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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. HaiPing 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..... KohnSham Theory .............. ...............50... ExchangeCorrelation 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 HIGHTEMPERATURE 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 NTYPE 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 11 Selected bulk properties of rutile TiO2. ................ ........................ ..............38 21 Calculated lattice constants, CPU times, convergence data, and total energies of rutile TiO 2. ............. ...............63..... 31 Comparison between the calculated structural parameters and experimental results for rutile TiO2............... ...............76.. 32 Positions of the Ti interstitial site in the Frenkel defect models shown in Figure 22. .....76 33 Calculated Schottky DFEs for rutile TiO2. ............. ...............76..... 34 Calculated Frenkel DFEs for rutile TiO2. ............. ...............76..... 35 Comparison of DFT calculated Frenkel and Schottky DFEs to published experimental and theoretical values for rutile TiO2. .............. ...............77.... 51 Calculated lattice parameters and TiO bond lengths for rutile TiO2 COmpared to the theoretical values and experimental values ................. .........__....... 117......... 52 Structural relaxation around defects. The relative changes from original average distances from perfect bulk are listed in percent. ................ ................. ......... .11 53 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 1010 atm), and oxidized condition (T=1200 K, SF 0.5eV, pO2= 10' atm). ................ ...............118........... . 61 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 11 Bulk structures of rutile and anatase TiO2........... ............... ......... ................39 12 Diagram of the TiO6 Octahedral structure in rutile TiO2.................. .......... .............39 13 Phase diagram of the TiO system. ................ .......................... ...............40 14 Calculated defect concentration in undoped TiO2 (A=0) at different temperature ranges as a function of pO2 USing reported equilibrium constants ................. ................ .40 15 The logarithm of weight change of rutile as function of logarithm of oxygen partial pressure. ............. ...............41..... 16 Thermogravimetric measurement of x in TiO2x aS function of oxygen partial pressure. ............. ...............41..... 17 Electrical conductivity measurement of TiO2x aS function of oxygen partial pressure. ...42 18 Electrical conductivity measurement of TiO2x aS function of oxygen partial pressure. ...42 19 Defect formation energies of (a) Con,, (b) Comet, (c) Con, o, (d) Vo defects as a function of Fermi level in the oxygenrich limit ................. ...............43......_.._.. 110 Defect formation energies as a function of the Fermi level, under the Tirich (left panel) and oxygenrich (right panel) growth conditions, respectively. ............. ................43 111 Calculated total DOS for TiO2 per unit cell compared to experimental UPS and XAS spectra for TiO2 (110) surface ................. ...............44........... .. 112 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.............. .44 113 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........... ... 114 Comparison of density of states (DOSs) between defective and perfect rutile TiO2.........45 115 Calculated total and partial density of states (DOSs) of anatase TiO2. ............. ................46 116 Calculated density of states (DOSs) of rutilestructured RuxTi 1xO2 with different Ru concentrations compared with experimentally determined spectra ................. ...............46 117 Calculated spin polarized density of states (DOSs) of the Codoped anatase (left) and rutile (ri ght) TilxCoxO2. ............. ...............47..... 118 Calculated DOS of the (a) relaxed defective structure with Cd (b) relaxed defective structure with Cd' and (c) unrelaxed defective structure with CdO ............... .................47 119 Total (A) and partial (B) density of states (DOS) for doped anatase TiO2 calculated by F LAPW .............. ...............48.... 21 Flowchart describing KohnSham calculation. ............. ...............64..... 22 Illustration of difference between allelectron scheme (solid lines) and p seudopotenti al scheme (dashed lines) and their corresponding wave functi ons..............64 23 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..... 31 The Schottky defect models considered in this study. ................... ... ............7 32 The Frenkel defect models considered in this study ................. ................ ......... .79 33 The densities of states of perfect and defective TiO2. The valanceband maximum is set at 0 eV. ........._._.. ...._... ...............80... 34 Possible octahedral Ti interstitial sites in rutile TiO2. ............. ...............80..... 41 Schematic illustration of the use of PBCs to compute defect formation energies for an isolated charged defect in a supercell approximation. ............. .....................9 42 Calculated defect formation energies and defect transition levels in different super cell s............... ..............9 43 Calculated defect formation energies for various charge states of the titanium interstitial in a 72atom supercell in TiO2 aS a function of the Fermi level (electron chemical potential) with and without application of the MakovPayne correction. ..........92 44 Calculated defect formation energies for intrinsic defects at 300 K and 1400 K with and without the MakovPayne correction under reduced conditions (pO21020)..............~93 51 Electrical conductivity of rutile TiO2 Single crystals as function of the oxygen partial pressure in the temperature range 12731773K ....._.__._ .... ... .__. ......._._.........1 52 The influence of system size on the defect formation energy of a single oxygen vacancy calculated by atomiclevel simulations using the empirical Buckingham potenti al ................ ...............120....... ...... 53 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.... 54 Ballandstick models showing relaxation around a titanium interstitial (a), an oxygen vacancy (b) and a titanium vacancy (c) in a TiO2 Supercell. ............. ................122 55 Crosssectional contour maps of structure (a) and charge density difference around a titanium interstitial of differing charges [Ti,"~ in (b) and Ti,""~ in (c)]. ............................. 123 56 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 57 Calculated defect formation enthalpies (DFEs) of defect complex [(a) Frenkel defect; (b) anionFrenkel defect; (c) Schottky defect] as a function of Fermi level at 1900 K when pO21010 ........._._._ ...............125._._._.. 58 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 59 Twodimensional 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.... 510 Contribution of vibrational energy and entropy to the defect formation energy of the indicated defects relative to the defectfree structure as calculated with the Buckingham potential. ............. ...............129.... 511 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). ............. ....................12 512 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 61 A 2x lxl supercell model for rutile TiO2 Structure. "X" shows the center of an 06 octahedral structure ................. ...............140................ 62 Total and partial DOS of pristine rutile TiO2. ................ ................. ......... .....140 63 Total and partial DOS comparison between pristine and defective TiO2 with a +4 charged Ti interstitial. ............. ...............141.... 64 Total and partial DOS comparison between pristine and defective TiO2 with a +2 charged oxygen vacancy. ............. ...............142.... 65 Valence density difference maps for: (a) (110) and (b) (110) lattice planes of pristine TiO2 Structure ................. ...............143................ 66 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 (110) 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 bandgap transition metal oxide due to its ntype 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 MakovPayne 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 electronicstructure 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 antiFrenkel 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 pseudostate 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 ntype doping of aluminum interstitials is limited by high concentrations of titanium interstitials and oxygen vacancies. Finally, the origin of shallow level ntype 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 shortrange cationcation 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 firstprinciples 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 oxygendeficient rutile TiO2 (1 10) surface [46]. 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 oxygendeficient 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 ntype 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 [914]. Individual defects and impurities at grain boundaries in oxides can be analyzed by high resolution transmission electron microscopy (HRTEM) [15], Zcontrast 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 Codoped 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 11. The structure studied in this dissertation is rutile TiO2. Table 11 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 12). 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 edgesharing 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 13). For example, orderdisorder transformations take place over the entire composition range of the TiO 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 TiO2x, 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 PO21 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 TiO2x In order to describe the point defects in terms of equations, it is important to have a notation system. The KroigerVink 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~), selfinterstitials (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 ntype defects, such as oxygen vacancies and titanium interstitials, are responsible for oxygendeficient 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' (12) VJ = Vj' + e (13) The corresponding defect equilibrium equations are also written as [Vf ]pg2 = K, (14) [V;]n= Kb[V ] (15) [Vj']n = Kc[Vj] (16) 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 (17) The concentration of electrons and the neutral, singly and doubly charged oxygen vacancies are related through the above four equations [equations (14)(17)]. 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= KapO21/2 if [Vo ] >> [Vox + Vo ], [Voltotai= (KaKb)1/2PO21/4 if [Vo ] >>[Vox +[ Vo], [Voltotai=( (/4KaKbKe) 1/3po2 1/6 Thus the concentration of oxygen vacancies in an oxygendeficient oxide may have an oxygen pressure dependence that ranges from pO21/2 to pO21/6 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: (18) Ti~ + 2OfX = V7, "+2V2' + TiO2 (19) ri1 O, = Vj' +O2(g) +2e' (110) The defect reaction equilibrium and charge neutrality can then be described as [Vr,""'][Ti:"' ]= K, (111) [Vr,""11][V ']2 = K2 (112) [Vh']n2 2Z = K3 (113) n +4[V,""'] = 2[VJ']i+ 4[Ti:"'] (114) 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) (115) Ti; = Ti,' + e' (11 6) Ti: = Ti:' + e' (117) Ti:' = Ti:" + e' (118) Ti: = Ti: + e' (119) Then we can obtain the defect reaction equilibrium equations as [Ti$]x 2 = KAq (120) [Tia']n =K,[Tilz] (121) [Ti:"]n = Kc[Ti,'] (122) [Ti'" ]n = KD 2i" ] (123) [Tia'"']n = KE 2'"]' (124) Also the charge neutrality principle requires n = [Ti:']+ 2[Ti:' ]+ 3[Ti'"]i+ 4[Tia'"']i (125) 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 oxygendeficient oxide may have an oxygen pressure dependence that is related to pO21/(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 ntype 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) (126) Tis + 20, = Ti,'" + 3e'+O, (g) (127) Ti"' = Ti:"' + e'. (128) It therefore follows that we get the following defect reaction equilibrium equations: [Vo']n2po? = KG! (129) [Ti," ]n po, = K,, (1 3 0) [Ti,'"']n = K,,[Ti'"] (131) Also the charge neutrality principle requires n = 2[Vo'] +3[Ti,'"] +4[Ti,'"']i. (132) The combination of the above equations [equations (129) to (132)] results in the following expression for pO21 po,o2 (133) K, + (K, + 3n K, +12n7KpK,) Equation (133) 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 (129) to (133). 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 ntype 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 14. 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 ntype conductivity over the entire range of pO2 and does not exhibits an np 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) (134) 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 (15), (16) and (121) to (124). 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, (135) n+[Al,'] = 2[Vo'] (136) As a result, the electrical conductivity will be proportional to pO21/4, HOt pO21/6. Thus it is possible that the relationship of defects to pO21/4 may reflect the presence of single charged oxygen vacancies, or trivalent Ti interstitials, or even trivalent 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 (137) n +[Al,,'] = 4[Ti:""] (13 8) On the other hand, selfcompensation 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, (139) 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) (140) Nb,O, = 2Nb,'"" + 10e'+ O, (g) (141) 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 TiO2x 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 oxygendeficient 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 PO21 n, it is found that plots of log x vs. 1/T do not yield a straightline 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 TiO2x 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 K1500 K), the weight change of TiO2 is proportional to pO21/6 (See Figure 15) [9]. This shows the existence of the fully charged oxygen vacancies. At low temperature (1250 K1300 K) the results show a transition in the oxygen pressure dependence from pO21/6 to pO21/2. 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 pO21/2 nor pO21/6 is a stable observation in this temperature range. Foirland also worked on thermogravimetric studies of rutile [10] (see Figure 16). He measured the weight change of the TiO2 aS a function of temperature (1 133 K1323 K) and oxygen pressure (13520 torr) and found the weight change to be proportional to pO21/6, 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 (110). 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 102 and 104 atm) over a temperature range of 1318 K1531 K. They found the weight loss to be approximately proportional to pO21/5, 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 16 and Figure 17). 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 17). They found that the value of n in pO21 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 microcrystalline TiO2 WAS reported to be independent of oxygen partial pressure [13]. The experimental study of GarciaBelmonte 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 ptype 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 (109 pO2 Pa >1019), TiO2 exhibits ntype behavior, as expected. At atmospheric oxygen pressures, highpurity 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 K1273 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 18). 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 14731700 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 Al203doped rutile TiO2 aS a function of oxygen partial pressure over the temperature range 950 1213 K [28]. The conductivity was found to change from ntype 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 ntype 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 MottLittleton methods, molecular dynamics (MD) simulations, and more theoretically rigorous first principles approaches. The maj or methods in the third class include HartreeFock (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 MottLittleton 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 semiempirical selfconsistent method [32]. They worked with titanium interstitials and oxygen vacancies and found donor levels in the range of 0.70.8 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 selfinteraction 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 firstprinciples 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 oxygenrich growth conditions the Co dopants would be formed primarily in neutral substitutional form, which conflicts with the experimentally observed behavior of Codoped samples (see Figure 19). Thus, they concluded that the growth conditions were most likely oxygen poor. When they considered oxygenpoor conditions they predicted roughly equal concentrations of substitutional and interstitial Co. NaPhattalung 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 ntype and ptype samples. Thus they believed that titanium interstitials are the strongest candidates responsible for the native ntype conductivity observed in TiO2. (See Figure 110) 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 postgrowth 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 Codoped 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 TiO bonding is essential for explaining ntype 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 112). 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 tightbinding study by Halley et at. [41], a linear muffintin orbital (LMTO) study by Poumellec et al. [42], a DFT study by Glassford and Chelikowsky [43], and a fullpotential linearized augmented plane wave (FLAPW) formalism by Asahi et at. [44]. For example, Halley et at. presented a tightbinding calculation to describe the electronic structure of a defective TiO2 with oxygen vacancies [41]. In Figure 113 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 O2pTi3d mixing in the valence band and a weaker O2pTi3d mixing in the conduction band. This contradicts the general assumption that TiO2 is an ionic compound. They also mentioned that the noncubic environment may allow Ti4pTi3d hybridization, which would explain the observed preedge and edge features in the Ti xray 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 114). NaPhattalung 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 115). Impurity doping is always important for the study of defects in semiconducting metal oxides. In order to achieve a high freecarrier 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 Ruinduced 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 116). Additionally, these states were found to be localized on the Ru with tz,like symmetry. In their DFT DOS calculations and absorption analysis for Codoped rutile and anatase TiO2, Weng et at. suggested that the pd exchange interaction between the O 2p and Co 3d electrons should be ferromagnetic, which means that intrinsic ferromagnetism should occur in the Codoped TiO2 Systems (see Figure 117). This is proven by their optical magnetic circular dichroism (MCD) spectra measurement. Errico et at. presented details about the electronic structure of the Cddoped TiO2 USing FLAPW methods and the results show that the presence of Cd impurity leads to the Cds levels at the bottom of the valence band, and impurity states at the top of the valence band (see Figure 118). 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 bandgap 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 119). Lastly, Umebayashi et at. reported the band structure of Sdoped 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 11. 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 (ax106, o1) Anisotropy of linear coefficient of thermal expansion at 30650K (ax106, 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 caxis, a=8.816x106+3.653x 109xT +6.329x1012xT2; Perpendicular to caxis, a=7.249x 106 +2. 198x109xT+1.198x1 012xT2. Dielectric constant Band gap (eV) Perpendicular to caxis, 160; Along caxis, 100; [coill 1.946i A titanium Oxygen [100] 1.983 A l01] 4~ ~~ [100] Rutiile Anatase S102.3018" loll q 0[ Figure 11. Bulk structures of tile and anatase TiO2 [2]. O Cr CY o 6 Figure 12. 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 13. Phase diagram of the TiO system [18]. TiO2x o TiO2. 2 1073 K, A 0) 1273 K. A Ing p(O,} [p in Pa] log p(O, [p in Pa] Figure 14. 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 10mB 10"8 10" ~ g17C. '~1077.C 18 16 R 12 1 8 LOGy 902 atr Figure 15. 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"2 101 108 1 OXYGEN PRESStJRE,atm. Figure 16. Thermogravimetric measurement of x in TiO2x aS function of oxygen partial pressure [23]. log p(O } [ p(O ) in Pa] Electrical conductivity measurement of TiO2x 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~" 10so rTo 10 0D' X) OXYGEN PRESSURE,atra Figure 17. Electrical conductivity measurement of TiO2x aS function of oxygen partial pressure [12]. r E C r s06 I Figure 18. 120 2 LL~ ~ 30 ~ L~ 1 0123 1 2 3 EF (eV) EF (eV) EF (eV) EF (eV) Figure 19. Defect formation energies of (a) Con (b) Co.,,t (c) Con Vo, (d) Vo defects as a function of Fermi level in the oxygenrich limit [35]. 6 I 6 (02 O T 4  4 2 1 2 4 1 4 Ti. 6  6  ITirich Orich 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 110. Defect formation energies as a function of the Fermi level, under the Tirich (left panel) and oxygenrich (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 111. 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 112. 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 113. 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 114. 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 115. Calculated total and partial density of states (DOSs) of anatase TiO2 [36]. 9 6 3 0 3 6 9 Energy (eV) Figure 116. Calculated density of states (DOSs) of rutilestructured RuxTilxO2 with different Ru concentrations compared with experimentally determined spectra [43]. !1 01 2 34 5 5 4 3 21 I1 2 34 Energy (eV) Energy (eV) Figure 117. Calculated spin polarized density of states (DOSs) of the Codoped anatase (left) and rutile (right) TilxCoxO2 [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 119. 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, selfinteraction, dipole interaction, and strong correlation effects. However, these problems are minimal in solidstate, total energy calculations of bulk TiO2, So these issues are not discussed in this dissertation. Overview of Density Functional Theory KohnSham Theory Solid state physics explicitly describes solids as a combination of positively charged nuclei and negatively charged electrons. If there are only timeindependent 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 groundstate energy, and H is the Hamiltonian operator. These latter two terms are described as H~~7 88 Y,+ e (22) E = T + V,2e + Ve = C(N Vi)+Icr,)+C1 (23) 1=1 2 1=1 I On the right side of equation (23) 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 electronnucleus attraction; and the third term, Ve, is the potential energy due to electronelectron interactions. This manybody problem, while much easier than it was before the BornOppenheimer approximation, is still far too difficult to solve. Several methods have been developed to reduce equation (22) to an approximate but more readily solvable form. Among these approaches, the ThomasFermiDirac 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 (HohenbergKohn theorem), and has been widely used in practice since the publication of the KohnSham equations in 1965 [49]. In the KohnSham formula, the groundstate energy of a manybody 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 oneelectron orbitals representing a noninteracting reference system as p(r)= I ((r) 2 The Hamiltonian and totalenergy functional can therefore be described as He = To ,+ Vse + Ve+ (24) E,[p]= T[p]+V.,+Ve,[p]+Vxc[p] (25) In equation (25), the groundstate energy is in the KohnSham 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 exchangecorrelation (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 noninteracting electrons, T, is known exactly. Kohn and Sham proposed an indirect approach to use this welldefined T (which is ThomasFermi 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 (25) 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 wavefunctions in the vicinity of the atomic core due to the orthogonalization to the inner electronic wavefunctions, 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 planewave basisset. 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 (25) is the electronelectron 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 (25) is the exchangecorrelation (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 LeeYangParr nonlocal correlation functional (B3LYP). Finally, the standard procedure to solve equation (25) is iterating until selfconsistency is achieved. A flow chart of the scheme is depicted in Figure 21. The iteration starts from a guessed electron density, po(r). Obviously using a good, educated guess for po(r) can speedup convergence dramatically compared with using a random or poor guess for the initial density. ExchangeCorrelation Functional The exchangecorrelation (xc) potential, Exc[p], is the sum of exchange energy and correlation energy as E, [p] = Ex [p]+ Ec[p] (26) 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 KohnSham energy, EKSIp], and the sum of the ThomasFermi energy, Hartree energy, Ulp], and exchange energy, Ex[p]. According to the second HohenbergKohn theorem, there should be an exact form of the exchangecorrelation 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 exchangecorrelation functional approximation and in fact the simplest of all, the localdensity approximation (LDA) was proposed by Kohn and Sham in 1965 as EMA[p]= Id3 HEG(p(r)). (27) 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 slowlyvarying 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 exchangecorrelation 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 (27), 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)) (28) 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 HartreeFockbased 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, hydrogenbonded 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 universallyapplicable Exc remains great challenge in DFT. Ongoing efforts to discover the next generation of density functionals includes developing orbitaldependent 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 planewave 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 allelectron calculation because the wavefunctions of valence electrons oscillate strongly in the vicinity of the atomic core due to the orthogonalization to the inner electronic wavefunctions (see Fig. 22). 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. 22). 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 normconserving pseudopotentials. There are many, widely used normconserving pseudopotentials. One of the most popular parameterizations is the one proposed in 1990 by Troullier and Martins [55, 56]. However in some cases, normconservation 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 normconservation constraint was abandoned, and a set of atomcentered 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 allelectron 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 planewave basis set for the expansion of the Kohn Sham wavefunctions, and pseudopotentials to describe the electronion interaction. A few pseudopotentials can be used in this study, such as normconserving pseudopotential generated using the optimization scheme of TroullierMartins (pspnc potential), normconserving 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 21 and the corresponding discussion). Two separate exchangecorrelation 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 PerdewWang functional [62]; PBE in form of PerdewBurkeErnzerhof functional [63]; and RPBE in form of Revised PerdewBurkeErnzerhof functional [64]). Based on our tests, the GGA of PerdewBurkeErnzerhof (PBE) is best suited towards our studies. The sampling of the Brillouin zone was performed with a regular MonkhorstPack kpoint grid. The MonkhorstPack 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 HellmanFeynman forces which is defined as the partial derivative of the KohnSham 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 Pulaylike 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 BroydenFletcher GoldfarbShanno (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 21 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 PerdewZunger functional, and GGA, in the parameterization of PerdewBurkeErnzerhof functional. As the pseudopotential, we took the ultrasoft pseudopotential (usp) and one normconserving pseudopotential optimized in the scheme of Lin et al. (recpot) [72]. The results in Table 21 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 normconserving 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 23. 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 KohnSham calculation would still differ from the true bandgap. The true bandgap may be defined as [47] Egp ; = n e (N 1)ef(N). (29) However the calculated KohnSham band gap for the difference between the highest occupied level and the lowest unoccupied level of the Nelectron system is = t+1I [ss (N 1s s _[ (tN +1)VTIs sNL E(N) (210) = E Axc Thus the Axc in equation (210) represents the shift in the KohnSham 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 KohnSham eigenvalues and defect transition levels (ionization levels). Since the KohnSham 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 Fermilevel position where the charge states ql and q2 have equal energy. This level can be observed in deeplevel transient spectroscopy experiment or be derived from temperaturedependent 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 (211) 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, MakovPayne compensation is applied in this study. Details of the MakovPayne 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 TiO short (A) 1.934 1.933 1.958 1.957 1.949 TiO 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 21. Calculated lattice constants, CPU times, convergence data, and total energies of rutile TiO2 LDA 500 eV cutoff recpot, realspace 4.605 2.991 1.967 1.979 130.78 All converged 1907.80 LDA GGAPBE 500 eV cutoff 500 eV cutoff GGAPBE 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) TiO short (a) TiO 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 21. 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 GGAPBE 400 eV cutoff usp, real space GGAPBE 400 eV cutoff usp, reciprocal Experiment [2] no converged yes , end Figure 21. Flowchart describing KohnSham calculation [47]. Figure 22. Illustration of difference between allelectron 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 23. 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 antiFrenkel 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, 7880]. For example, Baumard and Tani reported the electrical conductivity of rutile TiO2 doped with 0.043 at% Nb as a function of oxygen pressure in the temperature range 1273 K1623 K [78]. They found the cation Fenkel defect to be the predominant defect in Nbdoped 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 [2931, 81] and the results of electricalconductivity 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 [8386]. 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 PerdewBurkeErnzerhof functional (GGAPBE) 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 Brillouinzone sampling with 4 kpoints and planewave 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 BroydenFletcherGoldfarbShanno 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 31. 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, antiFrenkel and Schottky defects in bulk TiO2 aef calculated as E1F = E'O'(V, + Til) ErO'(bulk) (31) EfF = Ero'(Vo +o,) Eo' (bunlk) (3 2) E) = Ero'(V, + 2Vo) Ero'(bulk) + (p1, + 2Po,) (33) 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 antiFrenkel 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 defectfree 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 (34) 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 systemsize errors to the results. These systemsize 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 antiFrenkel defects than cation Frenkel and Schottky defects, only the latter two defect pairs are considered in the following calculations. Model Development Previous semiempirical selfconsistent 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 31. 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 selfdiffusion of Ti interstitials in TiO2. In this work, six Frenkel models, shown in Figure 32, are considered. The coordinates of the positions of the Ti interstitial atoms in all six models are listed in Table 32. 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 13) 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 46) 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 33. 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 34. 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 35 compares these calculated DFEs to DFEs reported in the literature that were calculated with DFT and semiempirical 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. (31) and (32), 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) (36) 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 semiempirical MottLittleton calculation also shows much higher DFEs [31]. This may due to differences in semiempirical potential parameters used for characterizing TiO2. The semiempirical 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 33. 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 wellknown 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 33 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 32 and 34 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 cdirection (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 46, 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 34. 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 31. Comparison between the calculated structural parameters and experimental results for rutile TiO2. (GGAPBE, generalized gradient approximation in the PerdewBurke Ernzerhof functional.) Approach a (A) c (A) TiO short (A) TiO long (A+) GGAPBE 340 eV 4.64 2.97 1.96 2.01 GGAPBE 500 eV 4.63 2.96 1.95 1.99 Experiment [39] 4.59 2.95 1.95 1.98 Table 32. Positions of the Ti interstitial site in the Frenkel defect models shown in Figure 22. 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 33. Calculated Schottky DFEs for rutile TiO2 Model Schottky DFE (eV) 1 3.51 2 3.01 3 4.98 4 5.47 Table 34. 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 35. 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 15.7217.52 Frenkel (distributed) 3.84  Schottky DFEs to published experimental Catlow semi empirical result [31] 5.258.22 11.1214.64 Space charge measurement [80] <6.6 <5.0 Model 1 Model 2 Model 3 Model 4 Figure 31. The Schottky defect models considered in this study. Model 3 Model 4 Model 5 Mlodel 6 Figure 32. 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 33. The densities of states of perfect and defective TiO2. The valanceband maximum is set at 0 eV. [001] me [00 Figure 34. Possible octahedral Ti interstitial sites in rutile TiO2 CHAPTER 4 CHARGE COMPENSATION INT TIO2 USING SUPERCELL APPROXIMATION Introduction Densityfunctional 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), planewave expansions and pseudopotentials have been extensively applied in the study of systems lacking full threedimensional 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 longrange interactions between defect and their periodic images in the neighboring supercells. The presence of this longrange interaction could dramatically change the evaluation of the defect formation energies. Giving TiO2 aS an example, it is believed that its ntype conductivity is partially due to the multivalence 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 nontrivial 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 bandgap 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 longrange interaction problem, normally a uniform electron gas background (j ellium background) is added to compensate for these artificial interactions (see figure 41). 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 localmoment 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 nonpolarizable 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 multicharge 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 PerdewBurkeEmzerhof form (GGAPBE) for the exchangecorrelation 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 Brillouinzone sampling is carried out using a 2x2x2 kmesh and a planewave cutoff energy of 400 eV is used. As the MakovPayne 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,). (41) 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. [9294], 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. (41), SF is the Fermi energy. In order to calculate the Fermi energy, we also calculate the valenceband maximum (VBM) [95]. The MakovPayne 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 ] (42) 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. (42) shows [96], the second correction usually counts for only 35% 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 MakovPayne 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 MakovPayne 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 MakovPayne 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 42(a)(c). In all three cases, the defect formation energies calculated with (without) the MakovPayne 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 infinitesupercell 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 MakovPayne 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 MPcorrected DFEs and E, for the 72, 24, and 12atom 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 MakovPayne approach gives the wrong sign of the correction on fully charged oxygen vacancies. In particular, as shown in figure 42(c), the uncorrected formation energies increase as the supercell decreases and, since the MakovPayne correction is always positive, this correction causes significant exaggeration of the formation energies. For example, in the case of the 72atom 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 42(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 xray 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 MakovPayne correction are calculated as a function of Fermi level under reduced conditions (PO2=1020) (See figure 43). 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 MakovPayne 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 MakovPayne (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 44). 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 MakovPayne 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 MakovPayne correction, our results clearly show the same trend. For example, figure 44(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 44(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 MakovPayne 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 MakovPayne correction. Summary TiO2 has been intensively studied as a wide bandgap transition metal oxide partially due to the multivalence nature of its cation. In this chapter, DFT calculations within the supercell approximation and MakovPayne 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 MakovPayne 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 MakovPayne 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 MakovPayne 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 41. Schematic illustration of the use of PBCs to compute defect formation energies for an isolated charged defect in a supercell approximation. (a) The longrange 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 longrange 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 50i 77 2.0 1.2 . / .5 722 1.5 Figure 42. Calulte defec fomaio enrge and deecraston level indifeen suerels (ac 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 43. Calculated defect formation energies for various charge states of the titanium interstitial in a 72atom supercell in TiO2 aS a function of the Fermi level (electron chemical potential) with and without application of the MakovPayne 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 Z212 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 44. Cacltddfc omto nrge o nrni eett300 K~ and iO; dr 1400 K wt andwihot he akvaye orrctonuner edce cndtios pO12 (b eetomto enries witou the Maoanecretinc()Dfc foraton negie wth heMaovanorcin CHAPTER 5 PREDICTION OF HIGHTEMPERATURE POINT DEFECTS AND IMPURITIES FORMATION IN TIO2 FROM COMBINED AB INITIO AND THERMODYNAMIC CALCULATIONS Introduction TiO2 can be easily reduced, which results in ntype 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 51. 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. Densityfunctional 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 ptype doping into ZnO of groupV elements such as N, P, and As [104, 105] or groupI 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 solidoxide 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 Tirich 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 atomicscale 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 selfconsistent set of DFEs are crucial input parameters for equilibrium, spacecharge segregation models [1 13]; models with accurate and selfconsistent 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 planewave expansions using the generalized gradient approximation in the PerdewBurkeEmzerhof form (GGAPBE) 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 Brillouinzone sampling is made on a MonkhorstPack grid of spacing 0.5nm l; a planewave 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 exchangecorrelation energy, and the calculated lattice parameters and TiO bond lengths are summarized in Table 51. 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 51. 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, (51) 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 52 shows how the DFE of an oxygen vacancy, as calculated from atomiclevel 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 52). However, particularly for small nxnxm (nym) supercells the DFEs deviate from this smooth curve (dashed curve in Figure 52). 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 coworkers [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( ). (_52) 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 (51) 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 zerotemperature, zeropressure DFT electronic structure calculation results and the hightemperature, finitepressure conditions of reallife applications. Charge Compensation The last term in equation (51) 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 longrange 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 MakovPayne approach is used, in which the background error is corrected to O(L3), 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 53(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 KohnSham eigenvalues do not account for the excitation state, this failure is still intimately related to a derivative discontinuity that arises in part from the exchangecorrelation 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) (53) 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 socalled '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 53(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 