed T (which is Thomas-Fermi energy, TTF[]) and combine the interacting kinetic ener gy terms in with the xc term, Vxc[]. This key co ntribution to the DFT theory lead W. Kohn to be one of th e winners of the 1998 Nobel Prize in Chemistry. The next term in equation (2-5) is the external potential energy, Vne, which is simply the sum of nuclear potentials centered at the atomic positions. In some simple cases, it is just the Coulomb interaction between the nu cleus and the electrons. However, in most cases, in order to

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52 describe the strong oscillation of the valence electron wave-functi ons in the vicinity of the atomic core due to the orthogonalization to the inner electronic wave-functions, enormous wave functions are needed in the calculation, especia lly for large Z atoms. In such cases, it is not feasible to calculate the Coulom b potential in the plane-wave ba sis-set. Under such conditions, the inner electrons can be considered to be almost inert and not significantly involved in bonding, which allows for the use of the pseudop otential approximation to describe the core electrons. The third term in equation (2-5) is th e electron-electron potential energy, Vee[], which is simply calculated as Hartree energy, which is the classic electrostatic energy of a charge distribution interacting with itsel f via Coulombs law. And the last term in equation (2-5) is the exchange-correlation (xc) potential. In the last 30 years, over one hundred xc functional approximations have been proposed in the literature. The most fam ous ones are the local density approximation (LDA), generalized gradient approximation (GGA), local spin density approximation (LSDA), local density approxima tion with Hubbard U term (LDA+U), exact exchange formalism (EXX), and numerous hybrid functionals such as Becke three parameter hybrid functional with the LeeYangParr nonlocal correlation func tional (B3LYP). Finally, the standard procedure to solve equati on (2-5) is iterating unt il self-consistency is achieved. A flow chart of the scheme is depict ed in Figure 2-1. The iteration starts from a guessed electron density, 0(r). Obviously using a good, educated guess for 0(r) can speed-up convergence dramatically compared with using a random or poor guess for the initial density. Exchange-Correlation Functional The exchange-correlation (xc) potential, Exc[ ], is the sum of exchange energy and correlation energy as

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53 ] [ ] [ ] [ C X XCE E E (2-6) The exchange energy, Ex[ ], is defined as the energy di fference between true electronelectron potential energy, Vee[], and direct Hartree energy, U[]. The correlation energy, Ec[ ], is defined as the difference between the ground state Kohn-Sham energy, EKS[], and the sum of the Thomas-Fermi energy, Hartree energy, U[], and exchange energy, Ex[ ]. According to the second Hohenberg-Kohn theorem, there should be an exact form of the exchange-correlation functional, Exc[ ], to calculate the ground state energy of any system. However, the explicit form of this functional remains unknown. The problem is that there is no way to independently determine if a new functi onal is the one and only exact form. Instead, new functional are developed and a ssessed by how well they perfor m, which involves a detailed comparison of the predicted prope rties, such as lattice parame ters, bulk properties, and band structure, with the experimental data. As the first exchange-correlation functional appr oximation and in fact the simplest of all, the local-density approximation (LDA) was proposed by Kohn and Sham in 1965 as )) ( ( ] [3r r d EHEG LDA XC (2-7) where HEG((r)) is the xc energy per unit volume of the homogeneous electron gas (HEG) of density and can be tabulated using the Monte Carlo method by Ceperley and Alder [50]. The xc energies for charged Fermi and Bose systems are calculated by fitting the Greensfunction Monte Carle data in an exact stochas tic simulation of Schrdi nger equations. A number of different parameterizations have been proposed for this function over the years, and it has been shown that the LDA is suitable for system s with slowly-varying densities. However, this approximation has some serious shortcomings, es pecially when it is used to study transition metals and metal oxides where correlation effects are important.

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54 The next step to improve the exchange-corre lation functional is to include the gradient correction to xc functional. Th e electron density gradients will help the approach describe systems where the electron density is not slowly varying. This is the starting point of the generalized gradient approximation (GGA). In this case, the functi onal has a similar form as in equation (2-7), but now depends not only on the density but also on its gradient,. The evaluation of the GGA xc potential is fa irly straightforwardly computed as )) ( 1 ))( ( ( ] [3 f r r d EHEG GGA XC (2-8) where the ) ( f is a Taylor expansion of gradient Comparing these two approximations, the LDA functional derived from electron gas data does work surprisingly well for many systems. However, it substantia lly underestimates the exchange energy (by as much as 15%) and gr ossly overestimates the correlation energy, sometimes by 100% due to the large error in the electron density. As result, it typically produces good agreement with experimental structural an d vibrational data, but usually overestimates bonding energies and predicts shorter equilibrium bond lengths than are found in experiments. In contrast, the GGA functional finds the right asymptotic behavior and scaling for the usually nonlinear expansion in the Taylor expansion. It show s surprisingly good agreement with Hartree-Fock-based quantum chemical methods. Ho wever, there is much evidence to show that GGA is prone to overcorrect the LDA result in ionic crystals, and it overestimates cell parameters due to the cancellation of exchange energy error in LDA. Since real systems are usually spatially inhomogeneous, the GGA approxim ation is typically more accurate in studies of surfaces, small molecules, hydrogen-bonded crys tals, and crystals wi th internal surfaces. Unfortunately, both of these xc approximations give poor eigenvalues and small band gaps in many systems due to the discontinuity in th e derivative of the xc energy functional.

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55 Finally, despite the success of the LDA and GG A they are far from ideal, and finding an accurate and universally-applicable Exc remains great challenge in DFT. Ongoing efforts to discover the next generation of density functionals includes developing orbital-dependent functionals, such as the exact exchange functional (EXX), and constructing hybrid functionals which have a fraction of the exact exchange te rm mixed with the GGA exchange and correlation terms. Pseudopotential Approximation In the DFT plane-wave calculation, the elect ron wave functions can be expanded using a series of plane waves. However, an extremely large plane wave basis se t would be required to perform an all-electron calcula tion because the wave-functions of valence electrons oscillate strongly in the vicinity of the atomic core due to the orthogonalization to the inner electronic wave-functions (see Fig. 2-2). This calculati on is almost impossible since vast amounts of computational time would be required. Fortunately, it is well known that the inne r electrons are strongly bound and are not involved significantly in bonding. Thus, the binding properties are almost completely due to the valence electrons, especially in metals and semi conductors. This suggests that an atom can be described solely on its valence electrons, which feel an effective inte raction (that is the pseudopotential) including both the nuclear attraction and the repul sion of the inner electrons (see Fig. 2-2). Therefore, the core electrons and nuclear potential can be replaced by a weaker pseudopotential that interacts with a set of modified valence wave functions, or pseudowavefunctions, that are nodeless and maxi mally smooth within some core radius. The pseudowavefunctions can now be expanded in a much smaller basis set of plane waves, saving a substantial amount of computer time.

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56 The concept of pseudopotential was first proposed by Fermi in 1934 and Hellmann in 1935 [52, 53]. Since then, various pseudopotential approximations have been constructed and proposed. Initially, pseudopotentials were parameteri zed by fitting to experimental data such as band structures. These were known as empirical ps eudopotentials. In 1973, a crucial step toward more realistic pseudopotentials was made by Topp and Hopfield [54]. They suggested that the pseudopotential should be adjusted such that it de scribes the valence char ge density accurately. Based on this idea, the modern ab initio pseudopotentials were cons tructed by inverting the free atom Schrdinger equation for a given reference el ectronic configuration. More importantly, the pseudo wave functions were for ced to coincide with the true valence wave functions beyond a certain distance, and to have the same norm (cha rge) as the true vale nce wave functions. The potentials thus constructed are called norm-conserving pseudopoten tials. There are many, widely used norm-conserving pseudopotentials. One of the most popular parameterizations is the one proposed in 1990 by Troullier and Martins [55, 56]. However in some cases, norm-conservation sti ll results in deep pseudopotentials and therefore requires large cutoff en ergies. As a result, the pseudopot ential is less transferable without gaining enough smoothness. In 1990, Vanderbi lt proposed an ultrasoft pseudopotential [57] where the norm-conservati on constraint was abandoned, and a set of atom-centered augmentation charges was introduced. In this cas e, the pseudo wave functions could now be constructed within a very large distance, allo wing for a very small basis sets. Vanderbilts ultrasoft pseudopotentials are most advantageous for the first row of the periodic table and transition metals. Its accuracy has been found to be comparable to the best all-electron firstprinciples methods currently available [58].

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57 Implementation and Benchmark Test For studies described in this dissertation, we employed DFT calculations as implemented in the CASTEP code [59]. CASTEP uses a plan e-wave basis set for the expansion of the KohnSham wavefunctions, and pseudopotentials to de scribe the electron-i on interaction. A few pseudopotentials can be used in this study, such as norm-cons erving pseudopotential generated using the optimization scheme of Troullier-M artins (pspnc potential), norm-conserving pseudopotential generated using th e optimization scheme of Lin et al. (recpot potential), and ultrasoft pseudopotential (usp pot ential) [60]. After several be nchmark tests, the ultrasoft pseudopotential was chosen (see Table 21 and the corresponding discussion). Two separate exchange-correlation energy approx imations can be employed in the study: the LDA as parameterized by Perdew and Zunge r [61], and three GGA functionals (PW91 in form of Perdew-Wang functional [62]; PBE in form of Perdew-B urke-Ernzerhof functional [63]; and RPBE in form of Revised Perdew-Burke-Ern zerhof functional [64]). Based on our tests, the GGA of Perdew-Burke-Ernzerhof (PBE) is best suited towards our studies. The sampling of the Brillouin zone was performed with a regular Monkhorst-Pack k-point grid. The Monkhorst-Pack grid method has been de vised for obtaining accurate approximations to the electronic potential from a filled electronic band by calculating the electronic states at special sets of k points in the Brillouin zone [65]. The ground state atomic geometries were obtained by minimizing the Hellman-Feynman forces which is defined as the partial derivative of the Kohn-Sham energy with re spect to the position of the ions [66, 67] using a conjugate gradient algorithm [68]. The gr ound state charge density and ener gy were calculated using a preconditioned conjugate gradient minimization algorithm coupled with a Pulay-like mixing scheme [69, 70].

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58 Since the simulation of condensed phases is concerned with a larg e number of electrons and a near infinite extension of wavefunctions, it is necessary to use a relatively small atomistic model. The effect of edge effects on the re sults can be decreased by implementing periodic boundary conditions (PBC), in which a supercell is replicated throughout space. By creating an artificially periodic system the periodic part of the wavefunction is allowed to expand in a discrete set of PWs whose wave vectors are the reciprocal lattice vectors of the crystal structure. In the supercell all the atoms are relaxed from their initial positions us ing the Broyden-FletcherGoldfarb-Shanno (BFGS) Hessian update method until the energy and the residual forces are converged to the limits that are set prior to running the DFT calculati on [71]. The BFGS method uses a starting Hessian that is recursively upda ted during optimization of the atomic positions. The main advantage of this scheme is its ab ility to perform cell optimization, including optimization at fixed external stress. In this study the convergence cr iteria for energy is 0.001 eV/atom and for residual forces is 0.10 eV/. Table 2-1 presents the results several benchm ark DFT calculations. This system consisted of one unit cell of rutile TiO2. It allowed us to gain a first impression of the capabilities and limitations of DFT as applied to the TiO2 system. The calculations were performed using both the LDA, in the parameterization of the Perdew-Zunger functional, and GGA, in the parameterization of Perdew-Burke-Ernzerhof func tional. As the pseudop otential, we took the ultrasoft pseudopotential (usp) and one normconserving pseudopotential optimized in the scheme of Lin et al. (recpot) [72]. The results in Table 2-1 show three important points. First, GGA always overestimated the cell parameters, and the LDA was accurate in determining the cell paramete rs when using recpot pseudopotential. However, LDA

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59 underestimated the cell parameters when using the usp pseudopotential. Ge nerally the universal feature of the LDA and GGA should be that the LDA tends to underestimate lattice parameters, which are then corrected by the GGA to values closer to the experi mental results. However, we also found that the underestimation/overestima tion of lattice parameter also depends on the pseudopotentials used in the calc ulations. For example, when using the norm-conserving recpot pseudopotential, both the LDA and GGA overestim ated the lattice parameters, although the parameters in LDA calculation was just slightly overestimated. Secondly, the calculations usi ng ultrasoft pseudopotential (u sp) with the LDA and real space were not well converged comparing with the other cases. The use of ultrasoft pseudopotentials were acceptable only in reciprocal space. Finally, there are no great differences in computational time among these various xc approximations. Thus, the best choice of approxim ations for use in our study is the combination of GGA + ultrasoft pseudopoten tial + reciprocal space. Application of Density Function al Theory in Defect Study Supercell Approximation In the DFT method, the supercell approximati on is the most common approach for perfect structure calculations and is also being widely used in defect structure calculations. In this approximation the artificial superc ell is composed of several primitive compound unit cells that contain the defect(s) or impurity atom(s) that are surrounded by hos t atoms. The entire structure is periodically repeated within the PBC condition. The symbolization used to describe the size of the supercell is l m n, where l, m, n is the number of repeated unit cell in the x, y, and z directions, respectively. Within the supercell, the relaxation of several shells of host atoms around the defect or impurity should be included. If the size of supercel l is large enough, the defects are considered to be well isolated. This is the idea behind the ideal dilute solution model.

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60 In order to check the influence of the supe rcell size on the defect formation energies (DFEs), we calculated the formation energy of a neutral oxygen vacancy for different supercells. Here we used three methods: DFT single point energy calculation, where the defect and its surrounding ions are not allowed to relax; DFT geometry optim ization calculation, where the defect and its surrounding ions are allowed to relax; and empi rical potential calculations implemented in the General Utility Lattice Prog ram (GULP) [73, 74]. GULP can be used to perform a variety of types of simulation on mate rials with tuned interatomic potential models employed in boundary conditions. The results are shown in Figure 2-3. When the system size increased from 111 to 223, the DFEs of geometry optimization calculations decreased from 6.09 eV to 5.06 eV. It is clearly shown that the la rger supercell leads to a more realistic DFE for a single, isolated defect system. For the single point energy calc ulation and Buckingham potential calculation, the predicted formation energies do not change significantly when the supercell size is increased from 111 to 334. Thus, the geometry optimization method takes into account lattice strain much better than the other two methods. Band Gap and Defect Levels Both the LDA and GGA approaches are well kno wn to give an underestimated value for the band gap of semiconductors and insulators. In fact, even if the true xc potential was known, the difference between the conduction and valenc e bands in a Kohn-Sham calculation would still differ from the true band-gap. The tr ue band-gap may be defined as [47] ) ( ) 1 (!N N EKS N KS N gap (2-9) However the calculated Kohn-Sham band gap for the difference between the highest occupied level and the lo west unoccupied level of the N-electron system is

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61 xc E N E N N N N N Egap KS N KS N KS N KS N KS N KS N KS gap )] ( ) 1 ( [ )] ( ) 1 ( [ ) ( ) (1 1 1 1 (2-10) Thus the xc in equation (2-10) repres ents the shift in the Kohn-Sham potential due to an infinitesimal variation of the density In another words, this shift is rigid and is entirely due to a discontinuity in the derivative of the xc energy functional. In chapter 5 we will discuss how to implement a lineup to shift the conduction band rigidly upw ard in order to match the experimental band gap. In most cases, point defects and impurities introduce defect le vels in the band gap or near the band edges of the semiconductors. The experiment al detection of these levels often forms the basis for the identification of the defect or impur ity. On the other hand, th ese defect levels can also be characterized theoreti cally by different methods such as Kohn-Sham eigenvalues and defect transition levels (ionizat ion levels). Since the Kohn-Sham eigenvalues do not account for the excitation aspect and thus cannot be directly compared with the experimental literature, only the defect transition levels are calculate d and discussed in this dissertation. The thermodynamic defect transition level (q1/q2) is defined as the Fermi-level position where the charge states q1 and q2 have equal energy. This level can be observed in deep-level transient spectroscopy experiment or be derived from temperatur e-dependent Hall data [75, 76]. The formula to calculate the thermodynamic transiti on level is shown as belo w (Ti interstitial is given as example)VBM i total i totalE q Ti E q Ti E q q ) ( ) ( ) / (1 2 2 1 (2-11) where ) (2q Ti Ei total and ) (1q Ti Ei total are the total energy of a supercell with a Ti interstitial with charge q2 and q1, respectively and VBME is the energy of valance band maximum.

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62 Charge State and Compensation In defect calculations usually a certain charge (electron or hole) is assigned to the defect in the supercell. This charge (electron or hole) is then completely delocalized over the supercell. Therefore, a neutralizing jellium background is applied to the unit ce ll for calculations of charged systems. The interaction of the defect with the jellium bac kground should counteract the interaction of the defect with its spurious peri odic images. However, the energy of such a system still converges very slowly as a function of the linear dimensions of the supercell [77]. In order to overcome this shortcoming, Makov-Payne compen sation is applied in this study. Details of the Makov-Payne correction and its use here is given in chapter 4.

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63 Table 2-1. Calculated lattice c onstants, CPU times, convergence data and total energies of rutile TiO2. LDA 500 eV cutoff recpot, real-space LDA 500 eV cutoff recpot, reciprocal GGA-PBE 500 eV cutoff recpot, realspace GGA-PBE 500 eV cutoff recpot, reciprocal Experiment [2] a ( ) 4.605 4.600 4.632 4.626 4.594 c ( ) 2.991 2.993 3.002 2.998 2.956 Ti-O short ( ) 1.967 1.966 1.972 1.970 1.949 Ti-O long ( ) 1.979 1.978 1.996 1.993 1.980 CPU time (s) 130.78 134.06 213.37 208.19 Convergence All converged All converged Stress(0.32 GPa) not converged All converged Total Energy (eV) -1907.80 -1907.84 -1915.18 -1915.23 Table 2-1. Calculated lattice c onstants, CPU times, convergence data and total energies of rutile TiO2. (continued) LDA 400 eV cutoff usp, realspace LDA 400 eV cutoff usp, reciprocal GGA-PBE 400 eV cutoff usp, realspace GGA-PBE 400 eV cutoff usp, reciprocal Experiment [2] a ( ) 4.550 4.547 4.634 4.630 4.594 c ( ) 2.924 2.927 2.964 2.964 2.956 Ti-O short ( ) 1.934 1.933 1.958 1.957 1.949 Ti-O long ( ) 1.952 1.952 1.997 1.996 1.980 CPU time (s) 650.86 73.49 503.68 127.26 Convergence Stress(0.2 1GPa) not converged All converged Stress(0.24 GPa) not converged All converged Total Energy (eV) -4962.27 -4962.25 -4973.64 -4973.59

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64 Figure 2-1. Flow-chart descri bing Kohn-Sham calculation [47]. Figure 2-2. Illustration of difference between all-electron scheme (solid lines) and pseudopotential scheme (dashed lines) and their corresponding wave functions [51].

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65 Figure 2-3. The influence of supercell size on the defect formation energy of a neutral oxygen vacancy as a function of supercell size, as calculated with DFT (both using single point energy, geometry optimization includi ng electronic relaxation and full atomic relaxation) and an empirical Buckingham potential. Formation Energy (eV) Supercell size

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66 CHAPTER 3 DFT CALCULATIONS OF INTRINSIC DEFECT COMPLEX IN STOICHIOMETRIC TIO2 Introduction In stoichiometric metal oxides, when a charged point defect is formed in the crystal, there should be another complementary point defect with opposite charge formed near the first defect to conserve the charge neutrality of the system These two defects together are called a defect complex. Krger and Vink proposed six possibl e basis types of defect complexes in a stoichiometric compound [19]. Among them, th e Schottky defect complex (cation and anion vacancies, for example, VTi+2VO in TiO2) and the Frenkel defect complex (cation vacancies and interstitials, called cation Frenkel defects, such as VTi+Tii in TiO2; or anion vacancies and interstitials, called anion Frenkel defe cts or anti-Frenkel defects, such as VO+Oi in TiO2) are the only ones have been found in oxides. Thus thes e two are the most co mmonly studied defect complexes in stoichiometric metal oxides, especially at high temperatures [31]. It is therefore important to understand de fect formation and diffusion mechanisms for Schottky and Frenkel defect complex in titanium dioxide. However there is no consensus in the literature on the relative stabi lities of these defects in TiO2. The space charge segregation measurements, thermogravimetric measurement a nd tracer impurity diffu sion experiments found that Frenkel formation energies were lower th an Schottky formation energies [9, 78-80]. For example, Baumard and Tani reported th e electrical conductivity of rutile TiO2 doped with 0.04-3 at% Nb as a function of oxygen pressure in the temperature range 1273 K-1623 K [78]. They found the cation Fenkel defect to be the predominant defect in Nb-doped rutile. Ikeda et al. present a quantitative study of space charge solu te segregation at grain boundaries in doped TiO2 using scanning transmission electron microsc opy (STEM) to measure aliovalent solute accumulation [80]. They determined the defect fo rmation energies at gr ain boundaries using bulk

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67 defect chemistry models and the experimental va lues of the space charge potential. For example, the defect formation energy (DFE) of a Ti vacancy at 1350 C ranged from 1.5 to 3.5 eV while the average value was 2.4 eV. At 1200 C the average value was down to 2.1 eV. In comparison, at 1505 C the DFE of a Ti vacancy ranged from 1.0 to 1.5 eV. These findings indicate that there is no strong dependence of the formation energy of Ti vacancies on temperature. Additionally, using the cation vacancy formation energy of 2.4 eV and the cation inters titial energy of 2.6 eV, a Frenkel formation energy of 5.0 eV was obtained. Based on the oxygen vacancy formation energy of 2.1 eV, a Schottky formation energy of 6.6 eV was obtained. Therefore they found a strong preference for Frenkel defect complex. However the defect models of Schottky and Frenkel defect complex cannot be simply described as the linear combina tion of these single oxyge n and titanium defects. In actuality, the above formation energy values are much higher th an the values calculate d by us for clustered Frenkel defects (of about 2 eV), but is much closer to the value calculated by us for the distributed Frenkel defect structure (of nearly 4 eV ). In contrast, several theoretical calculations found Frenkel DFEs were much higher than Sc hottky DFEs [29-31, 81] and the results of electrical-conduc tivity measurements were inconclusive [12, 82]. A related issue of current interest is the preferred interstitial diffusion path of cations in oxides. Experimental measurements of the diffusion of Li and B in TiO2 showed strong anisotropy, especially through the open channels along the [001] direction [83-86]. Thus it seems necessary to use more quantitative techniques su ch as first principles DFT calculations to obtain a full unders tanding of the relative stabilities of these de fect complexes in metal oxides such as TiO2. ab initio DFT calculations are used to calculate the DFEs of Frenkel and Schottky defects in rutile TiO2 and to study the diffusion of in terstitial Ti. Various defect

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68 configurations are considered to quantitatively assess the effect of structure on the DFE of each defect. Computational Details The approach is density functional theory (DFT) using the generalized gradient approximation in the Perdew-Burke-Ernzerhof f unctional (GGA-PBE) combined with nonlocal, ultrasoft pseudopotentials and plane wave expans ions in the CASTEP program [51, 87]. An ultrasoft pseudopotential for Ti is ge nerated from the configuration [Ne]3s23p63d24s2, where the 3s2, 3p6, 3d2, 4s2 electrons are explicitly treated valence electrons. An ultrasoft pseudopotential for O is generated from the configuration [He]2s22p4, where the 2s2, 2p4 electrons are explicitly treated valence electrons. The GGA calculations use Brill ouin-zone sampling with 4 k-points and plane-wave cutoff energies of 340 eV. A 2 2 unit cell is used to model bulk TiO2 and all the atoms are relaxed from their initial positio ns using the Broyden-Fletcher-Goldfarb-Shanno Hessian update method until the energy is converged to 0.001 eV/atom and the residual forces are converged to 0.10 eV/. The same convergence criteria are used for the atomic relaxation of defect containing structures. To check the accuracy of the calculations, we performed test calcula tions of the perfect bulk unit cell and the results are summarized in Tabl e 3-1. They show that cutoff energies of 340 eV and 500 eV are in good agreement with experi mental values. The error between the lattice parameters calculated from the ab initio DFT calculations and experi ment is less than 1.5%. Additionally, the total en ergy with a cutoff energy of 340 eV is converged to less than 1.6 meV/atom with respect to the values obtained for a cutoff energy of 500 eV. For this reason the computationally less intensive cu toff value of 340 eV is used in all subsequent calculations. The formation energy of Frenkel, anti-F renkel and Schottky defects in bulk TiO2 are calculated as

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69 ) ( ) ( bulk E Ti V E Etot i Ti tot F f (3-1) ) ( ) ( bulk E O V E Etot i O tot AF f (3-2) ) 2 ( ) ( ) 2 (O Ti tot O Ti tot S fbulk E V V E E (3-3) where Etot(VTi+Tii) is the relaxed total energy of a de fected unit cell containing one cation Frenkel defect pair of one Ti vacancy and one Ti interstitial. Etot(VO+Oi) is the relaxed total energy of a defect unit cell containing one anti -Frenkel defect pair of one oxygen vacancy and one oxygen interstitial. Etot(VTi+2VO) is the relaxed total energy of a defect unit cell containing one Schottky defect pair of one Ti vacancy and two O vacancies. Etot(bulk) is the energy of the defect-free system, and Ti and O are the chemical potentials of one Ti, one O atom, respectively. The Ti and O chemical potentials ar e not independent but are related at equilibrium by 22TiO O Ti (3-4) where TiO2 is the calculated total energy per TiO2 unit. It is recognized that the 2 unit cell, which is the largest unit cell that could be considered by us in this study, is a constraint on the system and may intr oduce system-size errors to the results. These system-size errors shoul d be comparable for the perfect and defective systems, however, and so should have only a small effect on the DFEs. Since our DFT calculations predict much higher formation energi es (about 7 eV) for anti-Frenkel defects than cation Frenkel and Schottky defects, only the la tter two defect pairs are considered in the following calculations.

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70 Model Development Previous semi-empirical self-c onsistent calculations found that the distribution of defects in TiO2 may not be random but rather spatially cl ustered [32]. Therefore, several Frenkel and Schottky defect models are built fr om the relaxed, bulk rutile TiO2 unit cell. In particular, four Schottky defect models are considered and are sh own in Figure 3-1. Each model consists of two O vacancies and one Ti vacancy, but the distan ces among the various vacancies vary in the different configurations. The two O vacancies and Ti vacancy are close to each other in model 1, where the Ti vacancy is separated from one O vacancy by 1.95 and is separated from the other by 3.57 and model 2, where the Ti vacancy is separated from both O vacancies by 1.95 The vacancies are more spread out in model 3 (separated by distances of 5.80 ) and model 4 (separated by distances of 5.47 ). In the case of the Frenkel defect models, seve ral possible interstitial sites are considered. Previous research shows that O migrates via a site exchange mechanism, wh ile the Ti interstitial diffuses via the 32c octahedral site [2]. There is an anisotropy of at least 108 to 1 in favor of Li and B diffusing along the [001] direc tion relative to the [110] direc tion [84]. It was consequently suggested that the equilibrium positions for inte rstitial cations must pr eferentially occupy the (100) planes. However, this hypothesis has not been tested for the self-diffusion of Ti interstitials in TiO2. In this work, six Frenkel models, shown in Figure 3-2, are consid ered. The coordinates of the positions of the Ti interst itial atoms in all six models are listed in Table 3-2. It should be noted that as a result of the periodic boundary conditions used in the calculations, the Ti interstitial at (2a, 3a/2, 5c/2) is the same as the Ti interstitial at (0, 3a/2, 5c/2). The first three Frenkel defect models (models 1-3) reflect the movement of a Ti atom from a lattice site to a neighbo ring octahedral interstitial site in the [100] or [ 010] direction to form the defect. The last three Frenkel models (models 4-6) reflect the movement of a Ti interstitial atom

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71 along the [001] direction. Therefore, in these models, the Ti intersti tials are placed in octahedral sites long the [001] direction. Results and Discussion The DFEs calculated for these four Schottky ar e summarized in Table 3-3. They indicate that in each case the individual point defects preferentially cl uster together in bulk TiO2, in agreement with the results of Yu and Halley [ 32]. Specifically, the formation energy of the Schottky defect in models 1 and 2 is 1.5 eV lowe r than the formation energy in models 3 and 4. This indicates that the Schottky defect prefers to form a clustered structure rather than spreading out across the lattice. The formation energy of the Schottky defect in model 2 is at least 0.5 eV lower than the next lowest formation energy in m odel 1. This indicates that the Schottky defect prefers to form a clustered conf iguration where the vacancies are closest to one another in a row to a clustered triangular structur e of Ti and O vacancies with so me larger separation distances. The DFEs of the six Frenkel de fects models are summarized in Table 3-4. In this case the DFE of the Frenkel defects in models 1, 2, and 3 are approximately the same, but are significantly lower than the DFEs of models 4, 5, and 6. These results indicate that the Frenkel defect prefers to exist as a clustered pair rather than as a combination of isolated Ti vacancy and Ti interstitial. Table 3-5 compares these calculated DFEs to DFEs reported in the literature that were calculated with DFT and semi-empirical methods or obtained from experimental results. In our work, the lowest Frenkel DFE is about 2 eV, whic h is much lower than the lowest Schottky DFE of about 3 eV. This finding agrees with space charge measurement resu lts that find that the Frenkel defect is more prevalent in rutile TiO2 than the Schottky defect [79, 80], although the results of other calculations [31, 34] show the Schottky DFE is much lower than the Frenkel DFE.

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72 The discrepancies in the literature results are most likely due to the configurations considered in the calculations and experimental data measurement analysis. In particular, Schottky and Frenkel DFEs can be calculated usi ng Eqs. (3-1) and (3-2), as we have done, or they can be determined by combining the DFE of an isolated TiO2 system with Ti vacancies with the DFE of an isolated TiO2 system with O vacancies or Ti interstitials. For example, the Schottky DFE can be calculated as the sum of the energies from two parts. The first part comes from the energy needed to extract a Ti atom from a supercell to form an isolated Ti vacancy, while the second part is twice the formation energy of an isolated O vacancy. Using this approach, the formulas to calculate the DFE of Schottky and Frenkel defects are ) 2 ( )] ( ) ( [ 2 ) ( ) (O Ti tot O tot tot Ti tot S fbulk E V E bulk E V E E (3-5) ) ( ) ( ) ( ) ( bulk E al Interstiti E bulk E V E Etot Ti tot tot Ti tot F f (3-6) Ikeda et al. separated the Frenkel DFE for TiO2 into individual terms by measuring solute segregation at a free surface and obtained a Ti vacancy formation energy of 2.4 eV and a Ti interstitial formation energy of 2.6 eV [80]. Therefore, they ob tained a Frenkel DFE of 5.0 eV. This is much higher than the value calculated by us for clustered Frenkel defects (of about 2 eV), but is much closer to the value calculated by us for the distributed defect structure (of nearly 4 eV), and would be expected to be closer still to a defect structure distri buted across a larger unit cell than we were able to consider here. The DFE calculated by the semi-empirical Mo tt-Littleton calculati on also shows much higher DFEs [31]. This may due to differences in semi-empirical potential parameters used for characterizing TiO2. The semi-empirical parameters used in Ref. [31] are determined from the properties of perfect crystals. For example, pa rameters can be fit to the cohesive energy, equilibrium lattice constant, a nd bulk modulus of a perfect crys tal. Consequently, these semi-

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73 empirical parameters may not be appropriate for the study of imperfect systems, including defect energy calculations, because the ions around the defects are in di fferent chemical environments from those in the perfect lattice. Temperature differences between our DFT cal culations and experimental measurements should also be addressed. Ou r DFT calculations are carrie d out at 0 K and under these conditions, Frenkel defects are pr edicted to be more likely to occur than Schottky defects. However this finding could change at high temp eratures. Although the cha nges in entropy are of the same order of magnitude, and therefore can be ignored when comparing different single point defects, the contribution of the change in entropy (TS) to the total free energy of complex defect system should not be igno red at very high temperatures (on the order of 1000 K). In the case of a Schottky defect, the change in entropy mu st be calculated for three vacancies, while in the case of a Frenkel defect, there are only two contributions to the change in entropy, that of one vacancy and one interstitial. Therefore the Scho ttky defect may be pref erentially stabilized at high temperatures by entropic contributi ons than the Frenke l defect in TiO2, which would reverse the findings of this theoretical study. The calculated total densities of states (DOS) for the perfect structure and those containing Schottky (model 2) and Frenkel (model 3) defects are shown in Figure 3-3. In the DOS of the perfect TiO2, the O 2s band is located between -17 and -15 eV, while the O 2p band is located between -6 and 0 eV and only th e lower conduction band is present. The band gap is calculated to be 2.11 eV, which is smaller than the experimental band gap of about 3 eV. This underestimation of the band gap is well-known to occur in DFT calculations using the GGA approximation [63]. In all other respects, the band struct ure is consistent with previous band structure calculations of perfect TiO2.

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74 Figure 3-3 shows that the DOS for Schottky m odel 2 and Frenkel model 3 are similar to the perfect structure in that no defect state is apparent. However the band gap calculated for the Schottly defect system is 2.17 eV, while that of the Frenkel defect system is 2.20 eV, both of which are larger than the band ga p of the perfect structure. It should be pointed out that these Schottky and Frenkel defects are formed from co mbinations of neutral vacancy and interstitial defects, rather than charged def ects that would be expected to introduce new occupied states in the band gap. The last issue to be considered by us is the pr eferential diffusion path of Ti interstitials in TiO2. Figures 3-2 and 3-4 shows several possible Ti diffusion paths through the rutile structure. The DFEs of the first three Frenkel defect models reflect the energy that a Ti atom needs to move from a lattice site to a neighboring octahedral s ite in the [100] or [010] direction and form the Frenkel defect. The results show that the impe dance in the [100] and [010] directions are approximately the same. The last three Frenkel models (models 4 to 6) are considered to understand the barrier to Ti interstitial diffusion along the [001] direction. It is well known that the rutile structure has open channels along the c-direction (a, 3a/2, z). The Ti interstitial position (a, 3a/2, c/2) is the center of an octahedron of O atoms, and is being considered as the most stable site for the location of the Ti interstitial. However there are also other equilibrium positions along this direction that can be considered as possible sites for a Ti interstitial, for example z = 3c/4. Bond length arguments suggest that site (a, 3a/2, c/2) might be the most stable equilibrium position. In Frenkel models 4-6, the Ti interstitials ar e placed in octahedral site s long the [001] direction at z = 3c/2, c/2 and c/4. The calculated DFEs for models 5 and 6 clearly show that site (a, 3a/2, c/4) has higher defect formation energy than site (a, 3a/2, c/2). This conclusion is cons istent with the findings of

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75 Huntington and Sullivan [84]. The difference in DFEs for models 3 and 4 (1.57 eV) is lower than the formation energy of the Frenkel defect in mode l 1 (2.02 eV). This finding indicates that it is much easier for one Ti interstitia l to move through the open channe l in the [001] di rection than it is to move from a lattice site in the [100] or [010] directions. This result is summarized in Figure 3-4. Summary Ab initio DFT calculations are used to determine the formation enthalpies of Frenkel and Schottky defects for several defect struct ure configurations in bulk rutile TiO2. The results show that both Frenkel and Schottky de fects prefer to cluster together rather than being distributed throughout the lattice. The Frenkel de fect is predicted to be more likely to occur in rutile at low temperatures than the Schottky defect, with a di fference is formation enthalpy of about 1 eV. The DOS for the Schottky and Frenkel models are also calculated. We find that their band features are quite similar to the DOS of th e perfect, defect free structure w ith only a small increase in the band gap predicted to occur. La stly, strong anisotropy in inters titial cation diffusion in TiO2 is supported by these calculations.

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76 Table 3-1. Comparison between th e calculated structural parameters and experimental results for rutile TiO2. (GGA-PBE, generalized gradient approximation in the PerdewBurke Ernzerhof functional.) Approach a () c () Ti-O short () Ti-O long () GGA-PBE 340 eV 4.64 2.97 1.96 2.01 GGA-PBE 500 eV 4.63 2.96 1.95 1.99 Experiment [39] 4.59 2.95 1.95 1.98 Table 3-2. Positions of the Ti interstitial site in the Frenkel defect models shown in Figure 2-2. Models Coordinated (x, y, z) 1 2a, 3a/2, 5c/2 2 a/2, 0, 5c/2 3 a, 3a/2, 5c/2 4 a, 3a/2, 3c/2 5 a, 3a/2, c/2 6 a, 3a/2, c/4 Table 3-3. Calculated Sc hottky DFEs for rutile TiO2. Model Schottky DFE (eV) 1 3.51 2 3.01 3 4.98 4 5.47 Table 3-4. Calculated Frenkel DFEs for rutile TiO2. Model Frenkel DFE (eV) 1 2.02 2 2.01 3 1.98 4 3.55 5 3.74 6 3.84

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77 Table 3-5. Comparison of DFT calculated Frenkel and Schottky DFEs to published experimental and theoretical values for rutile TiO2. Defect (eV) Current DFT result Dawson DFT result [34] Catlow semiempirical result [31] Space charge measurement [80] Schottky (clustered) 3.01 4.66 Schottky (distributed) 5.47 17.57 5.25-8.22 6.6 Frenkel (pair) 1.98 15.72-17.52 Frenkel (distributed) 3.84 11.12-14.64 5.0

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78 Figure 3-1. The Schottky defect mo dels considered in this study.

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79 Figure 3-2. The Frenkel defect m odels considered in this study.

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80 Figure 3-3. The densities of stat es of perfect and defective TiO2. The valance-band maximum is set at 0 eV. Figure 3-4. Possible oc tahedral Ti interstitial sites in rutile TiO2.

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81 CHAPTER 4 CHARGE COMPENSATION IN TIO2 USING SUPERCELL APPROXIMATION Introduction Density-functional theory (DFT) is a proven approach for the calculation of the structural and electronic properties of solid state materials. In particular DFT calculations combined with periodic boundary conditions (PBCs), plane-wave expansions and pseudop otentials have been extensively applied in the study of systems lacking full three-di mensional periodicity such as molecules, defects in bulk materials, a nd surfaces. The use of these approaches and approximations remove the influe nce of troublesome edge effects and allows a relatively small number of atoms to mimic much larger syst ems. For example, several DFT studies have examined the electronic structure of charge d titanium interstitials and impurities in TiO2 [45, 88]. However most of these computational approaches and approximations are originally developed for the calculation of perfect crystal structures. Consequently the use of these approaches and approximations do lead to technical difficulties in the study of charged defects. For example, although the supercell approximation accurately describes local bonding fluctuations between atoms, it also introduces artificial long-range interactions between de fect and their periodic images in the neighboring supercells. The pres ence of this long-range interaction could dramatically change the evaluation of the defect formation energies. Giving TiO2 as an example, it is believed that its n-type conductivity is partially due to the multi-valence nature of the cation. Specifically, ch arged defects, such as titanium interstitials with +3 and +4 charges and oxygen vacancies with +2 charges, have been shown experimentally to play a dominating role in a variety of bulk and surface phenomena in TiO2 [2, 12, 13, 23]. However, there is still little f undamental understanding of the pref erred charge states of point defects in TiO2 or in their transitions as a function of temperature. Therefore it is still non-trivial

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82 to determine the relationship between charge st ates and formation energies for even the most typical defects in TiO2. Such relationships are needed to allow the ready pred iction of defect transition levels that are essent ial to understand the optical propert ies of wide band-gap transition metal oxides. Even more surprisingly, there ar e DFT calculations that report the presence of titanium interstitials with the sole charge stat e (+4) as the predomin ant defect in the TiO2 [36]. This is problematic, as it is well known experimental ly that titanium inters titials with +3 and +4 charges are both the pred ominant defect in TiO2 even if oxygen vacancies are excluded from the structure. In order to overcome this artificial long -range interaction problem, normally a uniform electron gas background (jellium background) is added to compensate for these artificial interactions (see figure 4-1). Th e interaction of the defect with the jellium background should exactly counteract the interaction of the defect with its spurious period ic images. However, as Makov and Payne pointed out, the energy of this s upercell will still converg e very slowly as a function of the linear dimensions of the supercell [77]. Thus, a fe w approaches were proposed to correct the divergence of the Coulomb energy fo r charged defects. For example, Leslie and Gillan suggested a macroscopic approximation to consider a periodic array of point charges with a neutralizing background immers ed in a structureless dielectri c [89]. In addition, Makov and Payne derived a detailed indirect correction for charged defects in cubic supercells [77]. More recently, Schultz developed the local-moment co untercharge (LMCC) method [90], which uses the linearity of the Poisson equation to correct th e divergence of the charged defect energies. In contrast, Nozaki and Itoh direc tly treat charge distribution to keep the charged defect cell embedded in a perfect non-polarizable crystal [ 91]. Despite these more recent efforts, the Makov and Payne approach is still the most widely known and used approach.

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83 In this chapter, the application of Makov and Payne approach in the study of charged point defects in rutile TiO2 will be discussed. The effects of the corrections will be evaluated and compared for supercells of va rying size. More importantly, the defect formation energies obtained from the DFT calculations will be co mbined with thermodynamic data to study the influence of temperature on the relative stab ilities of intrinsic point defects in TiO2. The results indicate that although th e Makov and Payne approach may give an overestimated correction for the defect formation energies due to the fact that these defects are delocaliz ed in the system, it is still an appropriate approach to study defect levels in tran sition metal oxides such as TiO2 whose cations can exist in multi-charge states. However the dipole interactions, which should also be countered as a possible source of error, are not considered in the charge compensation. Computational Details The DFT calculations are perfor med using standard plane wa ve expansions within the generalized gradient approximation parameteri zed with the Perdew-Burke-Emzerhof form (GGA-PBE) for the exchange-corre lation potential [63]. All the cal culations are performed using the CASTEP program [59]. An ultrasoft pse udopotential for Ti is generated from the configuration [Ne]3s23p63d24s2, and an ultrasoft pseudopotentia l for O is generated from the configuration [He]2s22p4 [57]. The Brillouin-zone sampling is carried out using a 222 k-mesh and a plane-wave cutoff energy of 400 eV is used. As the Makov-Payne correction was originally developed for ideal c ubic ionic crystals, it is necessary to use a combination of unit cells to construc t a repeating supercell of that is as close to cubic as possible. Thus, three differe nt supercell models are considered: 112, 212 and 223. The corresponding numbers of atoms are 12, 24, and 72, respectively. While it is true that thermal lattice expansions w ould be expected to influence th ese results, it is also widely

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84 accepted that the energy difference in the calculati on of defect formation en ergies are relatively independent of these expansions. As these energy differences are what is important here, the lattice parameters are fixed to th e experimental values. Then all the atoms are relaxed from their initial positions until the energy is converged to within 0.001 eV/atom and the residual forces are converged to 0.10 eV/. In the charged defect calculations, all th e charges are compensated by a neutralizing jellium background charge. The defect formati on energies are calculated using the following equation that takes into account temperature, oxygen partial pr essure, and electron chemical potential (Fermi level) ) ( ) ( ) ( ) , ( ) , (F i i total total d fq P T n perfect E L q i E P T q i E (4-1) In Eq. (1), ) , ( L q i Etotal d is the total energy of the supercell containing defect i of charge state q as a function of supercell dimension L, ) ( perfect Etotal is the total energy of the corresponding perfect supercell, and ni is the number of atoms bei ng removed from the supercell or being added from the atomic reservoir. For example, ni=nO=1 for an oxygen vacancy and ni=nTi=-1 for a titanium interstitial. Following the approach of Finnis et al. [92-94], i(T,P) is the chemical potential of the defect atom i describe d as a function of temperature and oxygen partial pressure. Finally, in Eq. (4-1), F is the Fermi energy. In order to calculate the Fermi energy, we also calculate the valenc e-band maximum (VBM) [95]. The Makov-Payne correction formula for the elec trostatic energy of an isolated charged defect within PBC and a uniform jellium backgr ound in a cubic lattice can be described as follows [77] ] [ 3 2 2 ) , ( ) (5 3 2 L O L qQ L q L q i E q i Etotal d (4-2)

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85 where ) ( q i E is the extrapolated energies from an infinite supercell and the first correction term describes the elect rostatic energy of the point ch arge array in a uniform jellium immersed into a screening medium which has a dielectric constant and depends on the Madelung constant of the lattice ( )he second correction item desc ribes the interaction of the defect charge with the neutralizing jellium background and Q is defined as the second radial moment of the defect charge density. The third correction item de scribes the quadrupolequadrupole interaction to a high order of |L|-5. As Eq. (4-2) shows [ 96], the second correction usually counts for only 3-5% of the first correction item. And as the definition of the parameter Q still contain some ambiguity, here we only consider the first correction item in our calculations. An approximation, inherent in the Makov-Payne approach, is the use of a continuum dielectric constant to screen the in teractions, which should break down if the defect charges are not well localized. Results and Discussion To investigate the effect of supercell size on the Makov-Payne correction, we first calculate defect formation energies as a functi on of supercell size and compare the values for fully charged titanium inte rstitials, fully charged oxygen vacancies, and titanium vacancies with 2 charges with/without the Makov-Payne correction. The temperature is set to 0 K, the oxygen partial pressure is set to 0 atm, and the Fermi le vel is set at the midpoint of the experimental band gap (3.0 eV). All the calculated results are shown in figures 4-2( a)-(c). In all three cases, the defect formation energies calculated with (withou t) the Makov-Payne correction are well fit with red (black) straight lines. The exception is the calculated defect formation energy for titanium vacancies of -2 charge in the smallest supercell of 12 atoms. It is also noted that for each case, the lines nearly meet at the infinite-supercell limit (1/L 0). It is clear that in the case of the fully

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86 charged titanium interstitials and the titanium vacancies with -2 charges, the Makov-Payne correction improves the convergence of the defect formation energies. For example, in the case of the fully charged titanium interstitials, the energy difference between the MP-corrected DFEs and E for the 72-, 24-, and 12-atom supercells are 1.60 eV, 3.05 eV, and 3.85 eV, respectively. In the uncorrected case, the energy difference between the DFEs and E are 4.87 eV, 8.18 eV, and 10.45 eV, respectively. Additi onally, the larger supercell provides a better estimation of the charged defect energy since a smaller electrostatic correction is needed for the larger supercells. In contrast, the Makov-Payne approach gi ves the wrong sign of the correction on fully charged oxygen vacancies. In particular, as shown in figure 4-2(c), the uncorrected formation energies increase as the supercell de creases and, since the Makov-Payne correction is always positive, this correction causes significant exaggeration of the formation energies. For example, in the case of the 72-atom supercell, the overestimation of the correction compared to the extrapolated value from the infinite supercell, E, is 1.13 eV. This behavior can be attributed to the fact that, unlike the tita nium vacancies and interstitials, the fully charged oxygen vacancies are shallow level donor defects. As illustrated in fi gure 4-2(d), the defect tr ansition levels of fully charged oxygen vacancies are always shallow relativ e to the titanium vaca ncies and interstitials for supercells of the same size. This is in agreement with x-ray photoelectron spectroscopy results that also find evidence of shallow level oxygen vacancies in TiO2 [97] and similar problems have been reported for other semic onducting materials such as diamond and InP [98, 99]. After considering the effect of supercell size, we now invest igate the effect of the MakovPayne correction on the thermodynami c stability of charge states and the corresponding defect transition levels. The defect forma tion energies of a charged titaniu m interstitial before and after

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87 the application of the Makov-Payne correction are calculated as a function of Fermi level under reduced conditions (PO2=10-20) (see figure 4-3). As the Fe rmi level decreases, the thermodynamically preferred charge state of the interst itial changes from neutral to +1, +2, +3, and +4 charge states when the correction is a pplied. On the other hand, when the Makov-Payne correction is omitted in the calculation, an unstabl e transition occurs between the charge states. In particular, the +4 charge is predicted to be preferred over almost the entire range of the electron chemical potential, while the neutral an d +1 charged states o ccur near the conduction band minimum and are obviously unstab le relative to the +4 charge state. This is because a tiny change in the defect concentration leads to a sh ift of the Fermi level, and this will eventually cause a charge shift in the titanium interstitials fr om neutral or +1 to the +4. It is likely that the stability of the charge state is substantially affected by the el ectrostatic energy correction for TiO2. However, these results indicate that it is necessary to apply the Makov-Payne (or an alternative) correction while studyi ng defect levels of transiti on metal oxides that are not too shallow. Finally the defect formation energies for va rious charged intrinsic point defects as a function of temperature and electron chemical potential are calculated (see figure 4-4). These intrinsic defects include the titanium interstiti als, oxygen interstitials, titanium vacancies, and oxygen vacancies. The temperatures considered are 300 K and 1400 K. We compared the defect formation energies computed for these defects with and without the Ma kov-Payne correction. Experimental studies suggest that at low temp eratures oxygen vacancies are the most stable defect in rutile TiO2, while at temperatures as high as 1400 K, titanium in terstitials are dominant [12, 23].With the Makov-Payne corr ection, our results clearly show the same trend. For example, figure 4-4(c) indicates that at T=300 K, oxygen vacancies are more stable than titanium

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88 interstitials over almost all the Fermi levels considered in the reduced state. However, when the temperature goes as high as 1400 K, the reverse is tr ue (see figure 4-4(d)). Over almost the entire range of the band gap, the titanium interstitial with different charge states is predicted to be the most stable intrinsic defect in TiO2 at high temperature. Howe ver, without the Makov-Payne correction, the transition from oxygen vacancy to tit anium interstitial is not predicted to occur. Instead, titanium interstitials with +4 charge are predicted to be the predominate point defect in TiO2 at both of the temperature ranges considered here. This conflicts with the experimental finding that oxygen vacancies play an important role in TiO2 properties. Thus it is clear that the experimentally observed defect transition from oxyge n vacancies to titanium interstitials is well reproduced in our charged defect calculations when applying the Makov-Payne correction. Summary TiO2 has been intensively studied as a wide ba nd-gap transition meta l oxide partially due to the multi-valence nature of its cation. In this chapter, DFT calculations within the supercell approximation and Makov-Payne co rrection are carried out to de termine the preferred charge state of charged point defects in rutile TiO2. The first part of this study is to investigate the dependence of the defect formation energies on the supercell size and the electrostatic MakovPayne correction. The results show that the Makov-Payne correction improves the convergence of the defect formation energies as a function of supercell size for positively charged titanium interstitials and negatively ch arged titanium vacancies. However, in the case of positively charged oxygen vacancies, applying the Makov-Pa yne correction gives the wrong sign for the defect formation energy correction that is attributed to the delocalized nature of the charge on this defect in TiO2. Finally, we combine the calculated defect formation energies with thermodynamic data to evaluate the influence of temperature on the rela tive stabilities of these defects. These results indicate that when the Mak ov-Payne correction is app lied, a stable charge

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89 transition is predicted to occur for titanium inters titials. In addition, as the temperature increases, the dominant point defect in TiO2 changes from oxygen vacancies to titanium interstitials. Since this correction is more appropriate for the st rongly localized charges, its application to delocalized, shallow level defects should be treated with caution.

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90 Figure 4-1. Schematic illustration of the use of PBCs to compute defect formation energies for an isolated charged defect in a supe rcell approximation. (a) The long-range interaction between the isolat ed charged defect and its periodic image in the nearby supercell is shown with re d arrows. (b) The jellium background, shown as a uniform electron background, is applied to comp ensate for the arti ficial long-range interactions between the defect and its periodic defect images.

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91 Figure 4-2. Calculated defect formation energi es and defect transiti on levels in different supercells. (a)-(c) defect formation energies of a fully charged titanium interstitial (a), titanium vacancy with -2 char ge (b), and oxygen vacancy with +2 charge (c), as a function of supercell size by using DF T calculation with a nd without the MakovPayne correction. The straight lines are the formation energi es without the MakovPayne correction, and the dashed lines ar e the formation energies with the MakovPayne correction. (d) Defect transition levels of titanium interstitials from +3 to +4 (+3/+4), and titanium vacancies from -2 to -3 (-2/-3), and oxygen vacancies from +1 to +2 (+1/+2) in different supercells. The values on the lines are the defect transition levels with respect to the VBM, while the values under the lines are the total number of atoms of the corresponding supercell.

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92 Figure 4-3. Calculated defect formation energi es for various charge states of the titanium interstitial in a 72-atom supercell in TiO2 as a function of the Fermi level (electron chemical potential) with and without a pplication of the Makov-Payne correction.

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93 Figure 4-4. Calculated defect formation energies for intrinsic defects at 300 K and 1400 K with and without the Makov-Payne corr ection under reduced conditions (pO2=10-20). (a)(b) Defect formation energies without th e Makov-Payne correcti on; (c)-(d) Defect formation energies with the Makov-Payne correction.

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94 CHAPTER 5 PREDICTION OF HIGH-TEMPERATURE POINT DEFECTS AND IMPURITIES FORMATION IN TIO2 FROM COMBINED AB INITIO AND THERMODYNAMIC CALCULATIONS Introduction TiO2 can be easily reduced, which results in n-type doping and high conductivity. Experimental techniques, such as thermogravim etry and electrical conductivity measurements, have long been used to determine th e deviation from stoichiometry in TiO2 as a function of temperature and oxygen partial pressure. The analys es of these experiments rely on assumptions about the charges of the defects and their depe ndence on environmental conditions [12, 13, 23, 26, 100]. For reduced TiO2, the results for temperature below 1373 K are consistent with the presence of either titanium in terstitials with +3 charges or fully charged oxygen vacancies at various oxygen partial pressures, as illustrated in Figure 5-1. They also indicate that at moderate pressures, as the temperature is increased above 1373 K there is a transition from fully charged oxygen vacancies to fully char ged titanium interstitials. While the assumptions used in the analysis of the experiments are physically reasonable, further experimental refinement of defect stabilities has been ha mpered by the extreme sensitivity of the electronic and physical properties of TiO2 to minute concentrations of defects and impurities. Theoretical calculations have the advantage of absolute control of composition of the system under consideration and are thus well positioned to complement experimental data. Density-functional theory (DFT) calculations have been appl ied to study the defect formation and stability in various electronic ceramics. Fo r example, the stability of point defects in undoped ZnO, which is a promising fluorescence mate rial, such as Zn interstitials [101] and oxygen vacancies [102, 103] have been considered with this approach. In addition, several calculations have examined the efficiency of p-type doping into ZnO of group-V elements such

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95 as N, P, and As [104, 105] or group-I elements such as Li and Na [106]. DFT has also been applied to study oxygen vacancy formation and clustering in CeO2 and ZrO2 [107, 108], which are used to store and transpor t oxygen in solid-oxide fuel cell applications. For the actinide oxides, such as UO2 and PuO2, a few groups have applied DFT to examine defect complexes, such as oxygen interstitial clusters [109] and corresponding electronic structures [110]. In the case of TiO2, several ab initio studies have been applied to examine defect structure and stability. These studies ha ve focused, on, for example, Schottky and Frenkel defect complexes [34, 111], extrinsic point defects [35, 43, 112], and electronic structure of intrinsic point defects [32, 65]. Recent DFT studies of intrinsic defect formation energies in TiO2 [36, 37] find that there are no defect levels inside the band gaps of anatase TiO2, and that Ti interstitials with +4 charges are the predominant point de fect under Ti-rich conditions. However, these studies have been restricted to zero Kelvin and do not include electrostatic interaction corrections. Here, quantitative predictions of the stabilities of charged intrinsic and extrinsic point defects in rutile TiO2 are made using a judi cious combination of el ectronic structure and thermodynamic calculations. In particular, DFT ca lculations are used to obtain electronic structure energy information abou t both the pristine and defective atomic-scale systems [32, 34, 35, 111]. This information is then used in thermodynamic calculations to determine defect formation energies (DFEs). Importantly, a quant itative link is made to temperature and the oxygen partial pressure, which are the key paramete rs for controlling the type and concentration of dominant defects in TiO2 and other electronic ceramics [100] The resulting se lf-consistent set of DFEs are crucial input parameters for equ ilibrium, space-charge segregation models [113]; models with accurate and self-consistent DFEs will better predict defect density distributions in

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96 metal oxides and thus enhance the design of el ectronic, optoelectronic, and ionic conductor devices. Computational Methodology Electronic Structure Calculations The DFT calculations are performed with plan e-wave expansions using the generalized gradient approximation in the Perdew-Burke-Em zerhof form (GGA-PBE) as implemented in the CASTEP code [51, 59, 63]. The ionic cores are represented by ultrasof t pseudopotentials [57]. An ultrasoft pseudopotential for Ti is generated from the configuration [Ne]3s23p63d24s2, where the 3s2, 3p6, 3d2, 4s2 electrons are explicitly treated valence electrons. An ultrasoft pseudopotential for O is generate d from the configuration [He]2s22p4, where the 2s2, 2p4 electrons are explicitly treated valence electrons. The Brillouin-zone sampling is made on a Monkhorst-Pack grid of spacing 0.5nm-1; a plane-wave cutoff ener gy of 400 eV is used. A 223 unit cell is used to build a supercell for the perfect and defective bulk TiO2 calculations, and all the atoms in the supercell are relaxed to their equilibrium positions such that the energy is converged to 0.001 eV/atom and the residual forces are converged to 0.1 eV/. To check the applicability and accuracy of this combination of pseudopotentials and supercell size, calculations of perfect bulk TiO2 are performed using different approximations for the exchange-correlation energy, and the calculat ed lattice parameters and Ti-O bond lengths are summarized in Table 5-1. The results indicate that the LDA approach underestimates the equilibrium lattice parameters by about 1% [114, 115]. The suggestion for solving this problem is either to increase the cutoff energy at the pric e of more computationally intensive calculations, or to use the GGA approach. Here, the latter op tion is chosen. The results indicate that the GGA with a cutoff energy 400 eV is in good agreement w ith reported experimental values, as indicated in Table 5-1.

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97 Defect Formation Energies of Intrinsic Defects The Gibbs free energy of defect formation in TiO2 as a function of defect species charge state q, temperature T and oxygen partial pressure P is given as [75, 102, 116] F a a total total fq P T n perfect E q E P T q G ) ( ) ( ) ( ) , ( (5-1) Here Etotal(,q) is the relaxed total energy of the supercell containing the defect of charge state q obtained from the DFT calculations, Etotal(perfect) is the relaxed total energy of the supercell of the corresponding perf ect crystal, which is also obtained from DFT calculations. The value na is the number of atoms being removed from th e supercell to an atomic reservoir in the process of creating the defect. For example, na = nO = 1 for an oxygen vacancy, and na = nTi = -1 for a titanium interstitial. a( T,P ) is the chemical potential of the defect atom a as a function of temperature and oxygen partial pressure, which is obtained using a combination of DFT and thermodynamic values, as described in detail in the next section. Lastly, q F is the electron chemical potential associated with the charged defect, which can be thought of as the energy needed to move the appropriate number of el ectrons from infinity to the Fermi energy, F, following the approach of Zhang and Northrup [116] The Fermi energy is treated as a variable, and can be expected to depend on the charge associ ated with the majority defect in the sample. Vibrational entropies of formati on are neglected, but configurati onal entropies are treated with the usual ideal solu tion model [75, 102]. To estimate the effects of the limited superce ll size, Figure 5-2 shows how the DFE of an oxygen vacancy, as calculated from atomic-level simulations using an empirical potential, depends on the size of the supercell [117]. These empirical potential calc ulations were carried out by R. Behera in the Comput ational Materials Science Focu s Group at the University of Florida. The results show a smooth decrease in the formation enthalpy, normalized to the DFE of

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98 the 222 system, as the system size increases for n n n supercells (solid curve in Figure 5-2). However, particularly for small n n m (nm) supercells the DFEs deviate from this smooth curve (dashed curve in Figure 5-2). Regardless of the shape, however, the DFE is independent of system size for all the systems above the 666 supercell. Most rele vant to assessing the electronic structure result s is that the change in the DFE from the 222 to the 223 and 333 supercells accounts for approximately 66% of th e total change in DFE with respect to the 222 system. Most importantly, the small size of the si ze effect strongly indicate s that the results of the DFT calculations using the 223 supercell can be trusted to give the correct relative formation energies, within 0.04%. Thermodynamic Component The value of the chemical potential a(T,P) depends on the system environment. Following the approach of Finnis and co-workers [92, 93] the oxygen chemical potential is described in terms of temperature and oxygen partial pressure as ) log( 2 1 ) ( )] ( [ 2 1 ) (0 0 0 0 0 02 2P P kT T P T G P TO TiO f Ti TiO O (5-2) where TiO2 0 and Ti 0 are the chemical potentials of TiO2 and Ti, respectively, and are calculated using DFT, while Gf TiO2(T0,P0) is obtained from thermodynamic data [118]. O 0(T) is the difference of oxygen chemical potential between any temperature of interest and the reference temperature obtained from the therm odynamic data. Combining equations (5-1) and (52) allows for the determination of DFEs as a f unction of temperature and oxygen partial pressure. These relations are the key to this integrated appr oach, in that they provide the critical bridge between the zero-temperature, zero-pressure DFT electronic structure calc ulation results and the high-temperature, finite-pressure co nditions of real-life applications.

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99 Charge Compensation The last term in equation (5-1) treats the e ffect of defect charge and electron chemical potential on the defect formation energy. In the charged defect calculations, a specific charge is assigned to the defect in the supercell. However, an artificial long-range interaction between the defect and its periodic images is introduced in to the system. In order to overcome this, a neutralizing homogeneous backgrou nd charge is assumed and impl emented in the CASTEP code [59]. But as Makov and Payne pointed out, the to tal energy of this supe rcell still converges slowly [77]. Several approaches have been proposed to correct for this Coulomb energy error [77, 96, 119]. Here the Makov-Payne approach is used, in which the background error is corrected to O(L-3), where L is the dimension of the superc ell. The reader is referred to chapter 4 for additional details [120]. Results and Discussion of Intrinsic Defects Electronic Structure of Defects in TiO2 The first step of the integrated approach is to determine the electronic structure of the defects in an atomistic system with DFT calcula tions. We first calculate d the band structure of the perfect rutile TiO2. The GGA approach is well known to give an underestimated value for the band gap of semiconductors and insulators. Th e calculated band stru cture along the symmetry lines of the Brillouin Zone for perfect rutile TiO2 is shown in Figure 53(a). The band gap, Egap, at the point is 2.11 eV, which is much smaller th an the experimental value of 3.00 eV [2]. Although this may be explained by the fact that the Kohn-Sham eigenvalues do not account for the excitation state, this failure is still intimately related to a derivative discontinuity that arises in part from the exchange-correlation energy func tional, as shown by Perdew and Levy [121] and by Sham and Schlter [122]. By calculating the tota l energies of separate neutral, -1 and +1

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100 charged perfect supercells, another defin ition for the band gap of perfect rutile TiO2 is described as ) 0 ( 2 ) 1 ( ) 1 ( perfect E perfect E perfect E Etotal total total gap (5-3) Here, E (perfect,q) is the total energy of one perfect supercell with charge q. The band gap calculated by this equation for TiO2 is 2.54 eV. It is possible that this underestimated band ga p could affect the defect levels and formation energies of intrinsic defects in TiO2. Thus, a lineup is implemented in which the conduction band is rigidly shifted upward to match the experiment al band gap (the so-calle d scissor operator) [123]. The defect transition level (defect q1/q2) introduced by defects in the band gap or near the band edges is defined as the Fermi leve l position where the charge states q1 and q2 have equal energy [75]. The Fermi energy is one of the crit ical parameters in determining which of the alternative defects or their char ge states has the lowest form ation energy and should therefore predominate. Accurate defect levels from DFT calculations can help determine the photoluminescence spectrum data of TiO2. Applying the scissor operator, all the calculated transition levels that include positively charged states are scaled by a fraction k, the ratio of the experimental to the calculated data of the band gap. In this case, k = 3.00/2.54 = 1.18. All the negatively charged states remain unchanged. This result, thus, does not affect the DFT calculations themselves, but is used to adjust th e defect transition levels obtained from the DFT calculations. Figure 5-3(b) shows the results of calculations of the defect transiti on levels for an oxygen vacancy (VO), titanium interstitial (Tii), titanium vacancy (VTi), and oxygen interstitial (Oi) in TiO2 from calculations using a supercell containing a 223 unit cell. The results indicate that

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101 both donors (Tii and VO) and acceptors (VTi) have shallow defect transition levels except for the (Tii /) and (VTi/ ). These findings are in good agr eement with x-ray photoelectron spectroscopy data which showed that titanium interstitials with +3 charge states are 0.7-0.9 eV below the conduction band minimum, and that th ere is a shallow donor level located 0.87 eV below the conduction band minimum due to oxygen vacancies [97, 124]. These shallow levels indicate that the thermodynamic transitions are easily activated, making it difficult to determine whether TiO2 is an n-type or p-type oxide through an analysis of defect levels alone. Figure 53(b) also illustrates the surprisingly deep levels predic ted for the oxygen interstitial, which indicates that this defect prefers to remain in the neutral charge state, as is the case for other ntype transition metal oxides, such as ZnO [101]. Structural Relaxation The effect of point defects on la ttice strain is examined by re laxing all the atoms within the defect-containing supercell in the DFT calculation This relaxation can be analyzed in terms of the relative size of the point defect and the associ ated electrostatic charge effects. Figures 5-4(a) to (c) illustrate structural re laxation around a titanium intersti tial [in Figure 5-4(a)], an oxygen vacancy [in Figure 5-4(b)] and a titanium vacancy [in Figure 5-4(c)] in the TiO2 supercell. The results also indicate that the case of oxygen in terstitial is similar to the case of titanium interstitial. A detailed, quantitat ive summary of the relaxation re sults are given in Table 5-2. Within the rutile structure, there are two sets of first nearest ne ighbors that are labeled for the O6 octahedral site that are labeled 1NNa and 1NNb. In particular, for the fi rst nearest neighbor ions, those oxygen ions that have longer distance from the octahedral center are called 1NNa, while those that have shorter distan ce are called 1NNb. For example, in the presence of a titanium interstitial, the interstitial site at the center of the octahedral O6 structure has four first nearest

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102 neighbors (1NN) O ions at 2.24 (1NNa) and two 1NN O ions at 1.68 (1NNb). It also has four second nearest neighbors (2NN) Ti ions at 2.75 (2NNa) and two 2NN Ti ions at 2.32 (2NNb). From Table 5-2 we can see that, in the case of charged vacancies (VO and VTi), all the 1NN ions to the vacancy show similar strong outward relaxation (for example, ~14% for VO and ~7% for VTi), which can be attributed to the absence of the electrostatic attraction between the vacancy and 1NN ions. Additionally, all the 2NN ions show similar weak inward relaxations (of ~3%). This is consistent with the theoretical ob servation that isotropic relaxation behavior is found during vacancy formation in other oxide such as Al2O3 [95]. The distance between the vacancies and the NN ions is predicted to change only slightly when the charge on the vacancies varies from neutral to fully charged states. This is consistent with complete delocalization of the charge associated with the vacancies. By contrast, for the case of titanium and oxyge n interstitials, both the 1NN and 2NN ions relax in an anisotropic manner. For example, four of the 2NN titanium ions near a fully charged oxygen interstitial relax aw ay from the interstitial by ~3%, while the other two 2NN titanium ions relax towards the interstitial by ~20%. The larg er relative size of the in terstitials, especially the oxygen interstitials, is responsible for this an isotropic behavior. These results thus illustrate that the size and symmetries of the sites occupied by a given point defect significantly affect the relaxation of the surrounding atoms, while the ch arges associated with the point defects have only a weak influence on relaxation. Although charges associated w ith the point defects have only a weak influence on relaxation at low temperatures, the DFT calculations indicate that the bou nd charges on the point defects have a significant effect on the defect formation energy. To illustrate this, Figures 5-5

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103 shows cross-sectional contour maps of charge density differences surrounding a Tii and Tii interstitials at 0 K. A key feat ure, evident on comparing these two figures, is the substantial amount of charge residing on the Tii in the t2g orbitals. As the t2g orbitals are lower in energy than the eg orbitals for a Tii in an octahedral symmetry, when the temperature increases, these t2g orbitals will be the first to be occupied by any extra electrons in the system. It is therefore predicted that the Tii will be more stable than the Tii under these conditions. Furthermore, in both cases the interactions between the O 2p orbitals of the 1NN oxygen ions with the Ti interstitial 3d orbitals are predicted to be anti-bonding. Defect Formation Enthalpies The predicted defect formation enthalpies can be used as an input in thermodynamic equilibrium study to obtain def ect concentrations and to help understand doping behavior. Here, the defect formation enthalpies of various charge d defects are calculated using our approach. It is first observed that the relative stabilities of the point defects change dramatically when the temperature and Fermi level are allowed to vary. Figure 5-6 (a)-(c) shows the calculated DFEs of VO, Tii, Oi, and VTi in otherwise undoped TiO2 as a function of Fermi level F at the reduced state (log(pO2)= -20) at three different te mperatures. The results clearly indicate how the most stable defect charge states depend on the position of the Fermi level. The actual Fermi energy of a given TiO2 sample will vary with its microstructure, the level of impurities, and the presence of dopants. It is therefore necessary to determin e the DFEs over a physically reasonable range of Fermi energies to increase the re levance of the calculated values. When the Fermi level increases from 0 to 3 eV at room temperature and low oxygen partial pressure, the most stable intrin sic point defects are shown to be Tii , VO , VTi, and VTi, respectively [see Figure 5-6(a)]. In contrast, when the Fermi level is approximately equal to g/2

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104 at room temperature, the stable charge states for these same point defects are Tii , VO , Oi X, and VTi. The results also show that, as the Fermi le vel decreases, the thermodynamically preferred charge state of the titanium interstitials change s from neutral to +1, +2, +3, and +4 when the Makov-Payne correction [77] is applied. Intriguingly, the p-type defect, VTi, is predicted to be the most stable under conditions of F = 1.5-3.0 eV, T=300 K, and pO2=10-20 atm. Our results also show that extrinsic dopants, such as niobium substitutions, may ultimately compensate for the intrinsic, p-type behavior predicted for TiO2 at room temperature and lead to overall, extrinsic ntype behavior at low temperatur es. This is because the niobium ions carry an excess of charge when they substitute for the titanium, and this compensates for the intrinsic p-type behavior. Our findings are consistent with the experimental results of n-type doped TiO2 [100]. In contrast, at high temperature, pristine TiO2 naturally exhibits n-type behavior and doping simply enhances this behavior. Figure 5-6(a) also indicates that at T=300 K oxygen vacancies are more stable than titanium interstitials over almost all the Fermi levels consider ed in the reduced state. However, when the temperature incr eases to 1400 K in Figures 5-6(b) a nd (c), the reverse is true. Over almost the entire range of Fermi levels considered here, titanium interstitials of various charge states are predicted to be the most stable intrinsic defect in TiO2 at high temperature. Several prior conductivity measurement studi es [23, 100] have attributed th e slope of -1/5 of logarithmic dependence of the electrical conduc tivity on the oxygen pa rtial pressure to th e presence of fully charged titanium interstitials at temperatures higher than 1700 K and pO2=10-2 (region C in Figure 5-1). However, our calcula tions predict that at temperat ures around 1700 K at the nearatmospheric state (pO2=10-2), the coexistence of Tii and VO defects, rather than the presence of a single Tii defect, leads to the slope of -1/5 in the electronic conductivity experiments. This

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105 finding indicates that the expe rimentally ascertained titani um interstitial-oxygen vacancy transition at high temperatures in the reduced stat e occurs gradually between titanium interstitials with +3 charges and fully charged oxygen vacancies, rather than abruptly as suggested by Figure 5-1. However experimental studies may also suggest that the actua l measured Fermi level is far below the theoretical value at Eg/2 =1.5 eV for undoped TiO2. If this is the case, the presence of a single Tii defect supported by our DFT re sult is still consistent with the experiment results. In order to compare the results of our DFT calculations with the electrical conductivity measurements obtained from experimentally attain able oxygen partial pressures, Figure 5-6 (d)(f) shows the calculated DFEs of VO, Tii, Oi, and VTi in undoped TiO2 as a function of Fermi level F at another reduced state (log(pO2)= -10) at three di fferent temperatures. At first glance, the results show very similar trends as those in the first reduced state (log(pO2)= -20) as mentioned in Figure 5-6 (a)-(c). The only difference is that th e transition from titanium interstitials to oxygen vacancies occurs at even hi gher temperatures when TiO2 is in this reduced state (log(pO2)= -10). This indicates that either higher temperature or lower oxygen partial pressure could promote this titanium interstitials-oxyge n vacancies transition. In addition, we compare the oxygen partial pres sure range where this titanium interstitialoxygen vacancy transition occurs at F = 2.5 eV under three temp erature ranges: 1273K, 1773K and 2000K. Figure 5-6(g) shows that when the temperature increases from 1273K to 2000K, the pO2 where the transition occurs increases from around 10-37 atm to around 10-16 atm. This is consistent with the trend shown in Figure 5-1. Another important issue in the defect calcula tions concerns defect complexes. Here we calculated the DFEs of cation-Fre nkel and Schottky defect pairs us ing Eqs. (3-5) and (3-6). The formula to calculate DFEs of anti-Frenkel defects are listed below

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106 ) ( ) ( ) ( ) ( bulk E al Interstiti E bulk E V E Etot O tot tot O tot AF f (5-4) These defects pairs are treated as a combina tion of both neutral and charged defects, but the total charge states of the defect pairs are ke pt neutral. For example, the cation-Frenkel defect pair models can be considered in three different combinations. In the first combination, both the titanium interstitial a nd titanium vacancy are treated as neutral defects; in the second combination, the titanium interstit ial is treated as a defect with +2 charge while titanium vacancy treated as a defect with -2 charge; in the third co mbination, both defects are considered to be in their most preferred charge states, that is +4 and -4. Similar considerations are applied to the anti-Frenkel and Schottky defect models. The re sults are shown in Figure 5-7, which clearly shows that the formation energies of these defect complexes do not change with the Fermi level. Secondly the combination model of charged def ects gives lower DFEs than those of neutral defects. For example, the Frenkel combination of ) ' ' ( i TiTi V gives the lowest DFEs relative to the other two combinations of Frenkel defect models. Finally, it is interesting that the formation energies of Schottky defect complexes are much higher than the formation energies of cation-Frenkel and anti-Frenkel defect complexes. This is consistent with the results discussed in chapter 3. As already shown in Figure 5-6, the oxygen vacancies and titanium interstitials are predicted to have negative form ation energies for the reduced state at 1900 K. However after checking the formation of defect complexes such as Frenkel and anti-Frenkel defects, it is clear that the individual defects, such as an oxygen vacancy or a titanium interstit ial, can not be stable for a long time without the material decompos ing. Considering the oxygen vacancy as an example, it is predicted to have a negative form ation energy and therefore form automatically in the TiO2 system. However the predictive positive form ation energies of anti-Frenkel defects,

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107 which can be defined as the combination of one oxygen vacancy and one oxygen interstitial, will restrict the further formation of those single oxyg en vacancies. Therefore at high temperature the prediction of these defect complexes (Schottky, Frenkel, and anti-Frenkel defects) are necessary to understand defect formation mechanisms in TiO2 properly. On the other hand, although TiO2 is frequently used as typical n-type semiconducting oxide, p-type conductivity still needs to be discussed here. First, the DFEs for those single point defects in TiO2 in the oxidized state (pO2=102) are calculated using Eqs. (5-1) and (5-2) and the results are shown in Figure 5-8(a)-(c). It was found that at high temperature in oxidized state, the TiO2 is still predicted to be n-type, which is obviously in disagreement with intuition and experimental observations. The reason for this lies in the formula, Eq. (5-2), which is used to calculate the oxygen chemical potential. Using an oxygen vacancy as an example, theoretically the predicted oxygen vacancy formation energy should lie between two bounds: the titanium-rich bound and the oxygen-rich bound within any temperat ure range. However, it is found that at high temperature, for example, 1900 K as shown in Figure 5-8(g) the predicted formation energy of the oxygen vacancy extends outside of the titanium-rich bou nd and extends into the titanium-rich state. Thus, by using this formula at high temperature, p-type conductivity cannot be well predicted. On the other hand, the oxygen chemical po tential can also be calculated using the following formula, ) log( 2 1 2 1 ) (0 02P P kT g P TO O (5-5) Here, 02Og is the total energy of an isolated O2 molecule which is calculated using the DFT method. The DFEs for those single point defects in TiO2 in the oxidized state (pO2=102) are

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108 calculated using this equations are shown in Figure 5-8(d)-(f). This figure indicates that p-type conductivity is well repr oduced under these conditions. The dominant p-type defects are titanium vacancies, VTi, and oxygen interstitials, Oi. The titanium vacancies with different charge states are especially more stable than the other intr insic defects over almost all the Fermi levels considered in the oxidized state regardless of the temperature. This is consistent with the oxidized condition assumption. The most stable intrinsic def ects as a function of oxygen par tial pressure and temperature are also determined, yielding two-dimensional de fect formation diagrams [see Figure 5-9(a)-(c)]. The results show that, regardless of the Fermi level, the ordering of the most stable defects is almost the same as the temperature increases an d the oxygen partial pressure decreases: titanium vacancy oxygen vacancy titanium interstitial (oxygen interstitial). However the predicted preferred charge states of these defect s do change with the Fermi level. For example, titanium interstitials with +1, +3, +4 charges are predicted to be th e most stable intrinsic defects when the Fermi level is 2.5, 1.5 and 0.5 eV, resp ectively in the reduced state. This result further demonstrates that the de fect behavior of TiO2 is extremely sensitive to minute concentrations of defects and impurities. The most stable charge states and correspondi ng defect formation enthalpies of different intrinsic defects in TiO2 are listed for three typical conditions in Table 5-3. It is uncertain to what extent the equilibrium concentration of defects or impurities is kinetically achieved between the defects and the surrounding oxygen atmosphere under various experimental conditions. Therefore, the ability to consider all possible charge states in these calculations is a powerful tool to explore the relative stabilities of poi nt defects in electronic ceramics.

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109 Up until now, the calculations have ignored vibrational contributions to the relative stabilities of the defects under c onsideration. To quantitatively a ssess the effect of vibrations, empirical potential calculations are carried out at various temperatur es [73, 74]. Figure 10 illustrates the contribution of the vibrational energy and entropy to point defect stability over a wide temperature range (0-2000 K) as determined from the GULP program [73]. The results indicate that single vaca ncy defects are affected the least, while single interstitial defects are affected more by high-temperature vibrational co ntributions to energy an d entropy. In addition, defect complexes, such as Frenkel and Schottky def ects, are affected most strongly, especially at high temperatures. However, all these contributio ns are less than 30 meV/atom over the entire temperature range considered, with temperature st abilizing all the point defects relative to the defect-free structure. This gives us confidence in the approach outlined above for predicting the relative stabilities of defects from DFT and thermodynami c calculations without explicitly including vibrational effects. Extrinsic Impurities in Nonstoichiometric TiO2 Background When discussing defect chemistry of oxygen-deficient TiO2, the presence of various transition metal dopants are reporte d to have a significant effect on native defect concentrations [79]. In general, it is believed that intrinsic defects are more thermodynamically predominant at high temperatures, while at low te mperatures the doping of tria nd pentavalent impurity ions are what lead to the n-type conductivity [21]. These dopants in clude aluminum, gallium, chromium, vanadium, and niobium, among others. Sayle et al. used empirical potential calculations within the Mott-Littleton approximation [117] to identify the stable valence charge states of these transition metal impurities. They found that Al3+, Ga3+, Cr3+, and V3+ dissolve preferentially at substitutional sites in TiO2, with charge compensation provided by oxygen vacancies. In

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110 addition, Nb5+ was predicted to dissolve preferentially at substitutional sites with charge compensation provided by Ti3+ ions. On the other hand, it was found in diffusion measurements performed by Yan and Rhodes [22] that the rapid diffusion of small cations su ch as Al, Ga, Co and Fe resulted from an interstitial diffusion mechanism. These contradictor y results leave us with an important question: what is the best way to evaluate the effect of impurity doping under different temperature and atmospheric conditions? The importance of this question is well-illus trated by the example of aluminum, which is one of the most frequently mentioned dopants of TiO2 in the literature. The ionic radius of the aluminum ion with a +3 charge in a six-fold-coord inated site is 0.67 which is slightly smaller than the ionic radius of a titanium ion with a +4 charge (0.75 ). To complicate matter, depending on its valence charge, Al dopants can promote either n-type or p-type conductivity under specific circumstances. For example, when Al atoms occupy an interstitial site they promote n-type behavior, and when they substitute for a titanium atom on the regular lattice they produce p-type behavior. The experimental results of Slepetys and Vaughan indicated that aluminum dissolves both substi tutionally and inters titially over the temperature range 1473-1700 K [27]. This ambipolar (pand n-type) doping behavior of alumin um could seriously degrade the required n-type properties of TiO2 in its utilization devices such as varistors, oxygen sensors, and current collecting electrodes. Thus it is truly necessary to develop a predictive approach to determining the influence of Al dopants in TiO2 under a variety of conditions. Computational Details Our DFT calculations are performed using the generalized gradient approximation in the Perdew-Burke-Ernzerhof functi onal (GGA-PBE) combined with ultrasoft pseudopotentials and plane wave expansions in th e CASTEP code [51]. An ultrasoft pseudopotential for O is

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111 generated from the configuration [He]2s22p4, where the 2s2, 2p4 electrons are explicitly treated valence electrons. The configura tion of ultrasoft pseudopotentials generated for Ti and Al are [Ne]3s23p63d24s2 and [Ne]3s23p1, respectively. Those valence elec trons are explicitly shown in the configuration. The GGA calculations use Brillouin-zone sampling with 4 k-points and planewave cutoff energies of 400 eV. A 2 unit cell is used to model both pristine and defective TiO2. The other computational deta ils are as same as those in the previous chapters. To compare both the n-type and p-type doping effect of alum inum, we consider two different structure models that consist of a 2 3 supercell with either one aluminum interstitial or one aluminum substitutional defect. As discus sed in Section 5.3, the chemical potentials of elements within solid-state metal oxides depend greatly on environmental conditions. In the case of intrinsic defects in TiO2, the chemical potentials of Ti an d O exist in a range that is bounded by Ti-rich states, or O-rich states. In all case s, the chemical potentials of Ti and O are not independent but follow a simp le rule at equilibrium by 22TiO O Ti (3-3) When calculating the defect formation energies of impurity interstitials, only the chemical potential of that impurity needs to be consid ered. However, when calculating the formation energies of impurities that are substituted on the cation site, both the chemical potentials of the impurities and the cations need to be included in the calculation. For example, the formation energies of these impurities are calculated as F Al total i total Alq perfect E q Al E q Gi ) ( ) ( ) ( (5-6) F Ti Al total Ti total Alq perfect E q Al E q GTi ) ( ) ( ) ( (5-7) where Etotal(Ali, q) is the relaxed total energy of a def ect-containing superc ell with one Al interstitial, and Etotal(AlTi, q) is the relaxed total energy of a defect-containing supercell with one

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112 Al impurity ion that is substituted on the Ti site with an effective charge q. Etotal(bulk) is the energy of the defect-free system, and Ti, Al, and O are the chemical potentials of one Ti, one Al, and one O atom, respectively. Finally, F is the electron chemical potential. When we consider the chemical potentials in equations (5-6) and (5-7), the thermodynamic equilibration process between impurities and hos t ions is complex. Here, we assume two thermodynamic pseudo-states that can be used to describe the doping proc edure. The chemical potentials of Al and Ti can then be calculat ed using equations of two pseudo-states: the original pseudo-state 0 0 02 22TiO O TiO Ti Al AlG g (5-8) and the final pseudo-state 0 0 0) 3 ( 2 13 2 3 2Ti Ti O Al O O Al AlG g (5-9) In equation (5-8), the Al exists in a pse udo-state corresponding to a pure aluminum atom, which can be thought of as an Al atom that stays in a rese rvoir near the TiO2 system. In this pseudo-state, the Ti exists as an ion as in bulk TiO2. These two values are actually the upper bound state of Al and the lower bound state of Ti. This pseudo-state is most applicable before the Al dopes the TiO2 system. Following the doping of bulk TiO2 by Al, the final pseudo-state shown in equation (5-9) is most applicable. In this pseudo-state, aluminum ex ists as an aluminum ion and titanium exists in form of a pure titanium metal atom. These two values represent the upper bound state of Al and the lower bound of Ti, respectively.

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113 Through equations (5-6) to (5-9) we can compare the effect of doping TiO2 with extrinsic impurities. Results and Discussion of Alum inum Ambipolar Doping Effects We calculated the defect formation enthalpies (DFEs) of Al interstitials and Al substitutional defects at two temperatures, 300 and 1400 K. To illustrate the doping procedure, we compare these extrinsic DFEs with the intrinsi c DFEs in two stages. In the first stage, the DFEs of the impurities are compared with simila rly behaving intrinsic defects. For example, since the Ali is n-type dopant, we compared its DFEs with the DFEs of the intrinsic n-type defects Tii and VO. This comparison is shown in Figur e 5-9 (a) and (b). In the case of p-type AlTi, we compared its DFEs with the DFEs of the intrinsic p-type defects Oi and VTi. This comparison is shown in Figure 5-9 (c) and (d ). All these results are calculate d as a function of Fermi level and temperature in the reduced state (log(pO2)= -20). From Figure 5-9 (a) we see that at room temp erature, when Al is initially doped into TiO2 as an Al atom, the DFEs of Ali (EAli O) are much lower than the comparable intrinsic defects. In other words, Ali is expected to readily form in the system even if it is not the predominant defect. However, as original pseudo-state transforms to the final pseudo-state, th e Al dopant is oxidized to the Al ion. In the final pseudo-state, its DFEs (EAli F) increases significantly and is almost the same as the DFEs of Tii and VO. As a result, Ali becomes a normal dopant with no particular propensity to form in TiO2 at room temperature. At high te mperature (1400 K) the situation changes significantly (see Figure 5-9 (b)). Altho ugh the DFEs of the intr insic defects decrease relative to thei r values at 300 K, the DFEs (EAli F) are even lower and are predicted to preferentially form in TiO2 when Al dopants are present. Figure 5-9 (c) shows that in the or iginal pseudo-state the DFEs for AlTi are much lower than the DFEs of the similar intrinsic defects, VTi and Oi. However in the final pseudo-state, these

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114 DFEs increase dramatically and are higher than the comparable intrinsic defects. Thus these calculations predict that it is almost impossible to have Al substitute on the Ti lattice site in TiO2 at room temperature in the redu ced state. When the temperature increases to 1400 K in the case of the final pseudo-state (see Figu re 5-9(d)), the substitution of Al on the Ti site is more energetically favorable than at room temperature. In summary, in the first stage, when compar ing aluminum interstiti als with comparable intrinsic defects, the calc ulations predict that they are readily formed at high temperatures and it is almost impossible to have aluminum substitute on the titanium lattice site at room temperature. This provides some insight into the ambipolar doping effects of alumin um. However, without considering the charge comp ensation, the doping of TiO2 with aluminum is incomplete. In the second stage, the comp ensation effect of aluminum im purities with respect to the predominant intrinsic defects needs to be considered in the TiO2 sample. For example, when the current predominant intrinsic defects are n-type, such as Tii and VO, then the doped aluminum impurities tend to be substituted on the titanium sites. Figure 5-10 (a) and (b) shows that the DFEs of n-type and p-type defects are at almost the same level, so the formation of aluminum interstitials and substitutional defe cts are not greatly affected. In contrast, at the high temperature of 1400 K (Figure 5-10 (c), (d)), the p-type intrinsic def ects show relatively high DFEs and the n-type defects show relatively low DFEs. Thus we can image that the aluminum impurities would prefer to substitute on th e titanium lattice sites under such circumstances. As a result, compared with Ali, AlTi will become the predominant dopant a nd show a propsensity to form in TiO2 at high temperatures. In other words, when charge compensation, the n-type doping of Ali in TiO2 is predicted to be significantl y limited by high concentrations of Tii and VO and low concentrations of VTi and Oi at 1400 K.

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115 Summary In this chapter, the defect formation mechanis m of intrinsic defects and extrinsic impurities in rutile TiO2 are explored. The defect stability over a range of temperatures, oxygen partial pressures, and stoichiometries are evaluated by DFT calculations using a combined electronic structure and thermodynamic approach For the archetype material TiO2, the dominant point defects are predicted, and are found to be consistent with inferences from experimental electrical conductivity measurements; moreover the analysis clarifies a number of ambiguities in the experimental interpretation. First the structural relaxation and electronic structure calculat ion of dominant point defects at various charge states are calculated. The resu lts show that both donors (titanium interstitials and oxygen vacancies) and acceptors (titanium vacanci es) have shallow defect transition levels. Then the resulting defect formation enthalpi es at various charge states are used in thermodynamic calculations to predict the influe nce of temperature and oxygen partial pressure on the relative stabilities of these defects. The fa vored point defects are shown to be controlled by the relative ion size of the defects at low temperatures, and by charge effects at high temperatures. Thirdly, the ordering of the mo st stable point defects is predicted and found to be almost the same as the temperature increases and the oxygen partial pressure decreases: titanium vacancy oxygen vacancy titanium interstitial. And the experimentally observed transition in dominant point defects from oxygen vacancies to titanium interstitials is well predicted in the calculation. Finally, the ambipolar (pand n-type) doping behavior of aluminum in TiO2 is thoroughly studied. The concept of pseudo-states is proposed to describe the thermodynamic equilibrium procedure between impurities and host ions duri ng the doping procedure. It is predicted that

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116 when compared to similar intrinsic defects, aluminum interstitials readily form at high temperatures while aluminum substitutional defect s are not predicted to form at all at room temperature. However, when charge compensation is taken into account, AlTi becomes the predominant dopant in TiO2 at high temperatures. The n-type doping of Ali in TiO2 should thus be limited by by high concentrations of Tii and VO and low concentrations of VTi and Oi at the same temperature. Of considerable conceptual importance, the above combined approach provides the heretofore missing direct connec tion between electronic-structur e and atomic-scale phenomena on the one hand and the more complex, experiment ally relevant conditions on the other. It is therefore generally applicable for the systematic evaluation of defect formation in electronic ceramics under the full range of temperature and atmospheric conditions, such as are important for catalysis (low-temperature, high-pressure) a nd gas sensing (high-temperature, low-pressure) applications.

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117 Table 5-1. Calculated lattice parameters and Ti-O bond lengths for rutile TiO2 compared to the theoretical values and experimental values. Approach a () c () Ti-O short() Ti-O long() GGA-PBE 400eV 4.629 2.963 1.957 1.996 LDA 400eV 4.547 2.927 1.933 1.952 LDA 1000eV [114] 4.603 2.976 1.953 1.989 Experiment [115] 4.594 2.959 1.948 1.980 Table 5-2. Structural relaxa tion around defects. The relative changes from original average distances from perfect bulk ar e listed in percent. The cont ent in parentheses shows the neighboring order. First NN Relaxation Distance (), %change Second NN Relaxation Distance (), %change Tii original 2.237 (O1NNa), 0%; 1.676 (O1NNb), 0% 2.749 (Ti2NNa), 0%; 2.315 (Ti2NNb), 0% Tii X -9.5%; +14.6% +5.7%; +11.4% Tii -9.4%; +13.7% +6.0%; +11.1% Tii -8.5%; +13.4% +7.4%; +14.2% Tii -10.0%; +11.7% +7.7%; +16.3% Tii -10.8%; +10.9% +7.7%; +17.4% Oi original 2.237 (O1NNa), 0%; 1.676 (O1NNb), 0% 2.749 (Ti2NNa), 0%; 2.315 (Ti2NNb), 0% Oi X +8.0%; +9.3% +7.7%; -11.1% Oi +8.8%; +9.2% +7.0%; -10.5% Oi +8.4%; +35.1% +3.4%; -19.9% VO original 1.996 (Ti1NNa), 0%; 1.957 (Ti1NNb), 0% 2.964 (O2NNa), 0%; 2.795 (O2NNb), 0% VO X +13.9%; +14.2% -0.8%; -2.4% VO +12.8%; +12.9% -0.8%; -3.2% VO +13.8%; +13.6% -1.0%; -2.4% VTi original 1.996 (O1NNa), 0%; 1.957 (O1NNb), 0% 3.594 (Ti2NNa), 0%; 2.955 (Ti2NNb), 0% VTi X +4.4%; +7.6% -2.2%; -4.2% VTi +5.3%; +9.6% -2.2%; -3.7% VTi +4.9%; +8.5% -2.3%; -3.8% VTi +5.1%; +9.4% -2.3%; -3.6% VTi +5.6%; +9.7% -2.2%; -3.4%

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118 Table 5-3. Calculated de fect formation enthalpies for their mo st stable charge states of defects under three typical conditions: standard condition (T=300 K, F= 1.5eV, pO2= 1 atm), reduced condition (T=1700 K, F= 2.5eV, pO2= 10-10 atm), and oxidized condition (T=1200 K, F= 0.5eV, pO2= 105 atm). Standard condition Reduced condition Oxidized condition charge DFE (eV) charge DFE (eV) charge DFE (eV) VTi -4 1.222 -4 1.256 -2 5.124 VO +2 3.677 0 2.389 +2 1.098 Tii +3 5.643 +1 3.385 +4 0.729 Oi 0 2.570 -2 4.221 0 3.149

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119 Figure 5-1. Electrical conductivity of rutile TiO2 single crystals as func tion of the oxygen partial pressure in the temperature range 1273-1773K (Krger-Vink notation is used here.) The slope of conductivity in area A is proportional to pO2 -1/4, which is attributed to a predominance of Ti interstitials with +3 charges or oxygen vacancies of charge +1. The slope of conductivity in area B is proportional to pO2 -1/6, which is attributed to a predominance of fully charged oxygen vacancies. The slope in area C is proportional to pO2 -1/5, which is attributed to titanium inters titials with +4 charges. After Figure 6 from Blumenthal et al. [12].

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120 Figure 5-2. The influence of system size on the defect formation energy of a single oxygen vacancy calculated by atomic-level simu lations using the empirical Buckingham potential. The system sizes are multiples of the primitive rutile unit cell along the x, y and z directions respectively The defect formation enthalpies (DFEs) are normalized with respect to the DFE of the 222 system (DFE/DFE 222).

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121 Figure 5-3. Calcul ated band structure (a) and defect transition levels (defectq1/q2) (b) after the band gap lineup correction for TiO2. All the thermodynamic transition levels are calculated with respect to the valence band maximum regardless of their donor or acceptor character.

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122 Figure 5-4. Ball-and-stick m odels showing relaxation aroun d a titanium interstitial (a), an oxygen vacancy (b) and a titanium vacancy (c) in a TiO2 supercell. The oxygen ions are shown in red and the titanium ions are shown in ye llow. Relaxation directions of first and second nearest neighboring ions (1 NN, 2NN) are indicated by the blue and black arrows respectively. The vacancy defects are shown by dashed circles.

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123 Figure 5-5. Cross-sectional contour maps of structure (a) and charge density difference around a titanium interstitial of differing charges [Tii in (b) and Tii in (c)]. The blue circle indicates the charge density from the t2g orbitals of the titanium interstitial, the red circle indicates the charge density from the t2g orbitals of the nearest titanium ions, and the black circle indicates the charge de nsity from the upper valence band states of oxygen 2p orbital.

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124 Figure 5-6. Calculated defect formati on enthalpies (DFEs) of point defects (VO, Tii, Oi and VTi) as a function of Fermi level, oxygen partial pressure, and temperature [(a)-(f)]. (a)-(c) shows calculated DFEs at differe nt temperatures [T=300K in (a), 1400K in (b), and 1900K in (c)] in reduced state (log(pO2)= -20). (d)-(f) shows DFEs in another reduced state (log(pO2)= -10) in same temperature range. [T=300K in (d), 1400K in (e), and 1900K in (f)]. (g) shows the titanium interst itial-oxygen vacancy transition at different high temperatures in the reduced state.

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125 Figure 5-7. Calculated def ect formation enthalpies (DFEs) of defect complexs [(a) Frenkel defect; (b) anion-Frenkel defect; (c) Schottky defect] as a function of Fermi level at 1900 K when pO2=10-10.

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126 Figure 5-8. Calculated defect formati on enthalpies (DFEs) of point defects (VO, Tii, Oi and VTi) as a function of Fermi level, oxygen partial pressure, and temperature. (a)-(c) shows calculated DFEs at different temperatures [T=300K in (a), 1400K in (b), and 1900K in (c)] in oxidized state (log(pO2)=2) using original formula. (d)-(f) shows DFEs using corrected formula [T=300K in (d), 1400K in (e), and 1900K in (f)]. (g) shows DFEs of one neutral oxygen vacancy as function of oxygen partial pressure at two different temperatures (300 K and 1900 K) using origin al formula, and compare them with the DFEs calculated in two extreme c onditions (Ti-rich and O-rich). Formation Enthalpy (eV) Formation Enthalpy (eV) a) b) c) g) Formation Enthalpy (eV) Oxygen partial pressure log(p/p0) Ti-rich bound O-rich bound d) e) f)

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127 Figure 5-9. Two-dimensional defe ct formation scheme as a func tion of oxygen partial pressure and temperature calculated at three different Fermi levels [F= 0.5 eV in (a), 1.5 eV in (b), and 2.5 eV in (c)].

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128 Figure 5-10. Contribution of vi brational energy and entropy to th e defect formation energy of the indicated defects relative to the defect-free structur e as calculated with the Buckingham potential.

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129 Figure 5-11. Calculated defect formation enth alpies (DFEs) of aluminum impurities doped as interstitials (Ali) and substitutionals on the Ti site (AlTi) as a function of Fermi level and temperature [(a)-(d)] in the reduced state (log(pO2)= -20). (a)-(b) shows calculated DFEs for Ali at different temper atures [T=300K in (a), 1400K in (b)] comparing with intrinsic n-type defects (Tii, VO), (c)-(d) shows DFEs for AlTi at same temperature range. [T=300K in (c), 1400K in (d)] and comparing with intrinsic p-type defects (VTi, Oi). The formation enthalpies for impur ities are shown in a range, which is named as formation energy band with two pseudo-states. ( b ) 1400 K ( a ) 300 K ( d ) 1400 K ( c ) 300 K

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130 Figure 5-12. Calculated defect formation enth alpies (DFEs) of aluminum impurities doped as interstitials (Ali) and substitutionals on the Ti site (AlTi) as a function of Fermi level and temperature [(a)-(d)] in the reduced state (log(pO2)= -20). (a)-(b) shows calculated DFEs for Ali at different temper atures [T=300K in (a), 1400K in (b)] comparing with intrinsic p-type defects (VTi, Oi), (c)-(d) shows DFEs for AlTi at same temperature range. [T=300K in (c), 1400K in (d)] and comparing with intrinsic p-type defects (Tii, VO). The formation enthalpies for impurities are shown in a range, which is named as formation energy band with two pseudo-states. ( b ) 1400 K ( a ) 300 K ( d ) 1400 K ( c ) 300 K

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131 CHAPTER 6 ELECTRONIC STRUCTURE OF CHARGED IN TRINSIC N-TYPE DEFECTS IN RUTILE TIO2 Introduction It is well known that n-type behavior in metal oxides is related to the existence of shallow donor levels near the conduction ba nd that are created by intrinsi c point defects such as oxygen vacancies and cation interstitial s [101]. Experimental evidence of these shallow donor levels have been reported for rutile TiO2 [125]. However, the origin of this behavior is still under debate [32]. Although numerous calculations have been carried out to analyze the electronic structure of point defects in TiO2, most of the investigations have focused on oxygen vacancies [41] and impurities [38, 43, 45] with less emphasis on titanium interstitials [32, 37]. In particular, the effects of cation d-orbital occupancy and cation-cation interacti ons have not been widely examined as possible s ources of shallow donor levels. There are several proposed origins of n-type conductivity in rutile-structured metal oxides. One of these is related to the multi-valence character of the cations. For example, K l and Zunger [126] used density-functional theory (DFT) calculations to determine that both Sni 4+ and Sni 2+ have surprisingly low formation energies and can thus both act as sh allow donors in rutile SnO2. Also the nature of high dielectric constant s in transition metal oxides may lead to the suggestion that electrons can be easily trapped by oxygen vacanc ies, which has already been shown in HfO2, an n-type fluorite structured oxide [127]. Since TiO2 also has a high dielectric constant and can readily form charged, multi-vale nce titanium interstitials, either of these mechanisms may be used to explai n the shallow donor levels in TiO2. In this chapter, DFT calculations are carried out to explore the orig ins of shallow donor levels in TiO2. In addition to the mechanisms discussed above, we consider the effect of edgesharing TiO6 octahedral structures within the rutile structure on cation-cation orbital overlap in

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132 the presence of cation interstitials. Figure 6-1 illustrates a 211 rutile structure where the yellow and red spheres represent the Ti and O at oms, respectively. Each Ti cation is surrounded by a slightly distorted O6 octahedral structure that are centere d at (0, 0, 0) and (, ) and differ in orientation by a 90 rotation. These O6 structures form a hexagonal closed-packed sublattice with half of the octahedral sites fille d by cations [39]. In addition, the edge-sharing TiO6 octahedra form chains in th e [001] direction, unlike the [ 010] and [100] directions, where the neighboring octahedra only share corner with each other. Thus, the dominant cation-cation short range interaction is in the [ 001] direction for pr istine rutile TiO2, which can act as an open channel for the diffusion of self-interstitials and impurities [84]. However, the presence of titanium interstitials in the cen ter of an otherwise empty O6 octahedron (the X site in Figure 61) leads to the presence of edge-sharing TiO6 octahedral structures around the interstitital site in the (001) plane. The presence of these octahedral structures may signifi cantly enhance the shortrange cation-cation orbital overlap and lead to n-type conductivity. Computational Details DFT calculations are used to investigate the electronic properties of titanium interstitials and oxygen vacancies in rutile TiO2. All the calculations are pe rformed using the plane-wave basis CASTEP software [59]. The exchange-cor relation functional is parameterized in the generalized gradient approxima tion (GGA) in the form of Pe rdew-Burke-Emzerhof functional [63]. In addition, the interact ion between the core and valenc e electrons is represented by ultrasoft pseudopotentials (USPP) introduced by Va nderbilt [57]. In part icular, an ultrasoft pseudopotential for Ti is generate d from the configuration [Ne]3s23p63d24s2, where the 3s2, 3p6, 3d2, 4s2 electrons are explicitly treated valenc e electrons. Additionally, an ultrasoft

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133 pseudopotential for O is generate d from the configuration [He]2s22p4, where the 2s2, 2p4 electrons are explicitly tr eated valence electrons. A 223 unit cell (containing 72 atoms in the case of the pristine structure) is used to build a supercell modeling perfect and defective bulk TiO2. The Brillouin-zone sampling is made on a Monkhurst-Pack grid of spacing 0.5nm-1 and a plane-wave cutoff energy of 400 eV is used. All the atoms in the supercell are relaxed to thei r equilibrium positions, such that the energy is converged to 0.001 eV/atom and the residual for ces are converged to 0.10 eV/. In order to check the influence of atomic relaxation, the ch arge density difference is calculated for the defective structures before and after the atomic relaxation. Results and Discussion Analysis of the Density of States The total and partial density of states (DOS) of pristine TiO2 is calculated first. The total DOS is shown in Figure 6-2(a), where the value of energy equal to zero has been taken as the valence-band maximum (VBM). As the focus of this study is on bonding changes associated with defect formation, only the bands from -20 eV to +15 eV are examined. In this range there are four main energy panels: two valence band panels below zero and two conduction band panels above it. The calculated band gap is 1.98 eV, which is about 33% lower than the experimental value of 3.0 eV [128, 129] due to the well-known problems [39] associated with derivative discontinuities in th e exchange-correlation energy f unctionals in DFT [120, 121]. Below the VBM, the bands shown in the lower panel (-17.52 to 15.72 eV) result from the O 2s character with a width of 1.80 eV, as indicated in Figure 6-2(b). In contrast, the upper valence band (-5.32 to 0 eV) is primarily from the O 2p orbitals hybridized with the Ti 3d orbitals with a band width of 5.32 eV [see Figure 6-2(c)]. These numbers is in excellent agreement with the experimental values of 1.9 and 5.4 eV from x-ray photoelectron spectroscopy [130, 131] and

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134 ultraviolet photoelectron spectro scopy measurements [132], respect ively. We also note that in this upper valence band panel there are two majo r peaks with an energy separation of 2.49 eV that have their origin in the se paration between the nonbonding and bonding O 2p states [40]. Above zero, the lower conduction band, which results primarily from Ti 3d orbitals, consists of two sets of Ti 3d bands and has a width of 5.32 eV [see Figure 6-2(d)]. These two sets of Ti 3d bands are products of the crys tal-field splitting of the Ti 3d states in TiO6 octahedral structure into the triply degenerate t2g (from 1.96 to 4.23 eV, with a band width of 2.27 eV) and doubly degenerate eg states (from 4.23 to 7.28 eV, with a band width of 3.05 eV). It is important to note that the octahedral structure in rutile do es not have perfect octahedral symmetry, and this is the reason why the Ti 3d states have split into two degenerate states. However when a Ti interstitial is placed into the center of a nearby O6 octahedra (the X in Figure 6-1), a different splitting type is obs erved due to the distortion of the nearby O6 octahedra [133]. Additionally, a small component of the Ti 3p/4s hybridized orbitals remain in energy levels higher than the Ti 3d orbitals, from about 7 to about 13 eV. In order to examine the effect of edge-sharing TiO6 octahedral structures on the el ectronic properties of rutile TiO2 after the addition of a fully charged Ti interstitial, we calculate and compare the total and partial DOS structures for the defective and pristine structur es (shown in Figure 6-3). From Figure 6-3(a) we see that the conduction band minimum (CBM) of the sy stem with the Ti inters titial is higher in energy relative to the CBM of the pristine structur e. This is explained by the fact that even a small contribution from the Ti interstitials 3p/4s hybridized orbitals incr ease the energy of the CBM. It is also interesting to note that although the band width of this lower conduction band of about 5.30 eV is almost the same for the defect ive and pristine structures the band width of the

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135 triply degenerate t2g states increase by 15% to 2.61 eV for the defective system relative to the pristine structure. In other words, the lowe r conduction band becomes much broader and is therefore more effective at promoting the form ation of shallow donor levels. While there is no detailed experimental information a bout the band widths of the two Ti 3d states in the lower conduction band, electron energy loss spectroscopy data show a peak spacing between the t2g and eg states of 2.5 eV [134] and 3. 0 eV [135]. Our DFT calculations predict a peak spacing of 2.79 eV, which is in good agreement with the experimental values. To understand this phenomenon more thoroughly, th e partial DOS localized at specific Ti and O sites near the Ti interstitial are investigated. Figure 6-3 illustrates the partial DOS of the Ti interstitial (Tii) and a Ti ion (Ti3NN) far away from the interstitial site. From Figure 6-3(b) we can see that the eg states of the Ti interstitia l are expanded relative to the Ti states in the pristine structure. In addition, Figure 6-3(c) indicates that although the eg states of the Ti3NN is slightly increased relative to the states in the pristine st ructure, the band width associated with it is essentially unchanged (see Table 6-1). Since the first nearest neighbor ions of the Ti in terstitial are O ions, we also calculated the partial DOS localized on the 2p states of oxygen ions (O1NN) closest to the Ti interstitial site. The results are compared with those associated with the pristine structure a nd are shown in Figure 63(d). The most predominant difference lies in the upper edge of the upper valence band panel, which belongs to the nonbonding O 2p states. These nonbonding O 2p states have greater populations near the zero energy point or, in other words, the lo calization of the O 2p states decrease as a result of interactions with the Ti interstitial. This is cons istent with hybridization between the Ti interstitial 3d states and the nearby Ti 3d states and O 2p states. Hybridization

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136 enhances the short-range cation-ca tion orbital overlap and would be expected to lead to shallow, donor level n-type behavior in the TiO2. The total and partial DOS for defective rutile TiO2 structure with a fully charged O vacancy is also determined here, and the results are shown in Figure 6-4. As expected, the total DOS of this defective structure, shown in Figure 6-4(a), is not much different from the pristine structure except that the eg states in the lower conduction band ar e slightly wider. In other words, the existence of the O vacancy is less effec tive at lowering the conduction band than the Ti interstitial. In figures Figure [6-4 (b)-(d)] we inve stigate the partial DOS of an O ion (O2NN) and Ti ion (Ti1NN) that are near the O v acancy, and a Ti ion (Ti3NN) that is far away from the vacancy site. Figure 6-4(b) indicates that the nonbonding O 2p states at the upper edge of the upper valence band are less populated near zero, which is c onsistent with strong lo calization of these O 2p states. From Figure 6-4 (c ) we can see that the Ti1NN ions also undergo a large lower conduction band shift compared with the pristine structure. However, the partial DOS of the Ti3NN ion is almost the same as in the pristine structure. T hus, on the whole, O vacancies are less effective at lowering the conduction band than are Ti interstitials. In summary, the existence of Ti interstitials leads to the formation of edging-sharing O6 structures within the rutile stru cture. This enhances the probab ility of short-range cation-cation orbital overlap, leads to the elongation of the t2g band states, and, ultimately, shallow level n-type conductivity in rutile TiO2. Charge Density Difference Analysis To further understand th e electronic structure indicated in the DOS analysis, charge density difference calculations are performed for Ti in terstitials and O vacancies. This involves first calculating the valence charge de nsity difference in pristine TiO2, as shown in Figure 6-5, along

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137 the high-symmetry [110] and [ 0 1 1] directions. Figure 6-5 (a) shows the (110) plane, which contains two apical bonds that ar e indicative of a Ti state of yzd-like symmetry, while Figure 6-5 (b) shows the ( 0 1 1) plane, which contains four equatorial bonds and that are indicative of a Ti state of 2 2y xd-like symmetry. These symmetries are taken from the crystal splitting of the five Ti atomic d states in an O6 octahedral symmetry. The figure indi cates that there is considerable covalent bonding present in the Ti-O bonds, as i ndicated by a maximum or zero electron density midway between the Ti and O atoms. This finding is consistent with another DFT study that also found a significant degree of covalent character along the Ti-O bonds and regions of charge depletion around the Ti sites [39]. In Figure 6-6 we show the valence charge density difference contour map of a fully charged Ti interstitial and O vacancy before and af ter atomic relaxation. Figure 6-6(a)-(b) depict the valence charge density difference along the linear (twofold) coordination and rectangular (fourfold) coordination for a fully charged Ti interstitial in the O6 structure. From Figure 6-6(a) and (b) it can be seen that before atomic re laxation, the charge density difference on the Ti interstitial is strongly non-spherical from the Ti interstitial to the nei ghboring O ion sites. This would be expected to lead to si gnificant overlap of the unoccupied 3d states corresponding to the Ti interstitial and the valence electrons of near by Ti ions. However, after atomic relaxation the asymmetric 3d orbitals of the Ti interstitial become mo re symmetric. In fact, the entire charge density map becomes similar to that of the pristine structure. Thus, atomic relaxation smoothes strong asymmetr ic orbital overlap that is present in the unrelaxed defective structure. However in more realistic systems with multiple defects per system, there is a much higher probability of def ect clustering [32, 111]. This will constrain the amount of orbital overlap smoothing that can occu r during atomic relaxation. In these clustered

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138 defect systems, one would thus expect there to be sufficient orbital overlapping to cause the formation of shallow level n-t ype conductivity in rutile TiO2. In the case of O vacancies, Figure 6-6(c) show s that before atomic relaxation, there are also asymmetric orbitals around th e vacancy site, especia lly on the nearby Ti ions. In particular, the Ti states are slightly distorted but still exhibit 2 2y xd-like symmetry in the ( 0 1 1) plane. However, following atomic relaxation, hybridized Ti sp2 orbitals emerge around the three Ti sites nearest to the O vacancy site. Although so me hybrid orbitals are important for bonding in solid-state materials, in this case the formation of hybrid orbitals is only a tran sitional step in the bonding process. This is because these orbitals ar e eigenstates of neither the isolated atoms nor the solid state materials. The calculated DOS in Figures 6-2, 6-3 and 64 indicate that the Ti 3p/4s hybridized orbitals re main in higher energy levels than the Ti 3d/O 2p hybridized orbitals. Summary In this study, electronic structure, DF T calculations indicate the causes of n-type conductivity of rutile structured TiO2. The densities of states are calculated for pristine and defective structures, with fully charged titani um interstitials or oxygen vacancies. The lower conduction band of the defective structures is shifted up in energy relative to the perfect structure, which leads to a broader lower conduction band that more readily promotes the formation of shallow donor levels. In particular, the introduction of Ti in terstitials promotes the presence of edge-sharing TiO6 octahedral structures that e nhances short-range cation-cation orbital overlapping to a much more significan t degree than do oxygen vacancies. The results indicate that these edging-sharing octahedral st ructures play a central role in the electronic structure of rutile TiO2. These findings indicate a possibl e path towards control of doping efficiency in semiconducting transition metal oxides.

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139 Table 6-1. Calculated band gap a nd band width for perfect rutile TiO2 structure and defective structure with a fully charged titanium interstitial and with a fully charged oxygen vacancy. The experimental data of perfect structure from absorption spectra and from theoretical DFT calculations are shown as comparison. Band gap (eV) Lower conduction band (LCB) (eV) Width of t2g states in LCB (eV) Width of eg states in LCB (eV) Perfect structure 1.98 5.32 2.27 3.06 Tii 2.09 5.30 2.61 2.69 VO 1.99 5.32 2.52 2.80 Perfect structure theoretical [39] 2.00 6.20 2.90 3.30 Perfect structure theoretical [40] 1.78 5.90 2.60 3.30 Perfect structure Experiments [136] 3.00

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140 Figure 6-1. A 211 supercell model for rutile TiO2 structure. X shows the center of an O6 octahedral structure. Figure 6-2. Total and partial DOS of pristine rutile TiO2.

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141 Figure 6-3. Total and partial DOS compar ison between pristine and defective TiO2 with a +4 charged Ti interstitial.

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142 Figure 6-4. Total and partial DOS compar ison between pristine and defective TiO2 with a +2 charged oxygen vacancy.

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143 Figure 6-5. Valence density differe nce maps for: (a) (110) and (b) ( 0 1 1) lattice planes of pristine TiO2 structure. The red color represents negative charge density and the blue color positive charge density.

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144 Figure 6-6. Valence density difference maps be fore and after atomic relaxation for: (a) Ti interstitial along apical bond direction; (b) Ti interstiti al along four equatorial bond direction; and (c) oxygen vacancy in ( 0 1 1) lattice plane. The color scheme is the same as in Figure 6-5.

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145 CHAPTER 7 CONCLUSIONS It is well-known that titanium dioxide exhibits n-type conductivity and that this property is tightly connected to the point defects and impuritie s present in the structure. Consequently, there is great interest in trying to unde rstand defect structures and th e mechanisms by which they form in bulk TiO2, especially when they influence the mate rials conductivity in different ways that depend on environmental and temperature fluctuations. In the first part of this dissertation, the stru ctures of intrinsic defect complexes, such as Schottky and Frenkel defects, in bulk TiO2 are studied using the dens ity functional theory (DFT) method. Several defect structure models are built to understand the preferential configuration and distribution of these defects. The results indicate that both Frenke l and Schottky defects prefer to cluster together instead of being distributed throug hout the lattice. In add ition, the Frenkel defect is predicted to be more likely to occur in rutile at low temperatures than the Schottky defect, with a difference in formation enthalpy of about 1 eV The density of states (DOS) for the Schottky and Frenkel models are also calculated. The re sults show that their band features are quite similar to the DOS of the perfect, defect free structure, with only a small increase in the band gap predicted to occur. Lastly, strong anisot ropy in inters titial cation diffusion in TiO2 is supported by these calculations. In the second part of this dissertation, DF T calculations within the supercell approximation and Makov-Payne correction are carried out to determine the preferred electronic state of charged point defect s in rutile TiO2. When studying defect using th e supercell approximation, it is well known that this approximation introduces artificial long-range in teractions between a defect and its periodic images in the nei ghboring supercells. This phenomena impedes the accurate evaluation of defect formation energies In order to overcome this problem, the Makov-

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146 Payne correction is widely used in the study of defects. In this pa rt of the dissertation, the DFEs of charged intrinsic defects in TiO2 are calculated with and wi thout the Makov-Payne correction to quantify its effect on the results. These calc ulations show that th e Makov-Payne correction improves the convergence of the defect formation energies as a function of supercell size for positively charged titanium interstitials and negatively charged titanium vacancies. However, in the case of positively charged oxygen vacancies, applying the Makov-Payne correction gives the wrong sign for the defect formation energy correcti on of fully charged oxygen vacancies that is attributed to the delocalized nature of the charge on this defect in TiO2. The results also indicate that when the Makov-Payne correction is applied, a stable charge transiti on occurs for titanium interstitials. However, it is noted that the application of the Makov-Payne correction to delocalized shallow level defects should be treat ed with caution since th is correction is more appropriate for the strongly localized charges. In the third part of this dissertation, th e influence of temperature and oxygen partial pressure on the stability of intrinsic defects (including oxygen vacancies and interstitials, and titanium vacancies and intersti tials) and extrinsic impurities (including aluminum and niobium) are investigated using a new approach that integrates ab initio DFT and thermodynamic calculations. For the archetype material TiO2, the dominant point defect s are predicted, and are found to be consistent with inferences from e xperimental electrical c onductivity measurements. More importantly, a number of ambiguities in th e experimental interpretation are clarified. First the structural relaxation and electronic structure calculat ion of dominant point defects at various charge states are calculated. The resu lts show that both donors (titanium interstitials and oxygen vacancies) and acceptors (titanium vacanci es) have shallow defect transition levels. Then the resulting defect formation enthalpies at various charge states are used in

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147 thermodynamic calculations to predict the influe nce of temperature and oxygen partial pressure on the relative stabilities of these defects. The fa vored point defects are shown to be controlled by the relative ion size of the defects at low temperatures, and by charge effects at high temperatures. Also the ordering of the most stable point defects is predic ted. It is found to be almost the same as the temperature increases an d the oxygen partial pressure decreases: titanium vacancy oxygen vacancy titanium interstitial. Addition ally, the experimentally observed transition in dominant point defects from oxyge n vacancies to titanium interstitials is well predicted in the calculations. Al so it is found that the formation energies of Schottky, Frenkel, and anti-Frenkel defect complexes do not change with the Fermi level. At high temperatures the formation of these complexes will restrict the fu rther formation of single point defects, such as oxygen vacancies. In the area of impurity doping, the ambipolar (pand n-type) doping behavior of aluminum in TiO2 has been thoroughly studied. Th e concept of pseudo-states is proposed to describe the thermodynamic equilibrium procedure between impurities and host ions during doping. It is predicted that, when compared with competing intrinsic defects, the aluminum interstitials are easy to form at high temperatures, and it is almo st impossible to have aluminum substitutions on titanium lattice sites at room temperature. Howe ver, when charge compensation is taken into account, AlTi becomes the predominant dopant in TiO2 at high temperatures. The calculations also predict that n-type doping of Ali in TiO2 should be significantly limited by the low concentrations of VTi and Oi at the same temperature. Finally, the electronic structur e, DFT calculations are performed to understand the causes of n-type conductivity of rutile structured TiO2. The DOS are calculated for pristine and defective structures, with fully charged titanium interstitials or oxygen vacan cies. It is found that

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148 the lower conduction band of the de fective structures is shifted up in en ergy relative to the perfect structure, which leads to a broader lowe r conduction band that more readily promotes the formation of shallow donor levels. In particular, the introduction of Ti in terstitials promotes the presence of edge-sharing TiO6 octahedral structures that e nhances short-range cation-cation orbital overlapping to a much more significan t degree than do oxygen vacancies. The results indicate that these edging-sharing octahedral st ructures play a central role in the electronic structure of rutile TiO2. These findings also indicate a pos sible path towards control of doping efficiency in semiconducting transition metal oxides. The above combined approaches provide the heretofore missing direct connection between electronic-structure and atomic-scale phenomen a on the one hand and the more complex, experimentally relevant conditions on the other. It is therefor e generally applicable for the systematic evaluation of defect formation in electronic ceramics under the full range of temperature and atmospheric conditions, such as are important for catalysis (low-temperature, high-pressure) and gas sensing (high-temperature, low-pressure) applicati ons. By controlling the nature and concentration of point defects and impurities, one can image a new means of tailoring the conductivity of semiconducti ng transition metal oxides.

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PAGE 156

156 BIOGRAPHICAL SKETCH Jun He was born on October 5th, 1971, in Wuhan, Hubei, P.R.China. Jun He achieved his B.S and M.S from Department of Metallurgy, Wu han University of Science and Technology, in 1993 and 1997. He studied in a Ph.D. program in University of Science and Technology, Beijing until 2001 before he came to the United States. He started his Ph.D. study in the Department of Materials Science and Engineering with Dr. Susan B. Sinnott as his advisor. His Ph.D. dissertation title is First Principles Calculations of Intrinsic Defects and Extrinsic Impurities in Rutile Titanium Dioxide.


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FIRST PRINCIPLES CALCULATIONS OF INTRINSIC DEFECTS AND EXTRINSIC
IMPURITIES IN RUTILE TITANIUM DIOXIDE























By

JUN HE













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

2006

































Copyright 2006

by

Jun He


































To my parents and my wife









ACKNOWLEDGMENTS

I would like firstly to express my appreciation and respect to my advisor, Dr. Susan B.

Sinnott, who has been supporting, guiding and believing in me for over five years. I feel very

fortunate to receive her guidance and appreciate her openness and confidence in my ideas during

this research. I would also like to thank my current committee members (Dr. David Norton, Dr.

Eric Wachsman, Dr. Simon Phillpot and Dr. Hai-Ping Cheng) and former committee members

(Dr. Darryl Butt and Dr. Jeffrey Krause) for their assistance and participation on my supervisory

committee. Special thanks go to Dr. Micheal W. Finnis (Imperial College London, UK) and Dr.

Elizabeth C. Dickey (Pennsylvania State University) for their numerous guidance and helpful

suggestions.

I would also like to thank Dr. Sinnott's group and Dr. Phillpot' s group (former and current

members are acknowledged) for supporting and providing a pleasant working environment. My

research has been benefited from the discussions with Dr. Douglas Irving, Dr. Yanhong Hu,

Rakesh Behera, Wendung Hsu and Dr. Jianguo Yu.

I would also like to thank my friends, Lewei Bu, Hailong Meng and Qiyong Xu. Their

friendship has made these 5 years in UF wonderful.

Finally, I thank my family, without whom this thesis would not have been possible. I thank

my parents; their encouragement and love have accompanied me through this journey. I thank

them for their understanding and their belief in me. I am also blessed to have my wife with me

through this process.











TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....

LIST OF TABLES ................. ...............7..___ .....

LIST OF FIGURES .............. ...............8.....

AB S TRAC T ........._. ............ ..............._ 12...

CHAPTER

1 BACKGROUND ................. ...............14........_ .....


Defect Chemistry in Rutile TiO2 ..................... .... ......... ...............1
Oxygen Vacancies are the Only Predominant Defect in the System .............. ................19
Titanium Interstitials are the Only Predominant Defect in the System ................... ........20
Simultaneous Presence of Oxygen Vacancies and Titanium Interstitials in the
System ................ .. ..... ... ....._. .. ..... ..........2
Presence of Dopants and Impurities in the System ........._.._.. ...._.. ........_.._.....24
Experimental and Theoretical Studies of Defect Formation .............. .....................2
Experimental Studies of Defect Formation in Rutile TiO2 ................. ............. .......26
Theoretical Studies of Defect Formation in Rutile TiO2 ................. ........... ...........30O
Theoretical Studies of Electronic Structure ....__. ................. ............... 34.....
Sum m ary ................. ...............36........ ......

2 INTRODUCTION OF DENSITY FUNCTIONAL THEORY AND ITS
APPLICATION IN DEFECT STUDY .............. ...............49....


Overview of Density Functional Theory ................. ......... ...............50.....
Kohn-Sham Theory .............. ...............50...
Exchange-Correlation Functional ................. ...............52.................
Pseudopotential Approximation .............. ...............55....
Implementation and Benchmark Test................ ......... ............5
Application of Density Functional Theory in Defect Study ................. ........................59
Supercell Approximation............... .............5
Band Gap and Defect Levels ............_ ..... ..__ ...............60..
Charge State and Compensation............... ..............6

3 DFT CALCULATIONS OF INTRINSIC DEFECT COMPLEX IN
STOICHIOMETRIC TIO2................ ...............66.

Introducti on ................. ...............66.................
Computational Detail s .............. ...............68....
M odel Development .............. ...............70....
Results and Discussion .............. ...............71....












Summary ................. ...............75.................


4 CHARGE COMPENSATION IN TIO2 USING SUPERCELL APPROXIMATION. ..........81


Introducti on ................. ...............8.. 1..............

Computational Detail s .............. ...............83....
Results and Discussion .............. ...............85....
Sum m ary ................. ...............88.......... ......

5 PREDICTION OF HIGH-TEMPERATURE POINT DEFECTS AND IMPURITIES
FORMATION IN TIO2 FROM COMBINED AB INITIO AND THERMODYNAMIC
CALCULATIONS ................. ...............94.......... ......


Introducti on ................... .......... ...............94.......

Computational Methodology ................. ...............96.......... .....
Electronic Structure Calculations .................. .......... ...............96......
Defect Formation Energies of Intrinsic Defects ................. .............. ......... .....97
Thermodynamic Component ................. ...............98.................
Charge Compensation ..................... .. ...............99
Results and Discussion of Intrinsic Defects ................. ................ ......... ........ .99
Electronic Structure of Defects in TiO2............... ...............99..
Structural Relaxation ................. ...............101................
Defect Formation Enthalpies .................. ............ ...............103 .....
Extrinsic Impurities in Nonstoichiometric TiO2............... ...............109.
Background ................. ...............109................
Computational Details .................. ............. .......... ........ .........1
Results and Discussion of Aluminum Ambipolar Doping Effects ............... ... ............113
Sum m ary ................. ...............115...............

6 ELECTRONIC STRUCTURE OF CHARGED INTRINSIC N-TYPE DEFECTS IN
RUTILE TIO2 ................ ...............131................


Introducti on ........._._ ...... .. ...............131...

Computational Details .............. ...............132....
Results and Discussion .........._..........___ ...............133...

Analysis of the Density of States. ........._._.. ...... ...............133
Charge Density Difference Analysis ...._. ......_._._ .......__. .............3
Summary ........._._ ...... .. ...............138...

7 CONCLUSIONS .............. ...............145....


LIST OF REFERENCES ........._._ ...... .... ...............149..


BIOGRAPHICAL SKETCH ........._._ ...... .__ ...............156...










LIST OF TABLES


Table page

1-1 Selected bulk properties of rutile TiO2. ................ ........................ ..............38

2-1 Calculated lattice constants, CPU times, convergence data, and total energies of
rutile TiO 2. ............. ...............63.....

3-1 Comparison between the calculated structural parameters and experimental results
for rutile TiO2............... ...............76..

3-2 Positions of the Ti interstitial site in the Frenkel defect models shown in Figure 2-2. .....76

3-3 Calculated Schottky DFEs for rutile TiO2. ............. ...............76.....

3-4 Calculated Frenkel DFEs for rutile TiO2. ............. ...............76.....

3-5 Comparison of DFT calculated Frenkel and Schottky DFEs to published
experimental and theoretical values for rutile TiO2. .............. ...............77....

5-1 Calculated lattice parameters and Ti-O bond lengths for rutile TiO2 COmpared to the
theoretical values and experimental values ................. .........__....... 117.........

5-2 Structural relaxation around defects. The relative changes from original average
distances from perfect bulk are listed in percent. ................ ................. ......... .11

5-3 Calculated defect formation enthalpies for their most stable charge states of defects
under three typical conditions: standard condition (T=300 K, SF- 1.5eV, pO2- 1 atm),
reduced condition (T=1700 K, SF- 2.5eV, pO2- 10-10 atm), and oxidized condition
(T=1200 K, SF- 0.5eV, pO2= 10' atm). ................ ...............118........... .

6-1 Calculated band gap and band width for perfect rutile TiO2 Structure and defective
structure with a fully charged titanium interstitial and with a fully charged oxygen
vacancy. ............. ...............139....










LIST OF FIGURES


Figure page

1-1 Bulk structures of rutile and anatase TiO2........... ............... ......... ................39

1-2 Diagram of the TiO6 Octahedral structure in rutile TiO2.................. .......... .............39

1-3 Phase diagram of the Ti-O system. ................ .......................... ...............40

1-4 Calculated defect concentration in undoped TiO2 (A=0) at different temperature
ranges as a function of pO2 USing reported equilibrium constants ................. ................ .40

1-5 The logarithm of weight change of rutile as function of logarithm of oxygen partial
pressure. ............. ...............41.....

1-6 Thermogravimetric measurement of x in TiO2-x aS function of oxygen partial
pressure. ............. ...............41.....

1-7 Electrical conductivity measurement of TiO2-x aS function of oxygen partial pressure. ...42

1-8 Electrical conductivity measurement of TiO2-x aS function of oxygen partial pressure. ...42

1-9 Defect formation energies of (a) Con,, (b) Comet, (c) Con, o, (d) Vo defects as a
function of Fermi level in the oxygen-rich limit ................. ...............43......_.._..

1-10 Defect formation energies as a function of the Fermi level, under the Ti-rich (left
panel) and oxygen-rich (right panel) growth conditions, respectively. ............. ................43

1-11 Calculated total DOS for TiO2 per unit cell compared to experimental UPS and XAS
spectra for TiO2 (110) surface ................. ...............44........... ..

1-12 Calculated valence density difference maps for (a) (110) and (b) (-110) lattice planes.
(c) shows the experimental electron density map in (-110) plane in rutile TiO2.............. .44

1-13 Calculate density of states for the 8x8x10 rutile model with 0, 1, 5, and 10% oxygen
vacancies on a large energy scale showing the development of a tail of donor states
below the conduction band minimum ................. ...............45........... ...

1-14 Comparison of density of states (DOSs) between defective and perfect rutile TiO2.........45

1-15 Calculated total and partial density of states (DOSs) of anatase TiO2. ............. ................46

1-16 Calculated density of states (DOSs) of rutile-structured RuxTi 1-xO2 with different Ru
concentrations compared with experimentally determined spectra ................. ...............46

1-17 Calculated spin polarized density of states (DOSs) of the Co-doped anatase (left) and
rutile (ri ght) Til-xCoxO2. ............. ...............47.....










1-18 Calculated DOS of the (a) relaxed defective structure with Cd (b) relaxed defective
structure with Cd' and (c) unrelaxed defective structure with CdO ............... .................47

1-19 Total (A) and partial (B) density of states (DOS) for doped anatase TiO2 calculated
by F LAPW .............. ...............48....

2-1 Flow-chart describing Kohn-Sham calculation. ............. ...............64.....

2-2 Illustration of difference between all-electron scheme (solid lines) and
p seudopotenti al scheme (dashed lines) and their corresponding wave functi ons..............64

2-3 The influence of supercell size on the defect formation energy of a neutral oxygen
vacancy as a function of supercell size, as calculated with DFT (both using single
point energy, geometry optimization including electronic relaxation and full atomic
relaxation) and an empirical Buckingham potential. ............. ...............65.....

3-1 The Schottky defect models considered in this study. ................... ... ............7

3-2 The Frenkel defect models considered in this study ................. ................ ......... .79

3-3 The densities of states of perfect and defective TiO2. The valance-band maximum is
set at 0 eV. ........._._.. ...._... ...............80...

3-4 Possible octahedral Ti interstitial sites in rutile TiO2. ............. ...............80.....

4-1 Schematic illustration of the use of PBCs to compute defect formation energies for
an isolated charged defect in a supercell approximation. ............. .....................9

4-2 Calculated defect formation energies and defect transition levels in different
super cell s............... ..............9

4-3 Calculated defect formation energies for various charge states of the titanium
interstitial in a 72-atom supercell in TiO2 aS a function of the Fermi level (electron
chemical potential) with and without application of the Makov-Payne correction. ..........92

4-4 Calculated defect formation energies for intrinsic defects at 300 K and 1400 K with
and without the Makov-Payne correction under reduced conditions (pO2-10-20)..............~93

5-1 Electrical conductivity of rutile TiO2 Single crystals as function of the oxygen partial
pressure in the temperature range 1273-1773K ....._.__._ .... ... .__. ......._._.........1

5-2 The influence of system size on the defect formation energy of a single oxygen
vacancy calculated by atomic-level simulations using the empirical Buckingham
potenti al ................ ...............120....... ......










5-3 Calculated band structure (a) and defect transition levels E(defectql/q2) (b) after the
band gap lineup correction for TiO2. All the thermodynamic transition levels are
calculated with respect to the valence band maximum regardless of their donor or
acceptor character. ............. ...............121....

5-4 Ball-and-stick models showing relaxation around a titanium interstitial (a), an
oxygen vacancy (b) and a titanium vacancy (c) in a TiO2 Supercell. ............. ................122

5-5 Cross-sectional contour maps of structure (a) and charge density difference around a
titanium interstitial of differing charges [Ti,"~ in (b) and Ti,""~ in (c)]. ............................. 123

5-6 Calculated defect formation enthalpies (DFEs) of point defects (Vo, Ti,, O, and Vn)
as a function of Fermi level, oxygen partial pressure, and temperature [(a)-(f)] ............124

5-7 Calculated defect formation enthalpies (DFEs) of defect complex [(a) Frenkel
defect; (b) anion-Frenkel defect; (c) Schottky defect] as a function of Fermi level at
1900 K when pO2-10-10 ........._._._ ...............125._._._..

5-8 Calculated defect formation enthalpies (DFEs) of point defects (Vo, Ti,, O, and Vn)
as a function of Fermi level, oxygen partial pressure, and temperature. .........................126

5-9 Two-dimensional defect formation scheme as a function of oxygen partial pressure
and temperature calculated at three different Fermi levels [SF0.5 eV in (a), 1.5 eV in
(b), and 2.5 eV in (c)]. ............. ...............128....

5-10 Contribution of vibrational energy and entropy to the defect formation energy of the
indicated defects relative to the defect-free structure as calculated with the
Buckingham potential. ............. ...............129....

5-11 Calculated defect formation enthalpies (DFEs) of aluminum impurities doped as
interstitials (Al,) and substitutionals on the Ti site (Aln) as a function of Fermi level
and temperature [(a)-(d)] in the reduced state (log(pO2)= -20). ............. ....................12

5-12 Calculated defect formation enthalpies (DFEs) of aluminum impurities doped as
interstitials (Al,) and substitutionals on the Ti site (Aln) as a function of Fermi level
and temperature [(a)-(d)] in the reduced state (log(pO2)= -20). ............. ....................13

6-1 A 2x lxl supercell model for rutile TiO2 Structure. "X" shows the center of an 06
octahedral structure ................. ...............140................

6-2 Total and partial DOS of pristine rutile TiO2. ................ ................. ......... .....140

6-3 Total and partial DOS comparison between pristine and defective TiO2 with a +4
charged Ti interstitial. ............. ...............141....

6-4 Total and partial DOS comparison between pristine and defective TiO2 with a +2
charged oxygen vacancy. ............. ...............142....










6-5 Valence density difference maps for: (a) (110) and (b) (-110) lattice planes of
pristine TiO2 Structure ................. ...............143................

6-6 Valence density difference maps before and after atomic relaxation for: (a) Ti
interstitial along apical bond direction; (b) Ti interstitial along four equatorial bond
direction; and (c) oxygen vacancy in (1-10) lattice plane............... ..................4









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

FIRST PRINCIPLES CALCULATIONS OF INTRINSIC DEFECTS AND EXTRINSIC
IMPURITIES IN RUTILE TITANIUM DIOXIDE

By

Jun He

December 2006

Chair: Susan B. Sinnott
Major Department: Materials Science and Engineering

Titanium dioxide has been intensively studied as a wide band-gap transition metal oxide

due to its n-type semiconducting property. In this dissertation, first the defect formation

enthalpies of Frenkel and Schottky defects in rutile TiO2 are calculated. The results predict that

Frenkel defects are more energetically favorable than Schottky defects and both of them prefer to

cluster together in TiO2. The possible diffusion routes for interstitial Ti atoms are also

investigated.

Secondly, the dependence of defect formation energies on supercell size is investigated.

The results indicate that the electrostatic Makov-Payne correction improves the convergence of

defect formation energies as a function of supercell size for charged titanium interstitials and

vacancies. However this correction gives the wrong sign for defect formation energy correction

for charged oxygen vacancies. This is attributed to the shallow nature of the transition levels for

oxygen vacancies in TiO2.

Next, a new computational approach that integrates ab initio electronic-structure and

thermodynamic calculations is given and applied to determine point defect stability in rutile TiO2

over a range of temperatures, oxygen partial pressures, and stoichiometries. The favored point

defects are shown to be controlled by the relative ion size of the defects at low temperatures, and










by charge effects at high temperatures. The ordering of the most stable point defects is predicted

and found to be almost the same as temperature increases and oxygen partial pressure decreases:

titanium vacancy -oxygen vacancy -titanium interstitial. Also it is found that the formation

energies of Schottky, Frenkel, and anti-Frenkel defect complexes do not change with the Fermi

level. At high temperatures the formation of these complexes will restrict the further formation

of single point defects, such as oxygen vacancies. In the study of ambipolar doping behavior of

aluminum in TiO2, the concept of pseudo-state is proposed to describe thermodynamic

equilibrium procedure between impurities and host ions. It is predicted that at high temperatures

aluminum substitutional defects become the predominant dopant in TiO2 while n-type doping of

aluminum interstitials is limited by high concentrations of titanium interstitials and oxygen

vacancies.

Finally, the origin of shallow level n-type conductivity in rutile TiO2 is discussed. The

calculated densities of states for defective structures with fully charged titanium interstitials

show a broader lower conduction band, which may enhance short-range cation-cation orbital

overlap and thus lead to the formation of shallow donor levels.









CHAPTER 1
BACKGROUND

Transition metal oxides remain one of the most difficult classes of solids on which to

perform theoretical predictions using first-principles calculations. This is due to their complex

crystal structures and the fact that they usually exhibit a wide range of properties, including

acting as insulating, semiconducting, superconducting, ferroelectric, and magnetic materials.

As a prototypical semiconducting transition metal oxide, titanium dioxide (TiO2) has been

the focus of extensive experimental and theoretical studies for over four decades due to its

numerous technological applications [1, 2]. For example, TiO2 is widely used in heterogeneous

catalysis, as gas sensor, as a photocatalyst, as an optical coating, as a protective coating, as

biomaterial implants and as varistors in electric devices. Many of these applications are tightly

connected to the point defects and impurities introduced in the structure. These defects and

impurities can be found in the bulk, on the surface and at the grain boundary. Consequently,

there is great interest in trying to understand defect structures and the mechanisms responsible

for their creation.

Since the discovery of photolysis applications on TiO2 Surfaces by Fujishima and Honda

[3], it has been well established that surface defect states play an important role in surface

chemistry phenomena (such as mass transport and waste decomposition). For one thing, these

surface defects strongly affect the chemical and electronic properties of oxide surfaces. Recently

there has been considerable interest in using powerful instruments and techniques such as

scanning tunneling microscopy (STM) and transmission electron microscopy (TEM) to

investigate surface structures and defect diffusion on the TiO2 Surface. For instance, Diebold et

al. reported a series STM studies combined with theoretical calculations to determine the image

contrast in STM analysis of the oxygen-deficient rutile TiO2 (1 10) surface [4-6]. Based on their









observations, the local electronic properties induced by intrinsic and extrinsic defects on this

surface are discussed [7]. For example, they found in scanning tunneling spectroscopy (STS) that

oxygen-deficient defects do give rise to defect states within the band gap. Also their observations

indicated that those impurity atoms, which have a positive charge state and cause n-type doping,

may cause a localized downward band bending. In addition, Schaub et al studied the oxygen-

mediated diffusion of O vacancies on the TiO2 (1 10) surface using quantitative analysis of many

consecutive STM images [8]. All these findings called for a reinterpretation of the defect

chemistry of oxygen vacancies on the TiO2 (110) surface and opened the lead for further

experimental investigations as well as theoretical calculations for surface defect models.

In contrast, our fundamental understanding of defect formation and diffusion mechanisms

in the bulk and at grain boundaries is still unclear. One reason for this is that there are few

experimental techniques that can be used to explore the nature of defect formation in bulk

materials. Experimental techniques, such as thermogravimetry and electrical conductivity

measurements, have been used to study deviations from stoichiometry in bulk TiO2 aS a function

of temperature and oxygen partial pressure using reasonable assumptions since the early 1960's

[9-14].

Individual defects and impurities at grain boundaries in oxides can be analyzed by high

resolution transmission electron microscopy (HRTEM) [15], Z-contrast imaging, and electron

energy loss spectroscopy (EELS) in scanning transmission electron microscope (STEM) [16].

For example, Bryan et al. studied Co2+ and Cr3+ doped nanocrystalline TiO2 by HRTEM and

electronic absorption spectroscopy and found that the most important factor for activating

ferromagnetism in nanocrystalline Co-doped TiO2 is the creation of grain boundary defects,

which is identified as oxygen vacancies [17]. It has also been demonstrated that EELS is









sensitive to the changes in the oxygen concentration and could be used to probe the effect of

individual defects on the local electronic structure. Even though all these techniques are proven

to be most sensitive to the heavier elements in the crystal structure, the oxygen atom, which is

the element that in many cases plays the largest role in determining the electronic properties in

oxides, is the least well characterized in these experiments.

Although these experimental techniques have been successfully used for the study of

defect concentrations, further understanding of defect formation mechanisms is still limited due

to the extreme sensitivity of the electronic and physical properties of metal oxides to minute

concentrations of defects and impurities in the bulk material.

It is therefore important to understand the defect structure and formation mechanisms in

bulk metal oxides, such as TiO2, OSpecially when they influence the materials' conductivity in

different ways, depending on gas adsorption and temperature fluctuation. By controlling the

nature and concentration of point defects and impurities, one can image a new means of tailoring

the conductivity of semiconducting transition metal oxides. In this dissertation, the defect

structures of intrinsic defect complexes, such as Schottkey and Frenkel defect pairs in bulk rutile

TiO2, are Studied using the density functional theory (DFT) method. In addition, the influence of

temperature and oxygen partial pressure on the stability of intrinsic defects (including oxygen

vacancies and interstitials, and titanium vacancies and interstitials) and extrinsic impurities

(including aluminum, niobium) are also investigated using a new approach that integrates ab

initio DFT and thermodynamic calculations. Finally, the electronic structures of these defects are

calculated and compared with the electronic structure of the pristine structure.

Defect Chemistry in Rutile TiO2

Before giving a literature overview of experimental studies of defect formation in bulk

TiO2, an introduction to defect chemistry in this material is necessary.










Titanium dioxide has three maj or different crystal structures: rutile (tetragonal,

Dif, P42 / mnm a=b=4.594 A+, c=2.959 A+), anatase (tetragonal, Dif, 14, / amd, a=b=3.733

A, c=97.37 A) andIC brookite (rhombohedraIl, D~ -1 -Pbca, a=5.436 A, b=9.166 A~, c=5-.135 A)

[2]. Among these three structures, only rutile and anatase play important roles in applications

that make use of TiO2 and are, consequently, of great interest to researchers. Their unit cells are

shown in Figure 1-1.

The structure studied in this dissertation is rutile TiO2. Table 1-1 shows selected bulk

properties of this material. In the rutile structure, the basic building block consists of a titanium

atom surrounded by six oxygen atoms, TiO6, in a distorted octahedral configuration. In this

structure, the bond length between the titanium and the oxygen atoms for two apical bonds along

the linear (twofold) coordination is slightly longer than that of four bonds along the rectangular

(fourfold) coordination (see Figure 1-2). Also, the neighboring octahedral structures share one

corner along the <110> directions, and are stacked with their long axis alternating by 900. In the

<001> direction, there are edge-sharing octahedral TiO6 Structures connected by their edges with

the edges of neighboring octahedral structures.

As shown in early studies by Wahlbeck and Gilles [18], titanium oxide can occur in a wide

range of nonstoichiometric structures determined by temperature and oxygen pressure (see

Figure 1-3). For example, order-disorder transformations take place over the entire composition

range of the Ti-O solid solution depending on the temperature. As with most of the

nonstoichiometric oxides, the thermochemical and electronic properties of rutile TiO2 aef

directly influenced by the type and concentration of point defects.

The defect structure of rutile TiO2 has been studied since the early 1950's. For an oxygen-

deficient oxide like TiO2-x, the deviation from stoichiometry, x, has been studied by









thermogravimetry and electronic conductivity measurements. Usually, based on the assumption

that there is only one predominant defect residing in the oxide, the deviation x is described as a

function of oxygen partial pressure, x ac PO2-1 n. In addition, it was determined that n is very

sensitive to temperature [19, 20]. Thus we can say the deviation x can be described as a function

of both temperature and oxygen partial pressure in homogeneous TiO2-x-

In order to describe the point defects in terms of equations, it is important to have a

notation system. The Kroiger-Vink notation is employed in this dissertation [19, 20]. In this

notation system the type of defects is indicated by the combination of a maj or symbol, a

subscript and a superscript. The maj or symbol describes the type of defect and the subscript

shows its occupation site. The superscript is used to describe the charge state of this defect. The

charge can be described as the actual charges of defects, for example, +1, +2. However,

considering the contribution of defect charge to the whole perfect structure, it is more convenient

to assign an effective charge to the defect. Generally the zero effective charge is symbolized as a

cross (X) in the superscript, the positive effective charge is indicated by a dot (') and the negative

effective charge is shown by a prime ('). For example, the normal titanium and oxygen atoms on

the regular lattice sites have zero effective charge, and so are written as Ti~ and O, .

The intrinsic single point defects in TiO2 include vacancies (Vo and Vn~), self-interstitials

(O, and Ti,), and antisites (On and Tio). The oxygen vacancies with two, one or no electrons

localized around the vacancy site are written as Vf VJ and, Vj' respectively. The titanium

vacancies with four, three, two, one or no holes localized around the vacancy site are written as

V/, I ,, y 1", y il,'" I ,",respectively.


There are also numerous opportunities for dopants and impurities to be present in the

system. For example, aluminum, gallium, iron, magnesium, niobium, zinc, and zirconium have









been reported to appear in rutile TiO2 Samples [21, 22]. These impurities may occupy the regular

titanium site or an empty interstitial site. When an impurity ion M occupies a regular titanium

site, it is written as Mr,. On the other hand, if it occupies the empty interstitial site, it is written as

M,.~ The effective charge state will be discussed in the following chapter.

It is commonly suggested that the intrinsic n-type defects, such as oxygen vacancies and

titanium interstitials, are responsible for oxygen-deficient rutile TiO2. So, first let us only

consider the defect chemistry of oxygen vacancies in this system under low pO2.

Oxygen Vacancies are the Only Predominant Defect in the System

Ass~umeI a neutral oxygenl vac~anlcy (Vox is fome inc th1Lle structLure withI two eClectron

trapped around the vacancy site. Then, depending on the temperature, these two trapped

electrons may get excited one by one from the vacancy site. At the same time, the neutral oxygen

vacancy acts as an electron donor and becomes singly charged (Vo') or doubly charged (Vo ).

The defect reactions are then described by the following equations.

Of = VJ +1/202() (11

Vf = VJ + e' (1-2)

VJ = Vj' + e (1-3)

The corresponding defect equilibrium equations are also written as


[Vf ]pg2 = K, (1-4)

[V;]n= Kb[V ] (1-5)

[Vj']n = Kc[Vj] (1-6)

where n = [e'] defines the electron concentration.









Because the oxygen vacancies and the complementary electrons are the predominant

defects in the oxygen deficient oxide, the charge neutrality principle requires

n = [V2] +2[VJ']i (1-7)

The concentration of electrons and the neutral, singly and doubly charged oxygen

vacancies are related through the above four equations [equations (1-4)-(1-7)]. Then, by

combining these equations, expressions for each of the defects may be obtained. However we

need some assumption to solve these equations. The concentrations of oxygen vacancies are

given by the following limiting assumptions:

if [Vox] >> Vo ']+[Vo ], [Voltotai= KapO2-1/2

if [Vo ] >> [Vox + Vo ], [Voltotai= (KaKb)1/2PO2-1/4

if [Vo ] >>[Vox +[ Vo], [Voltotai=( (/4KaKbKe) 1/3po2- 1/6

Thus the concentration of oxygen vacancies in an oxygen-deficient oxide may have an

oxygen pressure dependence that ranges from pO2-1/2 to pO2-1/6

Titanium Interstitials are the Only Predominant Defect in the System

Assume the titanium interstitials are formed in the structure with simultaneous formation

of titanium vacancies by Frenkel defect reaction. The titanium vacancies may also help the

formation of oxygen vacancies through Schottky defect reactions. The entire defect reaction

scenario can be described as

TiT = Vr,""11+Ti: (1-8)

Ti~ + 2OfX = V7, "+2V2' + TiO2 (1-9)

ri1
O, = Vj' +-O2(g) +2e' (1-10)


The defect reaction equilibrium and charge neutrality can then be described as










[Vr,""'][Ti:"' ]= K, (1-11)

[Vr,""11][V ']2 = K2 (1-12)


[Vh']n2 2Z = K3 (1-13)

n +4[V,""'] = 2[VJ']i+ 4[Ti:"'] (1-14)

To finally obtain the relationship of these defects to the oxygen partial pressure, detailed

information of the equilibrium constants K1, K2, and K3 are HOCOSsary. However as Bak et al.

[14] pointed out, the equilibrium constants of Schottky and Frenkel defect reactions, K1 and K2,

are not yet available. The lack of data could be attributed to the fact that these two defect

reactions assume equilibrium at substantially higher temperatures than that of other defect

reactions reported in the literature. Therefore to overcome this, reasonable assumptions are made

here.

On the other hand, this case can be simplified from the analysis done by experimental

researchers as shown in the following condition. Assume that one neutral titanium interstitial is

formed in the structure with four trapped electrons, and then the charged interstitial forms step by

step.

Ti, + 2Of = Ti; + O2(g) (1-15)

Ti; = Ti,' + e' (1-1 6)

Ti: = Ti:' + e' (1-17)

Ti:' = Ti:" + e' (1-18)


Ti: = Ti: + e' (1-19)

Then we can obtain the defect reaction equilibrium equations as









[Ti$]x 2 = KAq (1-20)

[Tia']n =K,[Tilz] (1-21)

[Ti:"]n = Kc[Ti,'] (1-22)

[Ti'" ]n = KD 2i" ] (1-23)

[Tia'"']n = KE 2'"]' (1-24)

Also the charge neutrality principle requires

n = [Ti:']+ 2[Ti:' ]+ 3[Ti'"]i+ 4[Tia'"']i (1-25)

Therefore, the concentration of titanium interstitials is given by the following limiting

assumptions (according to experimental data, only titanium interstitials with +3 or +4 charge are

considered to be viable defects):


if [Tia']>> [Tia]+[Tiz"]+[Tia'"'], [Ti."]= n/3= 0.33(3K)4 02 4


if [Ti{"'] >> [Ti a'[;.]Ti"+[Ti;'"], [Ti{"'.]= n/4=0.25(4K') pO2 5

where K = KAKBKcKD, K' = KAKBKcKDKE.

Thus the concentration of titanium interstitials in an oxygen-deficient oxide may have an

oxygen pressure dependence that is related to pO2-1/(m+1), Where m is the charge of the titanium

interstitial.

Simultaneous Presence of Oxygen Vacancies and Titanium Interstitials in the System

Sometimes it is difficult to predict whether titanium interstitials or oxygen vacancies are

the predominant point defect in an n-type system. In fact, both defects may be important, at least

in certain temperature and oxygen partial pressure ranges. If this is the case, it is not easy to

predict the relationship of defect concentration to oxygen partial pressure. Here, let us assume









the oxygen vacancies with +2 charge and titanium interstitials with +3 charge are the

predominant defects in the system. Let us further assume that the titanium interstitials with +3

charge remain in equilibrium with titanium interstitials with +4 charge. The resulting total defect

reactions are therefore


O = o'+ 2e'+-O,(g) (1-26)


Tis + 20, = Ti,'" + 3e'+O, (g) (1-27)

Ti"' = Ti:"' + e'. (1-28)

It therefore follows that we get the following defect reaction equilibrium equations:


[Vo']n2po? = KG! (1-29)

[Ti," ]n po, = K,, (1 -3 0)

[Ti,'"']n = K,,[Ti'"] (1-31)

Also the charge neutrality principle requires

n = 2[Vo'] +3[Ti,'"] +4[Ti,'"']i. (1-32)

The combination of the above equations [equations (1-29) to (1-32)] results in the

following expression for pO21


po,-o2 (1-33)
K, + (K, + 3n K, +12n7KpK,)

Equation (1-33) allows for the determination of the relationship between n and pO2.

Consequently, the concentration of all possible maj or defects may be determined as a function of

pO2 USing Equations (1-29) to (1-33). However knowledge of these equilibrium constants is

required to solve these equations.









In considering this problem, Bak et al. presented their theoretical work on the derivation of

defect concentrations and diagrams for TiO2 based on experimental data of equilibrium constants

reported in the literature [14]. The fully charged oxygen vacancies and titanium interstitials with

+3 or +4 charge are all considered as possible maj or n-type defects in the system. Additionally,

the temperature considered in the defect diagram ranges from 1073 K to 1473 K. The resulting

defect diagram for undoped TiO2 is shown in Figure 1-4. They found that at low pO2, the slope of

-1/4 represents titanium interstitials with +3 charge. And at high pO2, the slope of -1/6 is

consistent with doubly charged oxygen vacancies. Also, the pO2 at which the concentration of

oxygen vacancies surpasses the concentration of titanium interstitials increases with temperature.

Finally, the concentration of titanium interstitials with +3 charge prevails over the titanium

interstitials with +4 charge at lower pO2, while at higher pO2 the titanium interstitials with +4

charge become the minority defects (the slope is assumed to be -1/4). These derived defect

diagrams therefore indicate that undoped TiO2 exhibits n-type conductivity over the entire range

of pO2 and does not exhibits an n-p transition in the temperature range of 1073 K 1473 K. This

is an interesting result. However the accuracy of their predicted defect concentrations highly

depends on the experimental equilibrium constants reported in the literatures, which actually

vary over a significant range during measurement. This is due to the practical problems during

the measurement of equilibrium defect concentrations. Since the determination of these intrinsic

electronic equilibrium constants is not the main obj ective of this dissertation, the reader is

referred to [14, 23] for additional details.

Presence of Dopants and Impurities in the System

In the presence of dopants and impurities in the system, the defect reactions rely on a few

additional complicating factors. These factors include the valences of the impurities relative to









the host compound, the lattice positions that the impurity atoms occupy, and the current

predominant native defects in the structure.

If a trivalent cation is dissolved in the structure, it may occupy the interstitial site or

substitute on the normal titanium site, depending on its ionic radius. Taking aluminum as an

example, the aluminum ion may occupy the interstitial site, and the resulting defect reaction is


Al203 = 2Al" +6e'+-O2(g) (1-34)


With the presence of the aluminum interstitial, the electron concentration will increase, and

the concentration of oxygen vacancies or titanium interstitials will decrease according to

equations (1-5), (1-6) and (1-21) to (1-24).

Since aluminum has a smaller ionic radius than titanium, it may substitute on the titanium

site. Assuming the fully charged oxygen vacancies are the predominant nature defects in the

system, then the defect reaction and the charge neutrality will be

2TzO2
Al203 4 2Alf,'+Vo' + 30, (1-35)

n+[Al,'] = 2[Vo'] (1-36)

As a result, the electrical conductivity will be proportional to pO2-1/4, HOt pO2-1/6. Thus it is

possible that the relationship of defects to pO2-1/4 may reflect the presence of single charged

oxygen vacancies, or tri-valent Ti interstitials, or even tri-valent impurities that occupy normal

titanium lattice sites.

If the fully charged titanium interstitials are the predominant defects in the system, then the

reaction and charge neutrality equation are

2A203 + 4Ti, a 4Alf,'+Tia'"' +3TiO2 (1-37)

n +[Al,,'] = 4[Ti:""] (1-3 8)









On the other hand, self-compensation may also need to be considered in the case of

aluminum. Aluminum can be considered to dissolve both substitutionally and interstitially due to

its small ionic radius. Therefore, the reaction can be written as

2Al,O, +3Ti 3l'+lX3i, (1-39)


In the case of pentavalent impurities, they are always treated as donors in TiO2. Taking

niobium as an example, the reactions can be written as


Nb,O, = 2Nb', + 2e'+-O,(g) (1-40)


Nb,O, = 2Nb,'"" + 10e'+- O, (g) (1-41)


In this case the electron concentration increases and the concentration of oxygen vacancies

or titanium interstitials decreases.

Experimental and Theoretical Studies of Defect Formation

Experimental Studies of Defect Formation in Rutile TiO2

Based on the above reasonable assumptions, the nonstoichiometry of rutile TiO2-x aS a

function of temperature and oxygen partial pressure has been studied using thermogravimetry

measurement techniques since the early 1960's. Kofstad first gave a detailed derivation of defect

formation in oxygen-deficient rutile TiO2 [9]. Afterwards, different defect models that assume

different defects are dominant were proposed for reduced rutile TiO2. However, there has been

no conclusive experimental evidence to indicate which of the above defect models is, in fact, the

best description of experimental systems. In fact, when the oxygen pressure dependence of x is

expressed as PO2-1 n, it is found that plots of log x vs. 1/T do not yield a straight-line relationship.

This suggests that the defect structure cannot be interpreted in terms of a simple model in which

either oxygen vacancies or titanium interstitials predominate. As a matter of fact,









nonstoichiometric rutile TiO2-x is thermodynamically stable over a wide range of oxygen partial

pressures and temperatures.

The first prestigious thermogravimetric study on nonstoichiometric TiO2 WAS reported in

1962 by Kofstad, showing that at low oxygen pressure and high temperature (1350 K-1500 K),

the weight change of TiO2 is proportional to pO2-1/6 (See Figure 1-5) [9]. This shows the existence

of the fully charged oxygen vacancies. At low temperature (1250 K-1300 K) the results show a

transition in the oxygen pressure dependence from pO2-1/6 to pO2-1/2. Neutral oxygen vacancies

were considered in the rutile, and their formation energy was measured to be 5.6 eV (129

kcal/mol). However, soon it was realized that neither the oxygen pressure dependence of pO2-1/2

nor pO2-1/6 is a stable observation in this temperature range. Foirland also worked on

thermogravimetric studies of rutile [10] (see Figure 1-6). He measured the weight change of the

TiO2 aS a function of temperature (1 133 K-1323 K) and oxygen pressure (13-520 torr) and found

the weight change to be proportional to pO2-1/6, which was interpreted as being due to fully

charged oxygen vacancies. He also measured a formation energy of 3.91 eV for oxygen vacancy

reaction shown in equation (1-10).

In contrast to these studies, Assayag et al. had a different explanation for the predominant

point defects in rutile TiO2 (cited in [9]). They measured the equilibrium oxygen pressure and

corresponding weight loss of a rutile sample while it was heated in oxygen (the oxygen pressure

was between 10-2 and 10-4 atm) over a temperature range of 1318 K-1531 K. They found the

weight loss to be approximately proportional to pO2-1/5, which was attributed to the existence of

fully charged titanium interstitials.

In 1967 Kofstad [24] proposed a new approach to study point defects in rutile TiO2 and

suggested that they simultaneously comprise both fully charged oxygen vacancies and titanium









interstitials with +3 and +4 charges, where the former is dominant at high pO2, and the latter are

predominant at low pO2 (See Figure 1-6 and Figure 1-7).

In the 1960s electrical conductivity measurement was also shown to be an important

experimental technique to explore defect structures in TiO2. For example, Blumenthal et at.

presented the results of their electrical conductivity measurements in the c direction over

temperature range 1273 K 1773 K and oxygen partial pressure range 1 10~1 atm [12] (see

Figure 1-7). They found that the value of n in pO2-1 n that were calculated from the slopes were

not integers, but varied from 4.2 to 4.8 at 1773 K at low oxygen partial pressures, and were

around 5.6 at high oxygen partial pressures at 1273 K. Thus, the nonstoichiometric defect

structure could not be described in terms of either a single predominant defect model or a single

predominant charge state for the defects. They also suggested that the conductivity of rutile in air

below 1223 K appeared to be impurity controlled due to the presence of aluminum rather than

due to intrinsic defects.

Recently, a conductivity experiment performed by Knauth et at. showed that titanium

interstitials were the dominant defect in TiO2. And more interestingly, the conductivity in nano-

and micro-crystalline TiO2 WAS reported to be independent of oxygen partial pressure [13]. The

experimental study of Garcia-Belmonte et at. also indicated that at high defect concentrations the

point defects may not be randomly distributed in the material but were instead clustered or

associated as a consequence of the interactions between the defects and incipient phase

separation [25]. This was proved by the existence of the crystallographic shear planes (CSP)

structure in reduced TiO2. The defect formation energy of a titanium interstitial was reported to

be 9.6 eV, and the formation of a titanium vacancy was reported to be 2.2 eV. They also found

that TiO2 exhibited a p-type regime when the oxygen partial pressure is high (105 > pO2 Pa > 10)









and the temperature is below 1200 K. In contrast, when the oxygen partial pressure is low (10-9

pO2 Pa >10-19), TiO2 exhibits n-type behavior, as expected. At atmospheric oxygen pressures,

high-purity rutile is believed to act as an excellent insulator.

Bak et al. also reported on charge transport in undoped polycrystalline TiO2 USing

electrical conductivity and thermopower measurements. The pO2 Tange WAS between 10 Pa and

70 kPa and the temperature range was 1173 K-1273 K [26]. They found that the slope of log o

vs. log pO2 at low partial pressures were -1/8.3, -1/6 and -1/4.7 at 1173, 1223 and 1273 K,

respectively. At high partial pressures the slopes were 1/7.6 and 1/11.9 at 1173 and 1223 K,

respectively (see Figure 1-8). These values are obviously different from the ideal n values

derived based on the above assumptions that fully charged oxygen vacancies (-1/6) or titanium

interstitials with +3 and +4 charges (-1/4 and -1/5) are the predominant ionic defects. This

departure indicates that there is more than one kind of maj or point defect taking part in

conduction as charge carriers in the studied ranges of pO2 and temperatures in TiO2.

For impurities studies, Slepetys and Vaughan measured the solubility of Al203 in rutile

TiO2 at 1 atm of oxygen over the temperature range 1473-1700 K [27]. They found that the

solubility increased from 0.62 wt% Al203 at 1473 K to 1.97 wt%/ Al203 at 1700 K. This means

that high temperature helps aluminum impurities to remain dissolved in the structure. Also at

1700 K the density of the sample with 1.60 wt% Al203 did not significantly change compared

with undoped TiO2. As a result, they suggested that aluminum dissolves both substitutionally and

interstitially. However, it should be noticed that their conclusions were only valid for the

atmospheric environment.

Considering the influence of the oxygen partial pressure, Foirland showed that the

solubility of Al203 inCreaSed with decreasing oxygen partial pressure in the temperature range









1173- 1473 K in the low partial pressure range (1 to 10- atm) [10]. His results also suggested

that aluminum dissolve in the interstitial sites. In the contrast, Yahia measured the electrical

conductivity of Al203-doped rutile TiO2 aS a function of oxygen partial pressure over the

temperature range 950 1213 K [28]. The conductivity was found to change from n-type to p-

type. This can only be explained by assuming that a large fraction of aluminum was dissolved

substitutionally in addition to existing as interstitial defects.

In summary, two possible temperature regimes have been considered in experimental

studies of defect formation mechanisms in rutile structures. The intrinsic defects are more

thermodynamically predominant at high temperatures, and the concentration of defects can

change over a range of oxygen partial pressures. At low temperatures, the doping of tri- and

pentavalent ions should lead to the n-type conductivity. However the impurity studies discussed

above were all performed at high temperatures, so behavior at lower temperatures is not as well

understood. Additionally, by using electrical conductivity measurement, only charged single

point defects were able to be considered. There is therefore no reported electrical conductivity

study of neutral defects or even intrinsic defect complexes such as Schottky and Frenkel defects.

Thus, a detailed computational examination of all these various findings is necessary to fully

understand the doping mechanisms responsible for all possible defects and impurities in

nonstoichiometric rutile TiO2.

Theoretical Studies of Defect Formation in Rutile TiO2

Although these experimental studies provide important information about the preferred

defect structures in bulk TiO2 and on TiO2 Surfaces, a full understanding of the various defect

structures and formation mechanisms for TiO2, even in the bulk, is still elusive. For this reason

theoretical calculations are employed for defect structures in TiO2 and play an important role that

is complementary to the experimental studies.









Most calculations use one of three classes of theoretical approaches: empirical and semi-

empirical methods such as tight binding and Mott-Littleton methods, molecular dynamics (MD)

simulations, and more theoretically rigorous first principles approaches. The maj or methods in

the third class include Hartree-Fock (HF) methods and density functional theory (DFT) methods.

All of these approaches have been applied to the study of defects in titanium dioxide. For

example, Catlow et al. performed an extensive series of Mott-Littleton calculations on TiO2 and

found that the Schottky defect was energetically more stable than the Frenkel defect in rutile [29-

31]. They also concluded that vacancy disorder will predominate in TiO2. In anOther important

study, Yu and Halley calculated the electronic structure of point defects in reduced rutile using a

semi-empirical self-consistent method [32]. They worked with titanium interstitials and oxygen

vacancies and found donor levels in the range of 0.7-0.8 eV for isolated defects in each case.

They also predicted the presence of defect clustering in nearly stoichiometric rutile with multiple

defects .

Although empirical studies are instructive and provide good insight into point defect

behavior in TiO2, these calculations cannot provide enough predictive and accurate information

about defect formation. In the worst case, they may lead to wrong conclusions. For example, the

potential parameters for atomistic simulations are generally determined from perfect crystal

properties such as cohesive energy, equilibrium lattice constants and bulk modulus. This may

cause discrepancies due to the complex nature of metal oxides such as TiO2 and the different

chemical environments in the perfect and defective lattices. Another problem is related to the

charge state of the defects in the system. In the empirical potential calculations, it is difficult to

deal with variable charges on defects and impurities. Thus the defects are usually treated as

either neutral or fully charged, which obviously does not include all possible states.









In last few years, DFT calculations using reasonably sized supercells have become more

popular and practical for the study of defects in transition metal oxides such as TiO2. For

instance, the active surface site responsible for the dissociation of water molecules on rutile TiO2

(110) was explored by Schaub et al. by using DFT calculations combined with STM experiments

[33]. Their results showed that the dissociation of water is energetically possible only at oxygen

vacancies sites. Dawson et al. also used the DFT method to study point defects and impurities in

bulk rutile [34]. Their results show that isolated Schottky and Frenkel defects are equivalent

energetically. The Schottky defect formation energy (DFE) was calculated using two different

approaches. In the first approach, the defect formation calculation involved calculating the

formation energy of the Coulombically bound Schottky trio Vn 2Vo in a 72 atoms supercell

(2x2x3 unit cell). The Schottky formation energy of this model was calculated to be 4.66 eV. In

the second approach, the Schottky formation energy was expressed as the sum of the energies of

one isolated titanium vacancy and twice the value of one isolated oxygen vacancy minus the

cohesive energy per unit cell. The isolated defect calculations were carried out on a 12 atom cell.

The Schottky formation energy was calculated to be 17.57 eV using this approach. The Frenkel

formation energy was also calculated through the combination of one titanium interstitial and

titanium vacancy in an unrelaxed 72 atom cell. The value of the Frenkel DFE was found to be

17.72 eV. While this study considered variations in the structures of the Schottky and Frenkel

defect complexes, the system size (of a 12 atom supercell) was quite small and may have

introduced self-interaction errors into the formation energy. An additional problem is that the

calculation of DFE of the Frenkel defect was carried out using an unrelaxed supercell because of

the high computational cost associated with allowing atomic relaxation. This is problematic, as it

is well known that atomic relaxation influences system energies.









Recently, a few DFT studies have been reported on the topic of intrinsic defect and

impurity formation in bulk TiO2. For example, Sullivan and Erwin reported their first-principles

calculations of the formation energy and electrical activity of Co dopants and a variety of native

defects in anatase TiO2 [35]. They found that under oxygen-rich growth conditions the Co

dopants would be formed primarily in neutral substitutional form, which conflicts with the

experimentally observed behavior of Co-doped samples (see Figure 1-9). Thus, they concluded

that the growth conditions were most likely oxygen poor. When they considered oxygen-poor

conditions they predicted roughly equal concentrations of substitutional and interstitial Co.

Na-Phattalung et al. also investigated intrinsic defect formation energies in anatase TiO2 at

0 K without considering the temperature influence and electrostatic interaction correction [36].

They found that the titanium interstitials (Ti,) has very low formation energy in both n-type and

p-type samples. Thus they believed that titanium interstitials are the strongest candidates

responsible for the native n-type conductivity observed in TiO2. (See Figure 1-10) They also

predicted that Vo has a higher formation energy than Ti,. However, after considering the lower

kinetic barrier needed to create Vo relative to the barrier to create Ti, from perfect TiO2, they

suggested that the post-growth formation of Vo is also possible, especially after the sample has

been heated for a prolonged time.

Cho et al. reported DFT calculations of neutral oxygen vacancies and titanium interstitials

in rutile TiO2 [37]. They calculated the DFEs for these two defects and found them to be 4.44 eV

and 7.09 eV for the oxygen vacancy and Ti interstitial, respectively. This indicates that the

formation of oxygen vacancy is energetically favored. However neither the charge state nor the

temperature/oxygen partial pressure was considered in this study. Weng et al. also performed









DFT calculations for Co-doped rutile and anatase TiO2. They found that the Co prefers to

substitute on the Ti site [38].

Theoretical Studies of Electronic Structure

A detailed understanding of Ti-O bonding is essential for explaining n-type

semiconducting behavior in rutile TiO2, Of CVen more generally, the electronic properties of

transition metal oxides. Detailed information about the band structure and density of states

(DOS) for perfect rutile has been reported by Glassford and Chelikowsky [39], and Mo and

Ching [40]. The calculated total DOS is in good agreement with experiment [39]. (see Figure 1-

1 1) Importantly, these calculations found a significant degree of covalent bonding in the charge

density contour map (see Figure 1-12). Since the focus of this dissertation centers on the

computational and theoretical study of defective structures, we will not discuss these studies in

detail. However, it should be indicated that the electronic structure of pristine rutile is a good

benchmark reference for the defective structure.

A lot of theoretical works have been devoted to the study of defective TiO2 Structures,

beginning with an early tight-binding study by Halley et at. [41], a linear muffin-tin orbital

(LMTO) study by Poumellec et al. [42], a DFT study by Glassford and Chelikowsky [43], and a

full-potential linearized augmented plane wave (FLAPW) formalism by Asahi et at. [44].

For example, Halley et at. presented a tight-binding calculation to describe the electronic

structure of a defective TiO2 with oxygen vacancies [41]. In Figure 1-13 they show the effects of

different concentrations of oxygen vacancies at randomly selected sites on the DOS. They found

donor states tailing into the band gap below the conduction band that increases with vacancy

concentration. In addition, Poumellec et at. reported electronic structure, LMTO method

calculations [42]. Their results show that there is a significant O2p-Ti3d mixing in the valence

band and a weaker O2p-Ti3d mixing in the conduction band. This contradicts the general









assumption that TiO2 is an ionic compound. They also mentioned that the non-cubic environment

may allow Ti4p-Ti3d hybridization, which would explain the observed pre-edge and edge features

in the Ti x-ray absorption spectrum.

Cho et al. reported DFT calculations on neutral oxygen vacancies and titanium interstitials

in rutile TiO2 [37]. They found that the oxygen vacancy does not give rise to a defect level

within the band gap while titanium interstitial create a localized defect level 0.2 eV below the

conduction band minimum (see Figure 1-14). Na-Phattalung et at. also investigated the native

point defects in anatase TiO2 USing DFT calculations and found that the defect states for Ti, and

Vo were predicted to be the Ti d states above the conduction band minimum (see Figure 1-15).

Impurity doping is always important for the study of defects in semiconducting metal

oxides. In order to achieve a high free-carrier concentration at low temperature, a high

concentration of dopant impurity is obviously required under conditions of thermodynamic

equilibrium. Generally the dopant concentration depends on temperature, oxygen partial

pressure, and the abundance of the impurity as well as the host constituents in the growth

environment.

For cation impurity doping, frequently studied impurities include ruthenium, cobalt, and

cadmium. For instance, Glassford and Chelikowsky studied Ru doping in rutile TiO2 USing DFT

[43]. They found that the Ru-induced defect states occur within the TiO2 band gap about 1 eV

above the O 2p band, which is in good agreement with absorption and photoelectrochemical

experiments (see Figure 1-16). Additionally, these states were found to be localized on the Ru

with tz,-like symmetry.

In their DFT DOS calculations and absorption analysis for Co-doped rutile and anatase

TiO2, Weng et at. suggested that the p-d exchange interaction between the O 2p and Co 3d









electrons should be ferromagnetic, which means that intrinsic ferromagnetism should occur in

the Co-doped TiO2 Systems (see Figure 1-17). This is proven by their optical magnetic circular

dichroism (MCD) spectra measurement.

Errico et at. presented details about the electronic structure of the Cd-doped TiO2 USing

FLAPW methods and the results show that the presence of Cd impurity leads to the Cd-s levels

at the bottom of the valence band, and impurity states at the top of the valence band (see Figure

1-18). In addition, the Cd was found to introduce fairly anisotropic atomic relaxation in its

nearest oxygen neighbors [45].

While considering that cation metal impurities often give quite localized d states deep in

the band gap of TiO2 and result is a recombination center of carriers, Asahi et at. suggested using

anion dopants, instead of cation dopants, in order to ensure photoreduction activity [44]. They

calculated the DOS of the substitutional doping of C, N, F, P, or S for O in the anatase TiO2

crystal using the FLAPW formalism. What they found is that the substitutional doping ofN was

the most effective doping because its p states contribute to the band-gap narrowing by mixing

with O 2p states. Furthermore, due to its large ionic radius and much higher formation energy, S

doping in the structure was believed to be difficult (see Figure 1-19).

Lastly, Umebayashi et at. reported the band structure of S-doped anatase TiO2 USing ab

initio DFT calculations and found that the band gap gets narrower due to the S doping into the

substitutional site. This obviously originates from the mixing of the S 3p states with the O 2p

states in the valence band [46].

Summary

This chapter gives a brief introduction to the experimental and theoretical studies of

defects and impurities in TiO2. The defect formation behavior and electronic property changes

are also discussed. It should be noted, however, that the influence of crystal structure on various









point defects is still poorly understood due to the experimental and theoretical complicacy

involved in such determinations. Additionally, many computational studies of TiO2 do not

consider the influence of temperature and oxygen partial pressure. Therefore it is truly necessary

to perform DFT calculations combined with thermodynamic calculations to obtain a more

thorough understanding of the defect chemistry in rutile TiO2.









Table 1-1. Selected bulk properties of tile TiO2.
Values


Atomic radius (nm)
Ionic radius (nm)
Melting point (Kelvin)
Standard heat capacity at 298K (J/mol oC)
Linear coefficient of thermal expansion at 0-
500K (ax10-6, o-1)
Anisotropy of linear coefficient of thermal
expansion at 30-650K (ax10-6, o -1)


O 0.066 covalentt); Ti 0.146 (metallic)
O(-2) 0.14; Ti(+4) 0.064
2143
55.06
8.19

Parallel to c-axis, a=8.816x10-6+3.653x 10-9xT
+6.329x10-12xT2;
Perpendicular to c-axis, a=7.249x 10-6
+2. 198x10-9xT+1.198x1 0-12xT2.


Dielectric constant

Band gap (eV)


Perpendicular to c-axis, 160;
Along c-axis, 100;







[coill
1.946i A titanium

Oxygen

[100]- 1.983 A










l01] -4~ ~~ [100]


Rutiile


Anatase


S102.3018"


loll q


0[


Figure 1-1. Bulk structures of tile and anatase TiO2 [2].


O


Cr
CY


o


6


Figure 1-2. Diagram of the TiO6 Octahedral structure in rutile TiO2.










t, *


1600 .

b' L rl

1200 .10f \__ .


TizO E
800

a+
"TiO Tit
400 I
0 0.4 0.8 1.2 1.6 2.0
O/TTi ratio

Figure 1-3. Phase diagram of the Ti-O system [18].




TiO2-x o -TiO2.
-2 1073 K, A 0) 1273 K. A-


Ing p(O,} [p in Pa] log p(O, [p in Pa]


Figure 1-4. Calculated defect concentration in undoped TiO2 (A=0) at different temperature
ranges as a function of pO2 USing reported equilibrium constants [14].



















































ri 1


e,A cock et aL
D ,Forland
a,M~oser et ol.
o,Kotstad
o,Atlas and Schlehman


10-mB 10-"8 10-"


~ g17C. '~1077.C









-18 -16 -R -12 -1 8
LOGy 902 atr



Figure 1-5. The logarithm of weight change of rutile as function of logarithm of oxygen partial
pressure [9].


\ X5" Pb'
n
O\n `


~13000C


1100"C


lo^3

O
r
Z
X

ID-"


'"F;'


1000"C


r
~
Y~r''


I i i~


10-"2 10-1 108 1


OXYGEN PRESStJRE,atm.


Figure 1-6. Thermogravimetric measurement of x in TiO2-x aS function of oxygen partial
pressure [23].























































log p(O } [ p(O ) in Pa]
















Electrical conductivity measurement of TiO2-x aS function of oxygen partial pressure
[26].


120 1500"C
'E 1200t












E lb2 i l l l l l ll il

-to 10~'( 10~" 10-so rTo 10 0D' X)
OXYGEN PRESSURE,atra



Figure 1-7. Electrical conductivity measurement of TiO2-x aS function of oxygen partial pressure
[12].


r
E
C
r



s06 I


Figure 1-8.





























120 2 LL~ ~ 30 ~ L~ 1 0123 1 2 3
EF (eV) EF (eV) EF (eV) EF (eV)


Figure 1-9. Defect formation energies of (a) Con (b) Co.,,t (c) Con Vo, (d) Vo defects as a
function of Fermi level in the oxygen-rich limit [35].







6- -I 6-
-(02 O T
4- -- 4-







-2 -1 -2-

-4 -1 -4--
Ti.

-6 -- -6 -
ITi-rich O-rich
g a~ I s a i l g I s t i a
0 12 301 2 3
Fermi Energy (eV) Fermi Energy (eV)


Figure 1-10. Defect formation energies as a function of the Fermi level, under the Ti-rich (left
panel) and oxygen-rich (right panel) growth conditions, respectively. The slope of the
line is an indication of the charge state of the defect. The band gap is set to be the
experimental band gap. The vertical dotted line is the calculated band gap [36].






















Energy (eV)


Figure 1-11. Calculated total DOS for TiO2 per unit cell compared to experimental UPS and
XAS spectra for TiO2 (110) surface [39].


I j:?rbl~ I I Io Ij:


Figure 1-12. Calculated valence density difference maps for (a) (110) and (b) ( 110) lattice
planes. (c) shows the experimental electron density map in ( 1 10) plane in rutile TiO2
[39].













0%





f~-L


EF EF
S400 Tiintersttial-
Bulk-----
300 I

20 -oo 1 I -


-4 -2 0 2 4 ,
Energy (eV)
Ti~eEF -

- i, ;;


-20) -X10
Enery (eV)


0 10


Figure 1-13. Calculate density of states for the 8x8x10 rutile model with 0, 1, 5, and 10%
oxygen vacancies on a large energy scale showing the development of a tail of donor
states below the conduction band minimum [41].


400 O vacancy -
Bulk ---

200 ~


--2 02
Energy (eV)


8

EF


4


"2


I


"


2
5
0'
m
1
% 0,5
Oii


Tia


-4 -2 0 2
Energy (eV)


4 6


-4 -2 02
Energy (eV)


4 6


Figure 1-14. Comparison of density of states (DOSs) between defective and perfect rutile TiO2.
(a) Total DOS for TiO2 with the oxygen vacancy compared with that of the perfect
crystal. The Fermi levels EF are Shown as vertical lines. (b) Partial DOS for the
supercell containing the oxygen vacancy. Tinear indicates one of three Ti atoms
neighboring the vacancy site and Tirar is the Ti atom furthest from the vacancy site.
(c) Total DOS for TiO2 with Ti interstitial compared with that of the perfect crystal.
(d) Partial DOS for the interstitial Ti atom Tiint and the Ti atom furthest from the
interstitial site [37].











Energy eV)
-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4


Figure 1-15. Calculated total and partial density of states (DOSs) of anatase TiO2 [36].


-9 -6 -3 0 3 6 9
Energy (eV)


Figure 1-16. Calculated density of states (DOSs) of rutile-structured RuxTil-xO2 with different
Ru concentrations compared with experimentally determined spectra [43].





































!-1 01 2 34 5 --5 4 3 2-1 I1 2 34
Energy (eV) Energy (eV)


Figure 1-17. Calculated spin polarized density of states (DOSs) of the Co-doped anatase (left)
and rutile (right) Til-xCoxO2 [38].


120. (a)










-8 -6 -
Enrg (V

Fiue11.Cluae O o h a eae eetv srcuewt b eae

defetiv stutr it d n (c) urlxddfciesrcuewt 4]






















-10 -5 0 5 10 -10 -5 0 5 10
E (eV)

Figure 1-19. Total (A) and partial (B) density of states (DOS) for doped anatase TiO2 calculated
by FLAPW [44].









CHAPTER 2
INTRODUCTION OF DENSITY FUNCTIONAL THEORY AND ITS APPLICATION IN
DEFECT STUDY

The success of density functional theory (DFT) is clearly demonstrated by the numerous

books, reviews and research articles that have been published in the last two decades reviewing

development of the theory and presenting results obtained with DFT [47]. Many researchers in

this field can be divided into three classes: those who developed the fundamentals of the theory

and/or new extensions and functionals, for example, W. Kohn, L.J. Sham, J.P. Perdew, and D.

Vanderbilt; those theoretical scientists who are concerned with numerical implementation, for

example, R. Car, M. Parrinello, M.C. Payne, and J. Hafner; and those application scientists the

vast maj ority who use the codes to study materials and processes that are important to different

research areas.

It is important that application scientists should have a sound knowledge of both the theory

and its applications, and understand its limitations and numerical implementation. Consequently,

in this chapter, a brief overview of density functional theory is provided in the first section. Then

in the second section, which is the main focus of this chapter, the application of DFT methods in

materials science is reviewed, with a particular emphasis on the study of point defects. Particular

attention is paid to the computationally technical aspects that are unique to defect calculation.

For example, questions about how to apply the supercell approximation to describe real

materials, how to deal with the band gap problem and interpret defect transition levels, and how

to consider charge states and charge compensation. It should be pointed out that there are some

additional issues that influence the results, such as overbinding, self-interaction, dipole

interaction, and strong correlation effects. However, these problems are minimal in solid-state,

total energy calculations of bulk TiO2, So these issues are not discussed in this dissertation.









Overview of Density Functional Theory


Kohn-Sham Theory

Solid state physics explicitly describes solids as a combination of positively charged nuclei

and negatively charged electrons. If there are only time-independent interactions in the system,

and the nuclei are much heavier than the electrons, the nuclei can be considered to be static

relative to the electrons. This greatly simplifies the system, which can be treated as an isolated

system with only N interacting electrons moving in the (now external) potential of the nuclei

while maintaining instantaneous equilibrium with them. This is the idea behind the Born-

Oppenheimer nonrelativistic approximation. Using this approximation, the classic Schroidinger

equation can be simplified and described as [48]

HfF = A'F (2 -1)

where E is the electronic energy, 7 is the wave function, E is the total ground-state

energy, and H is the Hamiltonian operator. These latter two terms are described as

H~~7 88 Y,+ e (2-2)


E = T + V,2e + Ve = C(--N Vi)+Icr,)+C1- (2-3)
1=1 2 1=1 I
On the right side of equation (2-3) there are only three terms that need to be evaluated. The

first term, T, is the kinetic energy of the electron gas; the second one, Vne, is the potential

energy of the electron-nucleus attraction; and the third term, Ve, is the potential energy due to

electron-electron interactions. This many-body problem, while much easier than it was before the

Born-Oppenheimer approximation, is still far too difficult to solve.

Several methods have been developed to reduce equation (2-2) to an approximate but more

readily solvable form. Among these approaches, the Thomas-Fermi-Dirac method, the Hartree-









Fock (HF) method, and the density functional theory (DFT) method are the most prevalent. Only

the DFT method will be discussed in this dissertation.

The DFT method was formally established in 1964 by the theoretical formulation proposed

by Hohenberg and Kohn (Hohenberg-Kohn theorem), and has been widely used in practice since

the publication of the Kohn-Sham equations in 1965 [49]. In the Kohn-Sham formula, the

ground-state energy of a many-body system is a unique functional of the electron density,

Eo = E[p] In addition, the electron density is parameterized in terms of a set of one-electron

orbitals representing a non-interacting reference system as p(r)= |I ((r) |2 The Hamiltonian


and total-energy functional can therefore be described as


He = To ,+ Vse + Ve+ (2-4)


E,[p]= T[p]+V.,+Ve,[p]+Vxc[p] (2-5)

In equation (2-5), the ground-state energy is in the Kohn-Sham form, EKsIp], which is

defined as the sum of the kinetic energy Tlp], the external potential energy Vne, the electron-

electron potential energy Vnelp], and the exchange-correlation (xc) potential energy Vxcho].

Although the exact form of Tlp] for a fully interacting set of electrons is unknown, the kinetic

energy of a set of non-interacting electrons, T, is known exactly. Kohn and Sham proposed an

indirect approach to use this well-defined T (which is Thomas-Fermi energy, TTFIp]) and

combine the interacting kinetic energy terms in with the xc term, Vxclp]. This key contribution to

the DFT theory lead W. Kohn to be one of the winners of the 1998 Nobel Prize in Chemistry.

The next term in equation (2-5) is the external potential energy, Vne, which is simply the

sum of nuclear potentials centered at the atomic positions. In some simple cases, it is just the

Coulomb interaction between the nucleus and the electrons. However, in most cases, in order to










describe the strong oscillation of the valence electron wave-functions in the vicinity of the

atomic core due to the orthogonalization to the inner electronic wave-functions, enormous wave

functions are needed in the calculation, especially for large Z atoms. In such cases, it is not

feasible to calculate the Coulomb potential in the plane-wave basis-set. Under such conditions,

the inner electrons can be considered to be almost inert and not significantly involved in

bonding, which allows for the use of the pseudopotential approximation to describe the core

electrons.

The third term in equation (2-5) is the electron-electron potential energy, V,,lp], which is

simply calculated as Hartree energy, which is the classic electrostatic energy of a charge

distribution interacting with itself via Coulomb's law. And the last term in equation (2-5) is the

exchange-correlation (xc) potential. In the last 30 years, over one hundred xc functional

approximations have been proposed in the literature. The most famous ones are the local density

approximation (LDA), generalized gradient approximation (GGA), local spin density

approximation (LSDA), local density approximation with Hubbard U term (LDA+U), exact

exchange formalism (EXX), and numerous hybrid functionals such as Becke three parameter

hybrid functional with the Lee-Yang-Parr non-local correlation functional (B3LYP).

Finally, the standard procedure to solve equation (2-5) is iterating until self-consistency is

achieved. A flow chart of the scheme is depicted in Figure 2-1. The iteration starts from a

guessed electron density, po(r). Obviously using a good, educated guess for po(r) can speed-up

convergence dramatically compared with using a random or poor guess for the initial density.

Exchange-Correlation Functional

The exchange-correlation (xc) potential, Exc[p], is the sum of exchange energy and

correlation energy as










E, [p] = Ex [p]+ Ec[p] (2-6)

The exchange energy, Ex[p], is defined as the energy difference between true electron-

electron potential energy, Veelp], and direct Hartree energy, Ulp]. The correlation energy, Ec[p],

is defined as the difference between the ground state Kohn-Sham energy, EKSIp], and the sum of

the Thomas-Fermi energy, Hartree energy, Ulp], and exchange energy, Ex[p].

According to the second Hohenberg-Kohn theorem, there should be an exact form of the

exchange-correlation functional, Exc[p], to calculate the ground state energy of any system.

However, the explicit form of this functional remains unknown. The problem is that there is no

way to independently determine if a new functional is the one and only exact form. Instead, new

functional are developed and assessed by how well they perform, which involves a detailed

comparison of the predicted properties, such as lattice parameters, bulk properties, and band

structure, with the experimental data.

As the first exchange-correlation functional approximation and in fact the simplest of all,

the local-density approximation (LDA) was proposed by Kohn and Sham in 1965 as


EMA[p]= Id3 HEG(p(r)). (2-7)

where SHEG@(p()) iS the xc energy per unit volume of the homogeneous electron gas (HEG)

of density p, and can be tabulated using the Monte Carlo method by Ceperley and Alder [50].

The xc energies for charged Fermi and Bose systems are calculated by fitting the Green's-

function Monte Carle data in an exact stochastic simulation of Schroidinger equations. A number

of different parameterizations have been proposed for this function over the years, and it has

been shown that the LDA is suitable for systems with slowly-varying densities. However, this

approximation has some serious shortcomings, especially when it is used to study transition

metals and metal oxides where correlation effects are important.









The next step to improve the exchange-correlation functional is to include the gradient

correction to xc functional. The electron density gradients will help the approach describe

systems where the electron density is not slowly varying. This is the starting point of the

generalized gradient approximation (GGA). In this case, the functional has a similar form as in

equation (2-7), but now e depends not only on the density p, but also on its gradient, Vp. The

evaluation of the GGA xc potential is fairly straightforwardly computed as


EG"'[p]= Jd'reHEG(p(r))(1+ f.(Vp)) (2-8)

where the f(Vp) is a Taylor expansion of gradient V p.

Comparing these two approximations, the LDA functional derived from electron gas data

does work surprisingly well for many systems. However, it substantially underestimates the

exchange energy (by as much as 15%) and grossly overestimates the correlation energy,

sometimes by 100% due to the large error in the electron density. As result, it typically produces

good agreement with experimental structural and vibrational data, but usually overestimates

bonding energies and predicts shorter equilibrium bond lengths than are found in experiments.

In contrast, the GGA functional finds the right asymptotic behavior and scaling for the

usually nonlinear expansion in the Taylor expansion. It shows surprisingly good agreement with

Hartree-Fock-based quantum chemical methods. However, there is much evidence to show that

GGA is prone to overcorrect the LDA result in ionic crystals, and it overestimates cell

parameters due to the cancellation of exchange energy error in LDA. Since real systems are

usually spatially inhomogeneous, the GGA approximation is typically more accurate in studies

of surfaces, small molecules, hydrogen-bonded crystals, and crystals with internal surfaces.

Unfortunately, both of these xc approximations give poor eigenvalues and small band gaps in

many systems due to the discontinuity in the derivative of the xc energy functional.










Finally, despite the success of the LDA and GGA they are far from ideal, and finding an

accurate and universally-applicable Exc remains great challenge in DFT. Ongoing efforts to

discover the next generation of density functionals includes developing orbital-dependent

functionals, such as the exact exchange functional (EXX), and constructing hybrid functionals

which have a fraction of the exact exchange term mixed with the GGA exchange and correlation

terms.

Pseudopotential Approximation

In the DFT plane-wave calculation, the electron wave functions can be expanded using a

series of plane waves. However, an extremely large plane wave basis set would be required to

perform an all-electron calculation because the wave-functions of valence electrons oscillate

strongly in the vicinity of the atomic core due to the orthogonalization to the inner electronic

wave-functions (see Fig. 2-2). This calculation is almost impossible since vast amounts of

computational time would be required.

Fortunately, it is well known that the inner electrons are strongly bound and are not

involved significantly in bonding. Thus, the binding properties are almost completely due to the

valence electrons, especially in metals and semiconductors. This suggests that an atom can be

described solely on its valence electrons, which feel an effective interaction (that is the

pseudopotential) including both the nuclear attraction and the repulsion of the inner electrons

(see Fig. 2-2). Therefore, the core electrons and nuclear potential can be replaced by a weaker

pseudopotential that interacts with a set of modified valence wave functions, or

"pseudowavefunctions," that are nodeless and maximally smooth within some core radius. The

pseudowavefunctions can now be expanded in a much smaller basis set of plane waves, saving a

substantial amount of computer time.









The concept of pseudopotential was first proposed by Fermi in 1934 and Hellmann in 1935

[52, 53]. Since then, various pseudopotential approximations have been constructed and

proposed. Initially, pseudopotentials were parameterized by fitting to experimental data such as

band structures. These were known as empirical pseudopotentials. In 1973, a crucial step toward

more realistic pseudopotentials was made by Topp and Hopfield [54]. They suggested that the

pseudopotential should be adjusted such that it describes the valence charge density accurately.

Based on this idea, the modern ab initio pseudopotentials were constructed by inverting the free

atom Schroidinger equation for a given reference electronic configuration. More importantly, the

pseudo wave functions were forced to coincide with the true valence wave functions beyond a

certain distance, and to have the same norm (charge) as the true valence wave functions. The

potentials thus constructed are called norm-conserving pseudopotentials. There are many, widely

used norm-conserving pseudopotentials. One of the most popular parameterizations is the one

proposed in 1990 by Troullier and Martins [55, 56].

However in some cases, norm-conservation still results in "deep" pseudopotentials and

therefore requires large cutoff energies. As a result, the pseudopotential is less transferable

without gaining enough smoothness. In 1990, Vanderbilt proposed an ultrasoftt" pseudopotential

[57] where the norm-conservation constraint was abandoned, and a set of atom-centered

augmentation charges was introduced. In this case, the pseudo wave functions could now be

constructed within a very large distance, allowing for a very small basis sets. Vanderbilt' s

ultrasoft pseudopotentials are most advantageous for the first row of the periodic table and

transition metals. Its accuracy has been found to be comparable to the best all-electron first-

principles methods currently available [58].









Implementation and Benchmark Test

For studies described in this dissertation, we employed DFT calculations as implemented

in the CASTEP code [59]. CASTEP uses a plane-wave basis set for the expansion of the Kohn-

Sham wavefunctions, and pseudopotentials to describe the electron-ion interaction. A few

pseudopotentials can be used in this study, such as norm-conserving pseudopotential generated

using the optimization scheme of Troullier-Martins (pspnc potential), norm-conserving

pseudopotential generated using the optimization scheme of Lin et al. (recpot potential), and

ultrasoft pseudopotential (usp potential) [60]. After several benchmark tests, the ultrasoft

pseudopotential was chosen (see Table 2-1 and the corresponding discussion).

Two separate exchange-correlation energy approximations can be employed in the study:

the LDA as parameterized by Perdew and Zunger [61], and three GGA functionals (PW91 in

form of Perdew-Wang functional [62]; PBE in form of Perdew-Burke-Ernzerhof functional [63];

and RPBE in form of Revised Perdew-Burke-Ernzerhof functional [64]). Based on our tests, the

GGA of Perdew-Burke-Ernzerhof (PBE) is best suited towards our studies.

The sampling of the Brillouin zone was performed with a regular Monkhorst-Pack k-point

grid. The Monkhorst-Pack grid method has been devised for obtaining accurate approximations

to the electronic potential from a filled electronic band by calculating the electronic states at

special sets of k points in the Brillouin zone [65]. The ground state atomic geometries were

obtained by minimizing the Hellman-Feynman forces which is defined as the partial derivative

of the Kohn-Sham energy with respect to the position of the ions [66, 67] using a conjugate

gradient algorithm [68]. The ground state charge density and energy were calculated using a pre-

conditioned conjugate gradient minimization algorithm coupled with a Pulay-like mixing scheme

[69, 70].









Since the simulation of condensed phases is concerned with a large number of electrons

and a near infinite extension of wavefunctions, it is necessary to use a relatively small atomistic

model. The effect of edge effects on the results can be decreased by implementing periodic

boundary conditions (PBC), in which a "supercell" is replicated throughout space. By creating an

artificially periodic system the periodic part of the wavefunction is allowed to expand in a

discrete set of PW' s whose wave vectors are the reciprocal lattice vectors of the crystal structure.

In the supercell all the atoms are relaxed from their initial positions using the Broyden-Fletcher-

Goldfarb-Shanno (BFGS) Hessian update method until the energy and the residual forces are

converged to the limits that are set prior to running the DFT calculation [71]. The BFGS method

uses a starting Hessian that is recursively updated during optimization of the atomic positions.

The main advantage of this scheme is its ability to perform cell optimization, including

optimization at fixed external stress. In this study the convergence criteria for energy is 0.001

eV/atom and for residual forces is 0.10 eV/A.

Table 2-1 presents the results several benchmark DFT calculations. This system consisted

of one unit cell of rutile TiO2. It allowed us to gain a first impression of the capabilities and

limitations of DFT as applied to the TiO2 System. The calculations were performed using both

the LDA, in the parameterization of the Perdew-Zunger functional, and GGA, in the

parameterization of Perdew-Burke-Ernzerhof functional. As the pseudopotential, we took the

ultrasoft pseudopotential (usp) and one norm-conserving pseudopotential optimized in the

scheme of Lin et al. (recpot) [72].

The results in Table 2-1 show three important points.

First, GGA always overestimated the cell parameters, and the LDA was accurate in

determining the cell parameters when using recpot pseudopotential. However, LDA









underestimated the cell parameters when using the usp pseudopotential. Generally the universal

feature of the LDA and GGA should be that the LDA tends to underestimate lattice parameters,

which are then corrected by the GGA to values closer to the experimental results. However, we

also found that the underestimation/overestimation of lattice parameter also depends on the

pseudopotentials used in the calculations. For example, when using the norm-conserving recpot

pseudopotential, both the LDA and GGA overestimated the lattice parameters, although the

parameters in LDA calculation was just slightly overestimated.

Secondly, the calculations using ultrasoft pseudopotential (usp) with the LDA and real

space were not well converged comparing with the other cases. The use of ultrasoft

pseudopotentials were acceptable only in reciprocal space.

Finally, there are no great differences in computational time among these various xc

approximations. Thus, the best choice of approximations for use in our study is the combination

of GGA + ultrasoft pseudopotential + reciprocal space.

Application of Density Functional Theory in Defect Study

Supercell Approximation

In the DFT method, the supercell approximation is the most common approach for perfect

structure calculations and is also being widely used in defect structure calculations. In this

approximation the artificial supercell is composed of several primitive compound unit cells that

contain the defects) or impurity atom(s) that are surrounded by host atoms. The entire structure

is periodically repeated within the PBC condition. The symbolization used to describe the size of

the supercell is lxmxn, where 1, m, n is the number of repeated unit cell in the x, y, and z

directions, respectively. Within the supercell, the relaxation of several shells of host atoms

around the defect or impurity should be included. If the size of supercell is large enough, the

defects are considered to be well isolated. This is the idea behind the ideal dilute solution model.









In order to check the influence of the supercell size on the defect formation energies

(DFEs), we calculated the formation energy of a neutral oxygen vacancy for different supercells.

Here we used three methods: DFT single point energy calculation, where the defect and its

surrounding ions are not allowed to relax; DFT geometry optimization calculation, where the

defect and its surrounding ions are allowed to relax; and empirical potential calculations

implemented in the General Utility Lattice Program (GULP) [73, 74]. GULP can be used to

perform a variety of types of simulation on materials with tuned interatomic potential models

employed in boundary conditions. The results are shown in Figure 2-3. When the system size

increased from 1xlxl to 2x2x3, the DFEs of geometry optimization calculations decreased from

6.09 eV to 5.06 eV. It is clearly shown that the larger supercell leads to a more realistic DFE for

a single, isolated defect system. For the single point energy calculation and Buckingham

potential calculation, the predicted formation energies do not change significantly when the

supercell size is increased from 1xlxl to 3x3x4. Thus, the geometry optimization method takes

into account lattice strain much better than the other two methods.

Band Gap and Defect Levels

Both the LDA and GGA approaches are well known to give an underestimated value for

the band gap of semiconductors and insulators. In fact, even if the true xc potential was known,

the difference between the conduction and valence bands in a Kohn-Sham calculation would still

differ from the true band-gap. The true band-gap may be defined as [47]

Egp ; = n e (N 1)-ef(N). (2-9)


However the calculated Kohn-Sham band gap for the difference between the highest

occupied level and the lowest unoccupied level of the N-electron system is












= t+1I [ss (N 1-s s _[ (tN +1)VTI-s sNL E(N) (2-10)
= E Axc

Thus the Axc in equation (2-10) represents the shift in the Kohn-Sham potential due to an

infinitesimal variation of the density. In another words, this shift is rigid and is entirely due to a

discontinuity in the derivative of the xc energy functional. In chapter 5 we will discuss how to

implement a lineup to shift the conduction band rigidly upward in order to match the

experimental band gap.

In most cases, point defects and impurities introduce defect levels in the band gap or near

the band edges of the semiconductors. The experimental detection of these levels often forms the

basis for the identification of the defect or impurity. On the other hand, these defect levels can

also be characterized theoretically by different methods such as Kohn-Sham eigenvalues and

defect transition levels (ionization levels). Since the Kohn-Sham eigenvalues do not account for

the excitation aspect and thus cannot be directly compared with the experimental literature, only

the defect transition levels are calculated and discussed in this dissertation.

The thermodynamic defect transition level s(ql/q2) is defined as the Fermi-level position

where the charge states ql and q2 have equal energy. This level can be observed in deep-level

transient spectroscopy experiment or be derived from temperature-dependent Hall data [75, 76].

The formula to calculate the thermodynamic transition level is shown as below (Ti interstitial is

given as example)


F(q, / q2)= E'oraz(Tii,,q) E'oraz(Ti,, q,) EVA1 (2-11)

where Etotai(Ti,, q,) and Etotal gi,, q, ) are the total energy of a supercell with a Ti interstitial

with charge q2 and ql, respectively and Erm is the energy of valance band maximum.










Charge State and Compensation

In defect calculations usually a certain charge (electron or hole) is assigned to the defect in

the supercell. This charge (electron or hole) is then completely delocalized over the supercell.

Therefore, a neutralizing j ellium background is applied to the unit cell for calculations of

charged systems. The interaction of the defect with the j ellium background should counteract the

interaction of the defect with its spurious periodic images. However, the energy of such a system

still converges very slowly as a function of the linear dimensions of the supercell [77]. In order

to overcome this shortcoming, Makov-Payne compensation is applied in this study. Details of the

Makov-Payne correction and its use here is given in chapter 4.








































a(A) 4.550 4.547 4.634 4.630 4.594
c (A) 2.924 2.927 2.964 2.964 2.956
Ti-O short (A) 1.934 1.933 1.958 1.957 1.949
Ti-O long (A+) 1.952 1.952 1.997 1.996 1.980
CPU time (s) 650.86 73.49 503.68 127.26-
Convergence Stress(0.2 All Stress(0.24 All-
1 GPa) not converged GPa) not converged
converged converged
Total Energy -4962.27 -4962.25 -4973.64 -4973.59-
(eV)


Table 2-1. Calculated lattice constants, CPU times, convergence data, and total energies of rutile
TiO2-


LDA
500 eV
cutoff
recpot,
real-space
4.605
2.991
1.967
1.979
130.78
All
converged

-1907.80


LDA GGA-PBE
500 eV cutoff 500 eV cutoff


GGA-PBE Experiment
500 eV cutoff [2]
recpot,
reciprocal


recpot, real-
space

4.632
3.002
1.972
1.996
213.37
Stress(0.32
GPa) not
converged
-1915.18


recpot,
reciprocal

4.600
2.993
1.966
1.978
134.06
All
converged

-1907.84


a (A,)
c (A)
Ti-O short (a)
Ti-O long (+)
CPU time (s)
Convergence


Total Energy
(eV)


4.626
2.998
1.970
1.993
208.19
All
converged

-1915.23


4.594
2.956
1.949
1.980


Table 2-1. Calculated lattice
TiO2. (COntinued)
LDA
400 eV
cutoff
usp, real-
space


constants, CPU times, convergence data, and total energies of rutile


LDA
400 eV cutoff
usp,
reciprocal


GGA-PBE
400 eV cutoff
usp, real-
space


GGA-PBE
400 eV cutoff
usp,
reciprocal


Experiment
[2]

























no






converged



yes ,

end



Figure 2-1. Flow-chart describing Kohn-Sham calculation [47].


Figure 2-2. Illustration of difference between all-electron scheme (solid lines) and
pseudopotential scheme (dashed lines) and their corresponding wave functions [51].











Defect formation energy
(neutral oxygen vacancy)
19.0

a? 18.5 + + + Buckingham potential
o Single point energy
18. Geometry optimization
LI. 7.



LE6.0
O




1Xlx1 1x1x2 2x2x2 2x2x3 3x3x3 3x3x4 4x4x4 4x4x6

Supercell size


Figure 2-3. The influence of supercell size on the defect formation energy of a neutral oxygen
vacancy as a function of supercell size, as calculated with DFT (both using single
point energy, geometry optimization including electronic relaxation and full atomic
relaxation) and an empirical Buckingham potential.









CHAPTER 3
DFT CALCULATIONS OF INTTRINTSIC DEFECT COMPLEX INT STOICHIOMETRIC TIO2

Introduction

In stoichiometric metal oxides, when a charged point defect is formed in the crystal, there

should be another complementary point defect with opposite charge formed near the first defect

to conserve the charge neutrality of the system. These two defects together are called a defect

complex. Kroiger and Vink proposed six possible basis types of defect complexes in a

stoichiometric compound [19]. Among them, the Schottky defect complex (cation and anion

vacancies, for example, I ,+2y, in TiO2) and the Frenkel defect complex (cation vacancies and

interstitials, called cation Frenkel defects, such as V ,+Ti, in TiO2, Or anion vacancies and

interstitials, called anion Frenkel defects or anti-Frenkel defects, such as Yo+O, in TiO2) are the

only ones have been found in oxides. Thus these two are the most commonly studied defect

complexes in stoichiometric metal oxides, especially at high temperatures [31].

It is therefore important to understand defect formation and diffusion mechanisms for

Schottky and Frenkel defect complex in titanium dioxide. However there is no consensus in the

literature on the relative stabilities of these defects in TiO2. The space charge segregation

measurements, thermogravimetric measurement and tracer impurity diffusion experiments found

that Frenkel formation energies were lower than Schottky formation energies [9, 78-80]. For

example, Baumard and Tani reported the electrical conductivity of rutile TiO2 doped with 0.04-3

at% Nb as a function of oxygen pressure in the temperature range 1273 K-1623 K [78]. They

found the cation Fenkel defect to be the predominant defect in Nb-doped rutile. Ikeda et al.

present a quantitative study of space charge solute segregation at grain boundaries in doped TiO2

using scanning transmission electron microscopy (STEM) to measure aliovalent solute

accumulation [80]. They determined the defect formation energies at grain boundaries using bulk









defect chemistry models and the experimental values of the space charge potential. For example,

the defect formation energy (DFE) of a Ti vacancy at 1350 oC ranged from 1.5 to 3.5 eV while

the average value was 2.4 eV. At 1200 oC the average value was down to 2. 1 eV. In comparison,

at 1505 oC the DFE of a Ti vacancy ranged from 1.0 to 1.5 eV. These findings indicate that there

is no strong dependence of the formation energy of Ti vacancies on temperature. Additionally,

using the cation vacancy formation energy of 2.4 eV and the cation interstitial energy of 2.6 eV,

a Frenkel formation energy of 45.0 eV was obtained. Based on the oxygen vacancy formation

energy of 42. 1 eV, a Schottky formation energy of 46.6 eV was obtained. Therefore they

found a strong preference for Frenkel defect complex.

However the defect models of Schottky and Frenkel defect complex cannot be simply

described as the linear combination of these single oxygen and titanium defects. In actuality, the

above formation energy values are much higher than the values calculated by us for clustered

Frenkel defects (of about 2 eV), but is much closer to the value calculated by us for the

distributed Frenkel defect structure (of nearly 4 eV). In contrast, several theoretical calculations

found Frenkel DFEs were much higher than Schottky DFEs [29-31, 81] and the results of

electrical-conductivity measurements were inconclusive [12, 82]. A related issue of current

interest is the preferred interstitial diffusion path of cations in oxides. Experimental

measurements of the diffusion of Li and B in TiO2 Showed strong anisotropy, especially through

the open channels along the [001] direction [83-86].

Thus it seems necessary to use more quantitative techniques such as first principles DFT

calculations to obtain a full understanding of the relative stabilities of these defect complexes in

metal oxides such as TiO2. ab initio DFT calculations are used to calculate the DFEs of Frenkel

and Schottky defects in rutile TiO2 and to study the diffusion of interstitial Ti. Various defect









configurations are considered to quantitatively assess the effect of structure on the DFE of each

defect.

Computational Details

The approach is density functional theory (DFT) using the generalized gradient

approximation in the Perdew-Burke-Ernzerhof functional (GGA-PBE) combined with nonlocal,

ultrasoft pseudopotentials and plane wave expansions in the CASTEP program [51, 87]. An

ultrasoft pseudopotential for Ti is generated from the configuration [Ne]3s23p63d24s2, where the

3s2, 3p6, 3d2, 4s2 electrons are explicitly treated valence electrons. An ultrasoft pseudopotential

for O is generated from the configuration [He]2s22p4, where the 2s2, 2p4 electrons are explicitly

treated valence electrons. The GGA calculations use Brillouin-zone sampling with 4 k-points and

plane-wave cutoff energies of 340 eV. A 2x2x3 unit cell is used to model bulk TiO2 and all the

atoms are relaxed from their initial positions using the Broyden-Fletcher-Goldfarb-Shanno

Hessian update method until the energy is converged to 0.001 eV/atom and the residual forces

are converged to 0.10 eV/A. The same convergence criteria are used for the atomic relaxation of

defect containing structures.

To check the accuracy of the calculations, we performed test calculations of the perfect

bulk unit cell and the results are summarized in Table 3-1. They show that cutoff energies of 340

eV and 500 eV are in good agreement with experimental values. The error between the lattice

parameters calculated from the ab initio DFT calculations and experiment is less than 1.5%.

Additionally, the total energy with a cutoff energy of 340 eV is converged to less than 1.6

meV/atom with respect to the values obtained for a cutoff energy of 500 eV. For this reason the

computationally less intensive cutoff value of 340 eV is used in all subsequent calculations.

The formation energy of Frenkel, anti-Frenkel and Schottky defects in bulk TiO2 aef

calculated as









E1F = E'O'(V, + Til)- ErO'(bulk) (3-1)


EfF = Ero'(Vo +o,) -Eo' (bunlk) (3 -2)

E) = Ero'(V, + 2Vo) Ero'(bulk) + (p1, + 2Po,) (3-3)

where Etor(yn+ Ti,) is the relaxed total energy of a defected unit cell containing one cation

Frenkel defect pair of one Ti vacancy and one Ti interstitial. Etor(Yo+O,) is the relaxed total

energy of a defect unit cell containing one anti-Frenkel defect pair of one oxygen vacancy and

one oxygen interstitial. Etor(yn+2Vo) is the relaxed total energy of a defect unit cell containing

one Schottky defect pair of one Ti vacancy and two O vacancies. Etor(bulk) is the energy of the

defect-free system, and IUn and puo are the chemical potentials of one Ti, one O atom,

respectively. The Ti and O chemical potentials are not independent but are related at equilibrium

by

ru~i + 2rUo = IUrio2 (3-4)

where punO2 is the calculated total energy per TiO2 unit.

It is recognized that the 2x2x3 unit cell, which is the largest unit cell that could be

considered by us in this study, is a constraint on the system and may introduce system-size errors

to the results. These system-size errors should be comparable for the perfect and defective

systems, however, and so should have only a small effect on the DFEs. Since our DFT

calculations predict much higher formation energies (about 7 eV) for anti-Frenkel defects than

cation Frenkel and Schottky defects, only the latter two defect pairs are considered in the

following calculations.









Model Development

Previous semi-empirical self-consistent calculations found that the distribution of defects

in TiO2 may not be random but rather spatially clustered [32]. Therefore, several Frenkel and

Schottky defect models are built from the relaxed, bulk rutile TiO2 unit cell. In particular, four

Schottky defect models are considered and are shown in Figure 3-1. Each model consists of two

O vacancies and one Ti vacancy, but the distances among the various vacancies vary in the

different configurations. The two O vacancies and Ti vacancy are close to each other in model 1,

where the Ti vacancy is separated from one O vacancy by 1.95 A+ and is separated from the other

by 3.57 A+, and model 2, where the Ti vacancy is separated from both O vacancies by 1.95 A+.

The vacancies are more spread out in model 3 (separated by distances of 5.80 A+) and model 4

(separated by distances of 5.47 A+).

In the case of the Frenkel defect models, several possible interstitial sites are considered.

Previous research shows that O migrates via a site exchange mechanism, while the Ti interstitial

diffuses via the 32c octahedral site [2]. There is an anisotropy of at least 10" to 1 in favor of Li

and B diffusing along the [001] direction relative to the [110] direction [84]. It was consequently

suggested that the equilibrium positions for interstitial cations must preferentially occupy the

(100) planes. However, this hypothesis has not been tested for the self-diffusion of Ti interstitials

in TiO2. In this work, six Frenkel models, shown in Figure 3-2, are considered. The coordinates

of the positions of the Ti interstitial atoms in all six models are listed in Table 3-2. It should be

noted that as a result of the periodic boundary conditions used in the calculations, the Ti

interstitial at (2a, 3a/2, 5c/2) is the same as the Ti interstitial at (0, 3a/2, 5c/2).

The first three Frenkel defect models (models 1-3) reflect the movement of a Ti atom from

a lattice site to a neighboring octahedral interstitial site in the [100] or [010] direction to form the

defect. The last three Frenkel models (models 4-6) reflect the movement of a Ti interstitial atom










along the [001] direction. Therefore, in these models, the Ti interstitials are placed in octahedral

sites long the [001] direction.

Results and Discussion

The DFEs calculated for these four Schottky are summarized in Table 3-3. They indicate

that in each case the individual point defects preferentially cluster together in bulk TiO2, in

agreement with the results of Yu and Halley [32]. Specifically, the formation energy of the

Schottky defect in models 1 and 2 is 1.5 eV lower than the formation energy in models 3 and 4.

This indicates that the Schottky defect prefers to form a clustered structure rather than spreading

out across the lattice. The formation energy of the Schottky defect in model 2 is at least 0.5 eV

lower than the next lowest formation energy in model 1. This indicates that the Schottky defect

prefers to form a clustered configuration where the vacancies are closest to one another in a row

to a clustered triangular structure of Ti and O vacancies with some larger separation distances.

The DFEs of the six Frenkel defects models are summarized in Table 3-4. In this case the

DFE of the Frenkel defects in models 1, 2, and 3 are approximately the same, but are

significantly lower than the DFEs of models 4, 5, and 6. These results indicate that the Frenkel

defect prefers to exist as a clustered pair rather than as a combination of isolated Ti vacancy and

Ti interstitial.

Table 3-5 compares these calculated DFEs to DFEs reported in the literature that were

calculated with DFT and semi-empirical methods or obtained from experimental results. In our

work, the lowest Frenkel DFE is about 2 eV, which is much lower than the lowest Schottky DFE

of about 3 eV. This finding agrees with space charge measurement results that find that the

Frenkel defect is more prevalent in rutile TiO2 than the Schottky defect [79, 80], although the

results of other calculations [31, 34] show the Schottky DFE is much lower than the Frenkel

DFE.









The discrepancies in the literature results are most likely due to the configurations

considered in the calculations and experimental data measurement analysis. In particular,

Schottky and Frenkel DFEs can be calculated using Eqs. (3-1) and (3-2), as we have done, or

they can be determined by combining the DFE of an isolated TiO2 System with Ti vacancies with

the DFE of an isolated TiO2 System with O vacancies or Ti interstitials. For example, the

Schottky DFE can be calculated as the sum of the energies from two parts. The first part comes

from the energy needed to extract a Ti atom from a supercell to form an isolated Ti vacancy,

while the second part is twice the formation energy of an isolated O vacancy. Using this

approach, the formulas to calculate the DFE of Schottky and Frenkel defects are

E) =,, Eror EfO'(bulk) + 2[Eror _~O EfO'(bulk)] + (pn + 2iPo) (3-_5)


E1 = E'O'(r E'O' (bul2k) + E'O'(Interstitialn2) Eto'(bulk) (3-6)

Ikeda et al. separated the Frenkel DFE for TiO2 into individual terms by measuring solute

segregation at a free surface and obtained a Ti vacancy formation energy of 2.4 eV and a Ti

interstitial formation energy of 2.6 eV [80]. Therefore, they obtained a Frenkel DFE of 5.0 eV.

This is much higher than the value calculated by us for clustered Frenkel defects (of about 2 eV),

but is much closer to the value calculated by us for the distributed defect structure (of nearly 4

eV), and would be expected to be closer still to a defect structure distributed across a larger unit

cell than we were able to consider here.

The DFE calculated by the semi-empirical Mott-Littleton calculation also shows much

higher DFEs [31]. This may due to differences in semi-empirical potential parameters used for

characterizing TiO2. The semi-empirical parameters used in Ref. [31] are determined from the

properties of perfect crystals. For example, parameters can be fit to the cohesive energy,

equilibrium lattice constant, and bulk modulus of a perfect crystal. Consequently, these semi-










empirical parameters may not be appropriate for the study of imperfect systems, including defect

energy calculations, because the ions around the defects are in different chemical environments

from those in the perfect lattice.

Temperature differences between our DFT calculations and experimental measurements

should also be addressed. Our DFT calculations are carried out at 0 K and under these

conditions, Frenkel defects are predicted to be more likely to occur than Schottky defects.

However this finding could change at high temperatures. Although the changes in entropy are of

the same order of magnitude, and therefore can be ignored when comparing different single point

defects, the contribution of the change in entropy (TAS) to the total free energy of complex

defect system should not be ignored at very high temperatures (on the order of 1000 K). In the

case of a Schottky defect, the change in entropy must be calculated for three vacancies, while in

the case of a Frenkel defect, there are only two contributions to the change in entropy, that of one

vacancy and one interstitial. Therefore the Schottky defect may be preferentially stabilized at

high temperatures by entropic contributions than the Frenkel defect in TiO2, which would reverse

the findings of this theoretical study.

The calculated total densities of states (DOS) for the perfect structure and those containing

Schottky (model 2) and Frenkel (model 3) defects are shown in Figure 3-3. In the DOS of the

perfect TiO2, the O 2s band is located between -17 and -15 eV, while the O 2p band is located

between -6 and 0 eV and only the lower conduction band is present. The band gap is calculated

to be 2. 11 eV, which is smaller than the experimental band gap of about 3 eV. This

underestimation of the band gap is well-known to occur in DFT calculations using the GGA

approximation [63]. In all other respects, the band structure is consistent with previous band

structure calculations of perfect TiO2.









Figure 3-3 shows that the DOS for Schottky model 2 and Frenkel model 3 are similar to

the perfect structure in that no defect state is apparent. However the band gap calculated for the

Schottly defect system is 2. 17 eV, while that of the Frenkel defect system is 2.20 eV, both of

which are larger than the band gap of the perfect structure. It should be pointed out that these

Schottky and Frenkel defects are formed from combinations of neutral vacancy and interstitial

defects, rather than charged defects that would be expected to introduce new occupied states in

the band gap.

The last issue to be considered by us is the preferential diffusion path of Ti interstitials in

TiO2. Figures 3-2 and 3-4 shows several possible Ti diffusion paths through the rutile structure.

The DFEs of the first three Frenkel defect models reflect the energy that a Ti atom needs to move

from a lattice site to a neighboring octahedral site in the [100] or [010] direction and form the

Frenkel defect. The results show that the impedance in the [100] and [010] directions are

approximately the same.

The last three Frenkel models (models 4 to 6) are considered to understand the barrier to Ti

interstitial diffusion along the [001] direction. It is well known that the rutile structure has open

channels along the c-direction (a, 3a/2, z). The Ti interstitial position (a, 3a/2, c/2) is the center

of an octahedron of O atoms, and is being considered as the most stable site for the location of

the Ti interstitial. However there are also other equilibrium positions along this direction that can

be considered as possible sites for a Ti interstitial, for example z = 3c/4. Bond length arguments

suggest that site (a, 3a/2, c/2) might be the most stable equilibrium position. In Frenkel models

4-6, the Ti interstitials are placed in octahedral sites long the [001] direction at z = 3c/2, c/2 and

c/4. The calculated DFEs for models 5 and 6 clearly show that site (a, 3a/2, c/4) has higher

defect formation energy than site (a, 3a/2, c/2). This conclusion is consistent with the findings of









Huntington and Sullivan [84]. The difference in DFEs for models 3 and 4 (1.57 eV) is lower than

the formation energy of the Frenkel defect in model 1 (2.02 eV). This finding indicates that it is

much easier for one Ti interstitial to move through the open channel in the [001] direction than it

is to move from a lattice site in the [100] or [010] directions. This result is summarized in Figure

3-4.

Summary

Ab initio DFT calculations are used to determine the formation enthalpies of Frenkel and

Schottky defects for several defect structure configurations in bulk rutile TiO2. The results show

that both Frenkel and Schottky defects prefer to cluster together rather than being distributed

throughout the lattice. The Frenkel defect is predicted to be more likely to occur in rutile at low

temperatures than the Schottky defect, with a difference is formation enthalpy of about 1 eV. The

DOS for the Schottky and Frenkel models are also calculated. We find that their band features

are quite similar to the DOS of the perfect, defect free structure with only a small increase in the

band gap predicted to occur. Lastly, strong anisotropy in interstitial cation diffusion in TiO2 is

supported by these calculations.










Table 3-1. Comparison between the calculated structural parameters and experimental results for
rutile TiO2. (GGA-PBE, generalized gradient approximation in the Perdew-Burke-
Ernzerhof functional.)
Approach a (A) c (A) Ti-O short (A) Ti-O long (A+)
GGA-PBE 340 eV 4.64 2.97 1.96 2.01

GGA-PBE 500 eV 4.63 2.96 1.95 1.99
Experiment [39] 4.59 2.95 1.95 1.98



Table 3-2. Positions of the Ti interstitial site in the Frenkel defect models shown in Figure 2-2.
Models Coordinated (x, y, z)
1 2a, 3a/2, 5c/2
2 a/2, 0, 5c/2
3 a, 3a/2, 5c/2
4 a, 3a/2, 3c/2
5 a, 3a/2, c/2
6 a, 3a/2, c/4


Table 3-3. Calculated Schottky DFEs for rutile TiO2-
Model Schottky DFE (eV)
1 3.51
2 3.01
3 4.98
4 5.47


Table 3-4. Calculated Frenkel DFEs for rutile TiO2.
Model Frenkel DFE (eV)
1 2.02
2 2.01
3 1.98
4 3.55
5 3.74
6 3.84









Table 3-5. Comparison of DFT calculated Frenkel and
and theoretical values for rutile TiO2-
Defect (eV) Current Dawson DFT
DFT result [34]
result
Schottky (clustered) 3.01 4.66
Schottky (distributed) 5.47 17.57
Frenkel (pair) 1.98 15.72-17.52
Frenkel (distributed) 3.84 -


Schottky DFEs to published experimental


Catlow semi-
empirical result
[31]

5.25-8.22

11.12-14.64


Space charge
measurement
[80]

<6.6

<5.0






















Model 1 Model 2


Model 3 Model 4


Figure 3-1. The Schottky defect models considered in this study.

































Model 3 Model 4


Model 5 Mlodel 6


Figure 3-2. The Frenkel defect models considered in this study.






















0
50
(b) Frenkel defect
100-

50

00


50


250 ~

200-

150 -

100 1

50 -


2
2

1

1


(a) Perfect


200

150 -

100

50-


(c) Sc hottity defect


-80 -60 -40 -20

Energy (eV)


Figure 3-3. The densities of states of perfect and defective TiO2. The valance-band maximum is
set at 0 eV.










[001]



me [00


Figure 3-4. Possible octahedral Ti interstitial sites in rutile TiO2-









CHAPTER 4
CHARGE COMPENSATION INT TIO2 USING SUPERCELL APPROXIMATION

Introduction

Density-functional theory (DFT) is a proven approach for the calculation of the structural

and electronic properties of solid state materials. In particular, DFT calculations combined with

periodic boundary conditions (PBCs), plane-wave expansions and pseudopotentials have been

extensively applied in the study of systems lacking full three-dimensional periodicity such as

molecules, defects in bulk materials, and surfaces. The use of these approaches and

approximations remove the influence of troublesome edge effects and allows a relatively small

number of atoms to mimic much larger systems. For example, several DFT studies have

examined the electronic structure of charged titanium interstitials and impurities in TiO2 [45, 88].

However most of these computational approaches and approximations are originally developed

for the calculation of perfect crystal structures. Consequently the use of these approaches and

approximations do lead to technical difficulties in the study of charged defects. For example,

although the supercell approximation accurately describes local bonding fluctuations between

atoms, it also introduces artificial long-range interactions between defect and their periodic

images in the neighboring supercells. The presence of this long-range interaction could

dramatically change the evaluation of the defect formation energies.

Giving TiO2 aS an example, it is believed that its n-type conductivity is partially due to the

multi-valence nature of the cation. Specifically, charged defects, such as titanium interstitials

with +3 and +4 charges and oxygen vacancies with +2 charges, have been shown experimentally

to play a dominating role in a variety of bulk and surface phenomena in TiO2 [2, 12, 13, 23].

However, there is still little fundamental understanding of the preferred charge states of point

defects in TiO2 Or in their transitions as a function of temperature. Therefore it is still non-trivial









to determine the relationship between charge states and formation energies for even the most

typical defects in TiO2. Such relationships are needed to allow the ready prediction of defect

transition levels that are essential to understand the optical properties of wide band-gap transition

metal oxides. Even more surprisingly, there are DFT calculations that report the presence of

titanium interstitials with the sole charge state (+4) as the predominant defect in the TiO2 [36].

This is problematic, as it is well known experimentally that titanium interstitials with +3 and +4

charges are both the predominant defect in TiO2 CVen if Oxygen vacancies are excluded from the

structure.

In order to overcome this artificial long-range interaction problem, normally a uniform

electron gas background (j ellium background) is added to compensate for these artificial

interactions (see figure 4-1). The interaction of the defect with the j ellium background should

exactly counteract the interaction of the defect with its spurious periodic images. However, as

Makov and Payne pointed out, the energy of this supercell will still converge very slowly as a

function of the linear dimensions of the supercell [77]. Thus, a few approaches were proposed to

correct the divergence of the Coulomb energy for charged defects. For example, Leslie and

Gillan suggested a macroscopic approximation to consider a periodic array of point charges with

a neutralizing background immersed in a structureless dielectric [89]. In addition, Makov and

Payne derived a detailed indirect correction for charged defects in cubic supercells [77]. More

recently, Schultz developed the local-moment countercharge (LMCC) method [90], which uses

the linearity of the Poisson equation to correct the divergence of the charged defect energies. In

contrast, Nozaki and Itoh directly treat charge distribution to keep the charged defect cell

embedded in a perfect non-polarizable crystal [91]. Despite these more recent efforts, the Makov

and Payne approach is still the most widely known and used approach.









In this chapter, the application of Makov and Payne approach in the study of charged point

defects in rutile TiO2 will be discussed. The effects of the corrections will be evaluated and

compared for supercells of varying size. More importantly, the defect formation energies

obtained from the DFT calculations will be combined with thermodynamic data to study the

influence of temperature on the relative stabilities of intrinsic point defects in TiO2. The results

indicate that although the Makov and Payne approach may give an overestimated correction for

the defect formation energies due to the fact that these defects are delocalized in the system, it is

still an appropriate approach to study defect levels in transition metal oxides such as TiO2 whose

cations can exist in multi-charge states. However the dipole interactions, which should also be

countered as a possible source of error, are not considered in the charge compensation.

Computational Details

The DFT calculations are performed using standard plane wave expansions within the

generalized gradient approximation parameterized with the Perdew-Burke-Emzerhof form

(GGA-PBE) for the exchange-correlation potential [63]. All the calculations are performed using

the CASTEP program [59]. An ultrasoft pseudopotential for Ti is generated from the

configuration [Ne]3 s23p63d24s2, and an ultrasoft pseudopotential for O is generated from the

configuration [He]2s22p4 [57]. The Brillouin-zone sampling is carried out using a 2x2x2 k-mesh

and a plane-wave cutoff energy of 400 eV is used.

As the Makov-Payne correction was originally developed for ideal cubic ionic crystals, it

is necessary to use a combination of unit cells to construct a repeating supercell of that is as close

to cubic as possible. Thus, three different supercell models are considered: lxlx2, 2xlx2 and

2x2x3. The corresponding numbers of atoms are 12, 24, and 72, respectively. While it is true

that thermal lattice expansions would be expected to influence these results, it is also widely









accepted that the energy difference in the calculation of defect formation energies are relatively

independent of these expansions. As these energy differences are what is important here, the

lattice parameters are fixed to the experimental values. Then all the atoms are relaxed from their

initial positions until the energy is converged to within 0.001 eV/atom and the residual forces are

converged to 0.10 eV/A.

In the charged defect calculations, all the charges are compensated by a neutralizing

jellium background charge. The defect formation energies are calculated using the following

equation that takes into account temperature, oxygen partial pressure, and electron chemical

potential (Fermi level)

AEi(i,qlT, P)= Ef'"' (i, q,L)- E'ore(perfect)+ n, ,(T, P)+qg(E,). (4-1)


In Eq. (1), Elftzi (i, q, L) is the total energy of the supercell containing defect i of charge

state q as a function of supercell dimension L, Etotai (perfect) is the total energy of the

corresponding perfect supercell, and n, is the number of atoms being removed from the supercell

or being added from the atomic reservoir. For example, n,=no=1 for an oxygen vacancy and

n,=ny,=-1 for a titanium interstitial. Following the approach of Finnis et al. [92-94], pu,(T,P) is the

chemical potential of the defect atom i described as a function of temperature and oxygen partial

pressure. Finally, in Eq. (4-1), SF is the Fermi energy. In order to calculate the Fermi energy, we

also calculate the valence-band maximum (VBM) [95].

The Makov-Payne correction formula for the electrostatic energy of an isolated charged

defect within PBC and a uniform jellium background in a cubic lattice can be described as

follows [77]

E,(i,qy)= Ear"iq,) qk 2nqQ +O[L ] (4-2)
2 EL 3 EL'










where E,(i, q) is the extrapolated energies from an infinite supercell and the first

correction term describes the electrostatic energy of the point charge array in a uniform j ellium

immersed into a screening medium which has a dielectric constant e and depends on the

Madelung constant of the lattice (a ). The second correction item describes the interaction of the

defect charge with the neutralizing j ellium background and Q is defined as the second radial

moment of the defect charge density. The third correction item describes the quadrupole-

quadrupole interaction to a high order of |L| 5. As Eq. (4-2) shows [96], the second correction

usually counts for only 3-5% of the first correction item. And as the definition of the parameter

Q still contain some ambiguity, here we only consider the first correction item in our

calculations. An approximation, inherent in the Makov-Payne approach, is the use of a

continuum dielectric constant to screen the interactions, which should break down if the defect

charges are not well localized.

Results and Discussion

To investigate the effect of supercell size on the Makov-Payne correction, we first

calculate defect formation energies as a function of supercell size and compare the values for

fully charged titanium interstitials, fully charged oxygen vacancies, and titanium vacancies with -

2 charges with/without the Makov-Payne correction. The temperature is set to 0 K, the oxygen

partial pressure is set to 0 atm, and the Fermi level is set at the midpoint of the experimental band

gap (3.0 eV). All the calculated results are shown in figures 4-2(a)-(c). In all three cases, the

defect formation energies calculated with (without) the Makov-Payne correction are well fit with

red (black) straight lines. The exception is the calculated defect formation energy for titanium

vacancies of -2 charge in the smallest supercell of 12 atoms. It is also noted that for each case,

the lines nearly meet at the infinite-supercell limit (1/L 0O). It is clear that in the case of the fully










charged titanium interstitials and the titanium vacancies with -2 charges, the Makov-Payne

correction improves the convergence of the defect formation energies. For example, in the case

of the fully charged titanium interstitials, the energy difference between the MP-corrected DFEs

and E, for the 72-, 24-, and 12-atom supercells are 1.60 eV, 3.05 eV, and 3.85 eV, respectively.

In the uncorrected case, the energy difference between the DFEs and E, are 4.87 eV, 8.18 eV,

and 10.45 eV, respectively. Additionally, the larger supercell provides a better estimation of the

charged defect energy since a smaller electrostatic correction is needed for the larger supercells.

In contrast, the Makov-Payne approach gives the wrong sign of the correction on

fully charged oxygen vacancies. In particular, as shown in figure 4-2(c), the uncorrected

formation energies increase as the supercell decreases and, since the Makov-Payne correction is

always positive, this correction causes significant exaggeration of the formation energies. For

example, in the case of the 72-atom supercell, the overestimation of the correction compared to

the extrapolated value from the infinite supercell, E,, is 1.13 eV. This behavior can be attributed

to the fact that, unlike the titanium vacancies and interstitials, the fully charged oxygen vacancies

are shallow level donor defects. As illustrated in figure 4-2(d), the defect transition levels of fully

charged oxygen vacancies are always shallow relative to the titanium vacancies and interstitials

for supercells of the same size. This is in agreement with x-ray photoelectron spectroscopy

results that also find evidence of shallow level oxygen vacancies in TiO2 [97] and similar

problems have been reported for other semiconducting materials such as diamond and InP [98,

99].

After considering the effect of supercell size, we now investigate the effect of the Makov-

Payne correction on the thermodynamic stability of charge states and the corresponding defect

transition levels. The defect formation energies of a charged titanium interstitial before and after









the application of the Makov-Payne correction are calculated as a function of Fermi level under

reduced conditions (PO2=10-20) (See figure 4-3). As the Fermi level decreases, the

thermodynamically preferred charge state of the interstitial changes from neutral to +1, +2, +3,

and +4 charge states when the correction is applied. On the other hand, when the Makov-Payne

correction is omitted in the calculation, an unstable transition occurs between the charge states.

In particular, the +4 charge is predicted to be preferred over almost the entire range of the

electron chemical potential, while the neutral and +1 charged states occur near the conduction

band minimum and are obviously unstable relative to the +4 charge state. This is because a tiny

change in the defect concentration leads to a shift of the Fermi level, and this will eventually

cause a charge shift in the titanium interstitials from neutral or +1 to the +4. It is likely that the

stability of the charge state is substantially affected by the electrostatic energy correction for

TiO2. However, these results indicate that it is necessary to apply the Makov-Payne (or an

alternative) correction while studying defect levels of transition metal oxides that are not too

shallow.

Finally the defect formation energies for various charged intrinsic point defects as a

function of temperature and electron chemical potential are calculated (see figure 4-4). These

intrinsic defects include the titanium interstitials, oxygen interstitials, titanium vacancies, and

oxygen vacancies. The temperatures considered are 300 K and 1400 K. We compared the defect

formation energies computed for these defects with and without the Makov-Payne correction.

Experimental studies suggest that at low temperatures oxygen vacancies are the most stable

defect in rutile TiO2, while at temperatures as high as 1400 K, titanium interstitials are dominant

[12, 23].With the Makov-Payne correction, our results clearly show the same trend. For example,

figure 4-4(c) indicates that at T=300 K, oxygen vacancies are more stable than titanium









interstitials over almost all the Fermi levels considered in the reduced state. However, when the

temperature goes as high as 1400 K, the reverse is true (see Eigure 4-4(d)). Over almost the entire

range of the band gap, the titanium interstitial with different charge states is predicted to be the

most stable intrinsic defect in TiO2 at high temperature. However, without the Makov-Payne

correction, the transition from oxygen vacancy to titanium interstitial is not predicted to occur.

Instead, titanium interstitials with +4 charge are predicted to be the predominate point defect in

TiO2 at both of the temperature ranges considered here. This conflicts with the experimental

finding that oxygen vacancies play an important role in TiO2 prOperties. Thus it is clear that the

experimentally observed defect transition from oxygen vacancies to titanium interstitials is well

reproduced in our charged defect calculations when applying the Makov-Payne correction.

Summary

TiO2 has been intensively studied as a wide band-gap transition metal oxide partially due

to the multi-valence nature of its cation. In this chapter, DFT calculations within the supercell

approximation and Makov-Payne correction are carried out to determine the preferred charge

state of charged point defects in rutile TiO2. The first part of this study is to investigate the

dependence of the defect formation energies on the supercell size and the electrostatic Makov-

Payne correction. The results show that the Makov-Payne correction improves the convergence

of the defect formation energies as a function of supercell size for positively charged titanium

interstitials and negatively charged titanium vacancies. However, in the case of positively

charged oxygen vacancies, applying the Makov-Payne correction gives the wrong sign for the

defect formation energy correction that is attributed to the delocalized nature of the charge on

this defect in TiO2. Finally, we combine the calculated defect formation energies with

thermodynamic data to evaluate the influence of temperature on the relative stabilities of these

defects. These results indicate that when the Makov-Payne correction is applied, a stable charge









transition is predicted to occur for titanium interstitials. In addition, as the temperature increases,

the dominant point defect in TiO2 changes from oxygen vacancies to titanium interstitials. Since

this correction is more appropriate for the strongly localized charges, its application to

delocalized, shallow level defects should be treated with caution.













+


(;a)







(t~)


Figure 4-1. Schematic illustration of the use of PBCs to compute defect formation energies for
an isolated charged defect in a supercell approximation. (a) The long-range
interaction between the isolated charged defect and its periodic image in the nearby
supercell is shown with red arrows. (b) The jellium background, shown as a uniform
electron background, is applied to compensate for the artificial long-range
interactions between the defect and its periodic defect images.





















at N o MP correctioln (d)b
so- winrMPcorrection I ihcort oR

O4- o 12 on
50--i 77 2.0 1.2 .



/ .5 722 1.5






Figure 4-2. Calulte defec fomaio enrge and deecraston level indifeen
suerels (a-c eetfrain nriso ul hrgdtanu itestal()




Paynecorrctin. (d) Dee ct~ trniinlvl f iaimitrttil rm+ o+
(+3/+4),and titaium vaace rm- o 3(2-) n xge aace rm+
to +2 (1+)idifrnsuecls. h auso h ie r h eettasto
levlswih rspcttotheVB, hie te ales ndr helins reth toalnube
of atm fte orsodngsprel






































Figure 4-3. Calculated defect formation energies for various charge states of the titanium
interstitial in a 72-atom supercell in TiO2 aS a function of the Fermi level (electron

chemical potential) with and without application of the Makov-Payne correction.


10








-5"L ,,3~


13


~





/r .i
-" tl I)
-- +2
___FT7-
+3
--~ ,
~
+4


-0 0.5 ;1 1.5 2
Fermi level (eV)


2.5 3


0.5 1 1.5 L
Fermi level (eV)


.


















2 V + 2-


a +4
-4 4 i


1 Z-212

-11






Ti


-6 -6
0 0.5 '1 1.5 2 2.5 :2 0 0.5 11 1.5 2 2.5
Ferm1 i lee (eI Fem lee (ev

Figur 4-4. Cacltddfc omto nrge o nrni eett300 K~ and iO-; dr 1400 K wt
andwihot he akv-aye orrctonuner edce cndtios pO-1-2

(b eetomto enries witou the Mao-anecretinc-()Dfc
foraton negie wth heMaov-anorcin









CHAPTER 5
PREDICTION OF HIGH-TEMPERATURE POINT DEFECTS AND IMPURITIES
FORMATION IN TIO2 FROM COMBINED AB INITIO AND THERMODYNAMIC
CALCULATIONS

Introduction

TiO2 can be easily reduced, which results in n-type doping and high conductivity.

Experimental techniques, such as thermogravimetry and electrical conductivity measurements,

have long been used to determine the deviation from stoichiometry in TiO2 aS a function of

temperature and oxygen partial pressure. The analyses of these experiments rely on assumptions

about the charges of the defects and their dependence on environmental conditions [12, 13, 23,

26, 100]. For reduced TiO2, the results for temperature below 1373 K are consistent with the

presence of either titanium interstitials with +3 charges or fully charged oxygen vacancies at

various oxygen partial pressures, as illustrated in Figure 5-1. They also indicate that at moderate

pressures, as the temperature is increased above 1373 K there is a transition from fully charged

oxygen vacancies to fully charged titanium interstitials.

While the assumptions used in the analysis of the experiments are physically reasonable,

further experimental refinement of defect stabilities has been hampered by the extreme

sensitivity of the electronic and physical properties of TiO2 to minute concentrations of defects

and impurities. Theoretical calculations have the advantage of absolute control of composition of

the system under consideration and are thus well positioned to complement experimental data.

Density-functional theory (DFT) calculations have been applied to study the defect formation

and stability in various electronic ceramics. For example, the stability of point defects in

undoped ZnO, which is a promising fluorescence material, such as Zn interstitials [101] and

oxygen vacancies [102, 103] have been considered with this approach. In addition, several

calculations have examined the efficiency of p-type doping into ZnO of group-V elements such










as N, P, and As [104, 105] or group-I elements such as Li and Na [106]. DFT has also been

applied to study oxygen vacancy formation and clustering in CeO2 and ZrO2 [107, 108], which

are used to store and transport oxygen in solid-oxide fuel cell applications. For the actinide

oxides, such as UO2 and PuO2, a feW grOups have applied DFT to examine defect complexes,

such as oxygen interstitial clusters [109] and corresponding electronic structures [110].

In the case of TiO2, SeVeral ab initio studies have been applied to examine defect structure

and stability. These studies have focused, on, for example, Schottky and Frenkel defect

complexes [34, 111], extrinsic point defects [35, 43, 112], and electronic structure of intrinsic

point defects [32, 65]. Recent DFT studies of intrinsic defect formation energies in TiO2 [36, 37]

find that there are no defect levels inside the band gaps of anatase TiO2, and that Ti interstitials

with +4 charges are the predominant point defect under Ti-rich conditions. However, these

studies have been restricted to zero Kelvin and do not include electrostatic interaction

corrections.

Here, quantitative predictions of the stabilities of charged intrinsic and extrinsic point

defects in rutile TiO2 are made using a judicious combination of electronic structure and

thermodynamic calculations. In particular, DFT calculations are used to obtain electronic

structure energy information about both the pristine and defective atomic-scale systems [32, 34,

35, 111]. This information is then used in thermodynamic calculations to determine defect

formation energies (DFEs). Importantly, a quantitative link is made to temperature and the

oxygen partial pressure, which are the key parameters for controlling the type and concentration

of dominant defects in TiO2 and other electronic ceramics [100]. The resulting self-consistent set

of DFEs are crucial input parameters for equilibrium, space-charge segregation models [1 13];

models with accurate and self-consistent DFEs will better predict defect density distributions in









metal oxides and thus enhance the design of electronic, optoelectronic, and ionic conductor

devices.

Computational Methodology

Electronic Structure Calculations

The DFT calculations are performed with plane-wave expansions using the generalized

gradient approximation in the Perdew-Burke-Emzerhof form (GGA-PBE) as implemented in the

CASTEP code [51, 59, 63]. The ionic cores are represented by ultrasoft pseudopotentials [57].

An ultrasoft pseudopotential for Ti is generated from the configuration [Ne]3s23p63d24s2, where

the 3s2, 3p6, 3d2, 4s2 electrons are explicitly treated valence electrons. An ultrasoft

pseudopotential for O is generated from the configuration [He]2s22p4, where the 2s2, 2p4

electrons are explicitly treated valence electrons. The Brillouin-zone sampling is made on a

Monkhorst-Pack grid of spacing 0.5nm l; a plane-wave cutoff energy of 400 eV is used. A

2x2x3 unit cell is used to build a supercell for the perfect and defective bulk TiO2 calculations,

and all the atoms in the supercell are relaxed to their equilibrium positions such that the energy is

converged to 0.001 eV/atom and the residual forces are converged to 0.1 eV/A.

To check the applicability and accuracy of this combination of pseudopotentials and

supercell size, calculations of perfect bulk TiO2 are perfOrmed using different approximations for

the exchange-correlation energy, and the calculated lattice parameters and Ti-O bond lengths are

summarized in Table 5-1. The results indicate that the LDA approach underestimates the

equilibrium lattice parameters by about 1% [114, 115]. The suggestion for solving this problem

is either to increase the cutoff energy at the price of more computationally intensive calculations,

or to use the GGA approach. Here, the latter option is chosen. The results indicate that the GGA

with a cutoff energy 400 eV is in good agreement with reported experimental values, as indicated

in Table 5-1.









Defect Formation Energies of Intrinsic Defects

The Gibbs free energy of defect formation in TiO2 aS a function of defect species a, charge

state q, temperature Tand oxygen partial pressure P is given as [75, 102, 116]


aGf(a(,q, T,P) a E"'o (a,qg) -E'"'" (perfect)+ +ny~ (T, P)+ +q, (5-1)

Here Erorai~a,q) is the relaxed total energy of the supercell containing the defect a of charge

state q obtained from the DFT calculations, Erorai~perfect) is the relaxed total energy of the

supercell of the corresponding perfect crystal, which is also obtained from DFT calculations. The

value no is the number of atoms being removed from the supercell to an atomic reservoir in the

process of creating the defect. For example, na = no = 1 for an oxygen vacancy, and n, = nnT = -1

for a titanium interstitial. IPa(T,P) is the chemical potential of the defect atom a as a function of

temperature and oxygen partial pressure, which is obtained using a combination of DFT and

thermodynamic values, as described in detail in the next section. Lastly, qq is the electron

chemical potential associated with the charged defect, which can be thought of as the energy

needed to move the appropriate number of electrons from infinity to the Fermi energy, 4,

following the approach of Zhang and Northrup [116]. The Fermi energy is treated as a variable,

and can be expected to depend on the charge associated with the maj ority defect in the sample.

Vibrational entropies of formation are neglected, but configurational entropies are treated with

the usual ideal solution model [75, 102].

To estimate the effects of the limited supercell size, Figure 5-2 shows how the DFE of an

oxygen vacancy, as calculated from atomic-level simulations using an empirical potential,

depends on the size of the supercell [117]. These empirical potential calculations were carried

out by R. Behera in the Computational Materials Science Focus Group at the University of

Florida. The results show a smooth decrease in the formation enthalpy, normalized to the DFE of









the 2x2x2 system, as the system size increases for nxnxn supercells (solid curve in Figure 5-2).

However, particularly for small nxnxm (nym) supercells the DFEs deviate from this smooth

curve (dashed curve in Figure 5-2). Regardless of the shape, however, the DFE is independent of

system size for all the systems above the 6x6x6 supercell. Most relevant to assessing the

electronic structure results is that the change in the DFE from the 2x2x2 to the 2x2x3 and 3x3x3

supercells accounts for approximately 66% of the total change in DFE with respect to the 2x2x2

system. Most importantly, the small size of the size effect strongly indicates that the results of

the DFT calculations using the 2x2x3 supercell can be trusted to give the correct relative

formation energies, within 0.04%.

Thermodynamic Component

The value of the chemical potential Pa,(T,P) depends on the system environment. Following

the approach of Finnis and co-workers [92, 93], the oxygen chemical potential is described in

terms of temperature and oxygen partial pressure as

1 1 P
IUo(T, P) [FU~0 o, -0 _lJ AG 'oT po)]+Ao\I:(T)+-O ~Ih /\ kT log( ). (_5-2)
2 2 P,

where #nTozo and pnTo are the chemical potentials of TiO2 and Ti, respectively, and are

calcuated singDFT, hile G/zo7 po) is obtained from thermodynamic data [118]. Apoo T)


is the difference of oxygen chemical potential between any temperature of interest and the

reference temperature obtained from the thermodynamic data. Combining equations (5-1) and (5-

2) allows for the determination of DFEs as a function of temperature and oxygen partial pressure.

These relations are the key to this integrated approach, in that they provide the critical bridge

between the zero-temperature, zero-pressure DFT electronic structure calculation results and the

high-temperature, finite-pressure conditions of real-life applications.









Charge Compensation

The last term in equation (5-1) treats the effect of defect charge and electron chemical

potential on the defect formation energy. In the charged defect calculations, a specific charge is

assigned to the defect in the supercell. However, an artificial long-range interaction between the

defect and its periodic images is introduced into the system. In order to overcome this, a

neutralizing homogeneous background charge is assumed and implemented in the CASTEP code

[59]. But as Makov and Payne pointed out, the total energy of this supercell still converges

slowly [77]. Several approaches have been proposed to correct for this Coulomb energy error

[77, 96, 119]. Here the Makov-Payne approach is used, in which the background error is

corrected to O(L-3), where L is the dimension of the supercell. The reader is referred to chapter 4

for additional details [120].

Results and Discussion of Intrinsic Defects

Electronic Structure of Defects in TiO2

The first step of the integrated approach is to determine the electronic structure of the

defects in an atomistic system with DFT calculations. We first calculated the band structure of

the perfect rutile TiO2. The GGA approach is well known to give an underestimated value for the

band gap of semiconductors and insulators. The calculated band structure along the symmetry

lines of the Brillouin Zone for perfect rutile TiO2 is shown in Figure 5-3(a). The band gap, Egap,

at the r point is 2. 11 eV, which is much smaller than the experimental value of 3.00 eV [2].

Although this may be explained by the fact that the Kohn-Sham eigenvalues do not account for

the excitation state, this failure is still intimately related to a derivative discontinuity that arises in

part from the exchange-correlation energy functional, as shown by Perdew and Levy [121] and

by Sham and Schltiter [122]. By calculating the total energies of separate neutral, -1 and +1









charged perfect supercells, another definition for the band gap of perfect rutile TiO2 is described




Egap = Erotal (perfect,-1) +Etotal (per~fect,+1)- 2E'tota (per~fect,0) (5-3)

Here, E (perfect,q) is the total energy of one perfect supercell with charge q. The band gap

calculated by this equation for TiO2 is 2.54 eV.

It is possible that this underestimated band gap could affect the defect levels and formation

energies of intrinsic defects in TiO2. Thus, a lineup is implemented in which the conduction band

is rigidly shifted upward to match the experimental band gap (the so-called 'scissor operator')

[123].

The defect transition level e(defec~tq1q2) introduced by defects in the band gap or near the

band edges is defined as the Fermi level position where the charge states ql and q2 have equal

energy [75]. The Fermi energy is one of the critical parameters in determining which of the

alternative defects or their charge states has the lowest formation energy and should therefore

predominate. Accurate defect levels from DFT calculations can help determine the

photoluminescence spectrum data of TiO2. Applying the scissor operator, all the calculated

transition levels that include positively charged states are scaled by a fraction k, the ratio of the

experimental to the calculated data of the band gap. In this case, k = 3.00/2.54 = 1.18. All the

negatively charged states remain unchanged. This result, thus, does not affect the DFT

calculations themselves, but is used to adjust the defect transition levels obtained from the DFT

calculations.

Figure 5-3(b) shows the results of calculations of the defect transition levels for an oxygen

vacancy (Vo), titanium interstitial (Ti,), titanium vacancy (Vr,), and oxygen interstitial (O,) in

TiO2 frOm calculations using a supercell containing a 2x2x3 unit cell. The results indicate that