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Supercell Electrostatics of Charged Defects in Periodic Density Functional Theory

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Supercell Electrostatics of Charged Defects in Periodic Density Functional Theory Supercell electrostatics of charged defects in periodic density functional theory Thomas Robert Durrant A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. Department of Physics and Astronomy University College London Gower street, London, WC1E 6BT October 21, 2019 I, Thomas Robert Durrant, 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 work. 2 Abstract Charged defects are often studied using density functional theory (DFT). However, the use of periodic boundary conditions (PBC) introduces a fictitious jellium back- ground required to maintain charge neutrality, along with periodic images of the defect. As a consequence, the calculations are found to be sensitive to the size of the supercell used. Several competing correction methods have been developed for bulk materials, but do not provide a completely satisfactory solution to this problem. One source of error in these corrections is suggested to be the simple density models employed. I propose a new electrostatic correction based on calculating electronic charge density differences between reference DFT calculations, and then solving Poisson equation for these charge models. For bulk materials, I demon- strate that this new correction is in agreement with the existing Lany-Zunger (LZ) method. This demonstrates that using more realistic density models is not sufficient to improve the corrections, but the new method does allow an improved understand- ing of potential alignment. I found that alignment errors are introduced by adding or removing atoms from the supercell to form the defect, and that these errors are related to their atomic radius. Secondly, the charge models I constructed are found to be in good agreement with the predictions of classical electrostatics, correspond- ing to both bound and free charges. I derive a new analytic charge correction based on this model, but it produces smaller corrections than anticipated. Further calculations demonstrate that the remaining unexplained finite-size er- ror is introduced by the finite-size dependence of the exchange-correlation energy of the screening bound charge, rather than on electrostatic grounds. Improved em- pirical corrections are constructed, which display differences between unrelaxed 3 Abstract and relaxed defects. The developed arguments are extended to surface slab models, where the observed errors are found to be dependent on the shape of the supercell employed. 4 Impact Statement Within academia, this thesis deals with several topics of fundamental interest. Firstly, the interplay between DFT and classical electrostatics enables an enhanced understanding of dielectric screening in real materials. The results of this thesis demonstrate how classical dielectric screening is achieved quantum mechanically. The section on Koopmans’ theorem (KT) contributes to the growing interest in this area. It seems clear that KT can be leveraged to develop improved hybrid functionals for materials, with better treatment of self interaction errors. However, the accuracy of these methods is damaged if the finite-size errors discussed in this thesis are not understood and corrected. The results of this thesis demonstrate that exchange-correlation plays an important role in the observed finite-size errors, and underlines the requirement that these two problems must be tackled consistently. Additionally, there is a significant amount of interest in extending our under- standing of charged defects to surfaces and 2D materials. It is shown that bound depletion charges are expelled to surfaces in these cases. Methods are developed in order to validate calculated formation energies for these kinds of system, and it is demonstrated that consistent charged defect formation energies can be calculated and compared between bulk and surface calculations. These methods can be devel- oped further in the future in order to understand in more detail the role surfaces may play in stabilizing the formation of charged defects. Outside of academia, charged defects play an important role in many techno- logical applications, including radiation damage, microelectronics, photovoltaics, catalysts and sensors. As the behavior of many materials in their desired appli- cation can be strongly effected by charged defects, understanding their behavior is 5 Impact Statement vital in order to bring new materials to market. First principles simulation allows in- sight into defect properties, but the need to use large supercells in order to produce reliable results can greatly increase the cost of such investigations. The methods developed in this thesis allow smaller supercells to be used reliably, increasing the availability of these calculations to industry. 6 Acknowledgements The work presented in this PhD thesis would not have been possible without the contributions of many people. I am especially indebted to my supervisor Prof. Alex Shluger for his constant encouragement and guidance. His enduring support on both a professional and personal level has been key to the completion of my thesis. I am grateful to group members past and present for the help they have given me. Mathew Watkins, Samuel Murphy, Jack Strand, Julian Gaberle, David Gao, Al-moatasem El-Sayed and Oliver Dicks deserve particular mention. I would like to thank Phil Hasnip for his help with Castep, in particular for his help in understanding the non-Coulombic energy. Without this key intervention, this project would have gone poorly. I am additionally grateful to Sanliang Ling for his help with cp2k, where his practical guidance has been invaluable. I would like to acknowledge both the EPSRC and the London Centre for Nan- otechnology for their financial support. Additionally, the ARCHER national super- computing service played a key role in enabling this research. I would also like to thank my friends for keeping my spirits high whilst I was finishing my writing, especially Doug and Esther Hogg, Andrew Crawley, Peter Mansfield, Rachel Haines and Paul Lee. All now know more about crystallographic defects than they where previously aware they wanted. Finally, I would like to thank my parents and my brother Jack for their constant help and encouragement over the course of my project. Without them I would not be where I am today. 7 Contents 0.1 List of publications . 33 0.2 Mathematical notation . 33 0.3 Key terminology . 33 1 Introduction 35 1.1 Charged point defects . 35 1.2 Thesis outline . 39 2 Established theory 42 2.1 Many-body quantum mechanics . 42 2.2 Density functional theory . 43 2.2.1 Total energy in a planewave basis . 44 2.2.1.1 Electrostatic terms and charged systems . 47 2.2.2 Planewave basis . 49 2.2.3 Brillouin zone sampling . 49 2.2.4 Gaussian planewave basis . 50 2.2.5 Exchange-Correlation functionals . 51 2.2.5.1 LDA functional . 51 2.2.5.2 PBE functional . 51 2.2.5.3 PBEsol functional . 52 2.2.5.4 PBE0 functional . 52 2.2.5.5 Auxiliary density matrix method . 53 2.2.5.6 PBE0 TC LRC functional . 54 2.2.6 Koopmans’ theorem . 55 8 Contents 2.3 Density functional perturbation theory . 56 2.4 Supercell defect formation energies . 56 2.4.1 Known finite-size scaling rules . 57 3 Defect electrostatics in the bulk 59 3.1 Electrostatic corrections . 59 3.1.1 The method of Makov and Payne . 59 3.1.1.1 Energy of a point charge . 61 3.1.1.2 Original derivation of the MP method . 61 3.1.2 The method of Dabo, Kozinsky, Singh-Miller and Marzari . 63 3.1.3 The method of Lany and Zunger . 64 3.1.4 The method of Freysoldt, Neugebauer and Van de Walle . 66 3.1.5 The method of Taylor and Bruneval . 67 3.1.6 The method of Murphy and Hine . 68 3.1.7 The method of Kumagai and Oba . 69 3.1.8 Discussion of present bulk methods . 69 3.1.8.1 Usage of present methods . 70 4 Development of the DEFECT SOLVER code 71 4.1 Electrostatic corrections . 71 4.1.1 Definition of electrostatic corrections . 71 4.1.2 Extension to dielectric materials . 72 4.1.3 Electrostatic libraries: PSPFFT ................ 74 4.1.4 Electrostatic libraries: DL MG ................ 75 4.2 Description of main subroutines . 75 4.2.1 get hartree isolated . 75 4.2.2 get hartree periodic . 76 4.2.3 get hartree isolated super . 77 5 Jellium and potential alignment 80 5.1 The jellium background and DFT . 80 5.2 Potential alignment and its causes . 83 9 Contents 5.2.1 Calculating the potential alignment in practice . 83 5.2.2 Model system: Gaussian charge in a box . 85 5.2.3 A new decomposition of potential alignment . 90 5.2.3.1 The DVAB component . 90 5.2.3.2 The DVBC component . 92 5.2.3.3 The DVCD component . 95 5.2.3.4 Discussion of all components . 96 5.2.4 Finite-size scaling of the DVAB component . 98 5.2.5 Model system: classical structureless atom . 102 5.2.6 Environment dependence of potential alignment . 103 6 Application of methods in the bulk 108 6.1 Overview of bulk calculations . 108 6.2 Case study: Vacancies in bulk diamond . 110 6.3 Case study: Vacancies in bulk MgO . 114 6.4 Case study: Vacancies in bulk SnO2 ................. 115 6.5 Figures of merit . 117 7 Structure of the bound charges 120 7.1 Classical electrostatics . 121 7.1.1 Electrostatics in vacuum . 121 7.1.2 Electrostatics within a dielectric . 122 7.1.3 The multipole expansion . 122 7.1.4 The polarization density . 123 7.1.5 Ideal problem: Point charge in an infinite dielectric . 125 7.1.6 Ideal problem: Point charge in a dielectric sphere . 126 7.2 Modeling of the defect charge .
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