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Ab-Initio Studies Into Intrinsic Piezoelectric Properties

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Ab-Initio Studies Into Intrinsic Piezoelectric Properties University of Leeds School of Chemical and Process Engineering Ab-Initio Studies into Intrinsic Piezoelectric Properties Supervisors: Author: Dr. Andrew Scott Joseph Anthony Hooper and Prof. Andrew Bell Submitted in accordance with the requirements for the degree of Doctoral Training in Advanced Particulate Materials September 2018 Intellectual Property and Declaration The candidate confirms that the work submitted is his own and that appro- priate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright mate- rial and that no quotation from the thesis may be published without proper acknowledgement. The right of Joseph Anthony Hooper to be identified as author of this work has been asserted by Joseph Anthony Hooper in accordance with the Copyright, Designs and Patents Act 1988. iii \If you feel yourself hitting up against your limit, remember for what cause you clench your fists. Remember why you started down this path, and let that memory carry you beyond your limit." - All Might v Abstract We formulate a method of understanding the intrinsic piezoelectric properties of a perovskite system on an atomistic level using a density functional theory (DFT) methodology implemented into the electron density code CASTEP. First we consider the basic unary perovskites; barium titanate, lead titanate and potassium niobate. Geometry optimisation, elastic compliance, linear re- sponse, and simulated strain calculations are performed and the polarisation and piezoelectric coefficients are calculated. We then define the partial piezo- k electric coefficient (δij) and demonstrate a way to generate the electron density shift, two novel methods in the understanding of intrinsic piezoelectric prop- erties. We then study the binary piezoelectric lead titanate, performing the same calculations and analysing the electron density shift and partial piezoelectric coefficient in order to identify new features of binary systems, the equalisa- tion between basic perovskite units and the \sawtooth" bonding asymmetry between the different B-site ions. Then the feasibility of these calculations is evaluated for bismuth ferrite, a material showing multiferroic properties. The conversion of non-orthonormal lattice axes to a cartesian coordinate system is addressed. We discuss the phonon and electric field calculations, and evaluate what is and is not possi- ble. We find that structural and electron calculations are possible and report the optimised geometry, the elastic compliance, total electron density and spin maps, and the electron density shifts. We identify the symmetry, non-locality, and rotational modes in rhombohedral bismuth ferrite and suggest future re- search based on these properties. vii Acknowledgements I am incredibly grateful to my supervisors Andy Bell and Andrew Scott for their help and support over many years, for all the discussions, introductions, knowledge, understanding, tears, and laughs. Without them this would never have been possible. I would also like to thank the other members of the Ferroelectrics and DFT groups for their advice and support. In ferroelectrics; Steven, Laura, Tom, Sam, Tim, Tim, and Anton have been incredibly helpful in finding data and understanding the subject materials. In DFT; the knowledge and expertise of Trevor and Che helped me understand not only how computer modelling works, but why it is essential. Many thanks to the CASTEP Development Group and the community mailing list for continued assistance and perseverance despite some very silly questions. Many thanks to the ARC help team for their assistance and perseverance despite some catastrophically silly questions. Mishti Oberoi has provided me with a great drive to improve throughout my research. Thank you, I love you. My family have supported and loved me unconditionally throughout this time. Their love has made this possible. And to my friends, too many to name who have helped in small ways, large ways, and all the ways in between. I owe you all. Thank you. ix Contents Declaration iii Abstract vii Acknowledgements ix Contentsx List of Figures xvi List of Tables xxiv Nomenclature xxxi 1 Introduction1 1.1 Introduction to Piezoelectrics...................1 1.1.1 What is a Piezoelectric?..................1 1.1.2 Applications of Piezoelectrics...............2 1.1.3 Lead-Free Piezoelectrics..................5 1.2 Introduction to Computing.....................6 1.2.1 Advances in Computational Power............6 1.2.2 Introduction to Scientific Computation..........8 1.3 Overview of the Project...................... 10 2 Scientific Information 11 2.1 The Science of Piezoelectrics.................... 11 2.1.1 The Direct and Inverse Piezoelectric Effect........ 11 2.1.1.1 Piezoelectric or Ferroelectric?.......... 15 2.1.2 The Perovskite Group................... 15 2.1.3 Intrinsic Piezoelectric Properties.............. 17 xi Contents xii 2.1.3.1 Piezoelectric Charge Constant.......... 17 2.1.3.2 Elastic Compliance................ 19 2.1.3.3 Dielectric Permittivity.............. 20 2.1.3.4 Electromechanical Coupling Coefficient..... 22 2.1.4 Relations Between Piezoelectric Constants........ 23 2.1.5 Barium Titanate as a Sample Material.......... 24 2.2 Density Functional Theory..................... 26 2.2.1 Ab-Initio Modelling Using DFT.............. 26 2.2.2 The Schr¨odingerEquation and Electron Density..... 28 2.2.2.1 Schr¨odingerQuantum Mechanics........ 28 2.2.2.2 Born-Oppenheimer Approximation....... 30 2.2.2.3 What is a functional?.............. 31 2.2.2.4 Electron Density, Hohenberg-Kohn Theorems, and Kohn-Sham Equations........... 31 2.2.3 Modern DFT and CASTEP................ 35 2.2.3.1 A CASTEP Calculation............. 37 2.2.3.2 Bloch's Theorem................. 38 2.2.3.3 The Exchange-Correlation Functional...... 39 2.2.3.4 Pseudopotentials................. 42 2.2.3.5 Plane Wave Energy Cut-Off........... 46 2.2.3.6 Monkhorst-Pack Grid.............. 48 2.2.3.7 Phonons...................... 51 2.2.3.8 Effective Charges................. 52 3 Literature Review 55 3.1 The Wu-Cohen Functional..................... 55 3.2 Calculating Properties of Piezoelectrics Using DFT....... 59 3.2.1 Structural Properties of Piezoelectric Perovskites.... 59 3.2.2 Band Structure Calculations of Barium Titanate..... 63 3.2.3 Vibrational Properties of Barium Titanate........ 66 3.2.4 Piezoelectric Properties of Barium Titanate....... 71 3.3 Sample Structures Review..................... 72 3.3.1 Lead Titanate (PTO).................... 73 3.3.1.1 First Principles Calculations of Lead Titanate. 75 3.3.2 Potassium Niobate (KNO)................. 78 3.3.2.1 First Principles Calculations of Potassium Nio- bate........................ 79 3.3.3 Lead Zirconate Titanate (PZT).............. 82 3.3.3.1 First Principles Calculations of Lead Zirconate Titanate...................... 85 Contents xiii 3.3.4 Bismuth Ferrite (BFO)................... 87 3.3.4.1 First Principles Calculations of Bismuth Ferrite 90 3.4 Summary.............................. 93 3.5 Research Aims........................... 94 4 Computational Equipment and Techniques 95 4.1 The CASTEP Code......................... 95 4.1.1 Local Implementation................... 95 4.1.2 HPC Implementation.................... 96 4.2 ARC HPC Facilities........................ 96 4.2.1 ARC1............................ 98 4.2.2 ARC2............................ 100 4.2.3 ARC3............................ 101 4.2.4 Hybrid Parallelism, Many Core Architecture, and Effec- tive Utilisation of New Technology............ 102 4.3 Methods and Algorithms Used in CASTEP Calculations.... 109 4.3.1 General Methods...................... 109 4.3.2 Geometry Optimisation Methods............. 110 4.3.3 Phonon Methods...................... 111 5 Piezoelectric Properties of Unary Perovskites 113 5.1 Introduction to Research on Unary Perovskites.......... 113 5.2 Performing the Calculations.................... 115 5.2.1 Ground State Calculations................. 115 5.2.2 Sample Structures..................... 119 5.2.3 Convergence Parameters.................. 120 5.2.3.1 Kinetic Energy Cut-Off............. 122 5.2.3.2 MP-Grid..................... 123 5.2.4 Structural Optimisation.................. 126 5.2.5 Linear Response Phonon Calculation........... 129 5.2.6 Dielectric Permittivity................... 131 5.2.7 Elastic Constants...................... 134 5.2.8 Strain Behaviour...................... 135 5.3 Analysis of Results......................... 138 5.3.1 A note on the calculation of d31 .............. 139 5.3.2 Derivation of an Equation for the Calculation of Electric Polarisation......................... 139 5.3.3 Spontaneous Polarisations of Sample Structures..... 141 5.3.4 Piezoelectric Coefficients of Sample Structures...... 142 5.3.5 Electron Density Mapping................. 145 Contents xiv 5.3.5.1 A Note on Software Compatibility....... 148 5.3.6 Electron Density Shift................... 149 5.3.7 The Partial Piezoelectric Coefficient............ 159 5.4 Conclusions............................. 162 6 Piezoelectric Properties of Binary Perovskites: Lead Zirconate Titanate 165 6.1 Introduction to the Sample Material............... 165 6.1.1 Building the Unit Cell................... 168 6.2 Performing the Calculations.................... 169 6.2.1 Convergence......................... 169 6.2.2 Unit Cell Geometry..................... 171 6.2.3 Linear Response....................... 176 6.2.4 Elastic Constants...................... 180 6.2.5 Piezoelectric Perturbations................
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