Ab initio studies of defect concentrations and diffusion in metal oxides Kilian Frensch Condensed Matter Theory Group Department of Physics Imperial College London A thesis submitted for the degree of Doctor of Philosophy 2011 \In physics, you don't have to go around making trouble for yourself - nature does it for you." - Frank Anthony Wilczek Declaration of Originality I certify that, except where appropriately referenced, the work is that of the author alone and has not been previously submitted to qualify for any other academic award. - Kilian Frensch, July 2011 Abstract This work presents a methodology for determining the concentrations and diffusion coefficients of point defects in metal oxides using ab initio calculations of defect for- mation energies and diffusion barriers, and the binding energies of defect-impurity clusters. The methodology is applied to analyse the long-standing mysteries sur- rounding the mechanism of self-diffusion in α-Al2O3. Al2O3 is a prototypical large band gap ceramic with extensive applications, many of which depend on its defect chemistry. In particular, point defect concentrations, that vary with temperature and impurity doping, govern diffusion properties such as creep, sintering, or the oxidation rate of Al-containing alloys. Experimental measurements of the self-diffusion coefficients in bulk alumina reveal three important truths that theory cannot reconcile, collectively termed the 'corundum conundrum'. First, large experimental activation energies for oxygen and aluminum diffusion and low theoretical formation energies imply unreasonably high diffusion barriers of ∼ 5eV. Second, aluminum diffusion is orders of magnitude faster than oxygen diffusion. Third, the oxygen diffusion coefficient is relatively insensitive to aliovalent doping, increasing by a factor of 100 on heavy Mg2+-doping, and decreasing by a similar amount on Ti4+-doping. We attempt to resolve this conundrum by calculating the formation energies and binding energies of a raft of native point defects and defect-impurity clusters as func- tions of temperature T and oxygen partial pressure pO2 , and the diffusion barriers of the native defects, using density functional theory. We then use a thermodynamic mass action approach to determine the concentrations of the defects and clusters, and the diffusion coefficients of the defects, as functions of T , pO2 , and the con- centrations of aliovalent dopants, [Mg2+] and [Ti4+]. In the process, we discover new ground-state defect structures for the aluminum vacancy and oxygen intersti- tial, and demonstrate that diffusion of aluminum vacancies and interstitials occurs by extended vacancy and interstitialcy mechanisms, and oxygen interstitials by a dumbbell interstitialcy mechanism, all of which yield much lower migration barriers than previous theory. Unfortunately, the results do not demonstrate the experimentally-found insensitivity of the oxygen diffusivity to aliovalent doping. This could be an artefact of approx- imations within density functional theory and our methodology, and we show that modest changes in the calculated binding energies lead to significant defect cluster- ing. This defect clustering could result in a buffering mechanism that can explain the insensitivity of the diffusivity to aliovalent doping, and may occur in other ionic materials. More accurate calculations, employing hybrid functionals or quantum Monte Carlo methods, may be necessary to elucidate this effect, but are currently computationally intractable for our purposes. 5 Acknowledgements This PhD has undeniably been the most challenging episode of my life this far. It would not have been possible without the immense support I received during the time. Above all, I would like to thank Nick, the most devoted and brilliant young scientist I have met, for his neverending help and advice. Much of this work is due to him. Matthew and Mike have been a fountain of knowledge, motivation, and inspiration, and the best supervisors a student could ask for. Arthur contributed crucial elements to this work at a pivotal stage, and was a driving force during the final year of my work. I would like to thank my friends, in particular Ola and Adam, for putting up with much of the frustration that accompanied the work and making the time all the more fun. I could thank Racheal for any one of a number of things over the past years, but each would only describe a small part of what makes her special. Suffice it to say that she is most definitely `the better half' and has contributed enormously to this thesis. My family has always supported me and I thank them for their guidance and love. My father has been an inspiration to me from a young age, and I dedicate this thesis to him, my mother, and Antonia, who have always been the nucleus of my motivation. Table of Contents 1 Introduction 11 2 The many body problem 15 2.1 Introduction . 15 2.2 The Hamiltonian . 16 2.2.1 The Born-Oppenheimer approximation . 18 2.3 Density Functional Theory . 19 2.3.1 The first Hohenberg-Kohn theorem . 19 2.3.2 The second Hohenberg-Kohn theorem . 20 2.3.3 The Kohn-Sham ansatz . 21 2.3.4 Exchange-correlation functionals . 25 2.3.5 Basis sets . 27 2.3.6 Pseudopotentials . 28 2.4 Quantum Monte Carlo . 30 2.4.1 Variational Monte Carlo . 30 2.4.2 Diffusion Monte Carlo . 32 2.4.3 Trial wave functions . 39 2.4.4 Wave function optimisation . 41 3 Point defects in solids 43 3.1 Introduction . 43 3.2 Defect species and notation . 45 3.3 Defect concentrations in the dilute limit . 47 3.3.1 The law of mass action . 47 3.3.2 Defect concentrations in the canonical ensemble . 48 3.3.3 Intrinsic and extrinsic defects . 54 3.4 Thermodynamics of defect formation energies . 57 3.4.1 Zhang-Northrup formalism . 57 3.4.2 Chemical potentials . 59 3.5 Density functional theory calculations of defect formation energies . 65 3.5.1 Vibrational effects . 65 3.5.2 Band gap corrections . 68 3.5.3 Finite-size effects . 71 7 Table of Contents 3.6 Diffusion kinetics of defects . 78 3.6.1 Mechanisms of diffusion . 79 3.6.2 Activation energies and diffusion barriers . 81 3.6.3 Fick's laws . 83 3.6.4 Diffusion coefficients from random walk diffusion . 86 3.6.5 Correlation effects . 88 3.6.6 Experimental methods . 93 4 Point defects and diffusion in Al2O3 97 4.1 Introduction . 98 4.2 Experimental background . 98 4.2.1 Oxygen diffusion . 99 4.2.2 Oxygen diffusion buffering in doped alumina . 102 4.2.3 Aluminium diffusion . 105 4.2.4 Anisotropy . 107 4.2.5 Alternative species . 108 4.3 Previous theoretical work . 109 4.4 Density functional theory calculations . 115 4.4.1 Geometry and basis set convergence . 115 4.4.2 Defect species and complexes . 122 4.4.3 Defect formation energies and cluster binding energies . 145 4.4.4 Chemical potentials . 152 4.4.5 Band gap corrections . 154 4.4.6 Finite-size scaling . 156 4.4.7 Vibrational contributions . 160 4.4.8 Diffusion barriers . 162 4.5 Defect concentrations . 179 4.6 Diffusion coefficients . 190 4.7 Impurity solubilities . 194 4.8 Quantum Monte Carlo calculations . 198 4.8.1 Trial wave functions . 199 4.8.2 Jastrow optimisation . 200 4.8.3 QMC band gaps . 202 4.8.4 Errors . 203 4.8.5 Diffusion barriers . 206 8 Table of Contents 5 Conclusion 209 A Point defects in solids 213 A.1 Arrhenius relationships of activation energies of diffusion . 213 A.2 Einstein equation from random walk diffusion . 214 A.3 Diffusion coefficients from random walk diffusion . 215 A.4 Correlation factors for native defect diffusion in alumina . 217 9 1 Introduction Physicists, and scientists in general, enjoy classifying things into distinct categories. It is therefore only natural that they classify themselves, creating the great divide in physics, between `experimentalists' and `theorists'. Of course, each group consid- ers itself superior to the other, occasionally even suggesting that the other merely exists to support it. However, physics is impervious to all this. It remains physics, regardless of who studies it in what way. Neither experimentalists nor theorists `invent' physics with their discoveries. In fact, both rely on the other for their own recognition, whether it be a theorist in need of experimental verification of their prediction, or an experimentalist in need of a theoretical explanation of their result. Ultimately, every physicist is actually part both, and if we wish to have a complete understanding of the world around us, experiment and theory must walk hand in hand. This study was motivated by the gap between experiment and theory in the under- standing of point defect diffusion in Al2O3, or alumina. For almost half a century, tracer diffusion studies on alumina doped with aliovalent impurities have shown that the concentrations of the native defects in alumina are strongly buffered. Theory has been unable to explain the cause of this buffering, in part due to the theoretical methods employed. This disparity between theory and experiment has been coined the `corundum conundrum' in a recent paper [80]. The current study attempts to re- solve this conundrum using a full ab initio, meaning from first-principles, approach. The results from this approach suggest that density functional theory calculations employing an LDA functional are unable to rationalise this long-standing conun- drum, and that further work using hybrid functionals or quantum Monte Carlo methods is necessary. Although results are presented for alumina, the theoretical 11 Chapter 1: Introduction formalism underlying the current study is general, and can be applied to a wide range of other metal oxides. In Chapter 2, the theoretical methods used in this study are presented: density functional theory (DFT) and quantum Monte Carlo (QMC).