Point Defects in Strontium and Barium Titanate from First Principles: Properties and Thermodynamics

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ABSTRACT

BAKER, JONATHON NEAL. Point Defects in Strontium and Barium Titanate from First Principles: Properties and Thermodynamics. (Under the direction of Douglas L. Irving.)

Density functional theory (DFT) calculations and point defect thermodynamics have been used together to investigate the strontium titanate (STO) and barium titanate (BTO) materials systems. While it had long been suspected that iron was the cause of brown coloration in Fe-doped SrTiO3, our calculations prove this assignment and resolve a discrep- ancy between the EPR and defect chemistry communities. The neutral iron substitutional is shown to exhibit two absorption processes, one associated with the conduction band and the other with the valence band, which explain the experimentally observed absorption spectrum. The interaction between the iron substitutional and an iron-oxygen vacancy complex were shown to be critical in correctly reproducing the oxygen pressure dependence of the coloration onset. Following this, we examined how and why the metal vacancy behavior differed in strontium and barium titanate. Oftentimes, point defects models are transferred directly between strontium titanate and barium titanate. At the same time, and in apparent disagreement with this, while the A-site vacancy is assumed to dominate in STO, there has been some disagreement about which type of metal vacancy dominates in BTO. We use DFT calculations to elucidate differences in the bond energies for the different sites in each material, which lead to the A-site dominating in STO, while combinations of the A-site, B-site, and B-site-oxygen vacancy complex dominate in BTO, depending on the conditions. We then examined the response of the vacancies to different processing conditions when doped with niobium and iron, two common dopants in these materials. Lastly, we investigate niobium doped STO as a test case for a framework for calculating hydrogen solubility on the basis of point defect energies, doping, and environmental conditions. Hydrogen incorporation is little-studied in these materials, and it is paramount to understand whether it plays a role as an ionically conductive species; this work is the ﬁrst step along that road. © Copyright 2018 by Jonathon Neal Baker

by Jonathon Neal Baker

A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Materials Science & Engineering

Raleigh, North Carolina

2018

APPROVED BY:

Elizabeth Dickey Ramòn Collazo

Carol Hall Douglas L. Irving Chair of Advisory Committee DEDICATION

To my wife Jennifer, for always loving me and encouraging me to do what I’m passionate about. To my parents, grandparents, aunts, and uncles, who have always fostered my passion for science. For my high school NJROTC instructor, LCDR Anthony Negron, USN (ret.) for teaching me self-discipline and showing me that I’m capable of anything if I try hard enough. To Mr. Smith and Mrs. Grooms, for teaching me introductory chemistry and calculus, and starting my academic journey. To Mrs. Ligon, for teaching me to write. To the Laders, for exposing me to foreign languages and cultures. To Dr. Kornev, Dr. Mefford, Dr. Kennedy, and Dr. Lickﬁeld: thank you for mentoring me during my time at Clemson University, and teaching me the basics of scientiﬁc research. To my sister Jackie: thank you for putting up with me. It can’t have been easy.

ii BIOGRAPHY

The author grew up in rural South Carolina with his sister Jackie, in northern Aiken county, and attended high school in the city of Aiken, where he graduated in 2009. His family has always encouraged his love of reading, science, programming, and enjoyment of the outdoors. During those formative years, he enrolled in NJROTC, where he made many lifelong friends and learned self-discipline, and took as many classes on science and math as his high school offered, while beginning to develop basic programming skills and a beginner’s knowledge of electronics. He also participated in an outreach event sponsored by the Clemson University Department of Materials Science and Engineering, where he was able to participate at a very basic level in materials science research as an intern, hosted by Dr. Konstantin Kornev. This enkindled his passion for materials science, as he saw it as the fascinating intersection of many different disciplines. He was also exposed to foreign cultures during this time, participating in a 3 week exchange trip to Germany, where he was hosted by a Turkish family who owned a döner shop. Although he hasn’t had much time for it recently, he enjoys taking long walks in the woods, bicycling, kayaking, photography, and target shooting. He is also an avid reader of science fiction. After graduating from high school, he attended Clemson University for the next four years, where he obtained his Bachelor’s degree in materials science, with a minor in chemistry. During that time, he worked as an undergraduate researcher for Dr. Olin Mefford IV, developing computer programs to simulate the heating rate of colloidal suspensions of magnetic nanoparticles. During the summers, he worked as an intern at Savannah River National Labs, getting exciting hands-on experience with corrosion research, scanning electron microscopy, vacuum lines, and PVD processes. While at Clemson, he met his wife, Jennifer. Upon graduating, they moved to Raleigh together, where Jonathon began graduate school at NCSU, working under the direction of Dr. Douglas Irving. During his first year there, he greatly expanded his knowledge of computers and programming while learning the basics of using density functional theory software to perform quantum mechanics simulations on solids, and developing a data storage and analysis solution to deal with the groups’ ever increasing amount of first principles data. Later, he and former undergraduate Brian Behrhorst re-engineered one of the group’s core tools, a point defect concentration solver for semiconductors, speeding it up by several orders of magnitude and making it

iii vastly more maintainable. Several other groups members have since built on this more maintainable framework to make substantial contributions of their own to this important tool. The author’s research focuses on point defects in barium and strontium titanate, and how it inﬂuences the optical and electronic properties of the material. This research forms the body of this dissertation.

iv ACKNOWLEDGEMENTS

The work presented in this thesis would not have been possible without help from numerous collaborators and friends, the support of friends and family, and generous funding and computing time provided by the Air Force Office of Scientific Research. I would like to thank Professor Douglas Irving for all of his advice and guidance during my graduate career; not only have I grown as a scientist during my time here, I have also grown as a person, and I feel that I have learned many life lessons that will serve me well the rest of my career. I would like to thank my colleagues Preston Bowes, Joshua Harris, Yifeng Wu, Kelsey Mirrielees, Dan Long, Nikki Creange, Brian Behrhorst, Ben Gaddy, and Changning Niu for always being willing to discuss research issues and making themselves available to bounce ideas off of. I would like to acknowledge contributions by Preston Bowes, Joshua Harris, Brian Behrhorst, and Ben Gaddy to software that I now use almost daily. I would also like to thank people who performed experiments complementing my simulations: Dan Long, Nikki Creange, Ali Moballegh, and Biya Cai. I would like to especially thank Professors Ramon´ Collazo and Elizabeth Dickey for always providing guidance and support during this endeavour. Preston Bowes deserves a special additional acknowledgement; in an attempt to split the workload during this project, I ran some of the underlying DFT calculations, and Preston ran others. This project is very much a joint effort between me and Preston. My research was directly funded with generous support from Air Force Office of Scien- tific Research Basic Research Initiatives FA9550-14-1-0264 and FA9550-17-1-0318, through Dr. Ali Sayir’s Aerospace Materials for Extreme Environments program. These grants also provided most of the hefty amounts of computing time required to perform this work. The computers themselves also merit an acknowledgement: The actual DFT calculations were performed on either the North Carolina State University High Performance Com- puter, or the Shepard, Armstrong, or Conrad supercomputers, administered through the Navy Department of Defense Supercomputing Resource Center and the Department of Defense High Performance Computing Modernization Program. During the initial phase of the project, we used the Labrea file server to store and process our results. The Labrea server was eventually replaced with a pair of redundant servers running newer hardware with more storage and more powerful, easy to maintain software: the R2D2 and C3PO file servers, which have safely and securely stored all of our data and performed most of our thermodynamics simulations since early 2017.

v TABLE OF CONTENTS

LIST OF TABLES ...... viii

LIST OF FIGURES ...... ix

Chapter 1 Introduction ...... 1

Chapter 2 Literature Review: A Historical Perspective ...... 6 2.1 Attention From the Condensed Matter Physics Community and Initial For- mulations of Defect Chemistry...... 7 2.2 Electrical Degradation and Electrocoloration...... 9 2.3 Empirical Defect Chemistry ...... 11 2.4 Mott-Littleton Based Defect Chemistry...... 12 2.5 Differences in Vacancy Behavior Between Strontium and Barium Titanate . 13 2.6 Degradation Revisited & Iron Defect Reaction Models...... 14 2.7 Hydrogen Incorporation...... 16 2.8 Thin Films...... 18 2.9 Defect Chemistry Modiﬁcations Near Surfaces ...... 18 2.10 Density Functional Theory Studies...... 19 2.11 Concluding Remarks ...... 19 2.12 Contrast to Semiconductor Community ...... 20

Chapter 3 Methodology ...... 22 3.1 Density Functional Theory...... 22 3.2 Anatomy of a plane-wave DFT calculation...... 24 3.2.1 Bulk Material Simulation...... 25 3.2.2 Bandstructure Simulation...... 26 3.2.3 Point Defect Simulation...... 27 3.3 Post-Processing...... 29 3.3.1 Extraction of Useful Parameters from Bandstructure...... 29 3.3.2 Treatment of Chemical Potentials ...... 32 3.3.3 Point Defect Formation Energies...... 34 3.3.4 Approximating Optical Transitions...... 36 3.3.5 Defect Solver...... 38 3.4 Data Management...... 40

Chapter 4 Brown Coloration of Iron doped Strontium Titanate ...... 43 4.1 Introduction ...... 44 4.2 Experimental Methods and Results...... 45 4.3 Computational Methods...... 46 4.4 Computational Results and Discussion...... 48

vi 4.5 Conclusions...... 52

Chapter 5 Mechanisms Governing Metal Vacancy Formation in BTO and STO .. 54 5.1 Introduction ...... 55 5.2 Methods ...... 57 5.3 Results & Discussion ...... 60 5.3.1 Characteristics of Vacancies in BTO and STO ...... 60 5.3.2 Origin of Vacancy Energy Differences...... 63 5.3.3 Vacancy Trends in Nb-doped BTO and STO ...... 64 5.3.4 Vacancy Trends in Fe-doped BTO and STO...... 67 5.4 Conclusions...... 70

Chapter 6 Calculating Hydrogen Solubility in Donor-doped STO ...... 71 6.1 Introduction ...... 72 6.2 Methods ...... 73 6.3 Results and Discussion...... 74 6.4 Conclusions...... 79 6.5 Supplemental Derivation: Finite Temperature Chemical Potential Space . . . 80

6.6 Supplemental Derivation: Calculation of µH from PH2 and PH2O ...... 82

Chapter 7 Conclusions and Future Work ...... 85

BIBLIOGRAPHY ...... 87

vii LIST OF TABLES

Table 5.1 TTL Locations of vacancies in BTO...... 61 Table 5.2 TTL Locations of vacancies in STO...... 62

Table 6.1 Ground State Energies for H2,O2, and H2O...... 84

viii LIST OF FIGURES

Figure 3.1 (a) illustration of the unit cell (blue cube), primitive cell (green cube), and lattice basis (blue vectors) of SrTiO3. (b) the [100] projection of an AB stack of SrTiO3 with embedded unit cell, primitive cell, and lattice basis...... 25

Figure 3.2 Density of States and electronic bandstructure of SrTiO3 along high symmetry paths in the ﬁrst Brillouin zone, with carrier band extrema marked with colored dots...... 30

Figure 3.3 0 K Chemical Potential Surfaces for AlN (a) and SrTiO3 (b) demonstrating the differences in accessible chemical potentials for binary and ternary compounds...... 33 Figure 3.4 Formation energy diagram for a fictitious defect with only two charge states...... 35 Figure 3.5 Configuration Coordinate diagram for the fictitious defect from fig. 3.4 ...... 36 Figure 3.6 High Level Overview of DFT Simulation Workflow...... 41

Figure 4.1 Absorption spectra taken for (a) electrocoloration and (b) annealing experiments...... 46

Figure 4.2 (a) Chemical potential space of SrTiO3, (b) Formation energy for select native and iron containing defects, (c) FeTi-VO binding energy 0 vs Fermi level, and Conﬁguration-Coordinate diagrams for (d) FeTi to 0 CB and (e) VB to FeTi optical transitions...... 49 Figure 4.3 Concentrations of native and Fe containing defects after annealing at 1073 (top left) and 1173 K (top right) and quenching to 300 K (bottom). Distinct compensation regimes are indicated by shaded regions la-

belled A-D. The brown color bars show annealing PO2 ranges which

will result in coloration. Gray lines show PO2 values used in experimental anneals...... 51

Figure 5.1 Native chemical potential spaces for (a) BTO and (b) STO. Traces have

been marked for ∆µ(O) sweeps along AO-rich and TiO2-rich boundaries of the ABO3 stability regions. Markers along the traces indicate 20 chemical potentials used for Fig. 5.2, corresponding to 10− (cir- 5 2 cle), 10− (hexagon), and 10 (diamond) atm of O2 at 1173 K. Shaded regions of the ∆µ(A) and ∆µ(Ti) axes show the variation in those

chemical potentials when sweeping ∆µ(O) along the TiO2-rich trace. 58

ix Figure 5.2 Defect formation energy diagrams for vacancies and vTi-vO in (a) BTO and (b) STO at markers along chemical potential traces in ﬁgures 5.1a and 5.1b, respectively. Solid and dashed formation energy lines correspond to respective traces in Fig. 5.1. At equilibrium, the Fermi level will always lie in an interval where all native defect formation energies are above 0...... 60

Figure 5.3 Defect formation energies for NbTi, NbSr, and vO in (a) BTO and (b) 15 STO at hexagonal markers (10− atm PO2 at 1173K) and middle of processing windows in ﬁg 5.1. µNb was determined self-consistently with Fermi level based on ﬁxed impurity content as described in the text...... 65 19 3 Figure 5.4 (a-f) High temperature defect concentration diagrams for 10 cm− Nb-doped (a-c) BTO and (d-f) STO after an anneal at 1173 K for dif-

ferent annealing conditions. Concentrations after: a TiO2-rich anneal are shown in (a) and (d); an anneal in the middle of the processing window are shown in (b) and (e); an AO-rich anneal are shown in (c) and (f)...... 66

Figure 5.5 Defect formation energies for FeTi, FeTi-vO, FeSr, and vO in (a) BTO and 15 (b) STO at hexagonal markers (10− atm PO2 at 1173K) and middle of processing windows in ﬁg 5.1. µFe was determined self-consistently with Fermi level based on ﬁxed impurity content as described in the text...... 68 19 3 Figure 5.6 (a-f) High temperature defect concentration diagrams for 10 cm− Fe-doped (a-c) BTO and (d-f) STO after an anneal at 1173 K for differ-

ent annealing conditions. Concentrations after: a TiO2-rich anneal are shown in (a) and (d); an anneal in the middle of the processing window are shown in (b) and (e); an AO-rich anneal are shown in (c) and (f)...... 69

Figure 6.1 (a) Chemical potential space of SrTiO3 at 1173 K balanced against Sr, Ti, SrO, and TiO2. Solid, dashed, and dotted lines indicate TiO2-rich, middle, and SrO-rich processing conditions. (b) and (c) Formation

energy diagrams for relevant point defects at endpoints of the TiO2- rich (b) and SrO-rich (c) traces, with gray shading marking band edge

positions at 1173 K. µH was taken at 1 atm of H2 or H2O gas at 1173 K, while µNb was solved for self-consistently based on the concentration. 75 19 3 Figure 6.2 Point defect concentrations for 10 cm− Nb doped STO equilibrated at 1173 K with ((d)-(f)) and without hydrogen ((a)-(c)) in TiO2-rich (((a),(c)), middle ((b),(e)), and SrO-rich ((c),(f)) conditions. Hydrogen

chemical potentials correspond to 1 atm availability of of H2 or H2O at 1173 K...... 77

x Figure 6.3 (a-c) Isodoping contours of hydrogen chemical potential and partial 19 3 pressure vs. oxygen partial pressure for STO doped with 10 Nb cm− at 1173 K for (a) TiO2-rich conditions,(b) middle conditions, and (c) SrO-rich conditions. The black line indicates the an availability of 1

atm gaseous H2 and H2O. (d) Temperature dependence of hydrogen solubility, evaluated against gaseous H2 and H2O at 1 atm. in STO 19 3 doped with 10 Nb cm− at TiO2-rich (solid lines), middle (dashed lines), and SrO-rich (dotted) chemical potential traces...... 78

xi CHAPTER

INTRODUCTION

Modern technology is, fundamentally, electronics. The reader no doubt commutes to work by car, train, or bus; these all use computers and analogue electronics in their control circuitry. The reader likely has a smart phone in their pocket, and this dissertation is probably being read on a computer or tablet. The building the dissertation is being read in doubtlessly has climate control, again controlled by a computer. Electronics are built from metals, semiconductors, and insulators. The behavior and performance of semiconductors and insulators are determined by the populations of their point defects, which are single-atom-scale changes in the crystal lattice, such as a missing or replaced atom, or an extra atom. Point defects are present in most materials at part- per-million to part-per-thousand levels, and most materials that haven’t been carefully puriﬁed have hundreds of types of them. These defects are not always detrimental; indeed, controlled amounts of charged point defects form the basis of silicon-based semiconductor technology. This dissertation focuses on predictive models of point defects in two speciﬁc types of insulator material, strontium titanate and barium titanate. Most commonly, these materials

1 are alloyed to form a material called barium-strontium titanate, which is used to produce capacitors called multi-layer ceramic capacitors (MLCCs), although barium titanate and strontium titanate are also used on their own to make more specialized components like variable capacitors, variable resistors, and positive-temperature-coefficient-of-resistance resistors. Current electronics circuitry makes wide use of MLCCs, which range in scale from fractions of a millimeter to several millimeters across. As an example of just how pervasive these are, modern cell phones (circa 2018) have several hundred of these capacitors on the main circuit board. The main mechanism by which point defects modulate the properties of the host material are by changing how easily a charge carrier can transition into the valence or conduction bands.[Kas06] This is typically discussed in terms of controlling the location of the Fermi level within the bandgap. Different types and amounts of point defects are desired for different applications. MLCCs are optimized to have as high a resistance and capacitance as possible, which is accomplished by introducing impurities that force the Fermi level to the middle of the bandgap and by alloying to adjust the bulk response of the material to an applied electric field. Other types of impurity can be detrimental to this goal, through pulling the Fermi level close to one edge of the bandgap, or by making mobile point defects more favorable, which migrate when the capacitor is charged. In contrast, variable capacitors exploit impurities that pull the Fermi level near the conduction band, to make it easier to tune an emergent phenomenon called a space-charge layer, which sustains a very high electric field over a very small region near the interface. Studying point defects in any material well enough to control the material’s properties has traditionally been a long and expensive process. Studying a single type of point defect takes many careful experiments to isolate the effects of the single defect of interest and accurately characterize its properties. And this process isn’t even the first step; first, well controlled synthesis and purification strategies must be developed and implemented. Only then can the point defects begin to be reliably characterized. To use a well known material as an example, it took over a decade of focused and expensive research from multiple labs for silicon to make the jump from research material to replacing germanium in most applications. Today, however, there is a golden opportunity to reduce the cost and time to research point defects controlling the properties of new semiconductors and insulators. Advances

2 in computer science, physics, and computers based on silicon technology have converged to make it possible to simulate bulk material properties and individual point defects on a timescale of days to weeks, using quantum mechanics. This type of well-established quantum-mechanical simulation is known as a density functional theory (DFT) simulation. The use of large scale parallel computing pervasive in modern supercomputers makes it possible to perform dozens or hundreds of these simulations at a time. Performing huge numbers of DFT simulations and then combing through the results to efficiently screen for desirable bulk materials properties has become routine in recent years (e.g. the materials genome project), but similar techniques focused on point defects have been slower to catch up. This is at least partly due to the fact that DFT simulations of individual point defects alone are insufficient to understand how they will change the material’s properties. Ideally, information on all of the point defects needs to be available, and then used in a thermodynamics model in order to capture the net response of the system. Such models are fairly poorly developed at this point. The work behind this dissertation has focused on developing techniques for efficiently studying large numbers of point defects, using the results in thermodynamics models, and applying those techniques to strontium and barium titanate. The thermodynamics models used to study the net response of a defect ensemble are based on generalization of earlier work in ultrawide bandgap materials.[Gad13a; Gad14b] Barium and strontium titanate have dozens of impurities spanning hundreds of possible defects present at levels high enough to affect its observable properties, so one of the first challenges was to develop a framework capable of sifting through the required volume of DFT simulations. This dissertation presents the results of three investigations using this data-handling/thermodynamics-modeling framework: 1. Understanding the brown coloration in iron-doped strontium titanate. This first foray used the initial 90 or so native and iron-related point defect simulations in our dataset, plus additional simulations for possible secondary impurities and optical transitions, to develop a model that explained the unusual coloration behavior of iron-doped strontium titanate. When oxidized, iron doped strontium titanate turns brown, and when it is lightly reduced, it turns clear. Our model extended the understanding of this phenomenon to include the role of the iron-oxygen vacancy first nearest neighbor complex in controlling the coloration onset, and provided firm evidence that an iron substituting on a titanium site was responsible for the two absorption processes that turned the crystal brown.[Bak17]

3 2. Extending our results to barium titanate. Barium titanate, as the other end-member of the barium-strontium titanate alloy, is at least as economically important as strontium titanate. It is often assumed that models developed for strontium titanate will be applicable with little to no modification to other titanate perovskites, especially barium titanate, and vice versa. However, our investigations do not support this assumption. This study focused on the metal vacancies in these materials, and shows that there are subtle but fundamental differences in the bonding behavior. This difference in bonding affects not just the favorability of vacancies, but probably many transition metal impurities as well, of which niobium and iron were specifically examined.[Bak18] 3. Exploring hydrogen incorporation in donor doped strontium titanate. Hydrogen is pervasive in processing these materials. It is present in aqueous chemistry based routes for purifying the powder, and it is used in furnaces to cheaply reduce oxygen pressure in bulk processing. It is also positively charged, highly mobile, and extremely challenging to study experimentally. This work focuses on developing a framework for calculating hydrogen solubility based on environmental conditions and the formation energies of its point defects, and then applying that framework to niobium-doped strontium titanate. Donor doped perovskites are often assumed to have less of a problem with hydrogen contamination than acceptor-doped perovskites. This work provides firm evidence that this rule of thumb holds true in strontium titanate as well, and explores the rich and varied behavior of hydrogen in donor-doped strontium titanate as processing conditions are varied. Between 1 and 2, much work was done on understanding the role of non-intentional and often unaccounted-for impurities in strontium titanate. The first investigation on this was headed up by another group member, Preston Bowes, and has been published in Applied Physics Letters.[Bow18] This focused on trying to reproduce the electrical behavior of a fairly well-characterized sample of strontium titanate from a widely cited article from the 1980s, and it showed that understanding and accounting for all of the impurities is essential to understanding the response of the system as a whole. While such a conclusion may seem obvious, there is a widespread lack of attention to unintentional impurities in the strontium and barium titanate research communities. Overall, this work has made some important contributions to the understanding of strontium and barium titanate. In doing so, it has pushed the envelope in terms of techniques available to study point defects in semiconductors and insulators in general using computer simulation. The dataset acquired over the course of these investigations has

4 much more data to give; many of the impurities simulated as background contaminants for these studies have their own unique properties on their own and in conjunction with other dopants, and we have only begun scratching the surface in terms of understanding them; some of them may even open up new strategies for commercial applications. Additionally, while this dissertation focuses on areas of primary authorship, the author has also made contributions to several other research areas during the course of this work, including helping to apply the data-handling/thermodynamics-modeling framework to study defects in aluminum nitride[Har18; Ald18] and contributing to development of methods for spatially resolving global sample electrical data[Bay16]. It is the hope of the author that the frameworks and strategies developed as part of this research eventually be applied to other materials systems by researchers around the world to greatly accelerate and enhance their own research.

5 CHAPTER

LITERATURE REVIEW: A HISTORICAL PERSPECTIVE

Strontium titanate has been a subject of research for almost 70 years, with the first literature appearing in the late 1940s. However, in many ways research on strontium titanate is still in its infancy, despite its long history. As a rough measure of research quantity with other well- known materials, a Google scholar search returns about 3.59 million results for “silicon", roughly 1.14 million results for “aluminum nitride", and about 83,000 results for “strontium titanate". This body of literature has established a decent engineering understanding of the material, sufficient for empirical tailoring of properties to enable mass production. Nevertheless, a lack of impurity concentration data has contributed to a proliferation of sim- plified defect models, complicating efforts to integrate the old data into more fundamental studies. Unless otherwise noted, none of the articles discussed in this section reported measuring impurity profiles. Systematic studies on strontium titanate grew out of efforts by multiple countries during World War II (1940s) to identify high energy density capacitor materials to replace

6 mica.[Ran04; Hae99; Dur50] The researchers quickly realized that rutile and the compounds

related to perovskite (CaTiO3) not only showed promise as a mica replacement, but also as an ideal material for radar waveguides. Building on earlier ideas by Goldschmidt[Gol26], Vegard[Veg21], and Pauling[Pau27], this ﬁrst wave of research was focused on empirically

exploring the dielectric and structural properties of TiO2 and mixtures of CaTiO3, MgTiO3,

BaTiO3, and SrTiO3.[Dur50; VH46; Meg46] Due to this focus, there was less emphasis at the time on point defect behavior and its effects on functional properties. This research

established the market dominance of BaTiO3 and (Ba,Sr)TiO3 for capacitors and microwave waveguides, a position it still enjoys today. There appears to have been less interest in unal-

loyed SrTiO3 at the time due to its lower dielectric constant and its lack of ferroelectricity.

2.1 Attention From the Condensed Matter Physics Commu- nity and Initial Formulations of Defect Chemistry

In the 1950s, many exciting areas of research were being explored simultaneously. In the earlier efforts, titanate crystals had all been produced by mixing oxide powders into a green body followed by isostatic pressing followed by firing. In the 50s, however, flame fusion techniques began to be applied to produce artificial gemstone and higher quality electronic materials.[Fei04] Solution processing methods for producing ceramic powders which could then either undergo flame fusion or isostatic pressing followed by conventional firing also came into usage.[Sab59] At the same time, float zone refining techniques were being applied to produce high purity germanium, and near the beginning of the next decade the Czochralski process was being applied to produce high quality silicon boules.[Fis12] Indeed, it could be argued that this decade marked the emergence of crystal growth as a distinct discipline.[Fei04] Researchers from several companies and institutions began studying strontium titanate for its own merits rather than just another member of the perovskite family or as a gemstone.[LJ53; Lev55; Fei04] Impurity data from one of the early National Lead Company 18 3 studies reported 9 impurities at concentrations above 10 cm− (with an additional 5 above 17 3 10 cm− ) from a flame fusion grown crystal.[Lev55] Kröger and Vink also published their widely cited work on treating defect formation in solids as a system of reactions, while Brouwer published his set of approximations to the Kröger-Vink reactions.[Krö54; Bro54; KV56; KV58]

7 There are two basic ideas behind the defect reaction model. The first is that all point defect populations in a crystal can be described as a spanning set of chemical reactions at steady state across all possible defect forms and charge states (i.e. one could describe 0 +1 a neutral oxygen vacancy becoming a singly ionized donor as VO VO + e−, with the V+1 e [ O ][ −] concentrations related as Keq = 0 ; and one could describe all possible defects and [VO] their charge states with a spanning set of such relations). The second is that, within such a spanning set of defect reactions, there will always be one or two “dominant" reactions for a given processing regime, and all others can be ignored for determining the charge balance. This latter approximation is necessary to actually fit the defect reaction constants once a defect reaction model has been assumed, since otherwise there would be too many unknowns. Some of the first work on impurities in strontium titanate appeared near the end of the 1950s.[Gan59] This work reported that iron doped strontium titanate crystals could be made to turn brown when oxidized and clear when reduced, in a reversible fashion, and discussed the absorption spectra that caused the brown color. The electron paramagnetic resonance (EPR) community began studying Fe-doped ¨ SrTiO3 in the late 50s. Müller’s dissertation work in 1958[M58] was concerned with deriving 1 and measuring (using EPR) the electron spin Hamiltonian of the Fe−Ti defect (often referred to as Fe+3 in the oxides community). Several years later, he published a paper discussing a second EPR signal appearing for Fe:STO.[Kir64] His conclusion was that the signal could only be caused by an iron substitutional directly adjacent to and strongly interacting with an oxygen vacancy. This description corresponds to what would now be referred to as a defect complex between the iron substitutional and the oxygen vacancy, specifically the +1 +3 (FeTi-vO) complex. At the time, he referred to this as an “axial FeTi signal". This signal, and how well it matched Müller’s spin Hamiltonian, was independently validated 4 years later by Baer et al. [Bae68]. Faughnan & Kiss [FK69] published their work on photochromism in transition metal 0 doped SrTiO3 in 1969. This work and its follow-up paper[Fau71] suggested that the FeTi defect was responsible for the brown coloration first reported on by Gandy [Gan59] a decade +1 earlier. The (FeTi-vO) defect complex was also reported in these papers, and the author reports detecting roughly equal concentrations of the substitutional and complexed forms of iron after processing in pure oxygen and 1 mmHg of air. Faughnan also performed some +1 work in collaboration with Müller where they detected FeTi by photochromic excitation

8 during an EPR experiment.[Mül71] Not long after that publication, Schirmer et al. [Sch75] +2 detected evidence of an (FeTi-vO) complex in oxidizing conditions in samples primarily doped with aluminum and magnesium. Near the same time, another group of authors detected neutral iron-oxygen vacancy complexes in primarily iron doped crystals which had been reduced.[Ber78] Müller seems to have changed his focus area in the 1980s to high temperature superconductors, work for which he would go on to win a Nobel prize. In 1964, Frederikse and coworkers collected Hall data on a number of nominally undoped and donor doped reduced strontium titanate crystals to study electron mobility and scattering mobilities.[Fre64] The authors concluded that longitudinal optical phonon scattering was the dominant scattering mechanism at room temperature and above. One of the ﬁrst simulations of the electronic bandstructure of strontium titanate was carried out using the Linear Combination of Atomic Orbitals method by Kahn & Leyendecker [KL64] in the same year. This study established that the valence band is primarily derived from oxygen 2p orbitals and that the conduction band is derived primarily from titanium 3d orbitals. In 1965, Paladino [Pal65] published his work on oxygen diffusion and oxidation in strontium titanate. In 1967, Walters & Grace [WG67] built on these two works and combined a simple defect model consisting of only oxygen vacancies and electrons with conductivity and thermopower results obtained in different hydrogen-containing annealing atmospheres. The authors also included a spectrographic analysis which showed very high silicon and 20 3 antimony levels, on the order of 10 cm− and iron and aluminum impurities on the order 19 3 of 10 cm− , although no attempt was made to account for them. While this was a primitive model, the work is notable because it was one of the ﬁrst such attempts, and defect chemistry models would become increasingly important in the coming years. Around the same time, many researchers began studying the phonon dynamics of perovskite crystals to try and explain the differences in phase transition sequences of the different perovskite family members. A representative and widely cited paper on this topic was published by Cowley [Cow64] in 1964.

2.2 Electrical Degradation and Electrocoloration

Tredgold and co-authors began publishing on changes to the electrical conductivity of barium titanate under an applied bias in the early 1960s.[BT60; Bra62; CT64] In their ﬁrst paper[BT60], the authors noted that the conductivity of their crystals tended to increase

9 with time under bias, and that the rate of change depended on the electrode choices. The observed increase was extremely rapid with copper, fast and steady with gold and silver electrodes, and slower with chromium, aluminum, and zinc. They hypothesized that the electrodes might have differences in oxygen permeabilities, or that the electrode material might diffuse a short way into the crystal, resulting in changes near the surface. Ultimately, they decided on the latter interpretation. In their next paper, they verified that gold diffuses into the material using a radioactive tracer experiment, but did not report similar tests for other electrode materials, or tests for the oxygen permeability hypothesis.[Bra62] Soon after this, they turned their attention to strontium titanate.[CT65] Because of the perceived similarity of barium and strontium titanate, the authors felt comfortable translating their prior findings in barium titanate to strontium titanate. In this work, the authors showed that strontium titanate crystals prepared with both flame fusion and a molten salt based method exhibited the same increase in conductivity over time as barium titanate. In 1971, Blanc & Staebler [BS71] attempted to unify the electrical degradation research with electrocoloration and photochromic research, while making extensive use of the EPR literature body. They reported specifically on a crystal doped with approximately 18 3 18 3 18 3 4 10 Ni cm− , 4 10 Mo cm− , and 5 10 Al cm− ; however they also discussed similar trends· in Fe, Mo,· and Al codoped samples.· They reported 2 color fronts propagating across the initially clear crystal from opposite ends when a strong electric field was applied. A brown color front propagated from the positive electrode (believed to be associated with

an NiTi-vO complex) and a blue color front (believed to be associated with molybdenum) propagated from the negative electrode, eventually meeting in the middle. These color fronts could be reversed by reversing the applied voltage. Modeling indicated that the colorless region had a much lower conductivity than either of the colored regions, and that the propagation of the conductivity and coloration changes was consistent with a process involving mass migration of oxygen vacancies. They believed that the conductivity proﬁle could be explained by analogy to a p-i-n junction, with the reduced region being n-type, the clear region insulating, and the oxidized region being p-type. They appear to have treated the impurities as free to associate and dissociate with oxygen vacancies, as their charge balance had separate terms for the individual defects, but no term for the complex. The authors note that their model breaks down at longer time scales, and that their temperature dependent conductivity measurements were non-linear with respect to

10 dopant concentration. They believed that this latter point indicated a more sophisticated impurity-oxygen vacancy interaction than they had considered.

2.3 Empirical Defect Chemistry

In 1974, one of the more comprehensive treatises on defect chemistry in perovskites was published by Seuter as a chapter in his thesis work.[Seu74]. Beginning with the idea of a defect reaction model as proposed by Brouwer and Kröger and Vink, Seuter explored the consequences of different sets of assumptions for dominant defect reactions in different

processing regimes for BaTiO3, invoking the Gibbs phase rule and charge neutrality as explicit constraints. Although he covered many possible defect chemistries, he noted that he had only considered the most simpliﬁed cases and had completely neglected defect complexes. Most of the later work on defect chemistry in barium and strontium titanate drew, whether directly or indirectly, on this work. Chan and Smyth published a considerably shorter work 2 years later: Defect Chemistry

of BaTiO3.[CS76] This paper introduced the Chan and Smyth’s extensive measurements on the high temperature conductivity of intentionally aluminum and niobium doped barium titanate after equilibration at different oxygen partial pressures and proposed qualitative defect chemistry models for the measurements. This was followed in 1978 by a more thorough article by Eror and Smyth[ES78] in which the authors sought to rigorously tie the observed conductivity measurements to a defect chemistry model. Their conclusion was that the defect chemistry was dominated by accidental acceptor impurities and oxygen vacancies. The same type of analysis for strontium titanate was ﬁrst published in 1981, by both Smyth and Eror.[BE81; Cha81]. Eror’s conclusion, based on similar defect chemical arguments as the 1978 work in collaboration with Smyth, was that the defect chemistry was usually dominated by accidental acceptor impurities. Their paper examined data for nominally Fe, Al, and Cr doped polycrystalline samples. The same year, Smyth published a similar, but ultimately much more widely cited paper on the same topic (but with undoped, Al, and Nb doped crystals).[Cha81] They both utilized a similar solution-based processing route and drew many of the same conclusions. An important difference is that the Chan article was one of the few in the ﬁeld to report an impurity analysis to determine what accidental impurities were in the crystal.[Cha81] However, the impurity levels were on the

11 same order as that obtained with high quality growth techniques from the 50s[Lev55], and the authors made no attempt to account for them in their discussion. Ironically, this paper has been cited by many subsequent authors as evidence that the behavior of the unintentional impurities is well understood. In the conclusion of the 1981 Chan article[Cha81], the high dielectric constant and the idea that the associating defects interacted at only a purely Coulombic level are used to justify neglecting defect complexes in the proposed model. This was at odds with the ﬁndings of the EPR community which noted extensive evidence of complexing. Despite this disparity, because the work of Smyth became so inﬂuential, this assumption that complexes could be neglected propagated through later literature on strontium titanate.

2.4 Mott-Littleton Based Defect Chemistry

In 1986, Lewis and Catlow published a widely cited paper which used a Mott-Littleton methodology to simulate defect formation enthalpies for some expected defects in barium titanate.[LC86] This paper tried to predict how effective different donors or acceptors would be at changing the free carrier population of the crystal in different conditions. They believed that the existence of defect complexes forming between some acceptors and oxygen vacancies could lower the effectiveness of those acceptors, by creating defect complexes which were either neutral or donors. Their model calculated defect incorporation energies using a Buckingham potential (this is a modification of the popular Lennard-Jones potential) to treat short range interaction and the Mott-Littleton approximation to treat long range interaction.[LC83] The need to explicitly treat the chemical potentials appears to have been avoided by using these defect energies in a defect reaction scheme. The defect chemistries they calculated were then related to other properties, such as conductivity. Their potential model is specific to metal oxides, and makes the assumption that oxygen- oxygen interactions are the same in all crystals and cation-cation interactions are purely Coulombic; instead it tries to capture the behavior of each metal’s interaction with oxygen in different coordination environments.[LC85] The short range interaction parameters between metals and oxygen were fit to reproduce the observable properties of each metal’s oxides. Obviously, this technique fails to capture changes to impurity-oxygen interaction specific to each oxide due to differences in electron localization during bonding. Neverthe- less, this research became a widely-cited reference because it offered insights that could

12 not be gleaned purely from experiment.

2.5 Differences in Vacancy Behavior Between Strontium and Barium Titanate

One of the assumptions of many researchers studying BaTiO3 and SrTiO3 is that, despite the different phase transition sequence and the different dielectric behavior, point defects behave essentially the same in both materials.[Bai90; DS15; MH97; CJ94] At the same time,

it is widely believed that the dominant metal vacancy in BaTiO3 is the B-site vacancy (vTi, vB) and that in SrTiO3 it is the A-site vacancy (vSr, vA). [MH97; Shi16; DS15] Compounding matters is the difﬁculty associated with direct observation of the effects of metal vacancies in these materials. Typically the dominant metal vacancy type is judged by the conductivity versus oxygen partial pressure slope in oxidized conditions, or arguments based on precipitation of competing phases. This has led to a shortage of data which has fueled debate on this topic.[Mor01; Smy02; Huy95; LC86] In 1976, Daniels[DH76] postulated that the A-site vacancy dominated in donor-doped

BaTiO3, based on investigations in SrTiO3 and PbTiO3, and developed and parameterized a model to explain his measured conductivity where the donor transitions from electronic compensation to barium vacancy compensation in oxidized conditions. This was disputed only a few years later by Jonker[JH82], who claimed that his experiments clearly supported dominance of the B-site vacancy in La-doped barium titanate, and a mixture of A-site and B-site compensation for niobium doping, based on phase diagram arguments. Lewis and Catlow soon weighed in on the matter with their Mott-Littleton models, which indicated that isolated titanium vacancies were more favorable than isolated barium vacancies.[LC83; LC86] Smyth initially published work which dismissed titanium vacancies as too unfavorable to form in large concentrations[CS76; CS84], while his later work strongly advocated that the titanium vacancy was the dominant metal vacancy in BaTiO3.[Smy02] More recently, work by Lee and Randall implied that both metal vacancies may be favorable. Their work suggested that variations in the processing conditions and precursor powder stoichiometry could shift the energy balance to favor either A site or B site vacancies.[Lee08a; Lee08b]

13 2.6 Degradation Revisited & Iron Defect Reaction Models

Beginning in the late 80s and continuing throughout the beginning of the new millennium, a research group cluster containing Rainer Waser and Joachim Maier began researching the electrical degradation problem using iron doped strontium titanate.[Was90a; Bie93; Was90b; Was90c; Bai90; Rod00; Rod01; Fle00] Waser, Härdtl, and Baiatu collaborated on a trio of papers (dc Electrical Degradation of Perovskite-type Titanates) in the early 90s which have become widely cited amongst those studying both the degradation phenomenon and the defect chemistry of barium and strontium titanate. The ﬁrst two papers focused on the

degradation of poly- and single-crystal SrTiO3 with various dopants. These polycrystalline samples were nominally doped with Al and La in various concentrations, and produced using solution chemistry based processing. These samples were universally processed and annealed in an oxygen atmosphere, and they reported some reproducibility issues in the second article in the series, with certain properties having a standard deviation of up to 25%, even before degradation. The ﬁrst paper explains differences in degradation in polycrystalline samples with different grain sizes and single crystals in terms of oxygen vacancy diffusion across grain boundaries being rate limiting to the degradation process.

The second paper has short discussions on behavior of Al, Ni, and La doped SrTiO3 samples, as well as a short section on internal electric ﬁeld of a BTO crystal as measured via optical Kerr effect, but principally concerns itself with the coloration front that co-occurs with

electrical degradation in Fe-doped SrTiO3. The electrocoloration portion of this study was self-described as serving as experimental validation of a degradation model presented in part three that attempted to link the defect chemistry, electrical degradation, and coloration. The third paper combined simpliﬁed defect reaction and drift diffusion models in a ﬁnite difference scheme to link the defect chemistry, electrical degradation, and coloration.[Bai90] This model used an acceptor ionization energy of 1.2 eV for all simulations, and in subsequent studies this number has become associated with the iron substitutional. Although this number predates a later and similarly widely cited study published by Joachim Maier with a slightly more exact value (1.18 eV)[Den95], both are likely from the same source due to the similarity of the number and the collaboration between the groups at the time.[Was90a; Bie93] Denk reports that the iron ionization energy of 1.2 eV was obtained by analysis of commercially grown strontium titanate samples purchased as part of his study. The value was obtained by analyzing his data with a number of assumptions.

14 A hole mobility was assumed, along with a defect chemistry containing only electrons,

holes, two charge states of FeTi (0 and -1), and a doubly ionized oxygen vacancy. It was further assumed that in oxidized conditions, all of the conductivity arose from free holes (in conjunction with the assumed mobility, this allowed the conductivity to be converted to 1 +2 a hole concentration) and that [Fe−Ti ]=2[vO ]. Together with the charge neutrality condition, this allowed calculation of defect concentrations (and through that, the ionization energy), given that the assumptions were valid. However, no impurity analysis was performed on the purchased sample to conﬁrm that iron was the only impurity present and, like Smyth, possible complexes with oxygen vacancies were neglected. Waser has remained active in the perovskites community. Some of his recent work includes work on resistive switching [Was09; Woj13] and EPR studies[Len11] on degraded strontium titanate crystals. Joachim Maier and coworkers studied local conductivity in poly- and single-crystal Fe

doped SrTiO3.[Rod00; Rod01; Fle00] Their polycrystalline samples were prepared in-house by mixed oxide sintering with precursor powders processed with solution chemistry, while single crystals were purchased from a German crystal supplier. Both sets of samples were nominally doped with ~0.2 mol% Fe, but no concentration proﬁles are provided. “Highly

pure" SrCO3 (the mineral celestite), TiO2 (rutile), and Fe(NO3)3 (H2O)9 (hydrated iron nitrate) as were used as precursor powders for the polycrystalline· samples. They studied the local conductivity by applying microcontacts and measuring the resistance between each pair of contacts. They performed such experiments for single crystal, polycrystalline,

and electrically degraded single crystals of Fe-doped SrTiO3. In 1995, Moos and Härdtl, other members of this research cluster, published some Hall effect data collected on nominally undoped strontium titanate crystals which had been reduced in hydrogen.[Moo95a] In the same year, they also published their work on using thermopower measurements on lanthanum doped strontium titanate to derive an effective density of states mass of electrons in strontium titanate.[Moo95b] They obtained a value of roughly 4.2, and also helpfully tabulated a list of other literature values for the effective mass, although it is not clear which of these are transport masses and which are density of states masses. They released another paper on lanthanum doped strontium titanate in 1996, this time focusing more on the defect chemistry and derivation of lanthanum’s defect reaction constants.[MH96] A year later, a much more comprehensive and more widely cited paper collating much of their data and many of their results was released.[MH97] After this, Härdtl appears to have started transitioning towards research into oxygen gas sensors for automo-

15 tive applications.[Men99; Moo00], retaining a focus on strontium titanate. He published several more papers on the topic and even took out a patent on defect chemistry based gas sensors. However, titanate compounds are still not used in this application[Moo11], probably due to a combination of the reproducibility problems endemic in the material system due to lack of impurity control and the market dominance of zirconia-based oxygen sensors, which drives up the relative cost of replacements.

2.7 Hydrogen Incorporation

Hydrogen is widely present during processing of strontium and barium titanate. It is used as a backfilling/reducing agent during annealing in the form of forming gas, a mixture of hydrogen and nitrogen or argon, to perform low oxygen partial pressure processing while maintaining atmospheric pressure in the furnace. These reducing conditions are necessary to prevent oxidation of so-called base metal electrodes, such as nickel, in multilayer ceramic capacitors. Exposure also occurs during the solution processing typically used on precursor powders. It is additionally present as water vapor in ambient air at volume fractions of around 1-2% over most of the planet, with the exception of deserts and mountainous regions.[McR80; Dai06] Despite this ubiquity, it is only very rarely considered as a possible confounding variable, and only a few authors have specifically examined even the solubility of hydrogen in these systems. Using different techniques and processing routes, various authors have estimated hydrogen content in strontium titanate after annealing in atmospheres containing steam 16 3 20 3 or H2 to range from 10 cm− to 10 cm− .[Web86; Sch99; TM11] Some authors claim that hydrogen solubility is fundamentally limited in acceptor-doped strontium titanate only by the amount of acceptor present, and that hydrogen can incorporate in excess of 80% of the acceptor concentration.[Kre99] High levels of dissolved hydrogen can significantly impact the electrical, optical, and ferroelectric properties of perovskites, so accurate information on its incorporation is crucial. The uncertainty in how much hydrogen incorporates in spite of its importance speaks to the difficulty of quantifying it. Additionally, little work has been performed examining hydrogen uptake in situations where hydrogen is not intentionally added to the processing atmosphere. Evidence from other proton conducting oxides and analogous changes to the IR spectra of these oxides and strontium titanate when annealed in hydrogen-containing atmospheres

16 suggests that hydrogen incorporation in strontium titanate is mainly accomplished through + 1 formation of an H interstitial bound to an oxygen atom (Hi ), and that hydrogen tends to be more soluble when acceptor-doping, and less soluble when donor-doping.[Was87; 1 Was88; Su13; NL97; Mün00] Given that Hi should become more favorable in acceptor- doped conditions and less favorable in donor-doped conditions because of changes in the chemical potential of the electron, this seems to make intuitive sense. However, more recent computational studies have shown that hydrogen can also form energetically favorable defect complexes with metal vacancies with similar near IR absorption features as the free interstitial.[Var14; Jir12] The hydrogen in these complexes is bound to an oxygen atom, just like the free interstitial, but it sits within a vacancy site instead of in the bulk crystal. These complexes are negatively charged and as such become more favorable in donor-doped material, which could indicate a more nuanced behavior. Computational studies have established that hydrogen moves through the crystal via a Grotthuss mechanism, where the hydrogen jumps between nearby sites, rather than a “vehicle" mechanism where it moves with its parent oxygen.[Kre96] In strontium titanate, hydrogen is believed to primarily transfer through inter-octahedral jumps, where the hydrogen ﬁrst rotates around its host oxygen and then jumps to one on a neighboring octahedra as the oxygens come close together during regular thermal motion. These studies predict, using different methods, the total activation energy of a hydrogen rotation-jump event to be around 0.5 eV in strontium titanate, although predicting the prefactor is considerably more 5 2 1 difﬁcult.[Bor11; Mün00] Münch gives a value of 1.25 0.9 10− cm s− for D(T) at 1290 K, with an activation energy of 0.5 .22 eV from his quantum± molecular· dynamics simulations. He does however note that his statistical± error is quite large due to the short integration time, the low number of protons, and the rareness of the event.[Mün99] Within the statistical uncertainty, this puts hydrogen on a similar footing as the oxygen vacancy in terms of how quickly it can diffuse through the crystal, based on estimates of the latter.[Den95; DS12] To the author’s knowledge, the only hydrogen defect chemistry diagram for strontium titanate was published by Waser.[Was87] This work was conducted with iron doped single crystals, and used water vapor to effect hydrogen incorporation. Integration of the IR signal corresponding the hydrogen was used to estimate hydrogen concentration after annealing in different conditions.

17 2.8 Thin Films

Beginning in the late 90s, there was an increasing focus on thin ﬁlm strontium titanate research. Prior to 2004, much of this work was focused on reﬁning production and processing

techniques of (Ba,Sr)TiO3 thin films for applications in electronically tunable microwave electronics such as phase shifters, mixers, and filters.[KP97] In 2004, Ohtomo and Hwang published their findings on a high-mobility electron gas formed at the interface of lanthanum aluminate and strontium titanate.[OH04] This finding, and disagreements about what causes the electron gas to form, have motivated and inspired over a decade of research examining different aspects of strontium titanate thin-film system. Despite this intensive research, many investigators still do not focus on impurity content in their thin films. In a fairly recent article from 2013, Kumar and co-authors report using pulsed laser deposition to deposit strontium titanate thin films alloyed with niobium for thermoelectric applications. However, their ablation target was only reported to be 99.99% pure, and their resulting thin film contained an unquantified amount of silicon, in addition to having up to 1% variation in cation stoichiometry.[Kum13] Some groups have begun to prioritize high purity and stoichiometric thin film production, a representative example being Susanne Stemmer’s group. They utilize a hybrid technique combining an effusion cell source for strontium, a plasma source for oxygen, and a metallo-organic source for titanium (titanium isopropoxide). This enabled separate purification of each element and separate fluence control of all 3 elements onto a lanthanum aluminate-strontium tantalate (LSAT) substrate. The resultant thin films have well-controlled stoichiometry and crystallinity, resulting in a high charge carrier mobility.[Jal09b; Jal09a; Son10] High quality, high purity thin films like this have the potential to enable great advances in this material if their use becomes more widespread.

2.9 Defect Chemistry Modiﬁcations Near Surfaces

Research on thick films and bulk materials continued concurrently with the thin films work. One of the more prolific contributors to both communities in the past two decades has been Roger de Souza, occasionally in collaboration with Waser or Maier. He is also one of the few authors to have reported an impurity profile with any of his work.[Kes15] Much of de Souza’s work is based around using molecular dynamics methods, sometimes in collaboration with

18 experiment, to study modiﬁcations to the defect chemistry near interfaces[DS09], defect association[Sch14], or oxygen vacancy behavior and migration[DSM08; Sch12; DS12; Kes15; WDS16]. His work is an important source of high quality information on the behavior of oxygen vacancies in these materials.

2.10 Density Functional Theory Studies

Paul Erhart sought to calculated the high temperature defect chemistry of barium titanate from density functional theory (DFT) calculations in the late 2000s.[EA07; EA08] However, his work has two limitations: it was performed at the GGA level, and it only included native vacancies and native vacancy complexes. As explained in the methodology section, DFT calculations performed at the GGA level do not properly reproduce electron self-interaction and any effects arising from it, such as the electron localization and the in gap energy levels of defects. To work around these limitations, he attempted to treat the impurities as an “effective acceptor" which was singly charged in all conditions and did not form any complexes. Real impurities in the material change their charge states and form complexes, and unfortunately treating the impurities this way negates many of the advantages of a first principles simulations-based model. This work is notable as being, to the author’s knowledge, the first attempt to model the defect chemistry of a perovskite entirely from DFT data. A few years later, Chris Van de Walle’s group did some DFT work in strontium titanate on the electronic structure, native vacancies, and hydrogen-containing defects.[Jan11; Jan14; Var14] The first article reported on changing electron mobility in response to different strain fields. The second article examined small polarons complexed with native vacancies, and attempted to explain two observed luminescence signals as arising from different configurations of polarons and oxygen vacancies. The last article listed refines some earlier GGA level calculations by performing them at the HSE level.[Vil07; Jir12]

2.11 Concluding Remarks

Lastly, despite the focus on iron in strontium titanate due to its relevance to my work, the large body of literature associated with it, and the consistent focus on iron through multiple decades of research, other elements have also been studied with the defect reaction

19 formalism in both barium and strontium titanate. A literature review collecting reported reaction constants for these impurities can be found in the work of Shi et al. [Shi16].

2.12 Contrast to Semiconductor Community

Meanwhile, research on germanium and silicon took a very different path from the development of titanate electroceramics. A pair of excellent historical reviews are available by Fisher et al. [Fis12] and Seidenberg [Sei97]. A chief difference between the development of electroceramics and semiconductor technology was that semiconductor manufacturers had their own R&D labs focused on fundamental physics of their materials and the point defects in those materials, while this was never really a high priority for electroceramics manufacturers. This emphasis on fundamental physics was expensive, but paid off rapidly. High purity germanium diodes were a mature technology by the 1950s because of a tremendous amount of money and research that went into developing high quality crystal growth techniques and characterizing point defects in the system. However, the germanium material system has a large disadvantage compared to silicon: it is not easy to grow an environmentally stable insulating layer on the surface. Once the researchers and companies involved realized how much easier development of integrated circuitry with silicon could be, they invested more money into research on silicon crystal growth and dopants to quickly reach parity with germanium technology. Later research was able to slingshot off of this early initial investment in high quality materials and characterization. This accelerated material system research, combined with ongoing work on general purpose electronic computing based ﬁrst on vacuum tubes, and later on germanium diodes, enabled rapid advancements and miniaturization in the ﬁeld of computing. Many of the approaches are still used today, especially in wide bandgap materials research, for describing defects in semiconductors, such as the use of Boltzmann or Fermi- Dirac statistics for describing defect ionization and carrier populations, were originally developed by this community. Indeed, the nitrides community indirectly evolved from the germanium and silicon communities. Much of the pioneering work on silicon came out of Bell Telephone Labs, and other companies were keen to copy their success. As a result, the pioneering work on LEDs and III-V materials was done at many companies including General Electric, Texas Instruments, RCA, and IBM, among others. Metallo- organic chemical vapor deposition (MOCVD) was developed relatively quickly, by 1968.

20 The community soon hit upon the idea of using 3 and even 4 component alloys to enable simultaneous tuning of lattice parameter and bandgap, and before long LEDs were available in every color, eventually including blue.[DK08]

21 CHAPTER

METHODOLOGY

This chapter discusses the major tools and methods used in this research. Sections 3.1 and 3.2 discuss, in general terms, what density functional theory is and how it works, the density functional theory package used, and how to apply it is for some common situations in this research. Section 3.3 discusses how to use the data from density functional theory calculations to model an ensemble of point defects in a host semiconductor. Finally, in section 3.4 discusses a data management tool developed to help deal with the huge data volume this research requires.

3.1 Density Functional Theory

Density functional theory, or DFT,works by self-consistently solving the orbitals of the Kohn- Sham system (eq. 3.1). Whereas Schrödinger’s equation describes a system of interacting electrons, the Kohn-Sham equation describes a system of non-interacting electrons, and treats the interactions between electrons as an interaction between the single electron and an effective many-electron potential. The validity and applicability of such a treatment

22 was shown by Hohenberg, Kohn, and Sham in work which would eventually win Kohn a Nobel prize.[HK64; KS65] With an exact form for the electron–many-electron potential, the description of quantum mechanics would become exact. However, although some terms in this treatment have known and exact forms, like the classical Coulomb kernel describing the single electron interacting with a mean electrostatic field, some of them do not. In practice many of the terms describing deviations from this mean field response are combined into a term called the “exchange-correlation" term, which is often modelled with a functional (a function with another function as input). This exchange-correlation term reproduces effects specific to Fermionic systems such as Pauli repulsion.

2 ħh 2 − + V φ = εφ (3.1) 2m0 ∇ One popular early class of functionals for treating this effective many-electron potential depended only on the local electron density, and hence was known as the “local density approximation", or LDA. A later, more successful set of functionals was developed which also took gradients in the electron density into account. These became known as “generalized gradient approximation functionals", or GGA functionals. One popular GGA functional within the condensed matter physics community is the functional developed by Perdew, Burke, and Ernzerhof (referred to as the PBE functional), which was itself based on an earlier functional known as the PW91 (Perdew-Wang ’91) functional.[Per96a; WP91] However, although these later GGA functionals are quite successful for calculating atomic geometries and bulk formation enthalpies, they still have problems properly reproducing bandgaps in many materials. This problem is believed to be related to an incorrect description of electron self-interaction, which also has severe implications for the accuracy of any point defect energy calculations performed with this set of functionals.[WN04; Fre14] There were a number of functionals developed after 1996 which attempted to better capture the electronic properties of solids. In 2003, Heyd, Scuseria, and Ernzerhof published a work in which they outlined a new functional which mixed a GGA functional with non- local Hartree-Fock exact exchange calculated up to a cutoff from each point r . By altering how much Hartree-Fock exact exchange is mixed in, this functional enables the electron self-interaction to be tuned such that the calculated electron bandgap matched the actual electronic bandgap in a given material.[Hey03; Hey06] It also greatly improves the accuracy of calculated of point defect properties.[Fre14] It has been used to great success for this purpose in many nitrides and oxides, including in this work.[Sac15; Col12; Gad13a; Bak17;

23 Gad14b; Bow18; Ald18; Har18] An additional approximation available in some DFT packages to improve performance without impacting accuracy is the pseudopotential approximation. This approximation allows calculations to be greatly sped up by not explicitly calculating wavefunctions for the the core electrons, which typically do not participate in bonding. Instead, they are treated as contributions to the spatial potential V (r ) in the Hamiltonian. The form of this spatial potential is ﬁt for each element to the core region of the solution of the full many- electron solution for the element’s isolated atom. It is important to choose pseudopotentials with enough electrons to accurately capture the bonding physics of the material and point defects being simulated. A more detailed overview of these concepts can be found elsewhere, such as in the dissertation work of Ben Gaddy.[Gad13b] The DFT program used in this research is the Vienna Ab Initio Simulation Package, or VASP.[KH93; KH94; KF96b; KF96a] According to its manual, VASP and CASTEP (another electronic structure code) share as a common origin a program written by Mike Payne at MIT. The serial version of VASP was developed in the early 1990s from this earlier program by Jürgen Hafner, Jürgen Furthmüller, and George Kresse at the University of Vienna. J. Holendar and George Kresse then parallelized the code in the late 90s. Since then it has incorporated functionality developed by many authors, and has become a standard for electronic structure codes due to its speed, ﬂexibility, and well-tested pseudo-potential library.

3.2 Anatomy of a plane-wave DFT calculation

Plane-wave based DFT software packages operate on inﬁnite periodic repetitions of the input geometry in 3 dimensional space. The initial atomic geometry is speciﬁed as a lattice

basis and a motif tiled onto that lattice. This has been illustrated with SrTiO3 in fig. 3.1, which shows one possible representation of the cubic unit and primitive cells tiled onto the lattice basis. The primitive cell for this lattice basis contains the corner strontium atom, the central titanium atom, and the three oxygen atoms forming corners of the green cube. The specified atomic geometry is combined with user-defined points in reciprocal space for performing the energy integration, and pseudopotentials for modeling fields approximating the effects of the core electrons. In a so-called single point energy calculation, a plane-wave representation of the wavefunctions of the explicitly modelled electrons is

24 Figure 3.1 (a) illustration of the unit cell (blue cube), primitive cell (green cube), and lattice basis (blue vectors) of SrTiO3. (b) the [100] projection of an AB stack of SrTiO3 with embedded unit cell, primitive cell, and lattice basis.

found via the variational principle by minimizing the total electronic energy of this periodic system. In a structural relaxation, single point energy calculations are performed in an inner loop while varying the coordinates of atoms in the motif in an outer loop until an energy minimum is found. Although there many types of specialized calculations possible, these two were the basis for the vast majority of the work presented here. While there are many subtleties involved in these calculations that are beyond the scope of this dissertation, three typical workﬂows for this project will now be discussed, with emphasis on the use of multiple functionals to accelerate calculations: simulation of bulk materials, simulation of bandstructure and determination of an appropriate exact exchange amount, and simulation of point defects.

3.2.1 Bulk Material Simulation

As one example of a typical workﬂow for this project, bulk material simulations are often required, whether for obtaining the ground state energy of a relevant thermodynamic phase, or for starting a more sophisticated bandstructure or point defect calculation. A starting point for the lattice and motif is often obtained from published X-ray crystallographic data, using a primitive cell of the material as the motif. After combining this with an appropriate reciprocal space mesh and pseudopotential selection, an initial relaxation of both the

25 lattice and the motif is performed at the PBE level. If no unexpected behavior (such as obviously incorrect geometries or an incorrect magnetic state) is found, then the calculation is repeated twice at the HSE level, with the same exact exchange amount as the point defect host material. The use of the same exact exchange amount in all project calculations is essential to allow the calculated energies to be directly compared against quantities like calculated chemical potentials. The first HSE relaxation allows both the lattice and the motif to relax, just like the PBE calculation, using the earlier calculation endpoint as a starting point. This saves a significant amount of computational resources as opposed to performing a full HSE structural relaxation without influencing the accuracy. The endpoint of the first HSE relaxation is then used as the starting point for the second, which only allows the motif to relax. This is done in order to eliminate what are known as Pulay stresses from the calculation; these unphysical stresses are an artifact resulting from using a basis set built up for calculations for one lattice in a different lattice.

3.2.2 Bandstructure Simulation

A second workflow example is a bandstructure calculation. A bandstructure calculation builds on a previous bulk material simulation of the material of interest. The final HSE geometry is used as input to a series of single point energy calculations with a special set of reciprocal space points. In addition to specifying a set of reciprocal space points to be used for energy integration, a number of additional “unweighted" points are chosen along high symmetry paths through the first Brillouin zone, so that the energy will be calculated at those points for each energy band, but not used in evaluating the total energy of the system. This allows the determination of a material’s bandstructure, and, through that, determination of bandgaps, the ideal (scattering- and electron-phonon-coupling-free) effective masses of the carriers along different crystallographic directions, and the 0 K and effective density of states. A standard set of high symmetry paths for different lattice symmetries to use in bandstructure calculations has been developed by the community.[SC10] Iterative bandstructure simulations are often performed with different exact exchange amounts, with the goal of “dialing in" an ideal exact exchange amount for the material of interest. Modifying the exact exchange amount modifies the bandstructure by modifying the exchange-correlation interaction. This has a very subtle effect on the band curvatures, and a much more dramatic effect on the separation between the conduction and valence bands. The relation between exact exchange amount and band separation is approximately

26 linear, so it is often possible to get close to the ideal amount in only three steps. This can then be ﬁne-tuned to match extrapolated 0 K bandgaps, if they are available. There are some authors who instead use an exact exchange fraction of 0.25 for every material they study. This exact exchange fraction is used for practical reasons, as it allows calculations performed in different materials to be more directly compared with each other. The amount is based on work by the authors of the HSE functional, where it was determined as a best-ﬁt value across many molecular and solid state systems.[Hey03; Per96b] This work uses a “dialed in" exact exchange amount, rather than using 0.25 as the exact exchange fraction, as 0.25 is not an optimal value for describing all systems.[Fre14]

3.2.3 Point Defect Simulation

Conﬁguration entropy drives the formation of point defects in solids. They can occur as a missing atom (vacancy), an extra atom (interstitial), or by another atom replacing a host atom (substitutional). Several isolated point defects may also interact and form a defect complex. Real point defects are also accompanied by a distortion extending through several layers of nearest-neighbor atoms. DFT simulation provides a route to understand the properties of point defects in solids, which is otherwise challenging to examine. To be most useful, these simulations should correspond to point defects in the dilute limit, where there is a low enough concentration of point defects that there are no interactions between them. Keeping in mind the periodic nature of plane-wave DFT atomic geometries, simulation of point defects thus requires some special considerations. First, the primitive cell is no longer a suitable choice for the motif, since, as an example,

replacing the Ti atom in the primitive cell of SrTiO3 with Fe would result in simulation of

SrFeO3 rather than SrTiO3. Thus, several repetitions of the primitive cell are instead used as the motif, with suitably extended lattice vectors. This is referred to as a supercell, and is typically described in terms of how many primitive cells were used in its construction (e.g.

a supercell consisting of 27 SrTiO3 primitive cells and 135 atoms, with 3 repetitions in each direction, would be described as a 3 3 3 supercell). The supercell should be large enough× × to capture the atomic distortions around the point defect, whether it is an isolated point defect or a complex. There are also several effects that the supercell should be large enough to prevent. Electronic interactions of a defect across the periodic boundary can lead to several non-physical effects when trying to extrapolate

27 defect energies to the dilute limit, such as Coulombic interactions artificially modifying the defect energy, or dispersion in what should be flat defect energy bands in reciprocal space. Strain interactions across the periodic boundary can also be a problem. To prevent them, it is typical to ensure that at least one fixed layer of atoms is present on the periodic boundary, while otherwise allowing as many atoms as possible around the defect to relax. In general, then, the larger the supercell, the better. However, the time required for a DFT calculation scales with the cube of the number of atoms. In practice, due to computational resource restrictions, the calculations relevant for this project are restricted to having fewer than 200 atoms in the motif at the HSE level. For this work, as in other state-of-the-art investigations of defects in strontium titanate, 3 x 3 x 3 supercells were used for isolated point defects and complexes between hydrogen and isolated vacancies[Iwa14; Var14; Jan14]; additionally, 3 x 3 x 4 supercells were used for first-nearest-neighbor defect complexes. The 3 x 3 x 3 primitive cell directly simulates regularly patterned defects present at 6.24 20 3 · 10 cm− in strontium titanate, and the 3 x 3 x 4 primitive cell simulates defect complexes 20 3 present at 4.68 10 cm− . The largest spherical “relaxation shell" that can be formed in these cells while· maintaining a fixed atom layer at the boundary is about 5 Å. Such a relaxation shell was used around every point defect discussed in this work. For first nearest neighbor complexes, one 5 Å shell was formed around each defect constituent, and for hydrogen-vacancy complexes, a 5 Å shell was simply formed around the vacancy, since the hydrogen atoms physically sit inside the vacancy void. However, three primitive cell-lengths (about 12 Å) in each direction is still too small to completely eliminate electrostatic interactions across the periodic boundary. An additional but related problem is that charged supercells have an additional potential term added in by the DFT software to deal with the otherwise-divergent electrostatic potential. This leads to different “potential alignments" for the calculated energies of different charge states of the same defect; this is essentially a floating reference problem. Fortunately, much work has been devoted to solving these two problems with post hoc correction schemes by researchers such as Freysoldt et al. [Fre14] and Kumagai & Oba [KO14]. This work uses a scheme implemented by Joshua S. Harris based on the work by Kumagai & Oba [KO14], using room temperature experimental dielectric constants to simultaneously correctly align the potentials and compensate for electrostatic interactions across the cell boundaries. This method was demonstrated to have good convergence with cell size for even high charge states across several materials[KO14].

28 The relaxations for point defects are typically started at the PBE level to get close to the energy minimum. Then, HSE level relaxations are performed on the results of the PBE relaxation. Initial defect charge state ranges are typically n+1 to n-1, where n is the predicted “ideal" charge state based on typical oxidation states of the involved species. This charge state range is then extended until reaching a thermodynamic transition level above the conduction band or below the valence band. On its own, this does not result in enough of a speedup to make large numbers of HSE simulations practical within our resource constrains. To further reduce the computation time associated with running HSE calculations on such large supercells, the defects are often centered so that the calculation can be run on a symmetry reduced subspace of the atomic geometry and reciprocal space mesh of the simulation. Relaxing within this subspace can restrict effects such as Jahn-Teller distortions from occurring. However, testing on defects expected to exhibit Jahn-Teller distortions have shown that the energy difference resulting from this is, at least for those defects, only on the order of hundredths of an eV, which is small enough to be safely ignored. For situations where higher precision is needed, such as validating coarse optical signatures calculated within the symmetry reduced subspace, the calculation can be continued without the symmetry reduction, which allows spontaneous symmetry breaking to occur.

3.3 Post-Processing

3.3.1 Extraction of Useful Parameters from Bandstructure

As discussed brieﬂy in the previous section, a number of useful quantities can be obtained from the bandstructure aside from the obvious valence band maximum and conduction band minimum. Ideal transport masses in the electron-phonon-free and scattering-free limit can be obtained, which can be used as inputs for physics based models for carrier transport. An accurate 0 K density of states can be directly obtained and used for numeric integration of the Fermi-Dirac integral with knowledge of the Fermi level. Additionally, an effective density of states can be constructed, which is far less costly to compute than direct numeric integration of the Fermi-Dirac integral, and which additionally captures some of the temperature dependence of the density of states. Figure 3.2 shows the calculated bandstructure and density of states for STO from our HSE

29 Figure 3.2 Density of States and electronic bandstructure of SrTiO3 along high symmetry paths in the ﬁrst Brillouin zone, with carrier band extrema marked with colored dots.

calculations with the ﬁnalized exact exchange amount of 0.2362, with carrier band extrema marked with green dots for the valence band maximum and yellow for the conduction band minimum. Depending on the context, the valence band maximum and conduction band

minimum are sometimes referred to as VBM or EVB, or CBM or ECB. The energy bands are 1 1 1 shown traversing paths between the high symmetry points Γ =(0,0,0); X =( 2 ,0,0); M =( 2 , 2 ,0); 1 1 1 and R=( 2 , 2 , 2 ). As calculated, the valence band maximum is not aligned to zero; the entire bandstructure is typically shifted such that EVB=0 to allow it be more easily read. To obtain the valence band maximum and conduction band minimum, the valence band is identiﬁed by electron counting, and the maximum energy of the valence band and the minimum energy of the conduction band are simply read out. For this calculation, these have values

of EVB=3.21 eV and ECB=6.46 eV Once the data has been properly organized, identifying the direct and indirect bandgaps also become trivial–the direct gap can be found by ﬁnding the minimum separation between the high symmetry path of the valence and conduction bands, while the indirect gap

is simply ECB-EVB. Values of 3.25 and 3.62 eV were obtained for Eg-dir and Eg-ind, respectively. These compare favorable to Van Benthem et al. [VB01] and Kok et al. [Kok15]. Van Benthem et al. [VB01] reported indirect and direct bandgaps of 3.25 and 3.75 eV,respectively. Although it is not explicitly stated, these appear to have been determined using the Tauc method with UV-vis absorption data collected with ellipsometry and reﬂection spectrometry. Mea-

30 surements by Kok et al. [Kok15] provided additional support for the reported indirect gap and probed its temperature dependence. The exact exchange amount was ﬁt using the indirect gap instead of the direct gap since multiple authors reported the same indirect gap. Bandstructure single point calculations can also be used as input for high resolution density of states calculations, by using the converged charge density and plane-wave expan- sions for the wavefunctions in a calculation with a much higher reciprocal space sampling density. While the bandstructure calculation shown in ﬁg. 3.2 only used a 6 6 6 weighted reciprocal space mesh, it was found to be necessary to use a 15 15 15× mesh× and mas- sively increase the number of energy bins for output to obtain a× density× of states that was directly usable for calculating carrier densities with the Fermi-Dirac integral. However, for situations where the Fermi level is far from the conduction band minimum or the valence band maximum, the electron concentration may be approximated much more cheaply using an effective density of states as[KK80]: (ECBM E f ) [n] = NC exp − − (3.2) kB T and the hole concentration as: (E f EVBM ) [p] = NV exp − − (3.3) kB T

In these equations, NC ,V is the effective density of states for electrons in the conduction band and holes in the valence band, respectively. These are deﬁned in equation 3.4 for

effective density of states masses mDOS∗ for electrons and holes, respectively.

3/2 mDOS∗ kB 3/2 NC ,V = 2 2 T (3.4) 2πħh

This deﬁnition comes from parabolic band theory, which posits that for non-degenerately doped cases, the density of states can be approximated as a geometric surface in reciprocal space whose curvature is the same as the curvature of the carrier bands at their extrema.

The density of states effective mass mDOS∗ is the geometric average of the inverse curvature of these surfaces along the principle crystallographic directions, multiplied by a degeneracy factor: 1/n 2 1 Y ∂ E − m h 2 g 2 (3.5) DOS∗ = ħ 2 n ∂ kn

31 The spatial degeneracy factor g is equal to the number of carrier surfaces appearing in the first Brillouin zone. As an example, in silicon, the conduction band minimum occurs at a point partway along the Γ -X vector, called the ∆ point, and the parabolic band surfaces are ellipsoids, because the surface curvature has 2 transverse components and a longitudinal component. Because there are six such surfaces in the first Brillouin zone, the degeneracy factor is 6. In strontium titanate, the conduction band minimum occurs at Γ and the valence band maximum occurs at R. There are band degeneracies at these points in both bands in our calculations, although because the impact of the steeper surfaces on the effective density of states is minimal, only the shallowest surfaces were used in calculating the effective densities of states. Although the true carrier band surfaces in STO are corrugated and highly nonlinear, the parabolic band approximations to these shallowest surfaces are spheres in reciprocal space, and there is only one sphere for each carrier band within the first Brillouin zone, making the spatial degeneracy factor g equal to 1 for both bands. Using

these approximations, we obtain density of states effective masses of approximately 6.2 m0 for the electron and approximately 11.8 m0 for the hole.

3.3.2 Treatment of Chemical Potentials

The chemical potential is the change in free energy when an atom of a given species is added to or removed from the system. Physically, it can modulated by varying processing conditions to control availability. In the grand canonical ensemble, it is also how theoretical predictions are connected to real experimental conditions. Within this ensemble, it is simple to place bounds on accessible chemical potentials in an arbitrary material, even one with o o competing phases. The chemical potential µ may be deﬁned as µi µi + ∆µi , where µ is the chemical potential of atom i in its thermodynamic reference phase≡ at 0 K. The number of elements in a host material determines the dimensionality of the accessible space for the native chemical potentials. Elemental materials only have a single component, and their native chemical potential is constrained to a 0 dimensional space (a single point). The chemical potentials in binary compounds can vary along a line; an increase in one chemical potential must be accompanied by a decrease in the other chemical potential. At 0 K, the length of this line is equal to the compound’s formation enthalpy. This is shown graphically in ﬁgure 3.3(a). Similarly, for a 3 member compound, the stoichiometry and formation enthalpy con-

32 Figure 3.3 0 K Chemical Potential Surfaces for AlN (a) and SrTiO3 (b) demonstrating the differences in accessible chemical potentials for binary and ternary compounds. strain the chemical potentials of atoms in the compound to a plane. Eq. 3.6 gives the 0 K plane, where n is the stoichiometry of element i in the host material. Furthermore, at 0 K, if

∆µi > 0, then the element will precipitate from the compound. This restricts the accessible chemical potentials lying on the plane to a triangle.[Bak17]

f X ∆Hc ompo und = ni ∆µi (3.6) i Additional constraints on accessible chemical potentials are imposed by competing (Eq. 3.7) and solubility limiting phases (Eq. 3.8), which may occur for 3-or-more-component systems as precipitates in the host material. The constrained space of accessible native chemical potentials for the SrTiO3 system is shown graphically in ﬁgure 3.3(b). The host compound is stable against competing and elemental phases in the white regions of this triangle, while the red and blue regions show areas where SrO and TiO2 may precipitate. In some situations, impurities may exceed their solubility limit because of kinetic limitations to diffusion, aggregation, and phase change. In such cases, the solubility constraints may be relaxed.

f X ∆Hc omp e t i ng > ni ∆µi (3.7) i

f X ∆Hs ol .l i mi t i ng > ni ∆µi (3.8) i

33 1 PO ∆µ ∆H T T ∆S T k T l n 2 (3.9) O = 2 ( ) ( ) + B P o − O2 Within this framework, the chemical potentials of ideal-gas-like species, such as the oxygen dimer, can be mapped to actual processing conditions using experimental data on internal enthalpy and entropy changes of the gas with temperature and pressure (eq. 3.9).[Per05; RS03; RS01; Ert12; WN04] Collections of such data are available in several publications such as Chase Jr. et al. [CJ86] and Ihsan [Ihs95]. This is a very general framework which is widely used not only for point defects[Sac15; Col12; Gad13a; Bow18; Gad14b; Ald18; Har18; Bak17], but has also been applied as part of strategies for calculating crystal growth preferences as a function of processing[Pai11], surface stability in different environmental conditions[Gad14a] and even the relative energy of various surface reconstructions[Dyc18]. The chemical potentials of the compound itself and elemental, competing, and solubility limiting phases can be accurately extended to finite temperatures in a similar fashion as ideal-gas-like species. However, this process requires accurate enthalpy and entropy data that extends almost to 0 K. In the case of strontium titanate, the chemical potential of oxygen at finite temperatures and pressures is already accurately captured due to the pressure to chemical potential mapping, and much of the rest of the defect chemistry is determined by impurities. Improving the accuracy of the native chemical potentials primarily affects strontium and titanium vacancies in STO, and the calculated chemical potentials of impurities sitting on those sites. The derivations for the finite temperature extension of this framework are presented in chapter 6, which, along with work by Harris et al. [Har18], motivated its development.

3.3.3 Point Defect Formation Energies

In the grand canonical thermodynamics ensemble, the formation enthalpy of a point defect D in charge state q can be written as 3.10.[WN04; Fre14]

f tot tot X corr ED q = ED q Eb ul k ni µi + q (E f + EVB ) + ED q (3.10) − − i

34 Figure 3.4 Formation energy diagram for a ﬁctitious defect with only two charge states.

equilibrium with respective chemical reservoirs, where µi is the chemical potential of species i and ni is the number of atoms exchanged between the bulk and reservoir. µi was corr discussed in-depth in section 3.3.2. ED q corrects for the finite size of the cell, as discussed previously, and is obtained for each charge state of each defect using a method implemented by Joshua S. Harris, based on the work of Kumagai and Oba with a relative permittivity of 300 for STO, and 1500 for BTO.[Fre14; KO14; Sam66; SG65] Defect formation energies are traditionally represented with a formation energy diagram, with the Fermi level E f taken relative to the valence band maximum EVB and used as f a free parameter. As the only free parameter at a given set of chemical potentials, ED q (E f ) has the form of a straight line, with the slope dictated by the charge q . One line is plotted for each charge state, and the lowest line determines the most thermodynamically favorable charge state. If the line changes slope within the bandgap, then the defect will change its charge state as the Fermi level crosses the point where the slope changes, which is known as a thermodynamic transition level (TTL). The TTL relative to the VBM can be easily solved for by finding the Fermi level where the energies are equal to each other using eq. 3.10 An example diagram for a fictitious defect with a neutral charge state in the lower half of the gap and a -1 charge state in the upper half of the gap is shown in figure 3.4.

35 3.3.4 Approximating Optical Transitions

Defect optical absorption and emission signatures can be approximated with ground state simulations in the limit that the excited electron or hole does not strongly interact with the defect post-excitation. This approximation builds on the defect formation energies and thermodynamic transition levels discussed in the previous section, and is often referred to as the Franck-Condon approximation.[WN04] It has been widely applied for accurately calculating point defect absorption and emission energies across multiple materials systems.[Col12; Gad13a; Gad14b; Bak17; Ald18; Har18] Consider the hypothetical point defect shown in figure 3.4. This point defect can exist in 2 charge states: 0 and -1. If the Fermi level is such that this defect is in the -1 charge state prior to any optical excitation (point A in the figure), then the optical excitation from -1 to 0 (path A-B) and subsequent relaxation from 0 to -1 (path C-D) can be represented with the configuration coordinate diagram in figure 3.5. If building on a dataset that already contains the defect ground state simulations, the absorption and emission energies can be solved for with as few as 2 single point DFT calculations for each thermodynamic transition level.

Figure 3.5 Configuration Coordinate diagram for the fictitious defect from fig. 3.4

36 The x-axis of this figure is a generalized one-dimensional coordinate representing the lattice distortions occurring for different defect charge states. The y-axis shows the energy change associated with changing charge state while the atomic geometry stays fixed. The lattice distortions around the defect are typically, but not always, displacements of atoms in the first few nearest-neighbor shells lying roughly along vectors connecting the atom to the defect center. These displacements change when the defect changes charge state in a similar fashion to what might be expected for a Raman breathing mode. In this example, an incoming photon excites an electron from the defect at point A into the conduction band, and the defect is now at point B. In the Born-Oppenheimer approximation, the electronic shells relax almost instantaneously in response to this process, so in situations where excitonic effects can be neglected (that is, where the excited electron does not interact strongly with the defect electron orbitals), point B is simply the atomic geometry of point A with an electron removed. Because this is a metastable configuration, the atomic geometry will eventually “catch up" to the electronic geometry, which is represented by path B-C. Then, as the defect comes back into equilibrium with the charge carrier reservoirs, it will eventually capture an electron from the conduction band and move from point C to point D and, if the process is radiative, this will be accompanied by emission of a photon. Point D is metastable, just like point B, so it will relatively quickly relax into the ground state geometry for that charge state, returning to point A. In this approximation, the zero phonon line, or ZPL, is equal to the distance from the relevant band edge and the defect TTL. In this example, the defect is absorbing a photon with loss of an electron to the conduction band, so the ZPL equals Eg ETTL . If it instead − involved an interaction with the valence band, the relevant ZPL would be ETTL . Along a given configuration coordinate curve (that is, for configuration coordinates of a given charge state), the energy difference between points can be evaluated with a simple difference. Using the labeled points in the diagram, the energy difference between B and C can be

evaluated as EB EC , and the same holds true for D and A. Once the ZPL has been found, the absorption and− emission energies can be solved for quite simply as (for defect-conduction

band transitions) Eabs = EZPL + (EB EC ) and Eemi = EZPL (ED EA). Every charge state may have up− to 2 transitions present− for− states in the gap, one involving the conduction band and the other involving the valence band. The equations for defect-valence band transitions may be solved for with the same methods as the defect- conduction band transitions, using the diagram as a guide, so long as the original state is

37 always represented on the bottom, and the excited state on top.

3.3.5 Defect Solver

The current defect solver is the third iteration of a program originally written by Ben Gaddy, a former group member.[Gad13a] That first version was specific to the aluminum nitride system and the code was still at a prototype stage of development; while the results were physically and mathematically sound, it was not designed to be general or fast. The second version was written by an undergraduate working for our group at that time, Brian Behrhorst, who was tasked with analyzing the code, simplifying the callstack to improve maintainability, and generalizing the program. Simplifications to the callstack led to dramatic speed improvements in this version. The third version was developed by the author based on the second version, but optimized for speed while maintaining the generality, and presented the functions and objects as a library. This made the code easier to maintain, modify, and incorporate into other programs. Support for quasi-Fermi levels was added in as part of this project based on the work of Lutz.[Lut96] Steady improvements to functionality, speed, and ease-of-use have been made since then. The core idea of the defect solver is to solve for the charge neutrality conditions at the chemical potentials and temperature where the defects are introduced, and then to solve for it again after the material has been quenched to its operating temperature. The equation relating defect concentration at equilibrium to the chemical potential (Eq. 3.11) is obtained by minimizing the system free energy with respect to number of particles, and obtaining the concentration from Stirling’s approximation to the configurational entropy. The formation f enthalpy of ∆HD q is shown in a previous section as equation 3.10.

f q ∆HD q ∆SXS [D ] NsitesNconﬁgexp − exp (3.11) ≈ kB T kB It is important to note that one of the main assumptions in the derivation of Eq. 3.11 is that contributions to the system free energy from the individual defect populations are dependent only on their concentrations, and not on other defect populations. This assumption will break down if the system exceeds the dilute limit. Equation 3.11 implicitly provides a system energy minimization constraint as a function of chemical potentials, including that of the electron. That is to say, defect concentrations calculated with eq. 3.11 are the lowest energy state possible given a set of chemical potentials

38 and a Fermi level. An electrostatic system energy minimum can then explicitly imposed: that the crystal must obey the condition of charge neutrality (Eq. 3.12).

X q p n + qi Di = 0 (3.12) − i At low temperatures, the total population across all charge states of a particular kind of defect is ﬁxed at the high temperature value and the distribution of charge states is adjusted to maintain charge neutrality at the lower temperature.

3.3.5.1 Sources of Uncertainty

Although in general trends are accurately captured with this framework, there are a few sources of uncertainty limiting the quantitative accuracy of the predictions. The ﬁrst con-

cerns the non-conﬁgurational entropy (∆SXS ), which consists of per-defect entropy terms such as electronic entropy, vibrational entropy, and magnetic entropy, among others. The magnitude of these entropic contributions is expected to be much smaller than the corresponding enthalpic terms[WN04] and the inﬂuence of neglecting them on predicted defect concentrations is at least partially compensated for by Fermi level movement and calculated impurity chemical potential. Because of this partial error cancellation, the extreme computational expensive of explicit vibrational entropy calculations at the HSE level, and the high number of unintentional impurities needing to be modelled, the per-defect entropy terms have not been calculated or included in the models presented here. The second source of uncertainty concerns the treatment of the electronic bandgap. The electronic bandgap changes with temperature, and the temperature dependent shrinkage is simple to measure experimentally, as was done by Kok et al. [Kok15] for strontium titanate. However, it is not as straightforward to measure the movement of the in-gap states relative to the band-edges with temperature. This can actually have a larger impact on the charge balance solution than the shrinkage of the bandgap itself, although not modeling the temperature dependence still gives satisfactory results for some properties such as the brown coloration associated with iron.[Bak17] To the author’s knowledge, there has only been one study which even partially addresses the separate movement of the conduction and valence bands, in silicon, and it did not examine the shift in defect levels with temperature, only the movement of the band edges.[Bey10] That experiment used soft x-ray spectrometry to measure changes in the energy difference between a core level, assumed

39 to be temperature independent, and the conduction and valence band edges. Despite the lack of data on these properties in STO and BTO, there are currently several schemes implemented in our code-base to move the in gap states with temperature in different ways in anticipation of other possible tests against experiment becoming available. When temperature dependent band edge movement has been included in our charge balance, the scheme has been explicitly described in the relevant work. There is also an informational limitation affecting simulation of defect complexes which are only weakly bound with respect to the average thermal energy at room temperature. Currently, defect complexes are not allowed to re-equilibrate with the populations of their constituent isolated defects when quenched. This is because most defect complexes are strongly bound and unlikely to dissociate. However, this approximation breaks down for weakly associated defect complexes containing highly mobile species. There are two dif- ficulties in explicitly calculating low temperature concentrations for the constituents of such complexes. The first is that the bulk chemical potentials becomes less well defined at low temperature, since the system is no longer fully in equilibrium due to kinetic effects; in simple re-ionization without re-equilibration, the poorly defined low temperature chemical potentials mathematically cancel and do not affect the quenched-in state. The second is that point defect equilibration itself may also become encumbered with kinetic effects at low temperature, and these can be challenging to simulate. When considering results obtained with the defect solver, we are careful to evaluate the potential impact of not accounting for such dissociation on our conclusions.

3.4 Data Management

Very few studies in STO report impurity profiles or sample purity. All of those that do show fairly high levels of background impurities, regardless of processing route. One of the first 18 3 studies on STO by Levin et al in 1955 reports 9 impurities above 10 cm− and 5 more above 17 3 10 cm− in his flame-fusion grown STO crystals.[Lev55] A much later study by Chan et al 17 3 in 1981 using solution processed SrTiO3 reported 16 impurities above 10 cm− , of which 3 18 3 were above 10 cm− . [Cha81] Accounting for each of those 16 impurities requires 10-20 simulations just to properly account for possible configurations and charge states, and even more simulations are required for native defects and to calculate optical absorption and emission energies.

40 Figure 3.6 High Level Overview of DFT Simulation Workﬂow.

Altogether, accounting for 16 impurities in STO requires between 300 and 500 simulations, without counting optical signature calculations. By contrast, in high quality silicon, the 14 3 system will typically only have one dopant present above 10 cm− , and the dopant is usually chosen because it occurs in only one configuration and one charge state. Dealing with the data for this many simulations is a non-trivial problem. Even organizing the thousands of files associated with the simulations is a challenge. Human error will always creep into tedious and repetitive tasks, such as setting up DFT simulations, calculating post-simulation corrections or optical transitions, or collating results from a large number of simulations. The possibility of human error is magnified in this case because the dataset is large and was collected over several years by multiple researchers. Removing the human element from this process greatly reduces the frequency of the types of difficult-to-notice errors that could lead to erroneous conclusions. To facilitate this, an SQL database and companion application were developed which automatically import VASP calculations, checks them for errors, and performs data analysis and postprocessing on the dataset. A high level overview of the entire workflow, highlighting the major role played by the database and its companion application, is shown in Fig. 3.6. DFT calculations completed on supercomputers are transferred to a highly redundant storage system, where the database application performs data extraction, error checks, and total dataset analysis. All of the data from this process is stored in the database and

41 is made available to both users and programs via the SQL interface. This process allows easy creation of input files for later stages of postprocessing, such as chemical potential space analysis, formation energy diagram generation, or determination of point defect concentrations for various processing conditions. The results of these analyses, in addition to allowing comparison with experiment, are then used to inform the next round of point defect simulations, which allows for a continuous high throughput simulation loop. The database is designed in two layers. The first layer is designed to efficiently store data and metadata about files. The schema for this layer are organized around a list of unique fingerprints for files, called hashes. A list of files and folders mirroring the imported filesystem is linked into this hash table to enable quick identification of duplicate files. These both then link into a general data table for storing arbitrary datatypes. This allows the system to store only one set of data for each unique file, which allows for increased extraction throughput and lower data storage requirements. The second layer is designed to store information about simulations, which are each composed of multiple files. This structure is organized around a list of simulations that the program detects. It contains information on, among other things: composition; calculated optical transitions; defect types, charges, and energies; and post-simulation corrections to account for the finite size of the supercells. A cluster analysis algorithm based around compositional cosines is used to determine what compound each simulation belongs to. This is then used in concert with an algorithm which calculates a similarity metric between two simulations based on the species and arrangement of atoms to autonomously map out the relationships between the simulations. These relationships are then used to calculate what kind of defect is in each simulation. Required user input has been reduced to designating an electronic structure simulation and the single point calculations for obtaining optical signatures within the Franck-Condon approximation; the rest of the process is automatic. This has reduced the amount of time spent collating data and checking it for errors from several days for each new batch of simulations to a matter of minutes, or several hours for a full re-import.

42 CHAPTER

BROWN COLORATION OF IRON DOPED STRONTIUM TITANATE

This chapter presents work I undertook to understand the point defects associated with the brown coloration that occurs upon oxidation of iron doped strontium titanate. The

original publication is titled "Defect mechanisms of coloration in Fe-doped SrTiO3 from ﬁrst principles" and is published in Applied Physics Letters Vol. 110, in 2017. Experimental data for this article was collected by Daniel Long and Ali Moballegh from Elizabeth Dickey’s group. Fruitful discussions were had with Ramòn Collazo and Lew Reynolds. DFT calculations for this work were performed by both the author and Preston C. Bowes. Computer time for the DFT calculations was provided by the DoD HPCMP,and the research was funded through Dr. Ali Sayir’s Aerospace Materials for Extreme Environments program via AFOSR BRI grants FA9550-14-1-0264, FA9550-14-1-0067, and a DoD NDSEG fellowship. To understand the underlying defect mechanisms governing the coloration of Fe-doped

SrTiO3 (Fe:STO), density functional theory calculations were used to determine defect

43 formation energies and to interpret optical absorption spectra. A grand canonical defect equilibrium model was developed using the calculated formation energies, which enabled 0 connection to annealing experiments. It was found that FeTi is stable in oxidizing conditions and leads to the optical absorption signatures in oxidized Fe:STO, consistent with experi-

ment. Fe:STO was found to transition from brown to transparent as PO2 was reduced during

annealing. The defect equilibrium model reproduces a consistent PO2 of this coloration

transition. Most critical to reproducing the PO2 of the coloration transition was inclusion of

a FeTi-VO ﬁrst nearest neighbor complex, which was found to be strongly interacting. The

coloration transition PO2 was found to be insensitive to the presence of minority background impurities, slightly sensitive to Fe content, and more sensitive to annealing temperature.

4.1 Introduction

Strontium titanate (STO) has a wide indirect bandgap (3.25 eV), a high dielectric constant, and poor bulk carrier mobilities. It is used industrially to manufacture capacitors and varistors.[Lev88; Uen03; MH97; Yan87; Fuj85] There is a growing research focus on this material for applications in thin-film capacitors, resistive random access memory, gas sensing, and high electron mobility two dimensional electron gas heterostructures.[Moo11; Woj13; Was09; OH04] It is also commonly used as a growth substrate for superconductors and other oxide thin films. [Hil89; Woj13] Ceramic single-crystal STO, commonly used for dielectric applications, has a significant number of background impurities present at levels of parts per million or higher.[Cha81] In an attempt to suppress the behavior of these unintentional impurities, a single dopant element is added at concentrations above the background impurities.[Cla00] Iron is commonly used because it is cheap, readily available, and makes the Fe-doped ceramic STO (Fe:STO) insulating through compensation. Under thermal and electrical stress the initial resistance of Fe:STO degrades over time.[Rod00] Experimental measurements of Fe:STO have found that resistance degradation is accompanied by the formation of a color front that transitions from dark brown at the anode (+ external potential) to transparent at the cathode (- external potential). [Was90c; Woj13; Len11] Similar absorption peaks to those found in electrocoloration have been observed in Fe:STO annealed at different oxygen pressures. Combined electron paramagnetic resonance (EPR) and absorption measurements provided indirect evidence that neutral iron

44 0 substituting on the titanium site (FeTi) is likely associated with the brown coloration in oxidized Fe:STO.[Mül71; FK69] This has led to the prevailing viewpoint that both coloration processes arise from shifts in oxygen vacancy concentration, which changes the ionization of iron substitutionals.[Was90c] The defect models typically invoked to understand the physical properties of Fe:STO 0 1 2 assume that FeTi, Fe−Ti ,VO, n, and p are the only significant species in the system and the interaction between FeTi and VO can be neglected.[Bai90; MH97; Den95; Cla00; Was90a; Wan16] These models then adjust the ionization of the Fe dopant until predictions fit a measured physical property. Nevertheless, a number of EPR studies have detected the presence of a first nearest neighbor iron oxygen vacancy complex, which suggests this interaction is non-negligible. [Kir64; Mül71; Bae68; Fau71; Len11] Despite the frequent detection of the complex in experiments, its influence on physical properties remains unclear. In this article a combination of experimental and ab initio techniques are used to understand the underlying defect mechanisms important to the coloration of Fe:STO annealed in different processing environments. These calculations explicitly account for the presence of the first nearest neighbor complex identified by EPR measurements. Iron is found to be primarily present in four different forms (complexes and charge states) across

accessible PO2 ranges, rather than the two forms typically assumed. Furthermore, direct 0 evidence is provided that the FeTi defect gives rise to two absorption shoulders which cause the brown coloration observed upon oxidation.

4.2 Experimental Methods and Results

The coloration behavior of Fe:STO with 0.01 wt% Fe from MTI Corp. was investigated experimentally. One set of samples was degraded while other samples were annealed in different partial pressures of oxygen. Absorption spectra taken from different regions of a degraded crystal subjected to a 100 V bias applied across the 2.5mm sample for 19 hours at 480 10 K are presented in Fig. 4.1(a). A separate sample∼ was taken through four ± anneal(1170 5 K)-quench(300 K) cycles alternating between air and a PO2 = 15 Pa in a tube furnace, with± optical spectra taken at the end of each cycle. Absorption spectra from this annealed sample are presented in Fig. 4.1(b). The color of Fe:STO was found to switch reversibly when annealed in the different environments from brown (air) to transparent

45 Figure 4.1 Absorption spectra taken for (a) electrocoloration and (b) annealing experiments.

(PO2 = 15 Pa). All spectra show a strong indirect band edge absorption at approximately 380nm (3.25 eV). The spectra of oxidized Fe:STO (anode and air annealed) show broad absorption shoulders centered at approximately 460 nm (2.70 eV) and 570 nm (2.18 eV). These peaks are consistent with previously reported experimental measurements.[Bie93; Gan59; Was90c]

4.3 Computational Methods

Density functional theory (DFT) calculations with screened hybrid exchange correlation functionals were used to understand the underlying defect mechanisms governing the coloration of Fe:STO. Vacancies, native and impurity interstitials, anti-sites, substitutional

impurities, and the ﬁrst nearest neighbor FeTi-VO complex were simulated. The background impurities simulated were chosen based on available composition data from MTI. All 17 3 impurities found in the crystals at a level greater than 10 cm− were simulated, as were 15 16 3 several that appear in concentrations of 10 -10 cm− , which includes S, Cl, Si, Al, Na, Mg, Ba, and Ni. Isolated point defects and nearest neighbor complexes were simulated using 3x3x3 (135 atoms) and 3x3x4 (180 atoms) repetitions of the primitive cell, respectively, with atoms within 5 Å of the defects free to relax. All calculations were performed with the HSE06[Hey03; Hey06] exchange correlation functional in VASP 5.3.3 with an exact exchange amount of 0.2362, collinear spin polarization, and a plane wave kinetic energy cutoff of 520 eV. [KH93; KH94; KF96b; KF96a] Defect

46 calculations and bulk electronic structure calculations used 2x2x2 and 6x6x6 Monkhorst- Pack reciprocal space meshes, respectively. The amount of exact exchange was selected to correct the underestimation of the bandgap common to traditional functionals. With the parameters presented here, a direct bandgap at Γ of 3.62 eV and an indirect bandgap between R and Γ of 3.25 eV was obtained, which are in good agreement with previous experiment and theory. [VB01; Jan14; Kok15] In addition to the improvement in the electronic structure, HSE06 also improves structural and thermodynamic predictions. These parameters led to a bulk lattice parameter of 3.901 Å and an enthalpy of formation of -16.53 eV, which are in close agreement with experiment.[OK73; Ihs95; JR11] Furthermore, HSE06 also improves the over delocalization of electronic charge that is common for traditional functionals.[Fre14] The formation energy of a point defect D in charge state q is determined by Eq. 1.

f t o t t o t X ED q = ED q Eb ul k ni µi + q (E f + VBM + ∆V ) (4.1) − − i

t o t t o t In this expression, ED q and Eb ul k are the total energies as determined by DFT of a super cell containing the defect D in charge state q and the total energy of the perfect bulk, respectively. In this grand canonical approach, the bulk is assumed to be in thermodynamic equilibrium with respective chemical reservoirs, where µi is the chemical potential of species i and ni is the number of atoms exchanged between the bulk and reservoir. o o Further, µi = µi + ∆µi where µi is the DFT energy per atom of the thermodynamic reference phase of i and ∆µi represents deviations from the 0 K reference phase. This allows a compact representation of accessible chemical potentials that accounts for stability against precipitation into reference phases or competing phases, and, in the case of impurities, solubility limiting phases.[Per05; RS03; RS01; Ert12; WN04] Accessible values for ∆µi are shown graphically for Sr, Ti, and O in Fig. 4.2(a). Bulk STO is thermodynamically stable in the thin white strip of this triangle. ∆V corrects for the ﬁnite size of the cell and was obtained using a method based on the work of Kumagai and Oba with a relative permittivity of 300.[Fre14; KO14; Sam66; SG65] The Fermi level E f was taken relative to the valence band maximum VBM and was used as a free parameter when plotting the formation energies. Results of the defect simulations were imported into a point defects database where initial post-processing was performed to extract defect properties (e.g. thermodynamic transition levels, optical signatures, etc.), while further analysis was performed by solving relevant

47 charge balance equations as done previously.[EA08; Gad13a] Defect configurational entropy is accounted for in the defect concentration expression prefactor through the number of identical configurations. Vibrational, electronic, and magnetic entropy are neglected due to the significant expense in calculating these for all charge states of each defect in our database and their expected small magnitudes relative to other terms in the formation energy expression. This is not expected to affect qualitative trends.[WN04]

4.4 Computational Results and Discussion

Formation energies for native vacancies and Fe defects are shown in Fig. 4.2(b). The forma-

tion energies of the Fe defects are plotted with µFe set relative to relevant solubility limiting phases at 0 K. The formation energies in the left pane are for highly reducing conditions while those in the right pane are for highly oxidizing conditions. Our results indicate that native interstitials and anti-sites have higher formation energies than vacancies in all processing regimes and are, therefore, not shown. The most favorable vacancies in reducing

and oxidizing conditions are VO and VSr, respectively. VTi is more favorable than VSr in oxidizing conditions when the Fermi level pins near the conduction band but this does not often occur in practice.[MH97; Cha81] The formation energies and thermodynamic transition levels for vacancies presented here are consistent with those of Janotti et al.[Jan14] Iron was found to prefer the titanium site, either as an isolated defect or in a ﬁrst nearest

neighbor complex with an oxygen vacancy. The isolated FeTi defect assumes all charge states

ranging from +1 to 2 within the bandgap. The predicted FeTi(0|-) midgap state is approximately 1.7 eV above− the valence band. In intermediate environments, the formation energy

of the FeTi-VO complex is close to or lower than either the FeTi or the VO. This indicates that it can be present in significant concentrations despite the configurational entropy penalty relative to the isolated defects. This is consistent with previous EPR measurements.[Kir64; Mül71; Bae68; Fau71; Len11] The complex adopts the +4, +2, +1, and 0 charge states as the Fermi level moves from the valence to the conduction band, skipping the +3 charge state. The strong interaction between the constituent defects leads to a large favorable (positive) binding energy across most of the Fermi level, as shown in Fig. 4.2(c), where the binding energy is defined as the difference between isolated defect formation energies and the

48 Figure 4.2 (a) Chemical potential space of SrTiO3, (b) Formation energy for select native and iron containing defects, (c) FeTi-VO binding energy vs Fermi level, and Conﬁguration-Coordinate 0 0 diagrams for (d) FeTi to CB and (e) VB to FeTi optical transitions.

formation energy of the complex. This is shown for FeTi and VO in Eq. 2.[Fre14; WN04]

E E E f E E f E E f E (4.2) b ( f ) = VO ( f ) + FeTi ( f ) (FeTi-VO)( f ) − The absorption and emission energies for each defect in the point defects database were calculated using the Franck-Condon approximation.[WN04] For each defect, both defect to conduction band (CB) as well as valence band (VB) to defect transitions were evaluated. The majority of the predicted absorption energies either lie close enough to the band edge to blend into its signal, or are in the infrared, where they would not show up in a UV/vis spectrum. Several defects with absorption peaks in the vicinity of those seen experimentally have high formation energies and were eliminated because of their resultant insigniﬁcant concentrations in the material. This down selection left four transitions in 0 the visible spectrum, all of which are related to iron. Two of these four, the (FeTi VO) 1 − to CB and Fe−Ti to CB transitions, do not satisfy the spin selection rule, and would thus be expected to be of very low intensity, if present at all. The only transitions remaining that satisfy the spin selection rule, were associated with a defect of low formation energy, and were in the vicinity of the experimentally observed peaks were both associated with 0 0 0 FeTi. The conﬁguration coordinate diagrams for FeTi to CB and VB to FeTi are shown in in 0 Fig. 4.2(d) and 4.2(e), respectively. The FeTi to CB transition (Fig. 4.2(d)) has an energy of 0 2.72 eV (456 nm), while the VB to FeTi transition (Fig. 4.2(e)) has an energy of 2.17 eV (571 nm). These predicted energies are in good agreement with the experimentally observed absorption shoulders. The calculated formation energies were used to obtain defect concentrations after an anneal and quench cycle. Defect concentrations are related to the defect formation energies

49 via an Arrhenius expression that depends on the Fermi level and chemical potentials.[Fre14] The Fermi level is the chemical potential of the electron in equilibrium and is determined by charge neutrality. Charge neutrality is satisﬁed when the sum of ionized donors and q holes is equal to the sum of ionized acceptors and electrons (i.e. p n + Σq D = 0). Native chemical potentials are ﬁxed by a path through chemical potential− space where the value

of ∆µO is connected to the oxygen partial pressure at the annealing temperature using data from the NIST JANAF thermodynamics tables.[CJ86; RS03; RS01; Ert12; WN04] The values for ∆µTi and ∆µSr must be within the white strip in Fig. 4.2(a). Here a path directly down the middle of this stability region was chosen and is shown in Fig. 4.2(a). While this is an arbitrary selection, it was found not to inﬂuence the major conclusions on coloration

and only changes the concentrations of background defects, such as VSr and FeSr. This leaves the dopant chemical potential, which can assume values of negative infinity up to the solubility limiting energy. Here, the known dopant concentration was used to back calculate the dopant chemical potential at that concentration. If growth was simulated, care should be taken to ensure that impurity chemical potential limits were not exceeded.[Col12; Gad13a; Gad14b] For annealing, it was assumed the temperature would not be high enough to change the dopant content of the crystal. During the anneal, defect concentrations were calculated as a function of chemical potential. At each chemical potential, the Fermi level was determined through charge neutrality. After the anneal and during the quench, the high temperature defect concentration was frozen and a new Fermi level required to maintain charge neutrality was determined. For this work, we simulated anneals at 1073 and 1173 K followed by an immediate 19 3 quench to 300 K with an iron doping level of 0.01 wt% Fe (2.77 10 cm− ). The predicted defect concentration profiles at annealing and quenched temperatures· are shown in Fig. 4.3. Based on these results, a significant fraction of iron is almost always present as a first nearest neighbor complex. At annealing temperatures, the results presented in Fig. 4.3 indicate that four distinct defect chemistry regimes form. In extremely reducing environments (Region A), the be- +2 ginnings of the classical VO :n compensation tail can be seen. In moderately reducing +1 conditions (Region B), electrons and the (FeTi-VO) complex compensate each other and are the dominant charged species. In the regime where most processing is done and out 1 +1 to highly oxidizing conditions (Region C), the Fe−Ti and (FeTi-VO) defects compensate 0 0 each other as majority defects while low concentrations of FeTi and (FeTi-VO) cross each

50 Figure 4.3 Concentrations of native and Fe containing defects after annealing at 1073 (top left) and 1173 K (top right) and quenching to 300 K (bottom). Distinct compensation regimes are

indicated by shaded regions labelled A-D. The brown color bars show annealing PO2 ranges which will result in coloration. Gray lines show PO2 values used in experimental anneals.

other with changing PO2 . Finally, in extremely oxidizing conditions (Region D), we see the 0 beginnings of a regime where the FeTi defect becomes dominant. As the ensemble is quenched to 300 K, the defect chemistry from the high temperature +2 regime is altered. Electrons and VO still compensate each other in extremely reducing 0 conditions (A). As PO2 is increased, (FeTi-VO) becomes the dominant defect (B). In the quenched region B, there is a change in the compensation of minority charged defects +2 +1 1 from VO :n to (FeTi-VO) :Fe−Ti . For quenched regions C and D, there is little change to the dominant defects present as compared to the high temperature compositions. While the dominant defects do not change character after quenching in the higher

PO2 regime, there is an abrupt change to the concentration of neutral defects present. Most notably, this occurs in the partial pressure window where samples were annealed

experimentally (gray lines in Fig. 4.3) and the PO2 of the abrupt drop changes with annealing temperature. Between the gray lines there is now a near vertical transition between the 0 0 (FeTi-VO) and FeTi. The sharp drop-off in concentration of these neutral defects is a result of a jump in the Fermi level, which is the primary mechanism for the material to maintain charge neutrality when quenched to low temperature. It is worth noting that complexes and isolated defects are not changing concentration, rather the charge on each defect is redistributed during the quench. The calculated Fermi level was found to always reside in the upper half of the bandgap when quenched to room temperature over the entire processing regime when Fe is the dominant defect. The calculated concentration proﬁles of the neutral defects are consistent with the

51 experimental coloration measurements and provide insight into the underlying physical mechanism associated with the coloration-transparency transition. The brown coloration 0 is associated with the presence of FeTi, which monotonically increases for a PO2 greater than the abrupt coloration transition. This monotonic increase is consistent with the optical

absorption measurements of Bieger et al. who annealed Fe:STO at increasing PO2 levels and observed an increased absorption, in accordance with Fermi’s golden rule.[Bie93] 0 Furthermore, the anneal at PO2 = 15 Pa is below the abrupt transition of (FeTi-VO) to 0 0 FeTi. Therefore, there should be no FeTi present in equilibrium and the sample should be transparent, consistent with the optical trends of Fig. 4.1(b) . The prediction of the coloration-transparency transition is also consistent with other experimental measurements in the literature.[Bie93; Was90c; Gan59] While this may not seem surprising, it is worth noting that these studies were performed with different iron concentrations and there is no guarantee that either the impurity proﬁles or the paths followed through phase space during annealing are similar to each other or to this study. It 0 0 was therefore explored how the position of the abrupt (FeTi-VO) to FeTi transition changes as a function of path through phase space, increasing doping levels, and varying concentrations of background impurities. While these factors had slight impacts on the location of

the coloration onset, the PO2 of the coloration transition remained on the same order of magnitude. Lower concentrations of Fe in STO have a more pronounced influence on the transition. As can be seen in Fig. 4.3, annealing temperature does have an influence on the position of the abrupt transition when quenched. The only other factor that had a signifi-

cant effect on the position of the PO2 transition was the inclusion of the (FeTi-VO) complex in the charge balance solutions. Omission of this complex led to results inconsistent with experimental observations, with the coloration onset shifting down to an annealing pres- 10 sure of approximately 10− Pa. The invariance of this transition to background impurities, path through phase space, and increasing Fe content explains why a consistent coloration onset is observed across multiple samples.

4.5 Conclusions

These calculations and experiments have extended existing Fe:STO defect chemistry models

to include the ﬁrst nearest neighbor FeTi-VO complex. This complex was found to have a large

favorable binding energy and its presence was essential in reproducing the experimental PO2

52 coloration transition. The PO2 coloration transition was found to be only slightly dependent on the path through phase space and to the concentration of background impurities.

53 CHAPTER

MECHANISMS GOVERNING METAL VACANCY FORMATION IN BTO AND STO

This chapter presents work I undertook to begin extending our results into barium titanate and to explore how transferable results were between barium titanate and strontium titanate. The work revealed some interesting differences in the metal vacancies, which are discussed in detail. It has been submitted under the title “Mechanisms Governing Metal

Vacancy Formation in BaTiO3 and SrTiO3" for publication and is currently under review. Fuitful discussions were had with K. J. Mirrielees, D. M. Long, N. C. Creange, C. A. Randall, and E. C. Dickey. DFT calculations for this work were performed by both the author and Preston C. Bowes. Computer time for the DFT calculations was provided by the DoD HPCMP. The research was funded with ﬁnancial support from a DoD NDSEG fellowship and Dr. Ali Sayir’s Aerospace Materials for Extreme Environments program through AFOSR BRI grants FA9550-14-1-0264 and FA9550-17-1-0318. Barium titanate (BTO) and strontium titanate (STO) are often treated as close analogues, and models of defect behavior are freely transferred from one material to the other with

54 only minor modiﬁcations. On the other hand, it is often reported that B-site vacancies (vB)

are the dominant metal vacancy in BTO while A-site vacancies (vA) dominate in STO. This difference precludes the use of analogous defect models for BTO and STO, begging the question: How similar are the defect chemistries of the two materials? Here, we address this question with density functional theory calculations using a state-of-the-art hybrid exchange correlation functional, which more accurately describes electronic structure and

charge localization than traditional functionals. We ﬁnd that vA is the dominant metal vacancy in STO, but that different combinations of vA, vB, and vB-vO complexes are present in BTO depending on processing and doping. Mechanistically, this occurs for two reasons: thermodynamic differences in the accessible processing conditions of the two materials and energy differences in the bonds broken when forming the vacancies. These differences can also lead to widely differing responses when impurity dopants are intentionally added. Therefore, the response of metal vacancy behavior in BTO and STO to the inclusion of niobium and iron, two typical dopants in these systems, is examined and compared.

5.1 Introduction

Barium titanate (BTO) and strontium titanate (STO) are central to established electronics technology and have been proposed as candidates for next generation applications. Current applications include thin ﬁlm and multilayer ceramics capacitors, positive temperature coefﬁcient resistors, varistors, and varactors.[Yan87; EA08; He04; MH97; Koz99] STO is used in research settings as an oxide growth substrate and a platform for forming a two-

dimensional electron gas (2DEG) with LaAlO3.[Woj13; OH04] Both materials have also been suggested as candidates for gas sensing, persistent RAM, and for use in power generation as thermoelectrics and fuel cell components.[Moo11; Woj13; Mut03; HP02] Additionally, these materials can be readily alloyed to form a solid solution of barium strontium titanate (BST), which is preferred over STO or BTO in epitaxial and high frequency applications due to BST’s combination of high energy density and tunability.[Nor04; Xu05; Dim07] The point defects in BTO and STO are often treated as analogues, and models of the mechanisms and defects governing physical properties are often used interchangeably with only minor modiﬁcations.[Bai90; DS15; MH97; CJ94] At the same time, it is widely believed

that the dominant metal vacancy in BTO is the B-site vacancy (vTi, vB) and that in STO it

is the A-site vacancy (vSr, vA). [MH97; Shi16; DS15] Compounding matters is the difﬁculty

55 associated with direct observation of the effects of metal vacancies in these materials. This has led to a shortage of data which has fueled debate on this topic.[Mor01; Smy02; Huy95; LC86] Here, ﬁrst principles methods are used to investigate how and why metal vacancies behave differently in the two materials and how it affects transferability of results between

the two materials. We find that vA dominate in STO, with vB only becoming favorable in donor doped, highly oxidizing conditions. Although BTO is a simple isovalent modification of STO with Ba substituting Sr on the A sublattice, vacancy behavior differs dramatically between the two systems. We find that there are three metal vacancy configurations with

similar formation energies in BTO (vA, vB, vB-vO) whose relative concentrations are sensitive to both doping and processing conditions. We highlight the differences in vacancy behavior by showing defect concentration diagrams computed for BTO and STO doped with niobium and iron, impurities commonly added to make these materials n-type (Nb) and insulating (Fe). We will show that the difference in metal vacancy behavior arises from two effects. First, the two materials have slightly different chemical environments for the same processing conditions. Because of this, it requires less energy to lose an A-site atom in STO than BTO

for equivalent processing conditions (i.e. TiO2-rich conditions at a given oxygen partial pressure). Second, differences in the electronic energy changes associated with removing

cations from each material enhance the favorability of vA in STO, and of vB and the vB- vO complex in BTO. The lower favorability (higher formation energy) of vA and higher

favorability of vB and the vB-vO complex in BTO brings the formation energies much closer together than in STO. This results in different combinations of these three defects depending

on processing conditions and doping in BTO, whereas vA dominates in STO. The paper is laid out as follows: In the following section (section 5.2), we discuss our computational methodology. In section 5.3.1, we discuss vacancy characteristics and differences in relative vacancy formation energies in BTO and STO. Section 5.3.2 explores the physical mechanisms behind the differences in vacancy formation energy in BTO and STO. Finally, sections 5.3.3 and 5.3.4 examine vacancy behavior in BTO and STO doped with niobium and iron, respectively.

56 5.2 Methods

Density functional theory (DFT) calculations with a screened hybrid exchange correlation functional (HSE06)[Hey03; Hey06] in VASP 5.3.3[KH93; KH94; KF96b; KF96a] were used to obtain the electronic structures of the host materials, formation energies of other thermodynamically relevant materials, and defect formation energies for vacancies, the ﬁrst nearest

neighbor divacancy complex (vTi-vO), and niobium- and iron-related defects. The use of this functional improves both the description of the electronic structure and the charge localization versus that of traditional functionals. This shortcoming in traditional functionals is believed to be mostly due to an incorrect description of electron self-interaction.[Fre14] The amount of exact exchange in each host material was chosen to correct the underprediction of the bandgap arising from this issue. The improved description of band energies and electron localization yields more accurate defect formation energies.[WN04; Fre14] Defect calculations and bulk electronic structure calculations were performed with 2 2 2 and 6 6 6 Monkhorst-Pack reciprocal space meshes, respectively. PAW pseudopotentials× × were× × used with 10 electrons explicitly modelled for strontium and barium, 12 for titanium, 6 for oxygen, 13 for niobium, and 8 for iron. Isolated defects were simulated in a 3 3 3 tiling of the primitive cell (135 atoms), while the nearest neighbor complex was simulated× × in a 3 3 4 tiling (180 atoms). All atoms within 5 Å of a defect site were free to relax. All calculations× × were performed with an exact exchange amount of 0.2362 for STO and 0.2519 for BTO, collinear spin polarization, and a plane wave kinetic energy cutoff of 520 eV. With these parameters, STO is predicted to have an indirect bandgap of 3.25 eV and lattice parameter of 3.901 Å, while BTO is predicted to have an indirect gap of 3.2 eV and a lattice parameter of 3.99 Å, consistent with experiment.[Kok15; VB01; SK05; OK73; She81] Competing and elemental reference phases for each material were simulated with the same parameters as the materials themselves. Calculations were performed in the cubic phase of STO and BTO. This is a natural choice, since the point defects incorporate during sintering and annealing, when both materials are cubic. Bader analysis was performed with the Voronoi approach as implemented and developed by the Henkelman group.[Hen06; Tan09; San07; YT11] Defect formation energies and concentrations are calculated within the grand canonical thermodynamic ensemble. This formalism allows results to be extended to real processing conditions via the chemical potentials. Eq. 5.1 gives the formation energy of a point defect

57 Figure 5.1 Native chemical potential spaces for (a) BTO and (b) STO. Traces have been marked for ∆µ(O) sweeps along AO-rich and TiO2-rich boundaries of the ABO3 stability regions. Markers along the traces indicate chemical potentials used for Fig. 5.2, corresponding to 10 20 (circle), − 10 5 (hexagon), and 102 (diamond) atm of O at 1173 K. Shaded regions of the (A) and (Ti) − 2 ∆µ ∆µ axes show the variation in those chemical potentials when sweeping ∆µ(O) along the TiO2-rich trace.

tot tot D in charge state q . ED q and Ebulk are the total energies obtained from DFT calculations of

the defective and bulk supercells, respectively. E f is the Fermi level relative to the valence corr band maximum EVB. ED q is a ﬁnite size correction based on the work of Kumagai and Oba, using experimental relative permittivity values of 300 for STO and 1500 for BTO.[KO14; SG65; Sam66] This method corrects both for potential alignment due to supercell charge and for electrostatic interactions across the periodic boundary, and has demonstrated good convergence with supercell size for even very high charge states across a variety of

solids.[KO14] ni and µi are the number of atoms added to or removed from the supercell to form the defect and the chemical potential of species i , respectively.

f tot tot X corr ED q = ED q Ebulk ni µi + q (E f + EVB) + ED q (5.1) − − i o o The chemical potential µ of species i may be deﬁned as µi = µi + ∆µi , where µi is the chemical potential of atom i in its thermodynamic reference phase at 0 K. For a 3 member compound, the stoichiometry, formation enthalpy (∆H f ), and elemental phases constrain the chemical potentials of atoms in the compound at 0 K to a triangle where ∆H f SrTiO3 = ∆µSr + ∆µTi + 3∆µO and ∆µi < 0. Competing BaO, SrO, and TiO2 phases further constrain

58 P f the chemical potentials, because they may begin precipitating if i ni ∆µi > ∆Hcomp phase, where ni is the stoichiometry of element i in the competing phase. [Ert12; RS01; RS03; WN04; Per05; Bak17] These relations are represented graphically in Fig. 5.1 for BTO (5.1a) and STO (5.1b). The host compounds are stable against competing and elemental phases in the white regions of these triangles, while red and blue regions show areas where AO and

BO2 compounds may precipitate. Other authors have discussed methods to map between oxygen chemical potential and experimental temperatures and pressures using thermodynamics databases, such as the JANAF tables.[CJ86; RS03; RS01; Ert12; WN04] Using this technique, the chemical potentials 20 5 marked in Fig. 5.1 were mapped to oxygen partial pressures of 10− (circle), 10− (hexagon), and 102 (diamond) atm at 1173 K. Since the diagrams in Fig. 5.1 are shown in terms of ∆µ

instead of µ, ∆µO will be in the same place for a given temperature and oxygen pressure on both diagrams. The formation energies may be used to calculate equilibrium defect concentrations using an Arrhenius expression, given chemical potentials and the Fermi level. In the grand canonical ensemble, these may be solved self-consistently for any given set of native chemical potentials and known impurity content by solving for charge neutrality, as has been done previously.[Bak17; Bow18; Har18; EA08; Gad13a] Defect configurational entropy is accounted for in the prefactor of the concentration expression. Vibrational, electronic, and magnetic entropy are neglected due to the significant computational expense associated with calculating them using hybrid functionals and their small expected magnitudes relative to the formation enthalpy and configurational entropy.[WN04] In this work, we employ this method to qualitatively highlight the differences in metal vacancy behavior when doping STO and BTO with donors and deep defects. While temperature dependent electronic and thermochemical properties (bandgap, specific heat, free energy, etc.) are available for STO well into cryogenic temperatures, less data in this temperature range is available for BTO. High quality data in this temperature range is a requirement for accurately extending DFT data to finite temperatures. So as to fairly compare the qualitative defect chemistry trends between the two materials and avoid introducing artifacts from incorrect temperature extrapolation of BTO properties, the 0 K bandgaps and 0 K chemical potential surfaces are employed here.

59 Figure 5.2 Defect formation energy diagrams for vacancies and vTi-vO in (a) BTO and (b) STO at markers along chemical potential traces in ﬁgures 5.1a and 5.1b, respectively. Solid and dashed formation energy lines correspond to respective traces in Fig. 5.1. At equilibrium, the Fermi level will always lie in an interval where all native defect formation energies are above 0.

5.3 Results & Discussion

5.3.1 Characteristics of Vacancies in BTO and STO

While there are many studies of point defects in BTO and STO at the PBE level, to the authors’ knowledge, there are only a handful at the HSE level, and most of them are focused on STO and in many cases solely on the oxygen vacancy.[Jan14; Mit12; Gry13; EM13; Bak17; Bow18; Var14; Iwa14; Cho11] Only some of these focus on charged defects, and two were written by the same authors as the present work.[Jan14; Var14; Mit12; Bak17; Bow18; Iwa14; Cho11] The defect formation energies and thermodynamic transition levels (TTLs) calculated here agree well with those vacancies examined by Janotti and Varley[Jan14; Var14], Iwazaki[Iwa14], and Choi [Cho11]. For the sake of discussion, equivalent processing conditions for BTO and STO are deﬁned here as an anneal at the same temperature and oxygen partial pressure, at one of the competing phase boundaries or at an equal fractional distance between them (i.e.

halfway between the TiO2- and AO-rich lines). The solid and dashed lines in ﬁgures 5.2(a) and 5.2(b) show calculated formation energies for vacancies and the divacancy complex for equivalent processing conditions at markers along the solid and dashed chemical potential traces in ﬁgures 5.1(a) (BTO) and 5.1(b) (STO), respectively. Oxygen vacancy formation

energies do not shift with changes in µTi and µA, so the formation energy of vO is plotted as one solid line in Fig. 5.2. Calculated TTLs for native defects are shown in Tables 5.1 and 5.2.

60 Table 5.1 TTL Locations of vacancies in BTO.

Defect TTL ETTL-EVB [eV] vO (+3 +2) 0.39 (+2 | +1) 3.15 | vBa (0 -1) 0.17 (-1 | -2) 0.19 (-2 | -4) 3.02 (-4 | -5) 3.09 | vTi (0 -1) 0.13 (-1 | -2) 0.40 (-2 | -3) 0.54 (-3 | -4) 0.61 | vO-vTi (+2 +1) 0.14 (+1| -1) 0.25 (-1 | -2) 0.45 (-2 | -3) 3.12 |

Consistent with a simple ionic picture, and as expected from both theoretical and experimental work, oxygen vacancies in both materials adopt the +2 charge state across most of the bandgap.[Jan14; Shi16; MH97] In both materials, this defect can transition to a +1 charge state when the Fermi level is near the CB by trapping an electron. Additionally, a transition to the +3 charge state can occur when the Fermi level is near the valence band by trapping a hole.

A-site vacancies (vSr and vBa) occur in the -2 charge state over most of the bandgap in both materials, consistent with the simple ionic picture used in canonical reaction

schemes.[Shi16] Both vSr and vBa have two nearly degenerate states near the valence band, allowing transitions to -1 and neutral charge states. Within the precision of our calculations, vSr has 3 degenerate states near the conduction band (-2 -5) allowing the defect to transition | to the -5 charge state very close to the CBM. In BTO, vBa exhibits similar states, but the degeneracy is lifted and shows up as separate (-2 -4) and (-4 -5) levels. Although the TTLs are very similar, the A-site vacancy is more favorable| in STO| than in BTO.

Our calculations predict that vTi occurs in the -4 charge state across most of the gap, consistent with literature on BTO.[Shi16] The TTLs for this defect are all located near the bandedge, and never occur as degenerate states. These vacancies are able to transition between the -4, -3, -2, -1, and 0 charge states. The more dramatic drop in formation energy

61 Table 5.2 TTL Locations of vacancies in STO.

Defect TTL ETTL-EVB [eV] vO (+3 +2) 0.33 (+2 | +1) 3.20 | vSr (0 -1) 0.15 (-1 | -2) 0.17 (-2 | -5) 3.10 | vTi (0 -1) 0.14 (-1 | -2) 0.36 (-2 | -3) 0.52 (-3 | -4) 0.72 | vO-vTi (+1 0) 0.04 (0 |-1) 0.22 (-1 | -2) 0.43 (-2 | -3) 3.20 |

of vTi as compared with vA when moving to more oxidizing environments is due to the larger

change in ∆µTi as compared to ∆µSr when increasing oxygen chemical potential. In fact,

the energy drop for vTi is twice the energy drop of vSr, due to the combination of host and competing phase stoichiometries. This is illustrated using shaded parallelograms in ﬁgures

5.1(a) and (b). Like vA, this defect has a noticeably different formation energy in BTO and

STO, despite the similar TTLs. However, unlike vA, it is more favorable in BTO.

The energetic favorability and prevalence of a vTi-vO complex in BTO was predicted at the PBE level by Erhart et al.[EA07] Similar to the PBE level ﬁndings for BTO, all in-gap states of this complex are clustered near the band edges in both materials. The primary charge state of this complex is -2, which is what would be expected based on formal oxidation states of the involved species. When the Fermi level is near the valence band the complex exhibits transitions of -1, 0 and +1 states in STO, and -1, +1, and +2 states in BTO. When the Fermi level is near the conduction band, the complex takes on the -3 charge state in

both materials. Like vTi, this defect is more favorable in BTO than STO despite the similarity in TTLs.

In STO, the formation energy of the A-site vacancy is always lower than that of the vTi-vO

complex, and is usually lower than that of vTi; these energy differences become larger when

moving from SrO-rich to TiO2-rich conditions. This leads to vSr being the dominant metal vacancy in STO in most conditions.

62 Interestingly, in BTO, the formation energy of vA in one competing phase limit is almost

exactly the same as the formation energy of vTi-vO in the other phase limit. In TiO2-rich conditions, the A-site vacancy is more favorable than the complex, while in BaO-rich

conditions, this is reversed. Halfway between TiO2-rich and BaO-rich conditions, vBa and vTi-vO have nearly identical formation energies over much of the gap. Because vTi, vBa,

and vTi-vO have such similar energies in all of these conditions, different combinations of them prevail depending on the exact location of the Fermi level and the exact position in processing space.

5.3.2 Origin of Vacancy Energy Differences

The differences in metal vacancy behavior arise for two reasons. First, there are several important differences between the chemical potential spaces of each material. Although the triangles are shown as being the same size, the STO triangle (Fig. 5.1(b)) is actually slightly larger. The stability regions for STO and BTO (white stripe) are in different places; the

stability region in STO is more negative on the ∆µTi and ∆µA axes than in BTO. Additionally, o o µBa is less negative than µSr. The cumulative effect of these differences is that, for equivalent processing conditions, it requires less energy to remove an A-site atom in STO than in BTO. Testing was performed to ensure that this effect was not due to changing the amount of exact exchange used in calculating thermodynamically relevant phases for BTO and STO. Nevertheless, differences in the thermodynamic space do not fully account for differ-

ences in the defect formation energies. For equivalent processing conditions, typical vA formation energies are between 0.6 and 0.8 eV lower in STO than in BTO, of which only 0.2-0.3 eV can be accounted for by differences in the chemical potential spaces. B-site vacancies and the B-site vacancy complex are typically 1.1-1.3 eV and 1.3-1.5 eV more favorable in BTO than in STO, and there are no signiﬁcant differences in the chemical potentials associated with titanium or oxygen atoms between the materials. These energy differences can be understood qualitatively by considering differences in the materials and the nature of the bonding on the different cation sites. A-site atoms primarily interact with the rest of the crystal ionically.[San15] All else being equal, the closer lattice spacing of STO should lead to a stronger ionic interaction with the lattice for Sr than Ba. However, the electronegativy of strontium is actually higher than barium. This, combined with the closer lattice spacing, leads to a lower effective charge on the strontium than might be expected. Bader charge analysis reveals that, while Ti and O have similar

63 charges in BTO and STO (+2.2 and +2.2, and -1.4 and -1.3, respectively), Ba and Sr do not; Ba has a nearly integer Bader charge of +2.0, while Sr has a charge of +1.7. The lower charge leads to a higher (less negative) ionic potential on the Sr site than on Ba site, as evaluated using the Madelung method. This means that less electrostatic work is associated with removing an Sr atom from STO as compared to removing a Ba atom from BTO. In contrast, the Ti–O octahedra are more covalent in nature, with the oxygen p-states forming the perovskite valence band and the empty Ti d-states forming the conduction band. Differences in the bond energy can be qualitatively treated using a Buckingham potential which has been used to describe Ti–O bonding in BTO and STO.[LC86; Akh95] Using this potential, it is found that Ti–O interactions are weaker in BTO than in STO. This is due to the larger lattice parameter and, thus, it requires less energy to remove Ti from BTO as compared to STO from a bond breaking perspective. These simpliﬁed approaches are consistent with and give energy differences within an order of magnitude of the respective differences obtained using the more sophisticated DFT calculations. In short, we have found that there are differences in the properties and formation energies of metal vacancy and metal vacancy complexes in BTO and STO. These differences have been attributed to the chemical potential space as well as subtle differences in bonding in the two bulks. The difference in bonding suggests that the properties of impurity dopants may also differ between the two materials. Furthermore, because of the differences in formation energies, the response of metal vacancies to doping may also be drastically different. For these reasons, the next two sections explore two typical dopants for these systems (niobium and iron) in terms of how they inﬂuence the behavior of the metal vacancies, and how their behavior differs between the two systems.

5.3.3 Vacancy Trends in Nb-doped BTO and STO

Metal vacancy compensation is important in donor-doped STO and BTO, because the higher Fermi level leads to lower formation energies and higher concentrations of metal vacancies. Niobium is a commonly used donor dopant in these systems. Formation energies for the primary forms of niobium in BTO and STO are shown in Fig. 5.3(a) and (b), respectively. 15 Native chemical potentials for this ﬁgure correspond to an oxygen partial pressure of 10− atm and temperature of 1173 K, halfway between the TiO2-rich and AO-rich condition. In 19 3 ﬁgures 5.3 and 5.4, STO and BTO systems doped only with 10 cm− Nb are considered. The impurity chemical potential and the Fermi level are solved self-consistently at each set

64 Figure 5.3 Defect formation energies for NbTi, NbSr, and vO in (a) BTO and (b) STO at hexagonal markers (10 15 atm PO at 1173K) and middle of processing windows in fig 5.1. was deter- − 2 µNb mined self-consistently with Fermi level based on fixed impurity content as described in the text. of native chemical potentials by using a fixed impurity concentration and imposing charge neutrality.

NbTi, NbA, and NbTi-vO ﬁrst nearest neighbor complexes were considered for this work. Our calculations show that niobium behaves similarly in STO and BTO. Niobium is found 1 to primarily incorporate as isolated NbTi. This defect has no in gap transitions in STO; in BTO, there is a (+5 +1) transition just above the valence band maximum. NbA defects were found to have unfavorably| high formation energies in all conditions studied. Additionally,

NbTi-vO complexes were found to have an unfavorable binding energy at all Fermi levels, indicating that they would spontaneously break apart. The valence configuration of an isolated Nb is 5s14d4, with all states having similar energies. When substituting for a titanium ion (isolated atom electron configuration 4s23d2) in STO or BTO, four of the electrons go into the O p-state derived valence band, while the fifth is ionized into the conduction band; there are no remaining electrons to participate in forming in gap states. This lack of in gap states probably contributes to the similarity of niobium’s behavior in BTO and STO. The fact that niobium is both a shallow donor and behaves similarly in BTO and STO makes it ideal for highlighting the differences in metal vacancy behavior in a less abstract manner than formation energy diagrams. We have therefore calculated the equilibrium 19 3 defect concentrations of 10 cm− Nb-doped BTO and STO at 1173 K for several processing regimes. Native chemical potentials are taken from the TiO2-rich traces (dashed lines in

65 Figure 5.4 (a-f) High temperature defect concentration diagrams for 1019 cm 3 Nb-doped (a-c) − BTO and (d-f) STO after an anneal at 1173 K for different annealing conditions. Concentrations after: a TiO2-rich anneal are shown in (a) and (d); an anneal in the middle of the processing window are shown in (b) and (e); an AO-rich anneal are shown in (c) and (f).

Fig. 5.1), the AO-rich traces (solid lines in Fig. 5.1), and traces halfway between them. The calculated defect concentration diagrams are shown in Fig. 5.4. The calculated concentration diagrams are largely similar to those constructed from traditional canonical reaction models for donor doping.[Shi16] Oxygen vacancy concentration increases when moving to more reducing conditions, while metal vacancy concentrations increase when moving to more oxidizing regimes. Donor doping induces a plateau in electron concentration over large regions of the processing space. This plateau begins to drop as the metal vacancy concentration becomes large enough to begin ionically compensating the donor.

Due to lower formation energy of vA in STO, it is the dominant metal vacancy in all processing conditions in donor doped STO (Fig. 5.4(d-f)). In the AO-rich limit, there is also

a large concentration of vTi present in STO. In donor doped BTO, due to the more balanced formation energies, a combination of vA, vB, and vB-vO complex occur depending on the processing conditions. In the TiO2-rich

limit (Fig. 5.4(a)), vA dominates. In the middle of the processing window (Fig. 5.4(b)), vTi becomes the dominant metal vacancy in oxidizing conditions, but all three metal vacancy

66 conﬁgurations are present, and in fact the vTi-vO complex is present at higher concentrations

than vTi in slightly reducing conditions. The divacancy complex is the dominant metal vacancy conﬁguration in all conditions in the AO-rich limit (Fig. 5.4(c)), with a lesser amount

of vTi also present. The predominance of strontium vacancies in STO, and the prevalence of

barium vacancies in TiO2-rich BTO and titanium vacancies and related defects in BaO-rich BTO, are consistent with conclusions drawn in prior experimental work.[BE82; Lee08a; Lee08b]

5.3.4 Vacancy Trends in Fe-doped BTO and STO

Iron is widely used to make STO and BTO more insulating. Previous work has shown that, in STO, iron exhibits TTLs deep in the gap which are responsible for the brown coloration observed in oxidized STO, and that isolated and complexed forms occur in similar concentrations, ionically compensating each other for large swathes of the processing window.[Bak17] This self-compensation pulls the Fermi level deeper into the gap than occurs with niobium doping. These characteristics make iron ideal for exploring metal vacancy behavior when the Fermi level is deep in the gap.

FeTi, FeA, and the ﬁrst nearest neighbor FeTi-vO complex were considered for this work.

Iron incorporates primarily as FeTi and FeTi-vO, while FeA is less favorable. Formation energy diagrams for the primary forms of Fe are shown in Fig. 5.5(a) and (b) for BTO and STO,

respectively. Note that Fig. 5.5 only shows the formation energies of FeTi and FeTi-vO in STO and BTO at a single point in processing space. As the oxygen chemical potential shifts, the relative favorability of the isolated and complexed forms change, due to the complex having

a dependence on both µTi and µO, while the isolated defect only has a dependence on µTi. As with niobium, native chemical potentials for this figure correspond to an oxygen 15 partial pressure of 10− atm and temperature of 1173 K, halfway between the TiO2-rich and AO-rich condition. In figures 5.5 and 5.6, STO and BTO systems doped only with 1019 3 cm− Fe are considered. The impurity chemical potential and the Fermi level are solved self- consistently at each set of native chemical potentials by using a fixed impurity concentration and imposing charge neutrality. The valence electron configuration of an isolated iron atom is (4s23d6). When substituting on a Ti site, 4 of the electrons behave like regular Ti electrons and drop into the valence band, leaving 4 remaining electrons that can form in-gap states. However, the differences in Ti–O bonding in BTO and STO lead to different TTL locations and different

67 Figure 5.5 Defect formation energies for FeTi, FeTi-vO, FeSr, and vO in (a) BTO and (b) STO at hexagonal markers (10 15 atm PO at 1173K) and middle of processing windows in ﬁg 5.1. was − 2 µFe determined self-consistently with Fermi level based on ﬁxed impurity content as described in the text.

relative favorability of the isolated and complexed forms of FeTi. This leads to substantial differences in defect concentration proﬁles. 2 In BTO, the isolated substitutional primarily occurs as Fe−Ti . There are three states clustered into two TTLs ((0 -2) and (+1 0)) near a Fermi level of 1 eV. However, isolated | | FeTi is only the minority form of Fe in BTO. The primary form is the FeTi-vO complex. This complex has two TTLs in the gap: a (+3 +2) transition near the valence band, and a (+2 0) transition near the middle of the gap. | | 1 2 In STO, the isolated substitutional primarily occurs as Fe−Ti . This can transition to Fe−Ti if the Fermi level is very close to the conduction band, or if the Fermi level is in the lower 0 +1 half of the gap it can transition to FeTi or FeTi . The FeTi-vO complex in STO exhibits (+4 +2) | and (+2 +1) transitions near the VB and a (+1 0) transition high in the gap. In STO, FeTi and | | FeTi-vO have more similar energies than in BTO. Overall, the TTLs are located deeper in the

gap for both FeTi and FeTi-vO in BTO than in STO. The resultant difference in defect proﬁles is pronounced. High temperature concentration diagrams for anneals at 1173 K are shown in Fig. 5.6(a-c) and (d-f) for BTO and

STO, respectively. Native chemical potentials from the TiO2-rich traces (dashed lines in ﬁg 5.1), the AO-rich traces (solid lines in ﬁg 5.1), and traces halfway between them were 0 2 used. In BTO, (FeTi-vO) is the dominant defect, but is electrically inactive. Instead, Fe−Ti is +2 compensated by large amounts of vO , and the electron concentration steadily decreases with increasing oxygen pressure, until it is surpassed by the increasing hole concentra- +1 tion. In contrast, in STO, the (FeTi-vO) complex is present across the entire regime and

68 Figure 5.6 (a-f) High temperature defect concentration diagrams for 1019 cm 3 Fe-doped (a-c) − BTO and (d-f) STO after an anneal at 1173 K for different annealing conditions. Concentrations after: a TiO2-rich anneal are shown in (a) and (d); an anneal in the middle of the processing window are shown in (b) and (e); an AO-rich anneal are shown in (c) and (f).

behaves as a poor donor. As conditions become more oxidizing, the material transitions +1 +1 1 from (FeTi-vO) :n compensation to (FeTi-vO) :Fe−Ti compensation. In both materials, the lower Fermi level with iron doping as compared to niobium doping leads to substantially lower metal vacancy concentrations. In BTO, metal vacancies 13 3 are calculated to be present at levels lower than 10 cm− for all three chemical potential traces, and there are no substantial differences in the defect proﬁles (ﬁg. 5.6(a-c)). In STO,

the impurity proﬁles are very similar, but the vSr concentration steadily increases when

moving from SrO-rich (fig. 5.6(f)) to TiO2-rich (fig. 5.6(d)) conditions. In SrO-rich conditions 13 3 (fig. 5.6(f)), vSr levels off at around 10 cm− . For the middle chemical potential trace (fig.

5.6(e)), the concentration of vSr increases with oxygen pressure before levelling off at around 16 3 10 cm− . In TiO2 rich conditions (fig. 5.6(d)), we calculate that vSr has a nearly identical 1 concentration profile as Fe−Ti . This slightly lowers the FeTi concentration relative to the 2 +1 complex, as less is needed to ionically compensate it with v−Sr present at similar levels. 0 Additionally, low levels of FeSr are present in the reducing regime in TiO2-rich conditions (fig. 5.6(d)), and steadily disappear as conditions become more oxidizing.

69 5.4 Conclusions

This work presents HSE-level native defect formation energies for BTO and STO, and establishes that there are fundamental differences in defect behavior in BTO and STO, due

to chemical and electronic differences between the materials. It was found that while vSr will be the dominant cation vacancy in all but heavily donor doped oxidizing conditions

in STO, different combinations of vBa, vTi, and vTi-vO will be present in BTO, depending on doping and processing conditions. The metal vacancies are more important in oxidized, donor-doped conditions. Because of the nature of the differences (different accessible processing windows, covalent bonding differences on the Ti sublattice, and ionic interaction differences on the A sublattice), it is not always appropriate to transfer native or impurity defect models between BTO and STO, or between these materials and other titanates, as is often done in literature. We believe this motivates careful defect characterization for each material.

70 CHAPTER

CALCULATING HYDROGEN SOLUBILITY IN DONOR-DOPED STO

This chapter presents work I undertook to develop and test a formalism for calculating hydrogen solubility in strontium titanate, but which is applicable to any semiconductor where hydrogen may be an issue. Niobium doped strontium titanate was selected as an initial test case because hydrogen is typically assumed to not incorporate very well into donor doped perovskites, based mostly on work in barium titanate and the perovskite cerates. If the results showed a low propensity to incorporate, then it would improve confidence in the new methods developed for this and provide firm evidence that the trend holds for strontium titanate as well, and if it turned out that hydrogen could incorporate very well, then that would also have been an important result. This manuscript is being prepared for submission. Fruitful discussions relevent to this work were had with Yifeng Wu. Computer time for the DFT calculations was provided by the DoD HPCMP,and the authors were financially supported from a DoD NDSEG fellowship and through Dr. Ali Sayir’s Aerospace Materials

71 for Extreme Environments program via AFOSR BRI grants FA9550-14-1-0264 and FA9550- 17-1-0318. Hydrogen contamination of strontium titanate (STO) during processing and usage is a known problem. However, it is relatively little-studied due to the difficulty of quantifying the amount of hydrogen that dissolves in the lattice. Here, we use hybrid exchange-correlation functional density functional theory calculations as input to a grand canonical thermodynamics framework to estimate hydrogen solubility and site preferences in donor-doped STO. Our results show, to the author’s knowledge, the first clear theoretical evidence that hydrogen contamination in donor-doped STO occurs at a low enough level to essentially ignore. But, this simple conclusion belies hydrogen’s rich behavior; unlike most dopants, it is able to easily change its incorporation site in response to changes in processing conditions. Overall, the findings are consistent with prevailing wisdom and suggest that the presented first principles approach could be used for systematic exploration of hydrogen’s impact as a function of doping and processing in this and other wide bandgap materials.

6.1 Introduction

Strontium titanate (STO) is a widely used material in industrial and research settings. It is often exposed to large amounts of hydrogen sources in several common processing routes and research applications, including high temperature reduction, solution processing, and water splitting. Concerns about the high reactivity of hydrogen led to a number of investigations on its incorporation. Using different techniques and processing routes, various authors have estimated hydrogen content in STO after annealing in atmospheres 16 3 20 3 containing steam or H2 to range from 10 cm− to 10 cm− .[Web86; Sch99; TM11] High levels of dissolved hydrogen can signiﬁcantly impact the functional properties of STO, and the wide estimated range speaks to the challenges of quantifying hydrogen uptake. Evidence from other proton conducting oxides and analogous changes to the IR spectra of these oxides and STO when annealed in hydrogen-containing atmospheres suggests that hydrogen incorporation in STO is mainly accomplished through formation of an H+ 1 interstitial bound to an oxygen atom (Hi ), and that hydrogen tends to be less soluble when donor-doping.[Was87; Was88; Su13; NL97; Mün00] However, more recent computational studies show that hydrogen can also form energetically favorable defect complexes with metal vacancies (metal vacancies are more favorable in donor-doped STO[Bak18]), indicat-

72 ing that hydrogen’s behavior may be more nuanced.[Var14; Jir12] Complicating matters further, studies on hydrogen solubility in donor-doped STO are scarce and offer conﬂicting estimates of hydrogen solubility.[Was88; Dai03] Here, we employ a grand canonical thermodynamics description of point defects, using hybrid exchange-correlation functional density functional theory (DFT) calculations as input, to directly examine hydrogen solubility in donor doped STO and its impact on the defect chemistry. Hydrogen can incorporate in STO in many ways because of its small size and unique bonding behavior. Due to the tremendous computational expense of examining all possible forms of hydrogen at a hybrid functional level, earlier theoretical work was used as a guide to the most favorable hydrogen incorporation sites.[Var14; Jir12; Iwa14]

6.2 Methods

Defect formation energies, ground state energies of relevant phases, and the electronic structure of STO were obtained from DFT calculations in VASP 5[KH93; KH94; KF96b; KF96a] with the HSE06[Hey03; Hey06] screened hybrid exchange correlation functional. Niobium is chosen as a prototypical donor in STO.[Bak18] Native defects, niobium related defects, and hydrogen related defects, including complexes between isolated vacancies and hydrogen, were considered for this work. Niobium-hydrogen complexes were not considered due to the positive charge states on the primary form of niobium and the mobile form of hydrogen. Defect calculations were performed with a 2 2 2 Monkhorst-Pack reciprocal space mesh and a 3 3 3 (135 atoms) tiling of the primitive× × cell. PAW pseudopotentials were used with 10 explicitly× × modelled electrons for strontium, 12 for titanium, 6 for oxygen, 13 for niobium, and 1 for hydrogen. All atoms within 5 Å of a defect were free to relax. An exact exchange amount of 0.2362, chosen to correct the underprediction of the bandgap endemic to traditional functionals, and a plane wave kinetic energy cutoff of 520 eV were used in all calculations. The calculated indirect bandgap of 3.25 eV and lattice parameter of 3.901 Å are in good agreement with experiment.[Kok15; OK73; VB01] Treatments of native chemical potentials in n-component materials have been extensively discussed elsewhere[Per05; RS01; RS03; Ert12; WN04], and the notation employed in our earlier work is used here. [Bak17; Bow18] However, this treatment is only strictly valid at 0 K, and was modified to account for finite temperatures due to the large expected impact of o temperature on solubility. Defining the chemical potential of species i as µi µi +∆µi and ≡

73 o DFT µi Ei , finite temperature effects can be included for the chemical potentials of the host ≡ f P compound X as: ∆HX +GX (T,P ) = i ni ∆µi . Stability conditions against precipitation of elemental phases j and competing phases k can be similarly updated: ∆µi < G j (T,P ), and P f i ni ∆µi < ∆Hk +Gk (T,P ), where ni is the stoichiometry of element i in compound X or k. A detailed derivation is provided in section 6.5. This leads to slightly different chemical potentials and thermodynamic boundaries at each temperature. The 1173 K surface is shown in fig 6.1(a), with shading where secondary phase precipitation may occur. Thermo- chemical tables and cryogenic heat capacity measurements were used to obtain Gibbs free energy changes with temperature and pressure for relevant elements and compounds.[CJ86; Ihs95; Gal02] Similarly, treatment of the bandgap contraction with temperature becomes important when trying to even qualitatively calculate the high temperature solubility of gasses in solids. As in previous work[Bow18], a single phonon model derived by Kok et al. [Kok15] was used to capture the bandgap shrinkage. As a first order approximation, we assume that the defect levels are rigid with temperature, and that the band edges shift around them with 2/3 of the movement in the valence band and 1/3 in the conduction band, consistent with earlier work.[EA08] The assumed band edge shifts between 0 K and 1173 K have been indicated with shading in figures 6.1(b) and 6.1(c).

6.3 Results and Discussion

Figures 6.1(b) and 6.1(c) show the formation energy diagrams for vacancies, NbTi,Hi,HO, vSr-

Hi, vTi-Hi, and the lowest energy conﬁgurations for vSr-2Hi and vTi-2Hi. After accounting for

hydrogen relaxation, there are 4 non-degenerate conﬁgurations for vSr-2Hi with formation energies within at most 0.28 eV of each other, and 2 non-degenerate conﬁgurations for vTi-2Hi within at most 0.3 eV of each other. Impurity chemical potentials and formation energies were calculated self-consistently with the Fermi level as in previous work[Bak17; 19 3 Bow18; Har18; Gad13a; Gad14b], with a niobium concentration of 10 cm− and assuming

1 atm of H2 or H2O at 1173 K. TiO2-rich and SrO-rich conditions are shown in ﬁg. 6.1(b) ﬁg. 20 2 6.1(c), respectively. Diamonds and circles mark reduced (10− atm O2) and oxidized (10 atm O2) conditions.

Niobium is found to occur mainly as NbTi and adopts the +1 charge state across the entire bandgap.[Bak18] While NbSr and NbTi-vO were also simulated, NbSr is prohibitively

74 Figure 6.1 (a) Chemical potential space of SrTiO3 at 1173 K balanced against Sr, Ti, SrO, and TiO2. Solid, dashed, and dotted lines indicate TiO2-rich, middle, and SrO-rich processing conditions. (b) and (c) Formation energy diagrams for relevant point defects at endpoints of the TiO2-rich (b) and SrO-rich (c) traces, with gray shading marking band edge positions at 1173 K. µH was taken at 1 atm of H2 or H2O gas at 1173 K, while µNb was solved for self-consistently based on the concentration.

high energy, and the binding energy of the complex indicates a strong preference to not associate.

We ﬁnd that Hi adopts the +1 charge state across the entire bandgap, at both low temperatures and the annealing temperature of 1173 K. Rather than sitting in octahedral or tetrahedral interstitial void as might be expected for a larger atom, hydrogen prefers to sit near an oxygen, slightly off-axis from the O–O bond with an O–H bond length of 0.97 Å, in line with prior work.[Mün00; Bor11; Jir12; Var14] Based on the conventional

understanding of hydrogen incorporation into perovskites, free Hi was expected to be the dominant hydrogen defect. However, our results indicate that this is only the case in donor doped STO over a narrow range of oxygen partial pressures. Instead, in oxidizing conditions, hydrogen usually prefers to reside within a metal vacancy void while remaining bonded to one of the ﬁrst nearest neighbor oxygens, while in heavily reduced conditions, the hydrogen replaces an oxygen atom and forms a multicenter bond (as are known to occur in other wide bandgap oxides[JW07]). Single or double hydrogen atoms sitting in the void of a strontium vacancy are referred

to as vSr-Hi and vSr-2Hi, respectively. Hydrogen is found to bond to the vacancy-coordinating

oxygen atoms with a similar bond length as the “free" Hi defect. For the 1 H conﬁguration, the H–O bond aligns with the [110] axis, while in the 2H conﬁgurations, the individual H–O bonds are angled slightly away from the [110] axis due to Coulombic repulsion. The vSr-Hi

complex occurs primarily in the -1 charge state. The vSr-2Hi complex occurs primarily in

75 the neutral charge state, can occur in 4 unique conﬁgurations, and is less favorable than

the vSr-Hi defect. Because of this relative unfavorability, complexes with higher numbers of hydrogen were not considered.

Hydrogen can also sit in the void of a titanium vacancy. As with vSr complexes, the hydrogen remains bonded to oxygen atoms coordinating the vacancy, maintaining a similar bond-length as the “free" interstitial. The H–O bond is aligned with the [100] axis for the 1 H conﬁguration; the bonds are instead angled slightly away from the axis in the 2 H

conﬁguration due to Coulombic repulsion. The vTi-Hi complex is found to primarily occur

in the -3 charge state, while the vTi-2Hi complex primarily occurs in the -2 charge state. The charge states for both sets of metal vacancy complexes are exactly what would be expected for H+ ions interacting ionically with the metal vacancies in their normal charge states. In reducing conditions, hydrogen favors the oxygen site, where it forms multicenter bonds with its neighboring atoms, as occurs in ZnO and MgO[JW07]; these bonds have

similar charge distributions to those of the replaced oxygen atom. Like Hi,HO occurs in the +1 charge state across the entire bandgap. Because of the dependence of formation energy on Fermi level, donor-type hydro-

gen related defects (Hi,HO) become more favorable when the Fermi level is low, while

the acceptor-type hydrogen related defects (vSr-Hi, vSr-2Hi, vTi-Hi, vTi-2Hi) become more favorable when the Fermi level is high, as occurs with donor doping. Like the formation energies and chemical potentials, defect concentrations were calculated self-consistently with the Fermi level as a function of oxygen and hydrogen chemical potential, as in prior work.[Bak17; Bow18; Har18; Gad13a; Gad14b] Vibrational, magnetic, and electronic entropies were neglected due to their small expected contribution to the defect formation energy relative to the configurational entropy and formation enthalpy. Because of this and the assumptions used in treating the bandgap, while solubility and concentration trends should be correctly reproduced by this model, the numbers themselves should only be treated qualitatively. 19 3 Figure 6.2 shows defect concentrations for STO doped with 10 Nb cm− at 1173 K without hydrogen (fig. 6.2(a-c)) and with a hydrogen availability of 1 atm H2 or H2O (fig. 6.2(a- c)) as a function of oxygen partial pressure, processed along the solid, dashed, and dotted lines displayed in figure 6.1(a). This simplistic hydrogen availability model was selected as a “worst case scenario" for evaluating how problematic hydrogen could be for donor-doped STO. Actual hydrogen availability is a complex and furnace-specific interaction between

76 Figure 6.2 Point defect concentrations for 1019 cm 3 Nb doped STO equilibrated at 1173 K with − ((d)-(f)) and without hydrogen ((a)-(c)) in TiO2-rich (((a),(c)), middle ((b),(e)), and SrO-rich ((c),(f)) conditions. Hydrogen chemical potentials correspond to 1 atm availability of of H2 or H2O at 1173 K.

PO2 ,PH2O of the starting air (1-2 vol% for typical relative humidities), and PH2 of the forming gas (typically 10-20 vol%).[Dai06; McR80] Even at such a high availability, the compensation remains fairly consistent in the presence and absence of hydrogen. In both cases (ﬁgs. 6.2(a-c) and 6.2(d-f)), the compensation

and electron concentration are controlled by the concentrations of NbTi and vSr except

in the most reducing conditions. In TiO2-rich conditions (figs. 6.2(a,d)), this interaction effectively eliminates the electron plateau. For conditions in the middle of the processing regime (figs. 6.2(b,e)), there is a short plateau in the electron concentration before it starts to die off. The plateau is broadest for SrO-rich conditions (figs. 6.2(c,f)) due to suppression

of the intrinsic metal vacancies (vSr).

In reducing conditions at 1173 K, the primary form of hydrogen is the HO defect. For the

middle processing trace (ﬁg. 6.2(e)), this gives way to Hi with increasing oxygen pressure,

and then to the vSr-Hi defect complex. The concentration proﬁles for HO and Hi are fairly consistent across the processing window. However, there are signiﬁcant differences in

the amount of Hi-metal vacancy complexes. In oxidized TiO2-rich conditions (ﬁg. 6.2(d)),

there is a signiﬁcant increase in the concentration of vSr-Hi and, to a lesser extent, vSr-2Hi.

77 Figure 6.3 (a-c) Isodoping contours of hydrogen chemical potential and partial pressure vs. oxygen partial pressure for STO doped with 1019 Nb cm 3 at 1173 K for (a) TiO -rich conditions,(b) − 2 middle conditions, and (c) SrO-rich conditions. The black line indicates the an availability of 1 atm gaseous H2 and H2O. (d) Temperature dependence of hydrogen solubility, evaluated against gaseous H and H O at 1 atm. in STO doped with 1019 Nb cm 3 at TiO -rich (solid lines), middle 2 2 − 2 (dashed lines), and SrO-rich (dotted) chemical potential traces.

Similarly, in oxidized SrO-rich conditions (ﬁg. 6.2(f)), there is an increase in the amount of vTi-Hi and, to a lesser extent, vTi-2Hi. Figures 6.3(a-c) presents lines of constant hydrogen concentration in donor-doped STO

processed in TiO2-rich (a), middle (b), and SrO-rich (c) conditions as a function of oxygen partial pressure and hydrogen chemical potential. Hydrogen availability corresponding to ﬁg. 6.2 is marked with black lines. Figure 6.3(d) shows the temperature dependence

of hydrogen solubility with 1 atm H2/H2O availability; the lines correspond to the solid, dashed, and dotted lines in fig. 6.1(a). In general, the concentration of a given hydrogen defect is found to scale linearly with total hydrogen concentration within a given processing regime, so if fig. 6.3 is used to determine the total amount of hydrogen, then fig. 6.2 can be used to find approximate defect concentrations. Section 6.6 contains a method for determining hydrogen chemical potentials in these plots for a given atmosphere, accounting

for H2 or H2O content. The hydrogen solubility in oxidized SrO-rich conditions may be

underestimated since vTi-vO-nH complexes were not simulated. For a given availability,

more hydrogen tends to be soluble near the TiO2-rich phase boundary than the rest of the

processing space. This is easy to see for the 1 atm H2/H2O availability lines (black) plotted in ﬁg. 6.3(a-c). The proton solubility is generally found to increase with temperature (shown in ﬁg. 6.3(d)), except for oxidized conditions near the SrO-rich boundary where this trend is reversed. This reversal occurs because of a combination of two factors. First, the Fermi

78 level is generally higher with lower temperature in donor-doped STO, to maintain charge neutrality against a ﬁxed donor concentration. This, combined with the higher Fermi level

and enhanced favorability of vTi related defects in oxidized SrO-rich conditions, increases

the concentration of vTi-Hi, and the overall hydrogen solubility, with decreasing temperature. On the whole, the positive temperature solubility dependence seen in most conditions is fairly weak, as can be seen in ﬁgure 6.3(d), and because of this a substantial fraction of the hydrogen incorporated at high temperature probably remains at low temperature, even neglecting low temperature permeability changes at the edge of the material.

The mobility of Hi is believed to be quite high compared to other mobile ionic species in STO, although it is still quite low compared to an electron. The net activation energy

for Hi transport has been estimated to lie between 0.4 and 0.5 eV[Mün00; Bor11], which is considerably lower than the most common estimates for the oxygen vacancy (0.6-0.7

eV).[DS12] This leads to a dramatically higher conductivity for Hi than for vO. As an example,

using De Souza’s value[DS12] for µvO at 500 K (a common temperature for ionic conduction

studies in STO), and making the assumption that the vO prefactor applies to Hi,Hi is found

to be between 16 and 170 times faster than vO. However, the electrons are considerably faster than either of these species, and in donor doped STO their current contribution dwarfs that of either ionic species in all conditions. Using the same assumptions, and taking

µe from Moos & Hardtl [MH97], electrons are between 60,000 and 620,000 times faster than the hydrogen interstitial at 500 K, and in most conditions there are also more of them present.

6.4 Conclusions

This work clearly establishes that, even at the high availability assumed for ﬁg. 6.2, hydrogen will not strongly contribute to the electrical properties of donor doped STO, through either compensation or ionic conduction, due to the high concentration of free electrons. Despite the minimal impact on bulk electrical properties, hydrogen is shown to exhibit a rich and varied response to processing conditions and temperature. This reﬁned model for hydrogen incorporation in donor-doped STO could guide future experiments and interpretation of results in this economically important system. This work also presents a robust theoretical method for examining hydrogen incorporation in other situations where it may be more

problematic. However, the uncertainty in estimates for the mobility of the Hi defect motivate

79 the need for more accurate mobility values, and more work is needed to understand how the hydrogen defect populations evolve upon quenching.

6.5 Supplemental Derivation: Finite Temperature Chemical Potential Space

We begin from the condition for chemical potential of a compound in equilibrium with its components, on a formula unit basis:

µSrTiO3 = µSr + µTi + 3µO

We then introduce the conditions necessary to thermodynamically prevent competing phase precipitation: that the chemical potential of component i in the compound must be less than the chemical potential of component i in some other phase. Ti referenced against

its elemental metals and rutile TiO2 as an example:

µTi(SrTiO3) < µTi(αTi,βTi,TiO2)

We then introduce a deﬁnition for µi which lets us break it into a term for the 0 K internal o energy obtained from DFT (µi ), and all other contributions (∆µi ), and substitute:

o µi µi + ∆µi ≡

o DFT µi Ei ≡

µo ∆µ µo ∆µ µo ∆µ 3µo 3∆µ SrTiO3 + SrTiO3 = Sr + Sr + Ti + Ti + O + O If we then substitute in the deﬁnition for the 0 K enthalpy of formation:

∆H f E DFT E DFT E DFT 3E DFT SrTiO3 = SrTiO Sr + Ti + O 3 − Then we obtain:

o o o o µSrTiO µSr + µTi + 3µO + ∆µSrTiO3 = ∆µSr + ∆µTi + 3∆µO 3 −

80 f ∆H ∆µ ∆µ ∆µ 3∆µ (6.1) SrTiO3 + SrTiO3 = Sr + Ti + O and for our phase stability limits, performing the same substitutions, using metallic Ti and rutile TiO2 as examples:

µTi(SrTiO3) < µTi(Ti)

o o µTi + ∆µTi(SrTiO3) < µTi + ∆µTi(Ti)

∆µTi(SrTiO3) < ∆µTi(Ti) (6.2)

µTi(SrTiO3) + 2µO(SrTiO3) < µTiO2

µo ∆µ SrTiO 2µo 2∆µ SrTiO < µo ∆µ Ti + Ti( 3) + O + O( 3) TiO2 + TiO2

f ∆µ SrTiO 2∆µ SrTiO < ∆H ∆µ (6.3) Ti( 3) + O( 3) TiO2 + TiO2 Equations 6.1, 6.2, and 6.3 then give a bounded n-plane of accessible chemical potentials. In previous work[Per05; RS01; RS03; Ert12; WN04; Bak17; Bow18], although different notations are typically used by each author, ∆µ values for the host, competing, and solubility limiting phases are typically assumed to be zero. ∆µ values for non-gaseous species are typically treated as varying between rich and poor limits of either elemental or competing phases. The chemical potential of gasses in equilibrium with the system are typically set as, in our notational convention, and using O2 as an example:

T,P ∆µO = ∆G 0 ∆H (T ) T ∆S(T,P ) | ≈ − 1 PO ∆µ ∆H T T ∆S T k T ln 2 (6.4) O 2 ( )O2 ( )O2 + B P o ≈ − O2 In eq. 6.4, values for the internal enthalpy and entropy changes are taken from thermodynamic references such as the NIST JANAF tables[CJ86] and are typically re-aligned such

81 that they go to 0 at 0 K and standard pressure.

In this case, treatment of ∆µTi and ∆µSr as free parameters is somewhat justiﬁed, due to the extremely low vapor pressure of these species in the system; their values will be

more strongly determined by the SrTiO3-TiO2-SrO equilibrium than equilibrium with their gaseous forms.

The form for ∆µO also suggests a form for ∆µSrTiO3 , ∆µTiO2 , ∆µSrO, ∆µSr(Sr), and ∆µSr(Ti). To be consistent with previous work, which, despite neglecting these values, seems to give satisfactory qualitative results, these terms must go to 0 at 0 temperature, and the evolution of the boundaries with temperature should reﬂect, in some way, an evolution f f from ∆H (0K ) to ∆G (T ). If these values are assumed to equal the internal Gibbs energy of each phase, re-aligned to evaluate to 0 at 0 K, then this derivation is both consistent with previous work at 0 K, and evaluation of the range spanned by any one chemical potential

between ni ∆µi (T,max) and ni ∆µi (T,min) equals the Gibbs free energy of formation of the compound at that temperature. Then, our equations 6.1, 6.2, and 6.3 become:

f ∆H G T ∆µ ∆µ 3∆µ (6.5) SrTiO3 + SrTiO3 0 = Sr + Ti + O |

T ∆µTi(SrTiO3) < GTi(Ti) 0 (6.6) |

f ∆µ SrTiO 2∆µ SrTiO < ∆H G T (6.7) Ti( 3) + O( 3) TiO2 + TiO2 0 | This formalism has been previously used to examine and understand the mechanism of the silicon compensation knee in AlN, where it was found to give much better agreement with experiment than using the 0 K chemical potential range and bounds.[Har18]

6.6 Supplemental Derivation: Calculation of µH from PH2 and

PH2O

Based on the weak temperature dependence of hydrogen solubility discussed in the main article, there is not a very large driving force to expel hydrogen from the STO at lower temperatures. As such, the amount incorporated will mainly depend on the hydrogen chemical potential in the gas phase during annealing. For simplicity, we consider two

82 regimes in the annealing environment: a regime where µH is dominated by H2, and a regime where it is dominated by H2O. In the ﬁrst regime, the hydrogen chemical potential is related

to its dimer partial pressure in the same manner as O2, discussed in the previous section. Thus, in this regime, using tabulated data from the NIST JANAF[CJ86] tables: 1 PH ∆µ ∆H T T ∆S T k T ln 2 (6.8) H 2 ( )H2 ( )H2 + B P o ≈ − H2 and

o µH = µH + ∆µH

where

o 1 DFT µH EH ≡ 2 2 In the second regime, µH is assumed to be controlled by the partial pressure of H2O, in the form of dry steam. In this case, the chemical potential of hydrogen may be determined as a function of the chemical potential of steam and oxygen, which are treated as independent in this approximation:

1 µH = µH2O µO (6.9) 2 − µO is determined as described in the previous section, as a function of temperature and

oxygen partial pressure. µH2O may then be determined in a similar fraction, where PH O ∆µ ∆H T T ∆S T k T ln 2 (6.10) H2O ( )H2O ( )H2O + B P o ≈ − H2O The steam partial pressure may be determined from the relative humidity and temperature of the starting airstream (prior to intake into the furnace) by using standard meteorological and psychrometric data. We have used the polynomial approximation of McRae [McR80] for this purpose. The reported expression gives the volume fraction of steam in the air. In the ideal gas approximation, this may be be freely converted to a partial

pressure, which can then be used in eq. 6.10. Then, µH2O may be determined the same way

as µH2 and µO2 :

83 µ EDFT ∆µ H2O = H2O + H2O

The table below provides ground state energies from DFT for H2,O2, and H2O, to allow

the reader to evaluate µH themselves, to examine hydrogen solubility for their speciﬁc experimental conditions. Note that chemical potentials calculated using these ground state energies will only be compatible with calculations based on DFT simulations with the same settings.

Table 6.1 Ground State Energies for H2,O2, and H2O.

DFT Gas Ei [eV] H2 -7.772 O2 -13.775 H2O -17.304

84 CHAPTER

CONCLUSIONS AND FUTURE WORK

The work reported in this dissertation represents significant milestones in the advancement of the state-of-the-art for strontium titanate specifically, and first principles based models of wide bandgap semiconductors in general. Our work proved the iron color centers as the source of the brown coloration, established confidence in the models, and reiterated the importance of defect complexes in this material. Our results on extending the results to BTO showed some fundamental differences in bonding between STO and BTO, and indicated that transferring results from STO directly to BTO may prove challenging. While some general rules of thumb may eventually be found to apply to all similar perovskites, the current results seem to show that there really is no substitute for extensively researching defects in each system individually. Lastly, a framework was developed for investigating hydrogen solubility and incorporation, which is a challenging problem across many materials systems, and showed that hydrogen in donor-doped STO is not soluble enough to affect the electrical properties. This work has also made considerable improvements to the defect solver prototype originally developed by Ben Gaddy. We now have a well-established, well-tested tool for

85 systematically using DFT defect calculations to explore defect behavior, and properties that depend on defect behavior, in real systems. In parallel with database systems designed to deal with high throughput DFT calculations for bulk systems, we have developed our own database system for dealing with high throughput DFT calculations for point defects. It addresses the need for a flexible, robust, and maintainable system for quickly postprocessing raw DFT calculations, and now forms a cornerstone of our workflow, not just in the perovskites systems, but also in our research on the nitrides. Both the defect solver and the defects database have made significant strides in capabilities, ease-of-use, generality, and stability, which are prerequisites for extended use by the materials community. However, in spite of these successes, there is still more to do. There are currently several active and planned areas of research to complement and expand our current findings. Coupled experimental and theoretical investigations aimed at better understanding the low and medium temperature conductivity of Fe-doped SrTiO3 are ongoing. With the framework for calculating hydrogen solubility seemingly validated, work can also progress in other doping regimes where hydrogen may be more pernicious, such as heavy acceptor doping. Finally, although not discussed here, significant strides have also been made by in developing a device-level model coupled to our defect solver, allowing extension of our analysis to interfaces and device-level phenomena.

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