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Electronic Structure Study of Copper-Containing Perovskites Electronic Structure Study of Copper-containing Perovskites Mark Robert Michel University College London A thesis submitted to University College London in partial fulfilment of the requirements for the degree of Doctor of Philosophy, February 2010. 1 I, Mark Robert Michel, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Mark Robert Michel 2 Abstract This thesis concerns the computational study of copper containing perovskites using electronic structure methods. We discuss an extensive set of results obtained using hybrid exchange functionals within Density Functional Theory (DFT), in which we vary systematically the amount of exact (Hartree-Fock, HF) exchange employed. The method has enabled us to obtain accurate results on a range of systems, particularly in materials containing strongly correlated ions, such as Cu2+. This is possible because the HF exchange corrects, at least qualitatively, the spurious self-interaction error present in DFT. The materials investigated include two families of perovskite-structured oxides, of potential interest for technological applications due to the very large dielectric constant or for Multi-Ferroic behaviour. The latter materials exhibit simultaneously ferroelectric and ferromagnetic properties, a rare combination, which is however highly desirable for memory device applications. The results obtained using hybrid exchange functionals are highly encouraging. Initial studies were made on bulk materials such as CaCu3Ti4O12 (CCTO) which is well characterised by experiment. The inclusion of HF exchange improved, in a systematic way, both structural and electronic results with respect to experiment. The confidence gained in the study of known compounds has enabled us to explore new compositions predictively. Interesting results have been obtained, and we have been able to identify new materials of potential interest, which represent clear new targets for future experimental studies. 3 Acknowledgements I’d like to dedicate this thesis to the loving memory of the most caring and selfless person I ever knew, my mother, Lauren Michel. Of course I wish that you were here to see this, but wherever you are, I hope that you are proud. During my time at UCL I have been fortunate enough to work with some amazing people of whom it would be rather churlish not to acknowledge. In particular I would like to thank Alastair, Paul, Kim, Aisha, Luis, Martin, Vladimir, Martijn, Gareth, Alexei, Scott, Steven and Antonio for their good friendship and generosity with their time. Most of all I would like to thank my supervisor, Dr. Furio Cora, for his unbelievable patience and incredible level of support, without which this PhD would not have been possible. I would like to acknowledge my family for their love and support throughout my studies. In particular I would like to thank my Grandmother for not giving up on me through school and my Uncle for enabling get to University. I am hugely grateful to Prof. Richard Catlow for giving me the opportunity to undertake this PhD and would like to thank EPSRC for funding. 4 Chapter 1 - Introduction .............................................................................................8 Chapter 2 - Crystalline Solids and Surfaces............................................................14 2.1 Crystal Lattices .....................................................................................................14 2.2 Crystal Structures..................................................................................................16 2.3 Symmetry, Point Groups and Space Groups.........................................................17 2.4 Miller Indices ........................................................................................................18 2.5 Reciprocal Lattice and Reciprocal Space..............................................................18 2.6 K-Points and the Brillouin Zone ...........................................................................19 2.7 Modelling Solids and Surfaces .............................................................................20 2.7.1 Bulk Calculations...........................................................................................20 2.7.2 Surface Calculations ......................................................................................21 2.8 Magnetic Structures ..............................................................................................22 Chapter 3 - Theoretical Methods..............................................................................24 3.1 Quantum Mechanics .............................................................................................24 3.2 Hartree-Fock Theory.............................................................................................25 3.2.1 Limitations of HF theory................................................................................32 3.3 Density Functional Theory (DFT) ........................................................................32 3.3.1 Hohenberg-Kohn Theorems and Kohn Sham Equations...............................36 3.3.2 Exchange and Correlation Functionals ..........................................................42 3.3.3 Limitations of DFT – the self interaction error..............................................44 3.4 Wavefunctions vs Densities ..................................................................................45 3.5 Hybrid DFT/HF Methods......................................................................................46 Chapter 4 - Basis Sets ................................................................................................49 4.1 Plane Wave Basis Sets ..........................................................................................49 4.2 Gaussian Basis Sets...............................................................................................51 4.3 Pseudopotentials....................................................................................................53 Chapter 5 - Choice of Computational Code ............................................................54 5.1 Self-consistent mixing techniques ........................................................................54 5.1.1 Linear Mixing ................................................................................................55 5.1.2 Broyden Mixing .............................................................................................56 5.2 Analysis of the Electronic Density........................................................................57 5.2.1 Population Analysis .......................................................................................57 5.2.2 Density of States ............................................................................................58 5 5.2.2.1 Mulliken Charges....................................................................................58 Chapter 6 - Perovskites..............................................................................................61 6.1 Structure and Distortions ......................................................................................61 6.2 Properties of Perovskites.......................................................................................67 6.2.1 Dielectric Properties.......................................................................................67 6.2.2 Ferroelectricity, Ferromagnetism and Multifeorroicity .................................68 6.2.3 Semi-Conductivity, Conductivity and High T Superconductivity.................69 Chapter 7 - AA3’B4Z12 Type Perovskites .................................................................71 7.1 Cu3CaTi4O12 (CCTO) ...........................................................................................71 7.1.1 Computational Study of Bulk CCTO.............................................................75 7.1.1.1 Computational Details.............................................................................76 7.1.1.2 Geometry Optimisations for the FM and AFM phases of bulk CCTO...78 7.1.1.3 Structural Properties................................................................................81 7.1.1.4 Electronic Properties ...............................................................................89 7.1.1.5 Conclusions on bulk CCTO ..................................................................100 7.1.2 Surface Calculations ....................................................................................101 7.1.2.1 Surface Relaxation ................................................................................106 7.1.2.2 TiO2 Termination of the 001 Surface....................................................110 7.1.2.3 Effect of Lattice Constant on Surface Electronic Properties ................113 7.1.2.4 Conclusions on the CCTO 001 surface.................................................115 7.2 CdCu3Ti4O12 (CdCTO) .......................................................................................116 7.2.1 Bulk CdCTO ................................................................................................116 7.2.1.1 Structural Properties..............................................................................118 7.2.1.2 Electronic Properties .............................................................................121 7.2.2 Surface Calculations ....................................................................................125
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