QUANTUM-MECHANICAL AB-INITIO CALCULATIONS OF THE PROPERTIES OF WURTZITE ZNO AND ITS NATIVE OXYGEN POINT DEFECTS Aneer Lamichhane A Thesis Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE August 2018 Committee: Alexey Zayak, Advisor Lewis Fulcher Marco Nardone Copyright c August 2018 Aneer Lamichhane All rights reserved iii ABSTRACT Alexey Zayak, Advisor ZnO is an abundant wide band gap semiconductor with promising applications in optoelec- tronic technologies. Electronic and optical properties of this material depend critically on the physics of various defects which are very common in ZnO. Controlling those defects is the key to the development of ZnO-based applications, which is still a challenging process. This master thesis work is primarily concerned in studying the pristine ZnO and its native oxygen point de- fects. The objective is to study, investigate, measure and correlate the electronic, vibrational and thermal properties of the pristine ZnO and its native oxygen point defects, along with drawing nec- essary inferences for creating substantial theories. Further, the mode of study is the first-principles calculations performed with density functional theory, implemented in the VASP code using the GGA-PBE and LDA+U as functionals. A short discussion of these calculations will be given. At last, we perform a comparative study with these functionals in their application to compute the electronic, vibrational and thermal properties of the pristine ZnO and its native oxygen point defects. iv Dedicated to all the flora and the fauna ! v ACKNOWLEDGMENTS At first, I would like to thank my supervisor Dr. Alexey Zayak for providing me an opportu- nity for being part in his research group. This thesis would not have been completed without his guidance, advice and benevolence. The lessons that I have learnt in his computational laboratory will definitely boost my scientific career. Likewise, I would like to thank my former and current research team members- Sajjad Afroosheh, Keshab Bashyal, and Chunesh Bhalla. I would like to express special thanks to Christopher K. Pyles and Shiva Prasad Bhusal for sharing their computer programming skills. I would like to acknowledge Dr. Lewis Fulcher for teaching valuable lessons of classical me- chanics and electrodynamics. I would also like to acknowledge Dr. Marco Nardone for teaching statistical mechanics. Besides solid state physics, these disciplines are the key theoretical elements of my thesis. Further, I would like to thank them for being my committee members. Moreover, I would like to thank Department of Physics and Astronomy for supporting me during my M.S. study. I would also like to thank Kimberly Spallinger for helping me to develop skills of technical English writings. At last, my deepest gratitude goes to my father, mother, brother and wife. vi TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION . 1 1.1 Elementary Survey of Computational Material Science . 1 1.2 Material of Study- Wurtzite ZnO and its Native Oxygen Point Defects . 4 1.3 Some Theoretical Background . 10 1.4 A Brief Overview on the Objectives of the Work . 17 CHAPTER 2 FUNDAMENTALS OF THE DENSITY FUNCTIONAL THEORY . 19 2.1 Electronic Structure Calculations . 19 2.2 Quantum Many-Body Problem and Born-Oppenheimer (BO) Approximation . 22 2.3 Thomas-Fermi-Dirac Theory . 23 2.4 The Hohenberg-Kohn (HK) Theorem . 25 2.5 The Kohn-Sham (KS) Ansatz . 27 2.6 Approximations to Universal Functional . 30 CHAPTER 3 A STUDY OF GGA, GGA + US AND LDA + U CALCULATIONS IN ZNO 33 3.1 Introduction . 33 3.2 Results . 35 3.3 Discussion . 37 3.4 Conclusion . 42 CHAPTER 4 COMPUTATIONAL STUDY OF ZNO AND ITS NATIVE OXYGEN POINT DEFECTS -I . 43 4.1 Introduction . 43 4.2 Effects of Defects on the Electronic Structure of ZnO . 44 4.3 Effects of Defects on the Vibrational Properties of ZnO . 52 vii 4.4 Conclusion . 60 CHAPTER 5 COMPUTATIONAL STUDY OF ZNO AND ITS NATIVE OXYGEN POINT DEFECTS -II . 61 5.1 Introduction . 61 5.2 Results of LDA+U approximations on the Electronic Structure . 63 5.3 Results of LDA+U approximations on the Vibrational Properties . 65 5.4 Conclusion . 68 CHAPTER 6 SUMMARY AND CONCLUSION . 69 BIBLIOGRAPHY . 72 APPENDIX A NORMAL MODES OF FREQUENCIES USING GGA . 77 APPENDIX B DATA OF THERMAL PROPERTIES USING GGA . 83 APPENDIX C NORMAL MODES OF FREQUENCIES USING LDA+U . 87 APPENDIX D DATA OF THERMAL PROPERTIES USING LDA+U . 93 viii LIST OF FIGURES Figure Page 1.1 The hexagonal unit cell of ZnO where the black atom is Zn and the red atom is O . 6 +2 1.2 A systematic diagram of formation of the Two ZnO (VO ).............. 8 + 1.3 A systematic diagram of formation of the One ZnO (VO )............... 8 0 1.4 A systematic diagram of formation of the Zero ZnO (VO)............... 9 0 + 1.5 The computed structures of- from left Zero ZnO (VO), One ZnO (VO ) and Two +2 ZnO (VO ) respectively, showing oxygen point defects. 10 1.6 The plot between ! versus k using equation (1:3:6), where M = 2m. 12 1.7 The dispersion curves of ZnO where the frequency is measured at T Hz. 13 0 + +2 1.8 The phonon density of states of Pristine, Zero (VO), One (VO ) and Two ZnO (VO ). 17 3.1 The crystal structures of ZnO used in the calculations where the gray ball is Zn and the red ball is O. 35 3.2 The schematic cartoons of different types of modes at Γ point where the arrow indicates the direction of vibration. 36 3.3 Comparison among the phonon dispersion curves of ZnO computed using the func- tionals GGA, GGA + Us (PAW 12 and PAW 20) and LDA + U . 37 3.4 The family of bell curves at different modes. 39 4.1 The crystal structure of the unit cell of ZnO (Zn as grey ball and O as red ball) used in the computation. 44 4.2 The crystal structure of 3×2×2 Pristine ZnO used in this computation. 45 4.3 The computed electronic band structures showing trapped electron states. 45 0 4.4 The state in Zero ZnO (VO) being windowed in the energy interval by two dotted lines. 46 0 4.5 The observed electron cloud in yellow color of the windowed Zero ZnO (VO) state. 47 ix + 4.6 The state in One ZnO (VO ) being windowed in the energy interval by two dotted lines. 47 + 4.7 The observed electron cloud in yellow color of the windowed One ZnO (VO ) state. 48 +2 4.8 The state in Two ZnO (VO ) being windowed in the energy interval by two dotted lines. 48 +2 4.9 No observed electron cloud of the windowed Two ZnO (VO ) state. 49 4.10 The electronic density of states of Pristine ZnO . 50 0 4.11 The electronic density of states of Zero ZnO (VO) . 50 + 4.12 The electronic density of states of One ZnO (VO ) . 51 +2 4.13 The electronic density of states of Two ZnO (VO ) . 51 0 + 4.14 Dispersion curves for Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and Two ZnO +2 (VO ) ......................................... 54 0 4.15 The variation of density of states of Zero ZnO (VO) with respect to Pristine ZnO . 55 + 4.16 The variation of density of states of One ZnO (VO ) with respect to Pristine ZnO . 55 +2 4.17 The variation of density of states of Two ZnO (VO ) with respect to Pristine ZnO . 56 4.18 The variations of the heat capacities with respect to Pristine ZnO. 57 4.19 Debye temperature of Pristine ZnO . 58 0 4.20 Debye temperature of Zero ZnO (VO) ........................ 59 + 4.21 Debye temperature of One ZnO (VO ) ........................ 59 +2 4.22 Debye temperature of Two ZnO (VO ) ........................ 60 5.1 The total density of state of Pristine ZnO using LDA+U. 62 5.2 The total density of state of Zero ZnO . 63 5.3 The total density of state of One ZnO . 64 5.4 The total density of state of Two ZnO . 64 0 + 5.5 Dispersion curves for Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and Two ZnO +2 (VO ) using LDA+U . 66 5.6 Debye temperature of Pristine ZnO . 67 x 0 5.7 Debye temperature of Zero ZnO (VO) ........................ 67 + 5.8 Debye temperature of One ZnO (VO ) ........................ 67 +2 5.9 Debye temperature of Two ZnO (VO ) ....................... 68 xi LIST OF TABLES Table Page 1.1 The compilation of various physical parameter of ZnO. 7 2.1 The mathematical approaches for many-body problems. 30 2.2 Jacob’s ladder for the XC energy approximations. 31 3.1 Phonon frequencies (cm−1) of ZnO at Brillouin Zone center Γ, calculated with GGA, GGA+U PAW12, GGA+U PAW20 and LDA+U . 36 3.2 The former data of ZnO frequencies (cm−1) calculated by various researchers. 38 3.3 Continuation of Table 4 . 38 3.4 Comparison of Phonon frequencies (cm−1) of ZnO at Brillouin Zone center Γ with the INS data . 40 3.5 Percentage error calculated with respect to the INS values. The + and − signs indicate overestimation and underestimation respectively. 40 4.1 The measurement of the phonon band gap in T Hz. 53 4.2 The enhancement of V DOS at low frequency due to defects. 56 4.3 The measurements of the heat capacities (J/K/mol) at the low temperatures. 57 5.1 The comparative measurements between GGA and LDA+U approximations . 68 xii LIST OF SYMBOLS AND ABBREVIATIONS BO Born-Oppenheimer DFT Density functional theory GGA Generalized-gradient-approximation GGA+U Generalized-gradient-approximation with Hubbard correction HF Hartree-Fock HK Hornberg-Kohn INS Inelastic neutron scattering KS Kohn-Sham LDA Local density approximation LDA+U Local density approximation with Hubbard correction LO Longitudinal optical + One ZnO Oxygen point defect with one charged state, (VO ) PAW Projected augmented wave PBE Perdew, Burke and Ernzerhof Pristine ZnO Pure ZnO, without any form of defects TDOS Total density of states TF Thomas-Fermi TFD Thomas-Fermi-Dirac TO Transverse optical +2 Two ZnO Oxygen point defect with two charged state, (VO ) UV Ultraviolet UV LED Ultraviolet light emitting diode VASP Vienna Ab initio simulation package XC Exchange-correlation 0 Zero ZnO Oxygen point defect with zero charged state, (VO) 1 CHAPTER 1 INTRODUCTION 1.1 Elementary Survey of Computational Material Science The history of modern material science can trace back its root in crystallography.