Quantum-Mechanical Ab-Initio Calculations of the Properties of Wurtzite Zno and Its Native Oxygen Point Defects

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

Alexey Zayak, Advisor

ZnO is an abundant wide band gap semiconductor with promising applications in optoelectronic 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 defects. 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 necessary 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 mechanics 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 ω 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 VDOS 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. Many scholars give credit to mineralogists for finding that most crystals are anisotropic to light. After the discovery of X-rays in 1895 by Wilhelm Rontgen¨ [1], there was a significant paradigm shift in the field of crystallography when the traditional method of investigating the crystal was replaced by the X-ray diffraction technique. Further, the impact of X-ray diffraction techniques was revolutionary as well as ubiquitous on the overall discipline of science; it can be verified by a large number of Nobel prizes awarded to scientists for X-ray diffraction studied in the twentieth century. In the year 1912, Max Von Laue became the first person to perform an experiment on X-ray crystallography. Based on the work of Paul Ewald on, ”The optical properties of a medium consisting of a regular arrangement of isotropic resonators”, [2] Laue had shown that a crystal can be used as a three-dimensional diffraction grating for X-rays. Furthermore, his X-ray crystallography experiment was successful in determining crystal and molecular structures, thus, setting a benchmark that the crystal can be modeled as the regular three-dimensional periodic arrangement of atoms. Since Von Laue treated the X-rays as a wave, there were some serious issues in determining the correct phase of the diffracted X-rays, hence, hindering the determination of the atomic positions and opposing precise measurement of bond lengths and angles in materials. In the years 1912-13, two physicists- father and son- William Henry Bragg and Lawrence Bragg solved the ”phase issue” by treating X-rays as a particle [3]. The duo proposed the simple idea of reflection in the linear chain of the atoms in the crystal, thus propounded the famous Bragg’s Law- 2dsinθ = nλ, where θ is the glancing angle, d is the interplanar distances and λ is the wavelength of the incident radiation [4]. The significance of Bragg’s Law is so profound that it is not only used to calculate crystal structure and arrangement of atoms but also used to measure the internal parameters such as the lattice constant, the bond length, the bond angles and other quantities. Further, this law implies that 2 the properties of the materials are dependent on the relative juxtaposition of atoms within them, therefore, providing evidence for variation of the physical and chemical properties of different al- lotropic forms of an element. After the series of X-ray diffraction studies on various organic and inorganic materials, William Henry Bragg wrote in the year 1925,”The properties of the metals must depend, in the first place, on the properties of the individual atoms, and, in the second place, on the atomic arrangement, which is in effect the state of crystallization” in his famous book ”Con- cerning the Nature of Things”, [5]. Later, in the year 1927, Davisson and Germer performed the experiment on the diffraction of electrons by a single crystal and Thompson performed the same experiment with a crystalline film [6]. Both of these experiments were successful in demonstrating the periodic structure of the crystal and of the wave-particle duality of the electrons. Henceforth, all these diffraction experiments were fruitful in establishing at least one fact that the atoms are arranged in three-dimensional periodic arrays in the crystal. Moreover, the diffraction experiment is extremely accurate in determining the structure. In the year 1950-53, Rosalind Franklin discovered the double helical structure of DNA molecule using X-ray diffraction [7]. Before 1912, many intellectual scientists had studied the solid materials as densely packed atoms, which interact with each other with an interatomic bond. There are different types of interatomic bonds like ionic, covalent, van der Waals, hydrogen and metallic. The nature of this interatomic bond is responsible for the mechanical (e.g, hardness and elasticity), electrical, optical, thermal and magnetic properties of all solid materials. Historically, the study of heat capacity is an ideal topic to disclose the atomic arrangement in the solid materials. In the year 1819, two French physicists Pierre Louis Dulong and Alexis Ther` ese` Petit had shown experimentally that the molar heat capacity of a large number of elements was equal to a constant value of thrice the universal gas constant [8]. Nearly after 87 years, Albert Einstein gave a theory to explain the heat capacity by attributing it to the vibrations of the solid. His theory was successful in predicting the upper limit of the heat capacity as that found by Dulong and Petit and the lower limit to be zero which is in agreement with the third law of thermodynamics. However, the heat capacity varies exponentially instead of linearly at the low temperature which is the major drawback of Einstein’s 3 theory of heat capacity. Einstein has modelled his solid to be densely packed atoms and these were treated as three- dimensional harmonic oscillators. It should be noted that the arrangement of atoms in Einstein’s solid is not taken into account in the theory. Again in the historic year of 1912, Debye gave a more pragmatic theory of heat capacity of solid which not only predicted the upper limit and lower limit of the heat capacity but also matched with the experimental data that the heat capacity varies linearly at the low temperature [9]. Nevertheless, Debye heat capacity fails at intermediate temperatures. In contrast, with Einstein’s solid, Debye modelled his solid to be an isotropic continuous medium. Both of these theories are successful and have their own limitations- Einstein gave the importance to the properties of the atoms in the solid whereas Debye gave the importance to the arrangement of the atoms in the solid. Physicists have encountered the same issue of the properties versus arrangement of the atoms in the Sommerfeld model of metals. After the revolutionary advent of X-ray crystallography, there was an inevitable urgency to model the solid by incorporating the periodic arrangement of the atoms besides the estimation of their properties. Additionally, the quantum mechanical treatment of the solid serves as the best model only when the Schrodinger¨ equation contains a periodic potential. Moreover, it was evident through the experiment that the atoms in the solid crystallize in the periodic pattern at low temperature and pressure. Finally, in the year 1940, solid state physics emerged as the separate branch of physics [10]. The majority of this discipline is focused on the crystal structure of the solid where the atoms are arranged periodically. Based on this arrangement, solid is classified as crystalline, non-crystalline or amorphous. This branch of physics is so diverse and evolving as the periodicity facilitates mathematical modelling both in the classical-statistical regime as well as the quantum-spectroscopy regime. When the dissociation energy is greater than the interatomic energy, the material no longer remains in the solid state. Physicists have been studying liquid state, plasma and other complex structures along with the solid state which led them to construct a new branch of physics. Between the year 1970-80, physicists have established the condensed matter physics [11], which is actually the super set of the solid state physics. The condensed matter physics provides a framework for 4 describing and determining what happens to large groups of particles when they interact via pre- sumably well-known forces [12]. This field is so diverse that it interacts with other disciplines of sciences such as biology, particle physics, chemistry, engineering and metallurgy. As a result, the discipline condensed matter physics is widely described as material science, thus, making the latter an interdisciplinary subject. After the Second World War, the role of computer simulation in testing various mathematical models generate attention in various disciplines of science. The integration of computer simulation is very pronounced in material science for computing the properties as well as visualizing and mimicking the structure of the material. The innovation of different varieties of software and computing tools has swiftly augmented theoretical research in material science. Many complex structures of crystal can be computed with great accuracy with the aid of a supercomputer. This structure involves the many body Schrodinger¨ problem and the analytical solution of it is mostly a cumbersome process. In the year 1964, Hohenberg and Kohn introduced ”Density Functional Theory”, [13] which approximates many body Schrodinger¨ problem with the numerical solution. Between the year 1989 to 2004, the development of computer simulation package VASP was built and upgraded, which is widely used in modeling material in the atomic domain. In the early twenty-first century and then, this simulation package is used as an important tool for theoretical scientists to calculate the electronic, optical, vibrational, thermal and magnetic properties of the various materials. Likewise, many other powerful computational tools are evolving in this century. As a result, many research journals in the field of material science are published with the aid of these computational tools and thus, creating a new discipline called computational material science. Today, it is one of the prime branches of science.

1.2 Material of Study- Wurtzite ZnO and its Native Oxygen Point Defects

Zinc oxide (ZnO) is a challenging material to study for any beginner in material science. The principal reason is that it is a multifunctional material with multiple properties, many of which are unique and distinct. It is one of the most studied semiconductors and thus investigation on this material provides a solid foundation to study any other material in future. Being a wide band gap of 3.37 eV at room temperature, it is widely used in optoelectronic devices in the short wavelength 5 and UV portion of the electromagnetic spectrum. It is primarily used in constructing UV LED. ZnO usually crystallizes in two forms- wurtzite lattice and rocksalt lattice. The wurtzite lattice is the most stable structure at the room temperature and possesses a hexagonal structure (space group P 63mc) with a non-centrosymmetric (lacking a center of symmetry) crystal. This kind of crystallographic structure makes ZnO as a piezoelectric material. Further, ZnO yields a variety of other properties whenever it is doped with a suitable material. Traditionally, it is used as a medicine for skin diseases and many industries use ZnO to make the cosmetic products. Moreover, it is used in chemical industries to synthesize materials as well as used as a catalyst in chemical reactions. On the other hand, the rock salt lattice usually occurs at a high pressure in the mantle and upper crust region of the earth and is the material of interest for geologists. This thesis is primarily concerned with wurtzite ZnO and the further discussion of rocksalt lattice is beyond the scope of it. Moreover, it should be understood that ZnO represents wurtzite lattice in all the later discussion of this thesis. The wurtzite ZnO has a non-centrosymmetric crystal structure with polar surfaces. Its structure can be visualized as two interpenetrating hexagonal close packed sublattices of zinc cations and oxygen anions shifted by the distance of the cation and anion bond in the c direction. The hexagonal unit cell of ZnO is shown by Figure 1.1. Its lattice constant are a = 3.25A˚ and c = 5.20A˚ with c/a ratio to be 1.6A˚ and u parameter to be 0.375 in fractional coordinates. The u parameter is the distance of bond between neighbouring atoms parallel to c axis. The bond between zinc and oxygen ions is mostly ionic having a bond angle of 109.070. Each zinc atom in the crystal is surrounded by four oxygen atoms forming a tetrahedral coordination and vice versa. Thus, its structure can be explained as an alternating planes composed of tetrahedral integration of zinc and oxygen ions in alternate ways. ZnO (Zn : 1s22s22p63s23p63d104s2; O : 1s22s22p4) is an II-VI semiconductor having a large band gap of 3.37 eV at room temperature. Due to its high band gap, it is used as an electronic devices with high breakdown voltage and can operate in high temperatures and electric ﬁelds. Fur- ther, this wide band gap makes ZnO sensible only to ultraviolet light and has photoluminescence 6

Figure 1.1: The hexagonal unit cell of ZnO where the black atom is Zn and the red atom is O . in this region. This optical property makes ZnO suitable for an UV LED. Further, there are various techniques developed to tune the band gap of ZnO suitable for an electronic transistor. ZnO is an n-type semiconductor and ample beneﬁcial outputs can be generated from it by doping. For instance, ZnO becomes ferromagnetic at room temperature when it is doped with Fe, Co or Ni. This ferromagnetic character makes ZnO as a promising material for spintronics application. The physical parameters of ZnO at room temperature (300K) are shown in Table 1.1. However, in real life, there is always deviation in the crystal lattice structure. Besides ionicity and lattice stability, the major cause of such deviation in a lattice structure is the defect or the deviation of the periodicity of atomic arrangement in the crystal. These defects are either point defects like zinc antisites and oxygen vacancies or extended defects like dislocations which increase the lattice constant of ZnO. The principal cause of defect in the crystal is the thermal agitation which increases anharmonicity in the lattice vibration and to maximize the entropy, atoms/ions may be delocalized from their lattice points. We are particularly interested in native oxygen point defects in ZnO, and the discussion of other kinds of defects are beyond the scope of this thesis. A defect in a crystal is a thermodynamic necessity. The oxygen point defect in ZnO is the delocalization of an oxygen ion from its lattice point. We are interested in the intrinsic or native oxygen point defects. Whenever an oxygen ion leaves its lattice point, it creates holes of charge plus two. As a result, the four zinc ions dangle within the lattice point causing anharmonicity in the 7 Table 1.1: The compilation of various physical parameter of ZnO.

Physical Parameter Value Crystal structure Wurtzite Lattice constant a = 3.25 A˚ and c = 5.20 A˚ Molar mass 81.3 g/mol Density 5.606 g/cm3 Melting point 2248K Bond length 1.977 µm Energy gap 3.4 eV direct Refractive index 2.008, 2.029 Dielectric constant 0 = 8.75 and ∞ = 3.75 Exciton binding energy 60 meV Effective mass electron = 0.24 and hole = 0.59 Electron-hole mobility 200 cm2V 1s1 Ionicity 62% Thermal conductivity 60 W/mK Breakdown voltage 5 × 106 V cm−1 Heat capacity Cp = 9.6cal/molK Heat of crystallization 62 KJ/mol Youngs modulus E (bulk ZnO) 111.24.7 GP a Bulk modulus, B (Polycrystalline ZnO) 142.23 GP a Born effective charge 2.10

overall lattice vibration. These holes in the lattice point create an electric potential in the crystal. Since, ZnO is an n-type semiconductor, there is always the availability of the mobile electrons in the crystal. As a consequence, if this Coulomb’s potential attracts two electrons, then the lattice point becomes neutral or zero charged; if it attracts one electron, then the lattice point charge becomes plus one. Otherwise, the lattice point charge becomes two plus if there is no interaction between holes and electrons. Henceforth, the native oxygen point defect in ZnO is classiﬁed as three kinds: (i) Oxygen point defect with zero charged state. (ii) Oxygen point defect with plus one charged state. (iii) Oxygen point defect with plus two charged state. Figures 1.2, 1.3, 1.4 explain their formations. Among these three kinds of oxygen point defects, 8

+2 Figure 1.2: A systematic diagram of formation of the Two ZnO (VO ).

+ Figure 1.3: A systematic diagram of formation of the One ZnO (VO ). 9

0 Figure 1.4: A systematic diagram of formation of the Zero ZnO (VO).

zero charged and two charged states are stable. In the one charged state, the hole traps one unpaired electron making this state paramagnetic and unstable. Further, for the sake of brevity,

0 we now will onwards call- an oxygen point defect with a zero charged state as Zero ZnO (VO), an + oxygen point defect with a one plus charged state as One ZnO (VO ) and an oxygen point defect +2 with a two plus charged state as Two ZnO (VO ).

0 + There is a signiﬁcant difference in the structural geometries of Zero ZnO (VO), One ZnO (VO ) +2 +2 and Two ZnO (VO ). That is, the gap near the vacancy site of Two ZnO (VO ) is larger than

0 + that of Zero ZnO (VO) or One ZnO (VO ) . Our computed structures show that the peripheral

+2 0 Zn ions displace outward maximum in Two ZnO (VO ) if we compared it with Zero ZnO (VO). This is obvious as there are two holes in the oxygen vacancy point which repels the peripheral Zn

+ ions outwards. The repulsion force is less in case of One ZnO (VO ) and almost absent in Zero

0 0 ZnO (VO). The literature has mentioned that the displacement in Zero ZnO (VO) is inward by 12 + +2 % and the displacements in One ZnO (VO ) and Two ZnO (VO ) are outward by 2 % and 23 % 10 respectively [14]. Our results agree with that literature as indicated by Figure 1.5

0 + Figure 1.5: The computed structures of- from left Zero ZnO (VO), One ZnO (VO ) and Two ZnO +2 (VO ) respectively, showing oxygen point defects.

1.3 Some Theoretical Background

This section provides some useful concepts on the lattice dynamics, phonon dispersion curves, phonon density of states and electronic density of states. The chapter 3 is entirely based on the concepts of lattice dynamics and measurement of phonon dispersion curves while the chapter 4 is based on the phonon density of states and electronic density of states. Thus, it is extremely important to include these subjects at the beginning of this thesis. Since, our material of study is ZnO, we begin this section by discussing with lattice dynamics of the diatomic crystal. Lattice Dynamics of the Diatomic Crystal: By the term, ”Diatomic”, we mean that the crystal constitutes of two kinds of atoms/ions. In our case, the crystal consists of a Zn ion and an O ion and the unit cell consists of two Zn ions and two O ions as shown in the Figure 1. Therefore, it is understood that this topic deals with the vibrations of ZnO crystal with two Zn ions and two O ions in the unit cell. We use simple model of the linear chain where Zn ions and O ions are

a lying alternatively and successively. Let the distance between a Zn ion and an O ion be 2 at rest, which implies that the distance between any two successive Zn ions and/or O ions is a. Further, we assume that the ionic force between Zn ion and O ion is simple harmonic. Let Un be the

th th displacement of the n Zn ion then on its left the displacement of the n O ion be un and on its 11 th right the displacement of the (n+1) O ion be un+1 as our model is that of linear chain. If J is the

th spring constant then the ionic force acting on the n Zn ion is: Fn = −J(2Un − un − un+1) and

th n O ion is: fn = −J(2un − Un − Un−1) [15]. But, using Newton’s II law, the ionic force acting

2 2 th ∂ Un th ∂ un on the n Zn ion is: Fn = M ∂t2 and on the n O ion is: fn = m ∂t2 where M and m are the mass of a Zn ion and an O ion respectively. Thus, we lead to second order differential equations:

∂2U M n = −J(2U − u − u ) (1.3.0.1) ∂t2 n n n+1

∂2u m n = −J(2u − U − U ) (1.3.0.2) ∂t2 n n n−1

which have the solutions in the form of sinusoidal waves as shown below:

˜ Un = Uexp(i[k(n + 1/2)a − ωt]) (1.3.0.3)

un =uexp ˜ (i[kna − ωt]) (1.3.0.4)

where, U˜ and u˜ are the amplitudes of the Zn ion and O ion respectively. After substituting the solutions into their respective differential equations, we get two equations which can be combined in matrix form as:

    2J − Mω2 −2Jcos(ka/2) U˜         = 0 (1.3.0.5) −2Jcos(ka/2) 2J − mω2 u˜

For non-trivial solution (U˜ 6= 0 and u˜ 6= 0 ), the determinant of the matrix has to be zero, i.e.,

2J − Mω2 −2Jcos(ka/2)

= 0 −2Jcos(ka/2) 2J − mω2

Once, we evaluate this determinant and after solving for ω2 we get 12

r 1 1 1 1 4sin2(ka/2) ω2 = J( + ) ± J ( + )2 − (1.3.0.6) M m M m Mm

The equation (1.3.6) is called the dispersion relation and the Figure 1.6 shows the plot of the

The phonon dispersion plot 0.3 Optical branch

0.25

0.2

0.15

0.1 Acoustic branch

0.05

0 -80 -60 -40 -20 0 20 40 60 80

Figure 1.6: The plot between ω versus k using equation (1.3.6), where M = 2m.

2π dispersion relation between ω and k. It is clear that the relation is periodic in k with a period k = a , −π π which implies that the behavior of this curve is completely described by the region a : k : a . This region is known as the Brillouin zone and represents the unit cell of the real crystal in reciprocal or phase space. Further, the equation (1.3.6) is invariant for time reversal as ω(k) = ω(−k). The branch for which ω = 0 when k = 0 is called the acoustic branch and ω 6= 0 when k = 0 is called the optical branch. The gap between acoustic branch and optical branch indicates the difference between the masses of two ions. The atoms/ions vibrate in phase yielding acoustic branch of

sound wave, i.e., ω = csk. For atoms vibrating not in phase yields optical branch and in an ionic crystal like ZnO, the relative motion between Zn ion and O ion induces electromagnetic wave of infrared frequency. In the case of 3 dimensions, the displacement can be in any of the three independent directions: one in the direction of wave vector (longitudinal) and two perpendicular 13 to the direction of the wave vector (transverse). Therefore, if there are s atoms in the crystal then the number of acoustic branch will be 3 and 3s-3 will be the number of optical branch where the total number of branches is the total number of the degree of freedom of atoms, 3s. The dispersion curves for ZnO unit cell containing two Zn ions and two O ions are shown in Figure 1.7. As the total number number of ions is 4 which gives 3 acoustic branches and 12-3=9 optical branches. Further, the Brillouin zone in 3 dimensional case has to be sampled in a certain symmetrical path which depends on the shape and size of its real crystal. For the hexagonal unit cell, the path in the Brillouin zone is Γ, K, M, Γ, A, where the Γ point is the center of the Brillouin zone.

Figure 1.7: The dispersion curves of ZnO where the frequency is measured at T Hz.

The Density of States: The density of states is an important concept in solid state physics. Many elementary properties of the materials are based on the density of states. In the chapter 4, we have used the notion of the density of states in calculating the electronic band gaps and heat

0 + +2 capacities of Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and Two ZnO (VO ). Therefore, it is important to include the concepts of electronic density of states and phonon density of states. (i) Density of States for Electrons: The electronic density of states is the central concept 14 on the band structure of solids. The materials can be divided in conductors, insulators and semiconductors based on the notion of density of states. In general, the electronic density of states D is deﬁned as the number of states (N) per unit energy (E) per unit volume (V ) of the real space [16]. Mathematically, 1 dN D = (1.3.0.7) V dE

At ﬁrst, we have to ﬁnd the number of states inside the sphere of radius k in the phase space. The

4 3 volume V of the sphere is V = 3 πk and the volume of the smallest unit cell in the phase space is 2π 3 given by Vs = ( L ) . Since, there can be two electronic states-spin up and spin down, the number of electronic states is given by

V k3L3 N = 2 = 2 (1.3.0.8) Vs 3π

Now, the dispersion relation of energy in the free electron gas model [17] is

2k2 E = ~ (1.3.0.9) 2m

which gives

(2Em)1/2 k = (1.3.0.10) ~ putting the value of k in equation (1.3.8) yields

L3(2m)3/2E3/2 N = (1.3.0.11) 3π2~3

Now, taking the derivative of equation (1.3.11) with respect to E and then substituting the resultant derivative in equation (1.3.7) yields

1 2m D = ( )3/2E1/2 (1.3.0.12) 2π2 ~2 15 which is the required expression of the density of states for electrons. The notion of the electronic density of states is extensively used in the chapter 4 for computing the band gaps of Pristine ZnO,

0 + +2 Zero ZnO (VO), One ZnO (VO ) and Two ZnO (VO ). (ii) Density of States for Phonons: The phonon density of states (D) is deﬁned as the number of modes (N) per unit frequency (ω) per unit volume (V) of the real space. Analogous to the electronic density of states, the density of states for phonons is given by

1 dN D = (1.3.0.13) V dω

1 dN dk or, D = (1.3.0.14) V dk dω

And, following the similar logic of equation (1.3.8), the number of modes in the phase space is given by

k3L3 N = (1.3.0.15) 6π2

differentiating equation (1.3.15) with respect to k yields

dN k2L3 = (1.3.0.16) dk 2π2

substituting equation (1.3.16) into equation (1.3.14) yields

k2 1 D = 2 (1.3.0.17) 2π vg

dω where, vg = dk is the group velocity. It is clear that the acoustic phonon only constitutes the density of states as the group velocity of the optical phonon is zero. And, the group velocity of the

acoustic phonon is the sound velocity cs. Using, ω = ck yields

3ω2 D = 2 3 (1.3.0.18) 2π cs 16 which is the required expression for the phonon density of states. The factor 3 implies that there are three acoustic modes per each k. In the chapter 4, we have used the notion of phonon density

0 + of states to calculate the Debye temperatures of Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and +2 Two ZnO (VO ). Figure 1.8 illustrates the phonon density of states of Pristine ZnO along with its native oxygen defects. The Debye temperature emerges from the Debye model of heat capacity which assumes that the solid crystal is nothing but a set of phonons in a box. And, the total energy of the phonons is given as

Z ωD 1 E = D(ω)V ω dω (1.3.0.19) ~ ~ω 0 exp( ) − 1 kB T where, D(ω)V is the number of modes per unit frequency in the entire space, ~ω is the energy of a phonon and 1 is the average number of phonons at each mode given by Bose-Einstein exp( ~ω )−1 kB T distribution [18]. The new variable ωD is the Debye frequency which is the largest frequency that can be excited at the given temperature. For the evaluation of the integral, the Debye frequency can be taken to be inﬁnity. However, for the correct representation of the heat capacity of a solid, it is better to measure ωD at a given temperature T . The heat capacity is merely the derivative of this total energy of the phonons with respect to the temperature. After evaluating the derivative of the equation (1.3.19), the heat capacity at higher temperature converges to the Dulong and Petit law and at lower temperature it is proportional to the cube of the temperature as shown below:

  3nkB, if, T >> TD Cv = 4  12π nkB T 3 3  ( ) , (CvαT ) otherwise.  5 TD

where, n is the total number of atoms which came into the above relation through the expression

6πn2c3 ω = ( s )1/3 (1.3.0.20) D V 17

Finally, TD is the Debye temperature, deﬁned as

~ωD TD = (1.3.0.21) kB

0 + +2 Figure 1.8: The phonon density of states of Pristine, Zero (VO), One (VO ) and Two ZnO (VO ).

1.4 A Brief Overview on the Objectives of the Work

The theoretical study of the material is immensely enhanced and influenced by the various simulation techniques developed as valid concepts, employed as perfect models and computed by powerful supercomputers. This thesis is about the study of ZnO and its native oxygen point defects, using the various simulation techniques based on the useful concept of, ”Density Functional Theory.” The perfect models of ZnO and its native oxygen point defects are created using various functionals and numerous calculations are run on the supercomputers to obtain the crude data. Fi- nally, these data are refined for possible investigation of the properties of our materials. We usually 18 tally the refined data with the experimental data if the similar kind of investigation has been done by the experimentalists. If not then these data are used to develop a hypothesis whose validity may be tested on the different but analogous materials in near future. This is the main objective of our work. The main objectives of this thesis are: (i) To study and understand the electronic, vibrational and thermal properties of ZnO and its native oxygen point defects.

(ii) To test the effectiveness of the simulations employed using the functionals GGA, GGA + US and LDA + U. (iii) To understand how the microscopic properties of the pristine material deviates with its intrinsic defect. (iv) To gain insight on the role played by the charged and uncharged lattice point on the overall chemistry of the material. (v) To study the correlation between atomic anomaly with the localization of the atomic vibration. (In the next chapter, we discuss thoroughly the density functional theory and its evolution. This chapter 2 is written in the aim of getting insight about the fundamentals of the density functional theory, which is crucial as our entire computations are based on it.) 19

CHAPTER 2 FUNDAMENTALS OF THE DENSITY FUNCTIONAL THEORY

2.1 Electronic Structure Calculations

Generally, electronic structure calculations are the measurement of the state of motion of the electrons around stationary nuclei. In terms of quantum theory, it is calculations of the wave functions of the electrons and their related energies. Almost every physical and chemical property of the material can be computed by the knowledge of its electronic structure. The knowledge of the electronic ground states give access to the properties like mechanical phenomenon of elasticity and stability, transport phenomenon of diffusivity, ionic conductivity and viscosity, electrical phenomena of polarizabilities and dielectric properties and lattice vibrations and several thermal phenomena, whereas the knowledge of excited electronic states give access to the properties like electronic conductivity and many optical phenomena. The former properties are called ground state properties and the latter are called excited state properties. Henceforth, the problem of calculating the electronic structure of the material is tremendously worthwhile and many scientists have been trying to purpose accurate models and theories after the role of the electron in material properties become clear. The calculations of the electronic structure are the major backbone problem of the material scientists. After the discovery of the electron in 1896-1897, the theory of electronic structure is the central issue for all the theoretical physicists. In the year 1897, J.J. Thomson not only discovered the electron at the Cavendish Laboratory in Cambridge [19] but also came up with the conclusion that the electron is a negatively charged particle. Based on this study, he proposed the ”plum pud- ding model” of the electronic structure which was soon modified by his student Ernest Rutherford. In 1910 Rutherford showed that the atom was made of a small positively charged nucleus and number of negatively charged electrons revolving around the nucleus making the atom electrically neutral [20]. His model was like the planetary model where the gravitational force is analogous to the electrostatic force. However, there were several experimental issues that are incompatible with 20 this model. Among them, the most important issue was the violation of Maxwell’s law which states that the accelerated charged particle radiates electromagnetic energy. An electron revolving around the nucleus should lose energy by emitting electromagnetic radiation and finally collapse into the nucleus making the atom unstable. In the year 1913, Neil Bohr tried to solve the contradiction of Rutherford model by simply stating that the physics of atomic scale could not be addressed by the classical mechanics and electromagnetism, but required a set of new laws which later became quantum mechanics [21]. Bohr postulated that the electron revolves around the nucleus only in specific orbit having specific energy and radius and that in such quantum orbits it does not emit radiation. This postulate was in accordance with the Planck’s theory of black body radiation and verification of the dependency of frequency absorbed/emitted by electrons with the difference in the energy levels of the orbits provide a robust foundation for the development of the quantum mechanics. Thus, it is always paradoxical that the search for the electronic structure creates quantum mechanics or the quantum mechanics is created for the search of electronic structure. Nevertheless, the electrons serve as the testing ground for the quantum mechanics. In the year 1926, Schrodinger¨ formulated the mathematical language of quantum mechanics which is the basis for the electronic structure calculation and it was soon applied to multi-electronic atoms (Heitler and London, 1927) and polyatomic systems (Bloch, 1928) [22]. With the quantum theory, many intrinsic properties of the electrons were discovered as well as tested experimentally. In 19300s, the band theory for independent electrons was formed [23] leading to the classification of the materials as conductors, insulators and semiconductors. The use of Schrodinger¨ equation in calculating the many-body systems is quite cumbersome and impractical in real life. For instance, in order to calculate the electronic configuration of

CO2, the Schrodinger¨ equation becomes a 66 dimensional problem (if we neglect the nuclear

dynamics) as the total number of electrons in CO2 is 22 and for three degree of spatial orientation

(neglecting spins), the conﬁguration is represented by the wave function ψ(r1, r2, ...., r66). Since, the presence of one electron in space inﬂuences the behaviour of the other electrons, the wave function cannot be expressed as the product of the wave functions of the individual electrons, i.e., 21

ψ(r1, r2, ...., r66) 6= ψ(r1) ∗ ψ(r2) ∗ ...... ∗ ψ(r66). Such difficulty in quantum mechanics is often described as the quantum many-body problem. Thus, the exact solution of Schrodinger¨ equation in the case of CO2 involves an equation in 66 degrees of freedom. Further, it is always complicated to understand the two body Coulomb interaction in this multiple number of electrons. As a result, the solution of Schrodinger¨ equation in quantum many-body problem must re- sort to some approximations. The first approximations to be used is Born Oppenheimer approximation which states that atoms have heavy nuclei and therefore can consider them as stationary sources of electrostatic potential. This approximation is vital to reduce the nuclei factor from the Hamiltonian function of the quantum multi-body system. After that, one may use either Hartree-Fock Method (HF) or Density Functional Theory (DFT). In HF method, one can treat the many body wave function as the combination of some standard base functions, i.e., ψ(1, 2, ..., N) = φ(1) ∗ φ(2) ∗ ...... ∗ φ(N) and can make the Hamiltonian function as separa- ble. Though this method does not treat the antisymmetric condition for electrons wave function, one can easily use the Slater determinant to incorporate the Pauli exclusion principle [24]. On the other hand, DFT approximates the many body wave function to electron probability density function which is only the function of spatial coordinates and thus reducing any N dimensional problem to a 3 dimensional problem. The mathematical details of this reduction of N dimensional problem to 3 dimensional function are shown in step-wise details subsequently in further sections. Density functional theory was proposed by Hohenberg-Kohn in 1960 [25] and due to the un- availability of the powerful computer between 1960-1990, the use of DFT in the calculation of electronic structure of the material is limited. However, after 1990, due to the advent of the powerful supercomputer, the DFT has revolutionized the field of material science by not only calculating the properties of the materials but also providing an alternate method of investigation instead of using traditional experimental ways. In the current time, DFT undoubtedly one of the techniques most used in computing electronic structures. The major benefits of DFT calculations can be summarized as follows: (i) DFT calculation is the first-principle or ab-initio calculation as it calculates the properties of the 22 material without using any adjustable parameters. (ii) DFT can obtain the structure of the materials beyond the capabilities of experiments. (iii) DFT predicts the properties on a microscopic scale and to such a depth which is currently inaccessible to experiments.

2.2 Quantum Many-Body Problem and Born-Oppenheimer (BO) Approximation

The Hamiltonian of a quantum many-body system consisting a set of atomic nuclei and electrons interacting via Coulombic electrostatic forces can be represented as:

where the indices I and J run on nuclei; M, R and Z represent the nuclear mass, position and atomic number of nucleus. Similarly, lower cases represent those properties of electrons. The terms belonging to the right hand side of the equations respectively are the kinetic energy of the nuclei, kinetic energy of the electrons, the potential energy of the nucleus-nucleus interaction, the potential energy of the electron-electron interaction and the potential energy of the electron- nucleus interaction.

If ψ(RI ; ri) is the total wave function of the system then everything about the system is known if we solve the time independent Schrodinger¨ equation, i.e.,

Hψ(RI ; ri) = Eψ(RI ; ri) (2.2.0.2)

However, this problem is difﬁcult to solve and impractical in real life and the cause of difﬁculty is already explained in the previous section of this chapter. The BO approximation treats this system as the electrons in the potential of the stationary nuclei by assuming nuclei are massive and slow in motion with respect to electrons. Thus, we can separate the total wave function of the system into the wave function of the nuclei (Π) and the wave function of the electrons in the potential of stationary nuclei (Ω), i.e., 23

ψ(RI ; ri) = Π(RI )Ω(RI ; ri) (2.2.0.3)

Thus, the Schrodinger¨ equation for electrons in the potential of stationary nuclei is

HeΩ(RI ; ri) = V (RI )Ω(RI ; ri) (2.2.0.4) where

And, the solution of equation (2.2.4) yields the energy eigenvalue V (RI ) which in turn can be used

to calculate the nuclear motion by integrating V (RI ) in the construction of Hamiltonian for nuclei. The process is illustrated as below:

n HnΠ(RI ) = E Π(RI ) (2.2.0.6)

where, H = −Σ ~ ∇2 + V (R ) n I RI I (2.2.0.7) 2MI

Hence, the importance of BO approximation lies in the separation of motion of electrons and nuclei. The picture of electrons dynamics in the static potential of the nuclei is the beginning point for DFT.

2.3 Thomas-Fermi-Dirac Theory

In the year 1927, the two physicists Thomas and Fermi independently used the electron density function instead of the wave function to determine the electronic structure of the many-body system. This is known as the Thomas-Fermi model [26]. The TF model is often regarded as a precursor to modern DFT. According to the TF model, the total energy of the electrons in a stationary potential (V (r)) of the nuclei is given as the functional of the electron density n(r). 24 Mathematically,

Z Z 1 ZZ n(r)n(r0) E[n(r)] = A n(r)(5/3)dr + n(r)V (r)dr + drdr0 (2.3.0.1) 1 2 |r − r0|

The ﬁrst term of the right hand side of the equation 2.3.1 is the kinetic energy of the noninteracting electrons in a homogeneous electron gas (HEG). It is merely the integration of the

(5/3) product of kinetic energy density (A1n(r) ) with the volume in the atomic units scale (~ = e =

4π 3 2 2/3 me = = 1) and A1 = (3π ) . The second term is the Coulomb interaction between nucleus 0 10 and electron and the last term is the Coulomb interaction between electrons as given by Hartree energy [27]. The TF model is incomplete as it does not incorporate the charge distribution related to the electron itself. Further, the model neglects the Pauli exclusion principle as there is no account of electron-electron interaction with respect to spins. Later in the year 1930, Dirac rectiﬁed

R (4/3) the model [28] by integrating a new term called local exchange term- A2 n(r) dr, where

3 3 1/3 A2 = − 4 ( π ) . Therefore, the total energy of the electrons in a stationary potential (V (r)) of the nuclei is given as

Z Z 1 ZZ n(r)n(r0) Z E[n(r)] = A n(r)(5/3)dr + n(r)V (r)dr + drdr0 + A n(r)(4/3)dr 1 2 |r − r0| 2 (2.3.0.2) Now, we seek such a distribution of electron density that would minimize the total energy E[n(r)] subject to the normalization conditions, which is, R n(r)dr = N and δ(E[n(r)] − µN) = 0, where µ is the Lagrange multiplier. After minimization with the normalization conditions, we 25 get

5 Z n(r0) 4 A n(r)(2/3) + V (r) + dr0 + A n(r)(1/3) − µ = 0 (2.3.0.3) 3 1 |r − r0| 3 2

The above equation (2.3.3) can be solved readily to obtain the ground state electron density. The TF model is the ﬁrst approach in density functional theory to use electron density instead of wave function and the equation (2.3.3) is lot simpler than the many-body Schrodinger¨ equation. However, this TF model is very crude and lacks the essential features to describe the shell structures of the atoms and bonding between atoms or molecules. Hence, this model cannot be used to describe electronic structures in matter.

2.4 The Hohenberg-Kohn (HK) Theorem

In the year 1964, Hohenberg and Kohn formulated and proved the theorem that forms the basis of DFT (please see [29], for detail study). The HK theorem shows that the energy of the system can be written exclusively in terms of the electronic density, thus revolutionizes the theory of electronic structure calculations by setting the measurable three dimensional spatial physical quantity, ”electron density”, as a basis instead of unmeasurable many dimensional non physical quantity, ”wave function”. The HK theorem is divided into two parts: Theorem I: The external potential is uniquely determined by the electronic density, besides the trivial additive constant.

Proof: We start the proof with the contradiction method. For that, we assume there exists two potentials V and V 0 for the same electronic density n and ﬁnally arrive at contradiction. Let ψ and ˆ E0 =< ψ | H | ψ > be the ground state wave function and ground state energy of the Hamiltonian

ˆ ˆ ˆ ˆ 0 0 0 ˆ0 0 H = T +V +Uee respectively. Similarly, let ψ and E0 =< ψ | H | ψ > be the ground state wave

ˆ0 ˆ ˆ0 ˆ function and ground state energy of the Hamiltonian H = T + V + Uee respectively. According to the variational principle 26

0 ˆ 0 E0 < hψ | H | ψ i

0 ˆ0 0 0 ˆ ˆ0 0 or, E0 = hψ | H | ψ i + hψ | H − H | ψ i

Z 0 0 ∴ E0 = E0 + ρ(r)[V (r) − V (r)]dr (2.4.0.1)

Similarly,

0 ˆ0 E0 < hψ | H | ψi

ˆ ˆ0 ˆ or, E0 = hψ | H | ψi + hψ | H − H | ψi

Z 0 0 ∴ E0 = E0 − ρ(r)[V (r) − V (r)]dr (2.4.0.2)

By adding equations (2.4.1) and (2.4.2), we get

0 0 E0 +E0 < E0 +E0, which is the contradiction. Thus, there exists an unique electronic density that determines the external potential. Hence, the ground state density determines all the properties of the system. Theorem II: The global minimization of the energy functional E[n(r)] = R n(r)V (r)dr + F [n(r)] by the exact

ground state density n0(r) is the exact ground state energy of the system if and only if F [n(r)] exists as a universal functional of the density, independent of the external potential V (r)..

Proof: For any wave function ψ, the energy functional E[ψ] can be written as:

E[ψ] = hψ | Hˆ | ψi (2.4.0.3)

This energy functional E[ψ] is the exact ground state energy of the system if and only if we can 27 minimize the wave function ψ in such a way that there is the conservation of the total number of particles. This is the variational principle and the required minimum wave function is often coined as, ”ground state” wave function ψ0 . According to HK theorem I, if ψ corresponds to a ground state with particle density n(r) and external potential V (r), then E[ψ] is a functional of n(r). By continuing from equation (2.4.3) E[ψ] = hψ|Hˆ |ψi

or, E[ψ] = E[n(r)] or, E[ψ] = R n(r)V (r)dr + F [n(r)]

or, E[ψ] > E[ψ0] R or, E[ψ] > n0(r)V0(r)dr + F [n0(r)]

∴ E[ψ] > E[n0(r)]

This implies that E[n(r)] = E[n0(r)] if and only if n(r) = n0(r); which means that by minimizing the total energy functional E[n(r)] with respect to the variation of electronic density n(r), one would ﬁnd the ground state energy of the system. Practically, this is still impossible as we do not know the universal functional F [n(r)]. It is often regarded that the universal functional is associated with the sum of the kinetic energy of the electrons and the interaction energy among

them i.e., F [n(r)] ≡ T [n(r)] + Uint[n(r)]. In the year 1965, Kohn-Sham proposed a methodology which overcome this difﬁculty of identifying the universal functional F [n(r)] [30], frequently known as Kohn-Sham (KS) Ansatz.

2.5 The Kohn-Sham (KS) Ansatz

First of all, the German term, ”Ansatz” means something like- educated guess for the unknown system. The Kohn-Sham (KS) Ansatz is therefore, a good guess in figuring out the exact ground state energy of the system in the real life. This ansatz applies the HK theorems into the practical use and simplifies the DFT calculations even in a personal computer. As a result, many electronic calculations are performed with the DFT, making it very popular in the investigation of the structure and properties of the matter. Walter Kohn was awarded with Nobel Prize for DFT in 1998. 28 The Kohn-Sham (KS) ansatz replaces the original many-body system of interacting particles with the fictitious system of noninteracting particles assuming that the both systems have exactly the same ground state density. It is a one to one correspondence between many body system of interacting particles with the real potential to the system of noninteracting particles having Kohn-

ˆ 1 2 Sham (KS) potential VKS. The Hamiltonian of this ﬁctitious system is HKS = − 2 ∇ + VKS, expressed in the atomic units. Now, a set of Schrodinger¨ like equations can be obtained with this Hamiltonian function

ˆ HKSψi(r) = εiψi(r) (2.5.0.1)

where, ψi(r) is the independent particle Kohn-Sham (KS) orbitals. Then, the Kohn-Sham (KS) orbitals are related to the electron density of N electrons by :

N X 2 n(r) = |ψi(r)| (2.5.0.2) i

The equation (2.5.2) is subject to the condition

Z n(r)dr = N (2.5.0.3)

Also, the non-interacting independent-particle kinetic energy TS[n(r)] is given by,

N 1 X Z T [n(r)] = ψ∗(r)∇2ψ (r)dr (2.5.0.4) S 2 i i i Finally, the universal functional F [n(r)] can be rewritten as

F [n(r)] = TS[n(r)] + EH [n(r)] + EXC [n(r)] (2.5.0.5)

where, EH [n(r)] is the Coulomb interaction between electrons(Hartree energy) as described in the

equation(2.3.1) and EXC [n(r)] is the XC energy. 29 Now, using the HK theorem II- the minimization of the energy functional E[n(r)] = R n(r)V (r)dr+ F [n(r)], subject to the constraint of conservation of the electrons number N can yield the ground state energy of the many-body system. Using the Lagrange minimization technique

Z Z δ{F [n(r)] + n(r)V (r)dr − λ( n(r)dr − N)} = 0 (2.5.0.6) where, λ is the Lagrange multiplier, given as

δF [n(r)] λ = + V (r) (2.5.0.7) δn(r)

Now, using the equation (2.5.5), we get

δT [n(r)] δE [n(r)] δE [n(r)] λ = S + H + XC + V (r) (2.5.0.8) δn(r) δn(r) δn(r)

δT [n(r)] or, λ = S + V (r) + V (r) + V (r) δn(r) H XC where, VH (r) and VXC (r) are the Hartree potential and XC potential respectively.

δT [n(r)] or, λ = S + V (r) (2.5.0.9) δn(r) KS

where, VKS(r) = VH (r)+VXC (r)+V (r) is the Kohn-Sham potential. Once we know VKS(r), we can calculate the Hamiltonian operator and use equation (2.5.1) for N times for the system having N electrons and thus ﬁnding the eigenstates and eigenvalues for each electron. In this way, the KS ansatz is a perfect tool in the theoretical sense. However, this tool is still appropriate in practice as VKS(r) is related to the ground state density via VXC (r). VXC (r) is still unknown as we do not explicitly know about EXC [n(r)]. In the next section, we discuss about various approximations of the quantity EXC [n(r)]. We have used Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) along with the Hubbard Corrections in this thesis. 30 2.6 Approximations to Universal Functional

In the previous sections, we have shown that how the difﬁculty of getting the solutions of many- body problem from Schrodinger¨ equation yielded the development of DFT over a period of time. This development can be re-summarized in very short but clear way in the tabulated format (see Table 2.1) as following:

Table 2.1: The mathematical approaches for many-body problems.

Wave function theory Density functional theory ˆ ˆ Schrodinger¨ equation: Hψ(ri) = Eψ(ri) Schrodinger-like¨ equation: HKSψi(r) = Eiψi(r) ˆ R Solution: E = minψhψ|H|ψi Solution: E = minn n(r)V (r)dr + F [n(r)]

Further, we have already explained why n(r) is simpler and superior to ψ(r) especially in real problems. However, the difﬁculty of explicitly knowing the parameter F [n(r)] makes the DFT calculations subject to errors. In order to address such errors, there are many approximations used to deﬁne F [n(r)]. The equation 2.5.5 tells that F [n(r)] can be determined if and only if EXC [n(r)] is determined. Thus, there is a direct relationship between the accuracy of the DFT with the approximations of XC energy. The Table 2.2 shows the well known Jacob’s ladder for the XC energy approximations [31]. In this thesis, we have used the following types of approximations to

determine EXC [n(r)]: (i) Local Density Approximation (LDA): The LDA approximation assumes the electronic density n(r) as homogeneous electron gas and at each point in this electronic gas there is an uniform XC energy. Using this approximation for n(r),

Z EXC [n(r)] = n(r)XC (n)dr (2.6.0.1)

where, XC (n) is the XC energy per particle of homogeneous electron gas of density n(r). The

XC (n) can be further decomposed as XC = X + C where the exchange energy X (n) is calcu- 31 Table 2.2: Jacob’s ladder for the XC energy approximations.

Simplicity Heaven of Chemical Accuracy Accuracy MP 2−like ↓ Ec Generalized ↑ ran- dom phase(B2P LY P, ...)

Hartree ↓ Ex HyperGGA ↑ (B3LY P , M06− 2X,...)

↓ ∇2n(r) MetaGGA ↑ (TPSS, M06− L,...)

↓ ∇n(r) GGA ↑ (BLY P , PBE, ...)

↓ n(r) LDA ↑ (SVWN) Earth of Hartree

3 3 1/3 lated analytically as Dirac functional as X (n) = − 4 ( π ) . And, the correlation energy C can be determined accurately from Quantum Monte Carlo calculations. The performance of LDA can be summarized using the nature of chemical bonding in various materials. For materials having hydrogen bonds and Van der Waals bonds, the LDA approximation is not sufficient and for ionic, covalent and metallic bonds, this approximation is generally good but mimic these materials with higher chemical bonding. Further, it is also not suitable for electron rich species or negatively charged materials. (ii) Generalized Gradient Approximation (GGA): If the density n(r) fluctuates such as in molecules then it is better to switch to GGA. In GGA, the functional depends both on the density and the gradient of the density. In other word, it is an improvement over LDA by considering 32 the gradient of the electronic density, i.e.., EXC = EXC [n(r), ∇n(r)]. Unlike from LDA, there are different parameterizations used for the exchange and correlation energy in order to account for the dependency of density as well as its gradient in GGA. Most of these parameterizations are semi-empirical, that is derived from fitting the experimental data. However, there are also some parameterizations which do not depend on the experimentally fitted data and include the corrections of the density gradient, making them valid for wide range of the systems. In this thesis, we have used parameter-free GGA functional developed by Perdew, Burke and Ernzerhof [32] and hence, known as PBE functional. Mathematically, PBE GGA functional is defined as

Z EXC [n(r)] = n(r)XC (n).Xdr (2.6.0.2)

where, X is the enhancement factor and it is advisable to look the reference [33], for the detail mathematical formulation of PBE. Hubbard Correction U: In strongly correlated systems where 3d, 4d, 4f and 5f electrons are localized and very close to Fermi energy, the LDA and GGA approximations fail as these approximations are independent of the electrons states. The electrons in these states are self- interacting and to account for this-orbital dependent potentials are introduced for these d and f electrons. The main aim of Hubbard correction U is to correct the effects of self interaction due to the localization of the states. The effective value of U is chosen in such a way that these localized states are pushed away from the Fermi level so that these states have no interaction with the bonding

states. In this thesis, we have used GGA + Us(of PAW 12 and 20) and LDA + U. (In the next chapter, we study the ground state properties of the Pristine ZnO using GGA,

GGA + US and LDA + U approaches. This chapter 3 is written in the aim of getting insight about the subtle differences in the computational capacities of these functionals.) 33

CHAPTER 3 A STUDY OF GGA, GGA + US AND LDA + U CALCULATIONS IN ZNO

3.1 Introduction

ZnO is a wide-band gap polar semiconductor with signiﬁcant optoelectronic, piezoelectric, thermal and transport properties which make it the material of choice for wide range of applications [34]. These properties are directly related to the consequences of vibration of nuclei around their equilibrium positions rather than the common model of static nuclei with electrons revolving around them. In other words, we can divide the properties of the solids in two categories- the one that are determined by the electrons and the other that are related to the movement of the atomic nuclei. The integration of thermal effects as well as the phase stability of the solids is governed by the vibration of the nuclei in them. As an illustration, the stable phase of any solid implies the minimization of the free energy. By deﬁnition G=H-TS, the free energy G has two components- one is enthalpy H and the other is entropy S. A very basic source of entropy is the thermal vibration of the nuclei. Henceforth, without understanding the lattice dynamics, it is impossible to explain any phenomena of the solid that incorporate temperature. Moreover, the propagation of sound waves in crystals and the interaction of light with the materials is directly associated with the lattice dynamics. We have already explained the lattice dynamics of the diatomic crystal in the chapter one where the vibration of nuclei around their equilibrium positions is described as a harmonic travelling wave. In more general terms, the coupled harmonic oscillations of the atomic nuclei are described as lattice waves. Each lattice wave is independent of each other and represents the distinct pattern of oscillations or modes with a common frequency, known as a normal mode. If we have s atoms per unit cell in the real crystal then any vibrations of the atoms in the crystal can be represented by 3s normal modes in the reciprocal crystal. The quanta of these normal modes of vibrations are called phonons, analogous to the photons which are the quanta of the electro-

th magnetic waves. If ωi is the frequency of a i normal mode, then the energy of the lattice wave

3 corresponding to this normal mode is Ei = (ni + 2 )~ωi, where ni is the number of phonons with 34 frequency ωi, distributed in 3-D energy level at a given temperature. The total energy of the crystal due to lattice vibration is merely the summation of this energy where i runs from 1, 2, .., 3s, and the whole lattice vibration can be modelled by a single lattice wave which is the superposition (linear combination) of all the normal modes of vibrations. The plots of ωi as a function of wave vector k are called dispersion curves. Further, we have already classiﬁed these dispersion curves based on the dependency of ωi with k as acoustic modes (ωi = 0, when k = 0) and optical modes (ωi 6= 0, when k = 0). Moreover, these modes can be divided as longitudinal and transverse modes based on the direction of the wave vector with respect to the direction of the wave propagation. The signiﬁcance of phonon dispersion relation is immense in solid state physics. The stability of the solid is directly related to its positive phonon modes. The measurement of phase and group velocity of acoustic and optical phonons plays an important role in the materialization of the physical properties of the solids. How sound propagates through the solids and the calculation of the refractive indices of the materials can be implemented from the knowledge of the phonon dispersion relation. In addition, the dispersion curves can be used to engineer the thermal conductivity of the material which has an important utilization in industrial sectors. For instance, in the experi-

dT mental sense, the thermal conductivity can be calculated as J = κ | dx |, where J is the heat ﬂux, dT κ is the thermal conductivity and dx is the temperature gradient. However, in a more fundamental P form, the expression for the thermal conductivity can be written as J = ki nki~ωi(k)vi(k)∇T where vi(k)∇T is the component of the phonon group velocity in the direction of the temperature gradient [35]. Thus, it is extremely necessary to measure the dispersion curves of phonon in order to understand the most important physical properties of bulk materials. In this chapter, we have used the ab-initio (ﬁrst principles) calculations based on the density functional theory (DFT) to calculate and measure the phonon dispersion curves of the Pristine

ZnO crystal. The functionals used in these calculations are GGA, GGA + Us with PAW 12 and 20 respectively and LDA + U. The role of these different functionals in the calculations of the same phonon dispersion curves (see Figure 3.3) may help in segregating out the effectiveness of them. Besides presenting the results in tables, we compare our results with the previous data obtained in 35 similar fields of research by various researchers (shown by Tables 3.2, 3.3). The first comparison of the results via the charts of normal distribution is often satisfactory but not flawless due to statistical bias. The next comparison is done with data from the inelastic neutron scattering (INS) (see Table 3.4), which is considered as the best method of computing phonon dispersion curves by the majority of the experimentalists. The computational tools used in this study are VASP, Phonopy, MATLAB, VESTA, JMOL.

3.2 Results

The ZnO used in this study is a 2×2×2 dimension super cell, whose structure is shown in Fig- ure 3.1. Further, it features uniaxial crystal structure with pronounced mass difference and strong bond polarity, which forces our study to incorporate LO-TO splitting in phonon band structure. This is accomplished by not neglecting dielectric tensors and Born effective charges. The piezoelectric properties of ZnO can be associated with dielectric tensors and Born effective charges. Finally, the simulated results of phonon calculations at the Γ point is shown in Table 3.1and the cartoons of the normal modes are shown by Figure 3.2.

ZnO-unitcell ZnO-2×2×2 supercell

Figure 3.1: The crystal structures of ZnO used in the calculations where the gray ball is Zn and the red ball is O. 36

(a) E2 (Low) Mode (b) E2 (High) Mode

Figure 3.2: The schematic cartoons of different types of modes at Γ point where the arrow indicates the direction of vibration.

Table 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

Mode(Γ) GGA GGA+U(P AW = 12) GGA+U(P AW = 20) LDA+U

E2(Low) 87.66 110.311 106.74 95.064 E2(High) 433.73 440.06 456.036 483.161 A1(TO) 382.70 387.40 397.60 426.956 E1(TO) 402.90 413.25 429.34 453.30 A1(LO) 538.83 579.06 585.32 596.97 E1(LO) 538.96 618.364 590.30 631.195 37

(a) Dispersion curves-GGA (b) Dispersion curves-GGA+U(PAW12)

Figure 3.3: Comparison among the phonon dispersion curves of ZnO computed using the functionals GGA, GGA + Us (PAW 12 and PAW 20) and LDA + U

3.3 Discussion

In this section, we have revised the former work related to the vibrational study of ZnO. The reference section mentioned at the end of this thesis includes such papers from which we have extracted the data for our ﬁnal conclusions. They are tabulated below. Now, for comparing our result with the former work, we construct the bell curve by evaluating the Gaussian distribution of previous data. Finally, we locate the position of our calculated value with the mean of these data in the bell curves (see Figure 3.4). By visual analysis of these curves, one can easily ﬁgure out the 38

Table 3.2: The former data of ZnO frequencies (cm−1) calculated by various researchers.

Ref [34] Ref [36] Ref [37] Ref [38] Exp − Exp. Raman− Infrared − Damen − Callender− Bairamov− Mode(Γ) (Raman) Spectro. Spectro. Raman − Raman − Raman − Spectro. Spectro. Spectro.

E2(low) 100 99 102,101,- 101 98 98,99 E2(high) 438 439 438,437,- 437 441 437.5 444,438,- 437.5 A1(TO) 380 382 380,379,- 380 380 381 378 378 E1(TO) 410 414 409,410,- 409.1,408.2,- 407 407 409.5 407,413,- 412 409.5 A1(LO) 584 574 574,576,- 574.5,577.1,- 574 576 579 570 E1(LO) 595 580 587,591,- 588.3,592.1,- 583 583 588 583,588,- 591 591

Table 3.3: Continuation of Table 4

Mode(Γ) Ref [39] Ref [40] Ref [41] Ref [42] Mean Standard Deviation

E2(Low) 98.40, 100.013 101 98.4 100 99.68 1.32 E2(High) 438.76, 437.962 437 438.8 438, 439 438.53 1.77 A1(TO) 378.276, 379.89 380 378.3 380, 379 379.56 1.135 E1(TO) 412.15, 409.73 407 412.1 410, 411 409.86 2.1 A1(LO) 573.46, 583.98 574 573.5 584, 575 576.407 4.22 E1(LO) 592.82, 595.24 583 592.8 595, 580, 575 587.71 5.79 39

Figure 3.4: The family of bell curves at different modes. 40 effectiveness of these functionals by mapping the calculations with the previous research. De- spite, the large standard error in the distribution of data and sufﬁciently small data sample, we can easily claim that the result of our work is satisfactory. However, the reliability of these comparisons may be in doubt. The phonon frequencies calculated with, perhaps the most powerful technique- the inelastic scattering of neutrons (INS) [39] are used for ﬁnal comparison. Unlike from other optical techniques, the INS method provides complete information of the phonon spectra for all the frequency range. The INS values show satisfactory agreement with our corresponding values, as shown in Table 3.4. Table 3.4: Comparison of Phonon frequencies (cm−1) of ZnO at Brillouin Zone center Γ with the INS data

Mode(Γ) INS GGA GGA+U(P AW = 12) GGA+U(P AW = 20) LDA+U

E2(Low) 98.40 87.66 110.311 106.74 95.064 E2(High) 438.76 433.73 440.06 456.036 483.161 A1(TO) 378.276 382.70 387.40 397.60 426.956 E1(TO) 412.15 402.90 413.25 429.34 453.30 A1(LO) 573.46 538.83 579.06 585.32 596.97 E1(LO) 592.821 538.96 618.364 590.30 631.195

Table 3.5: Percentage error calculated with respect to the INS values. The + and − signs indicate overestimation and underestimation respectively.

Mode(Γ) GGA GGA+U(P AW = 12) GGA+U(P AW = 20) LDA+U

E2(Low) −10.91% +12.10% +8.47% −3.4% E2(High) −1.15% +0.3% +3.94% +10.11% A1(TO) +1.17% +2.41% +5.10% +12.86% E1(TO) −2.24% +0.27% +4.17% +9.98% A1(LO) −6.04% +0.97% +2.06% +4.09% E1(LO) −9.08% +4.31% −0.42% +6.47%

It is an usual trend in statistics that the calculation yielding a percentage error less than 10% is considered to be a reliable measurement. Table 3.5 highlights the percentage errors due to the respective functionals. The GGA functional underestimates the INS measurements by maximum 41 of 10.91%. This maximum error results for the calculation of the lowest energy optical phonons

E2(Low); otherwise, the percentage error due to GGA is well below 10%. After introducing the

Us parameter to the GGA functionals, there is an overestimation of the INS measurements. In- terestingly, the GGA + Us also overestimate the lowest energy optical phonons by maximum of 12.10% keeping the reliable measurement for the remaining higher energy optical phonons. How- ever, the functional LDA + U underestimate the lowest energy optical phonons and overestimate to the remaining higher energy optical phonons. For a moment by neglecting the low lying E2 phonons, these variations of the phonons can be explained as per the nature of the functionals used in their computations. The LDA functional binds the atoms more tightly than the GGA, so that the bond length in the former is less than the latter. The introduction of the U parameter allows the atoms to be more localized. Further, the frequency of the optical phonon (ω) is related to the 1/2 zEb bond energy (Eb), atomic degree of freedom (z) and bond length (d) as ωα d [43]. In the case

of GGA, the parameters Eb is low and d is high, which indicates the reason for the underestimate

of phonons frequencies due to GGA. After we introduce Us parameters in GGA calculations, the

localization among atoms increases which produce a lower d and a higher Eb parameters. As a

result, there is an overestimation in GGA + Us calculations. By following the similar argument, it is not difﬁcult to explain the overestimation shown by the LDA + U calculation. The anomalous

behaviour of the low lying E2 phonons may be explained as their higher susceptibility towards the

anharmonicity nature of atomic vibrations [44]. The shifts in the frequency of E2 phonons may

be attributed to factors like anharmonicity, coupling of low lying E2 phonons with the acoustic

phonons or the decay of E2 phonons into a difference of two high-energy phonons or into the sum of lower- energy acoustic phonons [45]. Moreover, due to the long-range nature of the Coulomb interaction, the longitudinal optical phonon suffers extra polarization effect. This effect can be veriﬁed from the Table 7, where all these functionals calculate higher longitudinal optical phonon frequency compared to the transverse optical phonon. 42 3.4 Conclusion

Henceforth, this work expresses its validity by calculating the phonon frequencies of ZnO at its symmetric point. It adds the quarted sets of the new values of phonon frequencies at the different modes in the Γ point. It is further believed that this work might be helpful for understanding the bulk properties of ZnO. The computation of the phonon dispersion curves of ZnO using different functionals might help in tuning the heat ﬂow within the materials. This kind of study may assist in manipulating phonon behaviour, which can be used in the construction and optimization of the advanced materials. The properties such as thermal conductivity can be optimized based on the study of phonon behaviour. The above computational study is an approximation to the real crystal as we have neglected the effect of anharmonicity of the lattice vibration. Phonons decay and the coupling between low lying phonons with the acoustic phonons arise whenever there is an anharmonic lattice vibration. It is believed that the future work on the study of phonon dispersion will include such factors of anharmonicity. 43

CHAPTER 4 COMPUTATIONAL STUDY OF ZNO AND ITS NATIVE OXYGEN POINT DEFECTS -I

4.1 Introduction

0 In the very first chapter of this thesis, we have defined and discussed about Pristine, Zero (VO), + +2 One (VO ) and Two (VO ) ZnOs. The study of materials along with the defects is extremely important to understand their composite behavior. The deviation in overall properties of the material due to defects from its pristine nature yields not only an understanding of subtle phenomena about the dynamics in the atomic realm but also paves a path to engineer the material according to useful products. The effect of the deficiency of one atom from the lattice point or the addition of extra charge in the deficient lattice point is vital for understanding the microscopic origin of the macroscopic phenomena. As an illustration, the Pristine ZnO is suitable in producing UV LED as it has photoluminescence in this region. However, if this material is made more and more oxygen deficient then it can produce photoluminescence in the visible region [46]. Further, Pristine ZnO is an n-type semiconductor having a high band gap, and one can increase the conductivity by making the material more and more oxygen deficient which decreases the band gap. The primary objective of this chapter is to study the variation of the properties (both electronic

0 + +2 and vibrational/thermal) among Pristine, Zero (VO), One (VO ) and Two (VO ) ZnOs. These variations are feasible as the systems differ from each other by only one ion and/or one or two charged particles. For computational purposes, we create 3×2×2 super cell from the unit cell consisting of four Zn and four O ions whose crystal structure is shown by the Figure 4.1. This resulting super cell is designed as a Pristine ZnO. Its crystal structure is shown by the Figure 4.2.

0 The Zero ZnO (VO) is formed by removing one O ion from the Pristine ZnO. Further, we add one

0 positive charge and two positive charge in the lattice vacancy of Zero ZnO (VO) to form One ZnO + +2 (VO ) and Two ZnO (VO ) respectively. These three types of intrinsic oxygen defects are studied locally and ZnO is assumed to satisfy stoichiometrical as well as the conservational (mass/energy) 44

Figure 4.1: The crystal structure of the unit cell of ZnO (Zn as grey ball and O as red ball) used in the computation.

relation globally. The study of intrinsic oxygen point defects is signiﬁcant as many scholars believe the role of oxygen vacancy in n-type conductivity of ZnO [47]. We basically aim to study effects of defects on the electronic structure of ZnO, vibrational effects of defects and study of heat capacity in ZnO and its native oxygen point defects. The functional used in this ab-initio calculation is GGA. Despite the fact that DFT underestimates the band gap of the semiconductors, the results of the computation will have no serious issue for a comparative study of pure ZnO and its native oxygen point defects. This is something like - despite the error in measurement, the aspect ratio of the measurement is accurate. The computational tools used are VASP, Phonopy, Xmgrace, VESTA, VMD, JMOL and MATLAB.

4.2 Effects of Defects on the Electronic Structure of ZnO

The Pristine ZnO used in the study consists of 48 Zn ions and 48 O ions. Each Zn ion is bonded tetrahedrally to 4 O ions and vice-versa. When one O ion is removed, the four peripheral Zn ions dangle around the vacant oxygen lattice point. It was reported that the peripheral Zn ions in Zero

0 + +2 ZnO (VO) displaced inward by 12% and in the case of One ZnO (VO ) and Two ZnO (VO ) the displacement was outward by 2% and 23% respectively [14]. The four peripheral Zn ions form 45

Figure 4.2: The crystal structure of 3×2×2 Pristine ZnO used in this computation.

1 four dangling bonds by contributing 2 electron to the vacant lattice point. When the oxygen ion is removed, it creates a hole of charge 2 and the reason for the formation of four dangling bonds is to make this vacant lattice point neutral. These dangling bonds produce a symmetric state which

0 can be seen in the band gap. We have computed these states due to dangling bonds in Zero (VO), + +2 One (VO ) and Two (VO ) ZnOs and obtained similar results as those of the various scholars. As a proof, our computed results illustrate an idea similar to that of Figure 2 in article [14] which is shown at the top of this adjoining Figure 4.3.

Figure 4.3: The computed electronic band structures showing trapped electron states. 46 0 In the Zero ZnO (VO), the two electrons are trapped at the oxygen vacant point. As a result, the four peripheral Zn atoms move inward displacing the symmetric state towards the valence re-

+ gion. Likewise, the conﬁguration of One ZnO (VO ) traps one electron which causes the peripheral Zn atoms to displace the symmetric state slightly outward in between the valence and conduction

+2 regions. Similarly, there is no trap of the electron in Two ZnO (VO ) which results in the displacement of this state close to the conduction region. In order to verify the relation of the electron trapped in the oxygen vacant point to the displacement of the corresponding state, we select a small window in the energy interval of this state and then try to compute the presence or absence of the electronic cloud (see Figures 4.4, 4.5, 4.6, 4.7, 4.8, 4.9). The resulting computation further afﬁrms

0 our explanation of the displacement of these states in proportion of the systems-Zero (VO), One + +2 (VO ) and Two (VO ) ZnOs.

0 Figure 4.4: The state in Zero ZnO (VO) being windowed in the energy interval by two dotted lines. 47

0 Figure 4.5: The observed electron cloud in yellow color of the windowed Zero ZnO (VO) state.

+ Figure 4.6: The state in One ZnO (VO ) being windowed in the energy interval by two dotted lines.

Further, the presence of such states in the oxygen vacant defects of ZnO lower the band gap and increase visible light absorption. Experimentally, the lowering of band gap had been successfully 48

+ Figure 4.7: The observed electron cloud in yellow color of the windowed One ZnO (VO ) state.

+2 Figure 4.8: The state in Two ZnO (VO ) being windowed in the energy interval by two dotted lines. conﬁrmed by the enhancement of the photocurrent response when the ZnO was irradiated with the visible light [48]. Our DFT calculations also support lowering of the band gap even though 49

+2 Figure 4.9: No observed electron cloud of the windowed Two ZnO (VO ) state. it underestimates the band gap by more than 70%. The band gap of the Pristine ZnO is 3.4 eV at the Γ point. The valence band constitutes of the band of 2p orbitals and the conduction band constitutes the band of 4s states. The 3d states occupy the lower region of this conduction band. There is a repulsion between 3d states and 4s states signiﬁcantly and less covalent interaction of the 2p states with the 3d states. The overall dynamics of these states give the resultant value of 3.4 eV. The reason for the underestimation of the band gap by DFT is that it fails to estimate correctly the repulsion between 3d states and 4s states [49]. As a result, the covalency between 2p states with the 3d states is much higher. In our computation, the DFT prediction of the band gap of the Pristine ZnO using GGA approximation is 0.876eV . The decrease in band gap follow the order:

+2 + 0 Two ZnO (VO )>One ZnO (VO )>Zero ZnO (VO) (see Figures 4.10, 4.11, 4.12, 4.13). In the +2 case of Two ZnO (VO ), the system can add two electrons which make this material more electron

0 affinitive and thus the state disperses towards the conduction band whereas in Zero ZnO (VO), the material has no more electron affinity and therefore the state occupies the top region of the valence 50 band. The lack of electron affinity promotes strong covalency with the adjoining 3d states resulting

0 in the maximum decrement of the band gap of Zero ZnO (VO). On the other hand, the presence of + one unpaired electron in One ZnO (VO ) not only secures its tendency for acquiring an electron but also makes the material paramagnetic and therefore unstable.

Figure 4.10: The electronic density of states of Pristine ZnO

0 Figure 4.11: The electronic density of states of Zero ZnO (VO) 51

+ Figure 4.12: The electronic density of states of One ZnO (VO )

+2 Figure 4.13: The electronic density of states of Two ZnO (VO ) 52 4.3 Effects of Defects on the Vibrational Properties of ZnO

In the previous chapter, we have already stated that the vibrational dynamics of the atomic nuclei is the precursor of structural, mechanical, thermal, magnetic and optoelectronic properties of the materials. Further, the phonon is the smallest entity that can inﬂuence temperature dependent effects. The structural stability of the materials is associated with the positive vibrational modes of the phonons. The elasticity, refractive indices, thermal conductivity are the direct consequences of the distribution of the phonons. Likewise, the coupling among electrons and phonons give rise to the phenomena of the magnetic and optoelectronic properties of the materials. Moreover, we have already shown that the thermal properties like heat capacity have direct relation with the density of states of phonon. Therefore, it is always necessary to investigate the vibrational dynamics of the materials for understanding the subtle mechanisms of evolution of the properties in materials.

0 + We begin our study with the dispersion curves of Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) +2 and Two ZnO (VO ). In the case of Pristine ZnO, the total number of normal modes of vibrations

0 + +2 is 288 whereas in Zero ZnO (VO), One ZnO (VO ) and Two ZnO (VO ), the total number of normal modes is 285. This variation is obvious because the latter lacks one O atom. This also implies that

0 there is a difference in the phonon band gap between Pristine ZnO and the rest Zero ZnO (VO), + +2 One ZnO (VO ) and Two ZnO (VO ). However, the most important question would be whether

0 + +2 the phonon band gaps among Zero ZnO (VO), One ZnO (VO ) and Two ZnO (VO ) are same or different. If the gaps are same then it implies that the presence of electrons in the lattice vacancy point has no role in determining these gaps. Otherwise, the electrons that are trapped in the lattice

0 + vacancy point play a role in determining these gaps as Zero ZnO (VO), One ZnO (VO ) and Two +2 ZnO (VO ) have the same number of Zn atoms and O atoms. The phonon band gap refers the gap between the highest modes of low optical phonons and the lowest modes of high optical phonons. Technically, it is the difference of the modes of fre-

quency between A1(TO) and B1(Low) measured at the Γ point. We have computed the dispersion

0 + +2 curves of Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and Two ZnO (VO ) (Figure 4.14) and

−1 −1 0 have measured the band gap of 3.46 cm for Pristine ZnO, 3.47 cm for Zero ZnO (VO), 3.54 53 −1 + −1 +2 cm for One ZnO (VO ) and 3.15 cm for Two ZnO (VO ) (see Table 4.1). In polar materials the long range Coulomb interactions between ions make the dynamical matrix non-analytical and to correct for these non-analytical terms, we have included LO-TO splitting in our computation. These measurements have shown clearly that there is an effect of electrons present at the lattice vacancy point. The low optical phonon modes are contributed by Zn ions vibrations. Since four Zn atoms share a tetrahedral bond with one O atom. Due to the absence of this O ion, these four Zn ions dangle or are in strain. The rate of dangling will be low if the oxygen vacancy point is neutral

and increases by the absence of electrons in it. Thus, the variation of B1(Low) modes follows the

+2 + 0 order: Two ZnO (VO )>One ZnO (VO )>Zero ZnO (VO)> Pristine ZnO. However, the case is different for high optical phonon modes as these modes are a consequence of O ion vibrations. For

an oxygen vacancy lattice point devoid of electrons, the A1(TO) mode is minimum. It has been observed that this mode is maximum if there is one unpaired electron at the vacant lattice point. This is due to the fact that the transverse optical wave loses energy when it propagates through the vacant site and the presence of one unpaired electron in this vacant site favors and accelerates

the momentum of propagation. Moreover, the A1(TO) mode is a dipole induced mode. For a non

dipole induced mode like E2(High), the presence of oxygen vacancies decreases the frequency

and the effect of electrons in the vacant site is insigniﬁcant. Thus, the variation of A1(TO) modes

+ 0 +2 follows the order: One ZnO (VO )>Zero ZnO (VO)>Pristine ZnO> Two ZnO (VO ).

Table 4.1: The measurement of the phonon band gap in T Hz.

Materials B1(Low) A1(TO) Phonon band gap P ristineZnO 7.87 11.33 3.46 ZeroZnO 7.88 11.35 3.47 OneZnO 7.93 11.48 3.55 T woZnO 7.99 11.14 3.14

0 Next, we analyze the variation of vibrational density of states (VDOS) of Zero ZnO (VO), One + +2 ZnO (VO ) and Two ZnO (VO ) with respect to Pristine ZnO. Figures 4.15, 4.16, 4.17 highlight their respective variations in comparison with Pristine ZnO. 54

(a) Dispersion curves-Pristine ZnO (b) Dispersion curves-Zero ZnO

0 + Figure 4.14: Dispersion curves for Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and Two ZnO +2 (VO )

The comparative analysis of these VDOS has shown that there is an enhancement of the VDOS at low frequencies due to defects as indicated by Table 4.2. Further, we have seen clearly that there exists separate localized modes which are not vibrating in unison with the whole crystal.

+ In the case of One ZnO (VO ), the localized mode occurs at 14 numbered eigenstate whereas in +2 Two ZnO (VO ), it occurs at 15 numbered eigenstate (see Figures 4.16, 4.17). We have also seen

0 the localized mode in Zero ZnO (VO), but we failed to observe a sharp vibration of this mode and therefore, its eigenstate was not described (Figure 4.15). The occurrence of such localized modes 55

0 Figure 4.15: The variation of density of states of Zero ZnO (VO) with respect to Pristine ZnO

+ Figure 4.16: The variation of density of states of One ZnO (VO ) with respect to Pristine ZnO has decreased the frequency of normal modes of vibration because the defects increase the proportion of the phonons at the low frequency region and decrease the overall phonon group velocity.

0 + Finally, we would study the thermal properties of Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) 56

+2 Figure 4.17: The variation of density of states of Two ZnO (VO ) with respect to Pristine ZnO

Table 4.2: The enhancement of VDOS at low frequency due to defects.

Materials Frequency Bin Counts Frequency Bin Counts P ristineZnO 1 − 2 5 2 − 3 22 ZeroZnO 1 − 2 6 2 − 3 23 OneZnO 1 − 2 7 2 − 3 24 T woZnO 1 − 2 6 2 − 3 29

+2 and Two ZnO (VO ) and their variations. The data shown in Appendix B are the elaborate measurements of thermal properties of them. However, in this section we focus our attention in the heat capacity and its variation. The heat capacity is found by taking the temperature derivative of the equation 1.3.19. We are particularly interested in the variation of the heat capacities of Pristine

0 + +2 ZnO, Zero ZnO (VO), One ZnO (VO ) and Two ZnO (VO ) with the temperature. At the high temperature, the heat capacity approaches to a constant value as indicated by Dulong and Petit law. Therefore, we study the variations of the heat capacities at low temperatures using Debye model. It has been found that the variations of the heat capacities at the low temperatures follow the order: CO > CT > CZ > CP (see Figure 4.18) where the subscripts O, T, Z and P stand for One 57 Table 4.3: The measurements of the heat capacities (J/K/mol) at the low temperatures.

Temperature(K) CO CT CZ CP 10 6.10 6.23 4.889 5.26 20 55.18 53.86 47.20 44.69 30 166.19 164.40 153.53 148.31 40 298.89 296.97 285.96 279.65 50 424.96 422.87 413.53 407.26 60 537.17 534.92 527.97 521.92

+ +2 0 ZnO (VO ), Two ZnO (VO ), Zero ZnO (VO) and Pristine ZnO respectively. Table 4.3 shows the measurements at the low temperatures. At the temperature 10 K or less, we see that the variations do not follow the pattern or show some anomalies. These anomalous variations can be explained with respect to their zero-point energies. The zero-point energy of Pristine ZnO (539.53KJ/mol)

0 +2 is higher than Zero ZnO (VO) (529.45KJ/mol) and that of Two ZnO (VO ) (537.85KJ/mol) is + higher than One ZnO (VO ) (532.52KJ/mol).

Figure 4.18: The variations of the heat capacities with respect to Pristine ZnO.

0 + After that we plot the heat capacities of Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and Two

+2 3 ZnO (VO ) with respect to temperatures using Debye T law and then interpolate to the origin as the heat capacity is zero at 0 K temperature. Once, we have the slope values we can measure their respective Debye temperatures. As, we have already described Debye temperature mathemat- 58 ically in the ﬁrst chapter. The physical signiﬁcance of Debye temperature is that it is a temperature which demarcates the quantum regime with the classical regime of lattice vibration. That means, if T > TD all vibrational modes have an energy kBT and if T < TD all the high frequency modes disappear and the heat capacity decreases with respect to the decrease in the temperature. In our

measurements, the variation of Debye temperature follows the order: TDP > TDZ > TDT > TDO

0 +2 where the subscripts P, Z, T and O stand for Pristine ZnO, Zero ZnO (VO), Two ZnO (VO ) and + One ZnO (VO ) respectively. It should be noted that the negative values of the heat capacities shown in Figures 4.19, 4.20, 4.21, and 4.22 are just the image points and used for the ﬁtting purpose only. The heat capacity can never be negative.

Figure 4.19: Debye temperature of Pristine ZnO

The decrease in Debye temperature due to defects can be explained similarly as the defects decrease the phonon group velocity which in turn decrease the normal modes of vibration. Since, the Debye temperature is associated with the highest normal modes of vibration- the decrease in the normal modes of the vibration implies the lowering of the Debye temperature. Conversely, this implies that the less energetic phonons are present in the defect vacancy sites which can absorb more energy and yields higher heat capacity. Hence, there exists an inverse relation between heat capacity and Debye temperature. In summary, we can state that the presence of atomic anomalies 59

0 Figure 4.20: Debye temperature of Zero ZnO (VO)

+ Figure 4.21: Debye temperature of One ZnO (VO ) in the form of defects enhance the phonon or vibrational density of states at low frequencies which are marked by the occurrence of localized modes. The consequences of the presence of localized modes in the thermodynamical properties can be - reduction of the Debye temperature or excess of the heat capacity. 60

+2 Figure 4.22: Debye temperature of Two ZnO (VO )

4.4 Conclusion

In the long run, we have investigated the variation of the properties (both electronic and vibra-

0 + +2 tional/thermal) among Pristine, Zero (VO), One (VO ) and Two (VO ) ZnOs. Further, our study reveals how defects influence electrons and phonons dynamics. Since electrons are the carrier of electrical energy and phonons are the carrier of thermal energy, the tuning of the defects almost control every properties of the materials. This signifies the importance of the defects physics. We have shown that oxygen vacancy defects reduce the band gap and increase the heat capacity in ZnO. Further, the effect of charged defects versus uncharged defects on the overall properties of the materials are measured. However, we are unsure whether these effects are significant or not as the physics behind the influence of charged defects on the normal modes of the vibration is quite unclear. Our measurements have indicated that the charged defects lower the frequency of the low- lying phonon more than the uncharged defects. The future work may constitute about knowing the relationship between the charged defects with the normal modes of the vibration. (In the next chapter, we revise earlier computations of chapter 4 using LDA + U approaches. The chapter 5 is written in the aim of getting insight about the differences encountered in the properties of the materials as well as the computational capacities of these functionals.) 61

CHAPTER 5 COMPUTATIONAL STUDY OF ZNO AND ITS NATIVE OXYGEN POINT DEFECTS -II

5.1 Introduction

The principal objective of this chapter is to revise the former results of chapter 4 using the approximations LDA+U. In the previous chapter 4, we have computed the electronic and vibrational properties of Pristine ZnO along with its native oxygen point defects. The computations of those properties were carried by using the functional GGA (PBE). As we know that the GGA calculations not only consider electronic density functional but also its spatial variation; which make GGA more accurate than LDA. On the other hand, LDA being a simplest class of approximations to the XC energy functional which consider only the value of the electronic density at each point in space but not its spatial variation. Moreover, we have illustrated that the LDA binds the atoms tightly than the GGA by computing the Γ point frequencies in the chapter 3. However, these approximations fail to compute the correct value of the band gap in semiconductor. Our GGA calculations compute the band gap of ZnO to be 0.876 eV which underestimate the correct value of the band gap by more than 70%. The case with LDA calculations is quite extreme as it is less accurate than GGA. The failure of predicting the correct values of the band gap has been explained in the chapter 4 by stating that these approximations fail to estimate correctly the repulsion between 3d states and 4s states and fail to measure the correct covalency between 2p states with 3d states. DFT calculations can predict the correct measurements if it can reproduce correctly the high- frequency dielectric constant ∞ and the band gap. Generally, in ionic crystals the electron clouds are oscillating from their respective ions. This oscillation can be described as plasmons, analogous to the atomic nuclei oscillation is described by phonons. The correct modelling of plasmon frequency is vital as many electrical and optical properties of the material are associated to it. In theory, this can be achieved if one can able to reproduce ∞ correctly using some approximations to the XC energy functional. In LDA approximations, we add Hubbard term correction, which 62 correctly reproduce ∞ = 3.75 of ZnO. In the case of d and f electrons, the DFT fails to treat the on site Coulomb interactions of these localized electrons. The U parameter of Hubbard correction provides strength to the on site Coulomb interactions and the J parameter is associated to the on site exchange. We have found that using the U parameter to be 7.3 eV and the J parameter to be

constant as 0 can reproduce ∞ correctly in ZnO; which in turn ﬁxes the position of the d level better than GGA. As a result, the band gap in ZnO is predicted well by LDA+U than GGA. The band gap of ZnO using LDA+U was found to be 2.161 eV (see Figure 5.1). Nevertheless, this value still underestimates the correct band gap by more than 35%.

Figure 5.1: The total density of state of Pristine ZnO using LDA+U. 63 5.2 Results of LDA+U approximations on the Electronic Structure

We have done the same calculations using GGA and found that it underestimates the band gap by more than 70%. The true reproduction of dielectric constant at high frequency by LDA+U correctly represents the ground state electronic structure of ZnO. As a consequence, the LDA+U measures the band gap more accurately than the GGA. One thing is clear that both GGA and LDA+U computations have shown that the band gap decreases due to defects. However, the variations of band gaps with types of defects are quite awkward in the case of GGA. The order of

Figure 5.2: The total density of state of Zero ZnO

+2 + 0 variations in GGA is: Two ZnO (VO )>One ZnO (VO )>Zero ZnO (VO). This order should be:

0 +2 + Zero ZnO (VO)>Two ZnO (VO )>One ZnO (VO ) as we have seen that the trapped electron in the + case of One ZnO (VO ) forms the state quite middle of the band gap and the same state shifts close

0 to the upper valence band in the case of Zero ZnO (VO). The LDA+U calculates the band gap of the defects following the correct order as shown by Figures 5.2, 5.3, 5.4. 64

Figure 5.3: The total density of state of One ZnO

Figure 5.4: The total density of state of Two ZnO 65 5.3 Results of LDA+U approximations on the Vibrational Properties

Our computations for phonon frequencies at Γ point have indicated that the LDA+U underestimates feebly for low lying phonons and overestimates quite unfaithfully for high lying phonons. The frequencies of high lying phonons are dependent on the bond among the ions and since the LDA+U functional makes the ions attract strongly than the GGA, it is therefore obvious that the frequencies computed by the former functional are higher than the later functional. The computed frequencies due to LDA+U are shown in Appendix C. The variations in the phonon band gaps computed by LDA+U are slightly higher and follow the same order as GGA. Likewise, the heat capacities and Debye temperatures computed by LDA+U are higher than GGA. However, the variations in zero point energy of ZnO and its oxygen point defects follow peculiar order in both LDA+U and GGA computations. Their respective measurements along with other thermodynamical properties are listed in Appendix D. Generally, the frequencies of low lying phonons are dependent on the motion of Zn ions and the double charged oxygen defects disperse the peripheral Zn ions maximum, forming the localized vibration which is not in unison with the rest of the ions. As a consequence, the normal modes of

+2 vibration in the case of Two ZnO (VO ) are minimum. Thus, the usual order of frequency variation is: fP ristine>fZero>fOne>fT wo which implies that the corresponding variation of Debye temperature (see Figures 5.6, 5.7, 5.8, 5.9) follows: DP ristine>DZero>DOne>DT wo and heat capacity follows: CT wo>COne>CZero>CP ristine. 66

(a) Dispersion curves-Pristine ZnO, band gap 3.99 (b) Dispersion curves-Zero ZnO, band gap 4.23

0 + Figure 5.5: Dispersion curves for Pristine ZnO, Zero ZnO (VO), One ZnO (VO ) and Two ZnO +2 (VO ) using LDA+U

The LDA+U computations yield the similar order of variations. However, the GGA computations fail to discriminate the order between the charged defects in the correct way. It is interesting to note that both GGA and LDA+U establish the fact that, ’the presence of atomic anomalies in the form of defects enhance the phonon or vibrational density of states at low frequencies which are marked by the occurrence of localized modes. The consequences of the presence of localized modes in the thermodynamical properties can be - reduction of the Debye temperature or excess of the heat capacity.’ 67

Figure 5.6: Debye temperature of Pristine ZnO

0 Figure 5.7: Debye temperature of Zero ZnO (VO)

+ Figure 5.8: Debye temperature of One ZnO (VO ) 68

+2 Figure 5.9: Debye temperature of Two ZnO (VO )

5.4 Conclusion

Many results computed by LDA+U have similarity in order or pattern with GGA despite the measurements are higher in values than that of GGA. The measurements made by LDA+U are more consistent with the theoretical notion than GGA. Further, the LDA+U is more effective to study the charged defects than GGA. The following Table 5.1 highlights some of the measurements calculated by GGA and LDA+U approximations.

Table 5.1: The comparative measurements between GGA and LDA+U approximations

Measurements GGA LDA+U 0 0 Electronic-band-gap (eV) Pristine:0.876;VO:0.864; Pristine:2.164;VO:2.134; + +2 + +2 VO :0.871;VO :0.874 VO :1.654;VO :1.76 −1 0 0 Phonon-band-gap (cm ) Pristine:3.46;VO:3.47; Pristine:3.99;VO:4.23; + +2 + +2 VO :3.55;VO :3.14 VO :4.33;VO :3.80 0 0 Zero-point-energy (kJ/- Pristine:539.54;VO:529.45; Pristine:599.40;VO:596.97; + +2 + +2 mol) VO :532.52;VO :537.85 VO :602.45;VO :609.62 0 0 Debye-temperature (K) Pristine:381.93;VO:378.32; Pristine:418.25;VO:408.13; + +2 + +2 VO :372.87;VO :373.60 VO :401.51;VO :400.38 69

CHAPTER 6 SUMMARY AND CONCLUSION

In this thesis, we have used the computer modeling technique VASP for implementation of the ab initio methods to study the wurtzite ZnO and its native oxygen point defects. The DFT is used to compute the electronic as well as vibrational and thermal properties of Pristine ZnO, Zero

0 + +2 ZnO (VO), One ZnO (VO ) and Two ZnO (VO ). Under the DFT calculations, we have tested the various functionals like LDA, GGA, GGA+U, and LDA+U to figure out not only the variations of the efficacy of these functionals but also to disclose the properties of ZnO and its oxygen point defects and then compare the results with the corresponding experimental ones. Moreover, our materials of study differ from one another either by one ion or by one or two charges. Therefore, the computed results presented in this thesis have immense significance if one needs to know the variations of the properties due to deficiency of just one ion or by the excess of one or two charges. In the Chapter 1, the fair introduction of the subject matter and its background is presented. The concepts such as lattice dynamics and electronic as well as phonon density of states are built in this chapter as they are the principal concepts used entirely in this thesis. Further, this chapter also introduces ZnO and its native oxygen point defects as they are the materials of our investigation. We begin our study by mimicking and simulating the electronic structures of these materials and the process is based on the DFT, which is implemented in VASP codes. Thus, the Chapter 2 is entirely based on the evolution of DFT to the present day of its implementation. The many-body problem in physics is one of the historic problems and after the advance of quantum mechanics, the Schrodinger¨ equation is quite successful in solving the many-body problem. However, the solution involves solving many complex differential equations which make the use of Schrodinger¨ equation cumbersome, impractical and unrealistic for the physical many-body problem. The physicists are therefore looking for the alternative solutions of Schrodinger¨ equation which are not only straightforward and manageable but also practical for implementation in computer simulations. This search finally gave birth to the DFT and the Chapter 2 speaks thoroughly on the birth of the DFT to its present time evolution. 70 The DFT is like a big umbrella where different types of functionals exist below its canopy. We have implemented LDA, GGA, GGA+U and LDA+U functionals to study the vibrational property of the Pristine ZnO and its native oxygen point defects. Further, we have chosen the vibrational property to study the efficacy of these functionals as the vibrational property is the most fundamental ground state property and the proper means to test these functionals. The Chapter 3 is entirely based on the analysis of the results obtained from these functionals. All these results are compared with their corresponding experimental values and the cause for underestimation and overestimation are made. The significance of Chapter 3 is that it has added quartet sets of frequency values and made open to any time comparison with the experimental values. Likewise, the Chapter 4 is entirely based on the computations of electronic and vibrational as well as thermal properties of the Pristine ZnO along with its native oxygen point defects. This Chapter has tremendous significance as it measures the properties of the materials which are minutely different from one another either by the deficiency of one ion or excess of one or two charges. The role of defect physics in modeling the properties of the materials can be sensed from this Chapter. The functional used in these computations involve GGA and we redo the computations of Chapter 4 by using the functional LDA+U. The striking results obtained from the functional LDA+U are included in the Chapter 5. Finally, the last four appendices constitute the important data of Pristine ZnO and its native oxygen point defects obtained by using the functionals GGA and LDA+U respectively. In the conclusion, this thesis is quite successful in understanding the electronic, vibrational and thermal properties of ZnO and its native oxygen point defects. It has also tested the efficacy of the functionals GGA, GGA + Us and LDA + U. Further, this thesis is significant to understand how the microscopic properties of the pristine material deviate with its intrinsic defects. Moreover, this thesis is useful for gaining insight on the role played by the charged and uncharged defects on the overall chemistry of the materials. The most remarkable thing about this thesis is that it has shown the correlation between atomic anomalies with the localization of atomic vibration by bridging atomic anomalies with the Debye temperature. For future work, the anharmonic effects on the lattice vibration together with the phonons coupling shall be considered. We are also interested in 71 knowing the relationships between charged defects with the normal modes of vibration. Further, due to lack of time, we have not done analysis of the other thermal quantities like entropy, free energy etc., even though we have quantified these quantities. Their analysis will also remain as future work. 72

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APPENDIX A NORMAL MODES OF FREQUENCIES (in THz) USING GGA 77

Pristine ZnO Zero ZnO One ZnO Two ZnO -0.0423081758,-0.0462959555,-0.0412190924,-0.0562768455 -0.0361485350,-0.0271467254,-0.0282852453,-0.0286916724 -0.0130687870,-0.0178706666,0.0500501567,0.0664647656 1.8988343380,1.8782404068,1.6648304427,1.7296044118 1.8988343380,1.8802258439,1.7528026312,1.8036693341 1.9325425363,1.9480096419,1.8327048422,1.8042332403 1.9671324919,1.9652513087,1.8558370054,1.8434318685 1.9882133569,1.9793686547,1.9171659576,1.8728996210 2.0428084232,1.9983934819,1.9353329241,1.8950578742 2.0594394473,2.1129251608,1.9425801987,2.0370417283 2.0594394473,2.1480907896,2.0169389310,2.1415647664 2.3567238277,2.3746680431,2.0996789357,2.2372382650 2.3567238277,2.4172092480,2.1592936651,2.3014391590 2.4602397191,2.4573696324,2.2295846143,2.3123576360 2.5977463698,2.5134223742,2.3641168575,2.3346081268 2.6039325252,2.5585956638,2.3901048285,2.3976370608 2.6039325252,2.5762592039,2.4198623665,2.4148769103 2.6060084244,2.5929961334,2.4536135045,2.4262540702 2.6329749958,2.5995169397,2.5198644616,2.4369002792 2.6329749958,2.6157371286,2.5274431088,2.4839473282 2.6329749958,2.6406545905,2.5711162252,2.5264621443 2.6329749958,2.6519980304,2.5826005276,2.5282564596 2.6626631925,2.6965992070,2.5925173217,2.5632470403 2.6626631925,2.7043654754,2.6119553618,2.5801861494 2.7122337612,2.7437504782,2.6411990402,2.5941518163 2.7122337612,2.7590830553,2.6529307185,2.6340823321 2.7789483022,2.7761201621,2.6649039223,2.6418466193 2.7789483022,2.7924074001,2.6813270229,2.6909951440 2.8601058951,2.8126424653,2.7464275969,2.7284089529 2.8601058951,2.8215222390,2.7655908917,2.7432301935 3.0130045786,2.8920745383,2.7673031056,2.8070305051 3.0130045786,2.9453647137,2.7775811454,2.8307511661 3.0130045786,3.0056888088,2.8015775831,2.8416749628 3.0130045786,3.0338192847,2.9126936018,2.8757448274 3.0304621699,3.0444315512,3.0025081142,2.9622860200 3.0304621699,3.0496834627,3.0112221886,2.9685620086 3.0304621699,3.0514145167,3.0189515189,2.9845315115 3.0304621699,3.0667424983,3.0234580909,2.9963429437 3.0341475573,3.0706728529,3.0242308376,3.0232705343 3.0559398779,3.0783564233,3.0413701743,3.0238079918 3.0559398779,3.0827673231,3.0567213047,3.0498486671 3.0559398779,3.0977108217,3.0613973318,3.0594389764 3.0559398779,3.1040214133,3.0675365671,3.0762135647 3.0790002398,3.1509153208,3.0729124615,3.0951557014 3.1504190053,3.1564606457,3.0831186700,3.1073821116 3.1504190053,3.1711754341,3.0913375256,3.1432431347 3.1947655762,3.2146226530,3.1479693445,3.1527868948 3.1947655762,3.2324355795,3.1771136512,3.1631388126 3.2439409487,3.2392096216,3.1800240976,3.1863858593 3.3129185665,3.2536789779,3.1978448248,3.2042369039 3.3129185665,3.2871320253,3.1988935991,3.2293584875 3.3168631495,3.3000685181,3.2271310620,3.2405583050 78 3.3297359717,3.3143938483,3.2517081188,3.2566453574 3.3297359717,3.3164014618,3.2703671193,3.2877870823 3.3520299714,3.3496432738,3.2798393802,3.2903983538 3.3520299714,3.3719889117,3.3186552914,3.3152594844 3.3816664857,3.3861400969,3.3187571503,3.3171002186 3.3816664857,3.3932697145,3.3533808211,3.3237941618 3.3952482533,3.4022639981,3.3584526377,3.3338455137 3.3952482533,3.4037167137,3.3639425665,3.3626400228 3.4711417815,3.4262815411,3.3767454475,3.3649440283 3.4711417815,3.4296937031,3.3968365271,3.4168235952 3.4711417815,3.4848597868,3.3999196738,3.4263772016 3.4711417815,3.4915780254,3.4468812549,3.4332961590 3.4738807760,3.5092400639,3.4553445381,3.4460832424 3.5055895179,3.5168793561,3.4586414121,3.4678127018 3.5055895179,3.5222486320,3.4687196423,3.4951122326 3.5055895179,3.5335987811,3.4748710603,3.5180521240 3.5055895179,3.5439941047,3.4884131703,3.5390043548 3.7638771967,3.7279754938,3.5067646144,3.5846266871 3.8061591469,3.7529684273,3.6737048621,3.6730644108 3.8061591469,3.7893909941,3.6799797688,3.7244971855 3.8631009806,3.8077432834,3.7588673487,3.7373620146 3.8631009806,3.8345241900,3.8087356905,3.7850218068 3.8631009806,3.8484928044,3.8096289052,3.8067605335 3.8631009806,3.9009715607,3.8262463969,3.8326300633 4.0477303681,4.0271366609,3.8415473510,3.8545212339 4.0477303681,4.0537510603,3.9995507708,4.0146626667 4.0635187545,4.0576887782,4.0171723806,4.0255399262 4.0635187545,4.1034137237,4.0236311596,4.0280230904 4.0892888581,4.1174302598,4.0681201886,4.0351417996 4.0892888581,4.1230344117,4.0747569348,4.0974819933 4.6828275874,4.3344718409,4.0798904173,4.1698117602 4.6828275874,4.5869697837,4.5477771004,4.6078436962 4.7203730686,4.6127751286,4.5992446525,4.6798339408 4.7203730686,4.6325106989,4.6109993544,4.6849854001 4.7359205895,4.7134800593,4.6919377687,4.6927477567 4.7481668645,4.7316563715,4.7158738778,4.7063559890 4.7481668645,4.7389142101,4.7198915246,4.7180096936 4.7697622743,4.7406356293,4.7251927485,4.7432396160 4.7697622743,4.7761255153,4.7598954176,4.7726628339 4.9149704816,4.7829978463,4.7718682431,4.7962112923 4.9149704816,4.8775578313,4.8701845697,4.8729481416 4.9149704816,4.9131110510,4.8912707176,4.9185010233 4.9149704816,4.9349199402,4.9305125889,4.9262084793 4.9182612294,4.9353179207,4.9340665528,4.9708355803 4.9182612294,4.9520023335,4.9674872738,4.9765902477 5.1817243870,5.0207368060,4.9813982517,4.9988834117 5.1817243870,5.1445054589,5.1366540279,5.1459884142 5.2407462882,5.1470186074,5.1454014338,5.1505537185 5.2407462882,5.2670330482,5.2849281048,5.3081803366 5.4207624318,5.3407975542,5.3123668577,5.3388624877 5.4404531682,5.5025390583,5.5346192915,5.5068309949 6.3626058167,5.7610045272,5.5975414185,5.6131733524 6.3626058167,6.3012712432,6.3378657676,6.3525483978 6.3651458256,6.3157822287,6.3567605307,6.3710580348 79 6.3651458256,6.3311660045,6.3801384641,6.4160188098 6.3651458256,6.3568929852,6.4073298652,6.4538097585 6.3651458256,6.3818543930,6.4179780951,6.4736164093 6.3789835392,6.4223396614,6.4550725133,6.4738801641 6.3789835392,6.4420572541,6.4736614079,6.4934597590 6.6035275341,6.4949357637,6.5222930070,6.5606810956 6.6035275341,6.5992469523,6.6419985251,6.6915494905 6.6296290091,6.6148408186,6.6647520911,6.6931216391 6.6296290091,6.6828757418,6.7053548599,6.7423037865 6.9240275988,6.8172803482,6.8246389524,6.7627689071 6.9251047377,6.8532259263,6.9505349853,6.9903920102 6.9251047377,6.8864930885,6.9670546441,7.0369889700 6.9493433480,6.9095406618,6.9959814311,7.0531857951 6.9493433480,6.9250021360,7.0019006873,7.1028974073 7.1166407247,7.0942612326,7.1072612656,7.1432606393 7.1166407247,7.0963786883,7.1879061711,7.2849101488 7.1551594460,7.1264376775,7.1973027528,7.2954881195 7.1551594460,7.1596008598,7.2426398723,7.3046319102 7.3538064622,7.2125794981,7.2881985315,7.3511243930 7.3538064622,7.2136327901,7.2974962520,7.3747064063 7.3538064622,7.2881379588,7.3940176862,7.4348932613 7.3538064622,7.3366103396,7.4183947709,7.4587123823 7.4050626181,7.3700166019,7.4337416211,7.4593902845 7.4050626181,7.3730914703,7.4345160089,7.5027901741 7.4050626181,7.4024886112,7.4641260444,7.5223313399 7.4050626181,7.4094884756,7.4877036887,7.5720980936 7.4290453963,7.4419463711,7.5032504608,7.5886703830 7.4290453963,7.4507808727,7.5084494332,7.5900225796 7.4290453963,7.4743564374,7.5205298524,7.6016566539 7.4290453963,7.4799658832,7.5346667265,7.6449708616 7.5700503627,7.5180392102,7.6111145665,7.6510782606 7.5748430047,7.5530068355,7.6204241323,7.7069532992 7.5748430047,7.5650436109,7.6361026316,7.7170321431 7.5969147529,7.5825568361,7.6636403569,7.7177503143 7.5969147529,7.6064142355,7.6762108866,7.7651669947 7.6125337522,7.6085740464,7.6867702007,7.8195042260 7.8538269646,7.8094961529,7.8859836665,7.9883911238 7.8679963643,7.8766444570,7.9364674147,7.9936380489 11.3256168990,11.3479057839,11.4832670608,11.1419099546 11.9556125640,11.8879549660,11.9126589355,11.1911136188 12.0347442551,11.9753811627,12.0491452811,11.2641097123 12.0347442551,12.0348514016,12.1528517931,11.4581084525 12.2380332875,12.1250517174,12.1822048808,11.6811820766 12.2380332875,12.2754135388,12.2094489930,11.8145860395 12.4114753685,12.3030727174,12.3522769087,11.9460008115 12.4275945404,12.3165033085,12.3969845549,12.0230834075 12.4320616080,12.3423326665,12.4175916603,12.1667400598 12.4448456435,12.3878622594,12.5240108032,12.1992755267 12.5204193105,12.4264011210,12.5894209168,12.2534695604 12.5204193105,12.4797541481,12.6202272844,12.3158211062 12.7486585885,12.6206099155,12.6649146286,12.5313810052 12.7486585885,12.6844102168,12.6762214512,12.5791689784 12.7486585885,12.6850071445,12.7759502167,12.6450471069 12.7486585885,12.7486242389,12.8578468465,12.7152356332 80 12.8934883678,12.7935293477,12.8614026123,12.7739628269 12.8950737974,12.8414759616,12.9191837310,12.8139570883 12.9472442476,12.8526916757,12.9461339466,12.8230458264 12.9472442476,12.9007372630,13.0273386646,12.8458514426 13.0313922794,12.9212738095,13.0411739010,12.9201530683 13.0854927552,12.9716097789,13.0480496721,12.9385015857 13.0854927552,12.9882265778,13.1139282862,13.0958120383 13.0854927552,12.9971730695,13.1615832157,13.1287217468 13.0854927552,13.0336244267,13.1628886840,13.1323752646 13.0964920965,13.0475578682,13.2002069403,13.1696962645 13.1058917920,13.0565943834,13.2606829065,13.2556456522 13.1058917920,13.0647075212,13.2616984080,13.2915920056 13.1110912295,13.0938019610,13.2672254740,13.2957995587 13.1110912295,13.0938784900,13.3087487973,13.3236499637 13.1962674056,13.1714909720,13.3177175607,13.3515694804 13.1962674056,13.1770287561,13.3564383905,13.3780340692 13.2073297945,13.1885297630,13.3583649375,13.4262105099 13.2073297945,13.1933755704,13.3806583201,13.4517114634 13.2267831407,13.2370092896,13.3888639649,13.4584252645 13.2267831407,13.2608200477,13.3952406777,13.4640688353 13.2473398512,13.2641876061,13.4040566046,13.4752189555 13.2473398512,13.2768545525,13.4420510870,13.5162135752 13.3646538785,13.2983499206,13.4562260727,13.5623930694 13.3646538785,13.3236364435,13.4740527724,13.5829257470 13.3646538785,13.3359789084,13.5361397245,13.5844799080 13.3646538785,13.3434623748,13.5397851750,13.6243575046 13.3798749278,13.3505132621,13.5457142318,13.6470235412 13.3798749278,13.3791279753,13.5783744544,13.6936593489 13.3798749278,13.3851869131,13.5887070412,13.6992642982 13.3798749278,13.4162035275,13.6040440804,13.7052634271 13.4232776615,13.4301074236,13.6271880090,13.7464372316 13.4232776615,13.4603643498,13.6779607821,13.7599409215 13.4429278801,13.4937633905,13.6854673073,13.7972342277 13.4429278801,13.5386902977,13.7076463338,13.8124887853 13.5460828367,13.5521777374,13.7265524130,13.8423591879 13.5575827992,13.5619254315,13.7406047072,13.8530476288 13.6479165090,13.6291000933,13.7886418359,13.8654783827 13.6479165090,13.6353217659,13.8228465526,13.9139507090 13.6536039969,13.6553882173,13.8243859822,13.9280985177 13.6536039969,13.6621817981,13.8784654118,13.9875657364 13.6669743423,13.6884734167,13.8817423970,14.0023012998 13.6669743423,13.7167934312,13.9123739644,14.0201189802 13.6913007195,13.7336462957,13.9310892164,14.0457043258 13.6968591188,13.7436906471,13.9378103002,14.0640920970 13.8276406366,13.7766893044,13.9750116525,14.1044708157 13.8276406366,13.8036509252,13.9761110041,14.1427775928 13.8276406366,13.8249394738,14.0063476650,14.1925188861 13.8276406366,13.8655847970,14.0270018566,14.2112050162 13.8448532838,13.8675397759,14.0917411205,14.2487226492 13.8448532838,13.8730638397,14.1155333716,14.3072789470 13.8448532838,13.8975047330,14.1225777778,14.3119174423 13.8448532838,13.9190347041,14.1352659066,14.4026315583 13.8802830932,13.9469489545,14.1430048930,14.4164351070 13.8802830932,13.9619505233,14.2215687326,14.4672042345 81 13.8864559786,14.0012769101,14.2601394455,14.5389585893 13.8864559786,14.0744502461,14.2607618552,14.5657608703 13.9586650970,14.1094001331,14.3083863716,14.6022249153 13.9586650970,14.2244263043,14.4238809812,14.6042675441 14.2269162831,14.2306077162,14.4354663934,14.6505414194 14.2269162831,14.2441658868,14.4362102048,14.6776265472 14.2269162831,14.2659887765,14.4436503753,14.6878337512 14.2269162831,14.3179819002,14.5025157032,14.8535862979 14.2825060690,14.3901859606,14.5215047079,14.8762357575 14.3848480107,14.4263634737,14.7033369286,14.9825739311 14.3848480107,14.4631513727,14.7295190187,14.9873515458 14.6372295568,14.6427271526,14.7381796040,15.0272094069 14.7185855431,14.6846813107,14.8864060576,15.0919512003 14.7185855431,14.7771350845,14.9517161275,15.1195093820 14.8650145149,14.8329388562,14.9858256905,15.2122515870 14.8650145149,14.8381787860,15.0311903559,15.2915005454 14.9275135730,14.8820745286,15.0644062933,15.2974084774 14.9275135730,14.9025040743,15.1126710023,15.3081992461 14.9275135730,14.9488650253,15.1190738183,15.3924649255 14.9275135730,15.0027096048,15.2503260757,15.4350058459 15.0907567422,15.0126863150,15.2780537310,15.4587384579 15.0907567422,15.1243822405,15.2974854438,15.4670223542 15.1097157835,15.1583597500,15.3006137582,15.4877174097 15.1097157835,15.1613976463,15.3169817753,15.5030551966 15.2637667664,15.2728211065,15.3419277764,15.5373446102 15.2637667664,15.2773309467,15.3856702229,15.5739329708 15.2851259583,15.2932346513,15.4270900266,15.5924224469 15.2851259583,15.2971714215,15.4485266217,15.6421640094 15.3430528279,15.3322353306,15.4842185517,15.7469805920 15.3430528279,15.3970787624,15.5081306255,15.7749642546 15.4270702098,15.4076464088,15.5389053314,15.7883996706 15.4270702098,15.4094056801,15.5421201866,15.7977549053 15.4270702098,15.4787044578,15.5786330567,15.8438882946 15.4270702098,15.4832851452,15.6447220655,15.9009358184 15.5052018881,15.5633875353,15.6896142151,15.9424289268 15.5052018881,15.6166262906,15.6923271515,15.9746750458 15.6004849231,15.6302504441,15.7344620997,15.9850714675 15.6004849231,15.6843359034,15.7542612319,16.0387153117 15.6176108270,15.6855553962,15.7929260688,16.0699626997 15.6176108270,15.6870413627,15.8290900336,16.0892686766 15.6993509590,15.7740423562,15.8874431314,16.0988929611 15.6993509590,15.8701301556,15.9371641184,16.1144635182 15.7730458756,15.8753294241,15.9658310827,16.1430962396 16.0257295284,15.9708749342,16.0037999671,16.1778413364 16.0747185466,16.0238724048,16.0127164450,16.1883653356 16.0747185466,16.0252751862,16.0465284597,16.2030307555 16.0747185466,16.0262239702,16.0727688236,16.2194503219 16.0747185466,16.0581942732,16.0759641147,16.2810390159 16.0854702672,16.0883313035,16.0987074894,16.3018700365 16.0881069250,16.0906715330,16.1539806317,16.3304749563 16.0881069250,16.0995293461,16.1780500075,16.3704080211 16.0881069250,16.1016999794,16.2042476287,16.3826450549 16.0881069250,16.1139791982,16.2234559956,16.4285103920 16.0926742893,16.1388341027,16.2353385739,16.4399924224 82 16.0926742893,16.1457251863,16.2580582877,16.4693610019 16.0972511507,16.1578932780,16.2704980691,16.5410968539 16.0984293249,16.1710494426,16.2777130400,16.6479468080 16.0984293249,16.1800258582,16.2964081620,16.6684350429 16.1129252928,16.1908452485,16.3227542873,16.6751747149 16.1352640301,16.2069030130,16.3610982356,16.7045580823 16.1352640301,16.2464467866,16.3693864370,16.8205984551 16.1525552757,16.2605914028,16.3714513610,16.9755461850 16.1525552757,16.2671974304,16.4271803174,17.3901078025 16.1807624991,16.2765227497,16.4325206368,17.4447301273 16.1982559316,16.2978835770,16.4611710510,17.4950215073 16.2028576490,16.3127058345,16.5076267968,17.5776101196 16.2456920628,16.4119015585,16.5204313465,17.6192472556 16.2456920628,16.4171656529,16.5373201888,17.6747882301 16.2456920628,16.4871395088,16.5637085674,17.7500351891 16.2456920630,16.5454377295,16.5883762344,17.7947739722 16.3993751221,16.5848247062,16.6274713608,17.8174694601 16.3993751221 16.4910709389 16.4910709389 Appendices

APPENDIX B DATA OF THERMAL PROPERTIES 83 USING GGA Pristine ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 96 num_modes: 147456 num_integrated_modes: 147344 zero_point_energy: 539.5399588 high_T_entropy: 8271.0681917 thermal_properties: - temperature: 0.0000000 free_energy: 539.5399588 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 539.5399588

- temperature: 10.0000000 free_energy: 539.5324126 entropy: 2.3559990 heat_capacity: 5.2668203 energy: 539.5559726

- temperature: 20.0000000 free_energy: 539.4618307 entropy: 14.6118731 heat_capacity: 44.6952266 energy: 539.7540682

- temperature: 30.0000000 free_energy: 539.1587532 entropy: 50.5742665 heat_capacity: 148.3163627 energy: 540.6759812

- temperature: 40.0000000 free_energy: 538.3676295 entropy: 111.0642462 heat_capacity: 279.6524145 energy: 542.8101994

- temperature: 50.0000000 free_energy: 536.8849716 entropy: 187.3813085 heat_capacity: 407.2654336 energy: 546.2540370 84 Zero ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 95 num_modes: 145920 num_integrated_modes: 145912 zero_point_energy: 529.4545797 high_T_entropy: 8219.6718722 thermal_properties: - temperature: 0.0000000 free_energy: 529.4545797 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 529.4545797

- temperature: 10.0000000 free_energy: 529.4505849 entropy: 1.6310553 heat_capacity: 4.8897643 energy: 529.4668954

- temperature: 20.0000000 free_energy: 529.3862913 entropy: 14.3598774 heat_capacity: 47.2033028 energy: 529.6734888

- temperature: 30.0000000 free_energy: 529.0780932 entropy: 51.9320062 heat_capacity: 153.5999980 energy: 530.6360534

- temperature: 40.0000000 free_energy: 528.2648103 entropy: 114.1019369 heat_capacity: 285.8630996 energy: 532.8288878

- temperature: 50.0000000 free_energy: 526.7445211 entropy: 191.8197956 heat_capacity: 413.5348170 energy: 536.3355108 85 One ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 95 num_modes: 145920 num_integrated_modes: 145912 zero_point_energy: 532.5241970 high_T_entropy: 8228.2772567 thermal_properties: - temperature: 0.0000000 free_energy: 532.5241970 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 532.5241970

- temperature: 10.0000000 free_energy: 532.5192385 entropy: 2.0134932 heat_capacity: 6.1097986 energy: 532.5393735

- temperature: 20.0000000 free_energy: 532.4397802 entropy: 17.4914677 heat_capacity: 55.1861571 energy: 532.7896095

- temperature: 30.0000000 free_energy: 532.0790283 entropy: 59.3386573 heat_capacity: 166.1947954 energy: 533.8591880

- temperature: 40.0000000 free_energy: 531.1720976 entropy: 125.2651592 heat_capacity: 298.8911490 energy: 536.1827039

- temperature: 50.0000000 free_energy: 529.5255810 entropy: 205.7397346 heat_capacity: 424.9659940 energy: 539.8125677 86 Two ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 95 num_modes: 145920 num_integrated_modes: 145912 zero_point_energy: 537.8529242 high_T_entropy: 8211.4099238 thermal_properties: - temperature: 0.0000000 free_energy: 537.8529242 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 537.8529242

- temperature: 10.0000000 free_energy: 537.8475479 entropy: 2.1459394 heat_capacity: 6.2356803 energy: 537.8690073

- temperature: 20.0000000 free_energy: 537.7678632 entropy: 17.2666214 heat_capacity: 53.8632805 energy: 538.1131956

- temperature: 30.0000000 free_energy: 537.4126961 entropy: 58.4599817 heat_capacity: 164.4041899 energy: 539.1664955

- temperature: 40.0000000 free_energy: 536.5173127 entropy: 123.8540048 heat_capacity: 296.9735262 energy: 541.4714729

- temperature: 50.0000000 free_energy: 534.8871862 entropy: 203.8830522 heat_capacity: 422.8783945 energy: 545.0813388 Appendices

APPENDIX C NORMAL MODES OF FREQUENCIES(in THz)USING LDA+U 87

Pristine ZnO Zero ZnO One ZnO Two ZnO -0.0945603544 -0.0988626392 -0.0874875935 -0.1049045983 -0.0660932297 -0.0928478891 -0.0795519719 -0.0371843391 0.1964719939 -0.0375807343 -0.0351313642 -0.0349700472 2.1249420509 2.0927111781 1.9032665231 1.9195448877 2.1602056466 2.1184696027 1.9888743376 1.9895307827 2.1659505224 2.1359939276 2.0142138843 2.0142996375 2.1659505224 2.2366391501 2.1130041021 2.0678511238 2.3512377864 2.2401342967 2.1627272606 2.1248800136 2.4151197203 2.2555799470 2.1642495686 2.2192476500 2.5494551652 2.3873800316 2.2590217386 2.2268993363 2.5494551652 2.4324917347 2.3131729339 2.2798461130 2.8409753436 2.6713412596 2.4210470515 2.4370721899 2.8409753438 2.7267233210 2.4810600525 2.4905202491 2.8783911805 2.7583877925 2.6166490314 2.5619483701 2.9398284024 2.8365485242 2.6768579565 2.6037269795 2.9398284024 2.8524686803 2.7320521154 2.6489534654 3.0178460690 2.8900495435 2.7512511204 2.6840666845 3.0209954474 2.9472478094 2.8211046854 2.7054427645 3.0209954474 2.9634396825 2.8693901311 2.7511920420 3.0209954474 2.9983577767 2.8756994733 2.7730078751 3.0209954475 3.0061872419 2.8862921150 2.8078656350 3.0552046761 3.0171324772 2.9015836150 2.8123774486 3.1617309659 3.0943731888 2.9355090826 2.8548511996 3.1617309659 3.0963392842 2.9758551700 2.8570422484 3.2562787917 3.1391793911 2.9826938849 2.9001238523 3.2562787917 3.1418802752 3.0067054724 2.9558162598 3.2865133826 3.1442146629 3.0094101350 3.0434340667 3.2865133826 3.1672339598 3.0913512078 3.0530780300 3.3560170443 3.2406710904 3.1253532037 3.1032871074 3.3560170443 3.2517280816 3.1317255874 3.1637041152 3.5969338063 3.2914381622 3.1571158986 3.2000883197 3.5969338063 3.3810960172 3.1626677305 3.2085259392 3.5969338063 3.4210343146 3.2507205316 3.2179938502 3.5969338063 3.4636627189 3.3261587295 3.2911685220 3.6143915628 3.4850847270 3.4332198533 3.3457763157 3.6143915628 3.4997992259 3.4392672492 3.3490480921 3.6143915628 3.5245749901 3.4459372237 3.3824237474 3.6143915631 3.5289063838 3.4543247598 3.4024743616 3.6483323718 3.5373178851 3.4729616821 3.4333572367 3.6483323718 3.5561708572 3.4749527625 3.4451096227 3.6483323718 3.5605426364 3.4996704529 3.4547135922 3.6483323718 3.5661403278 3.5044183854 3.4929770334 3.6738570715 3.5748275084 3.5155839206 3.5073310058 3.7127486008 3.6183241702 3.5182281603 3.5181343473 3.7127486008 3.6199164659 3.5383813333 3.5330692973 3.7178440241 3.6257772886 3.5771751518 3.5761313508 3.7583065258 3.6720792680 3.5879563710 3.5773927227 3.7583065258 3.7029110125 3.6185484419 3.6130907455 3.7844437932 3.7329107543 3.6489892733 3.6264616699 3.8888025203 3.7737863787 3.6985749795 3.6993025248 3.8888025203 3.7881173687 3.7087721367 3.7134229019 3.8891161682 3.8045984436 3.7498736255 3.7253156016 88 3.8891161682 3.8438534029 3.7547257918 3.7355200333 3.9304851246 3.8603549678 3.7737175601 3.7554194399 3.9304851246 3.8632654071 3.8031680107 3.7734147504 3.9465755527 3.8720131110 3.8211207222 3.7906054136 3.9465755527 3.8834326228 3.8243478988 3.7980186801 3.9599831351 3.9014201462 3.8338226242 3.8173837659 3.9717660849 3.9138117423 3.8383389219 3.8379782342 3.9717660849 3.9544260251 3.8604166136 3.8414747533 4.0469847086 3.9765099380 3.8779147904 3.8815894904 4.0469847086 3.9896204971 3.9021070588 3.8940131055 4.0469847086 4.0071476145 3.9394488279 3.9379748984 4.0469847086 4.0330183437 3.9624920837 3.9517240111 4.1228706863 4.0396459771 3.9782744293 3.9663158840 4.1541112100 4.0690480160 3.9930122296 3.9824706604 4.1541112100 4.0705300723 3.9990829780 4.0027802081 4.1541112100 4.0730002357 4.0248368102 4.0078945005 4.1541112100 4.0974641977 4.0645499690 4.0736219518 4.4718139257 4.3723398862 4.0678497667 4.0972565663 4.4718139257 4.3993599596 4.3185175235 4.3008847539 4.5207981950 4.4625218180 4.3417674729 4.3745078958 4.5953448176 4.4766073435 4.4183815803 4.4121119902 4.5953448176 4.5173757601 4.4747038233 4.4410906076 4.5953448176 4.5702676346 4.5040899938 4.4626045837 4.5953448176 4.5966867295 4.5171117824 4.4943596284 4.7348270844 4.6568835477 4.5495811776 4.5165275435 4.7348270844 4.6944505905 4.6009388290 4.6212396395 4.8408454211 4.7052153216 4.6669325171 4.6430156274 4.8408454211 4.7745261658 4.6732980224 4.7010382649 4.8683599878 4.7893786482 4.7219759719 4.7086552868 4.8683599882 4.7961908501 4.7245564192 4.7345421333 5.1910030517 4.8503846602 4.7846476200 4.8042375419 5.1910030517 5.2601254732 5.2645189390 5.2471814955 5.5503198120 5.4022544484 5.3351728607 5.3351178246 5.5503198120 5.4309189544 5.4142614377 5.5124239896 5.6183010050 5.4682254996 5.4677599101 5.5255742780 5.6183010051 5.4804864458 5.4813150152 5.5429259671 5.6215815938 5.5517423745 5.5335083237 5.5782533678 5.6215815938 5.6255322260 5.5847830221 5.5843029687 5.6659913088 5.6297675886 5.6130431355 5.6176298465 5.6659913088 5.6536970249 5.6592134031 5.6472077908 5.7064188696 5.6659818221 5.6633118881 5.6549715107 5.7276902889 5.6826498018 5.6788635927 5.7122088383 5.7276902889 5.7775509801 5.7678776328 5.7961312313 5.9175190633 5.8226003802 5.8386091509 5.8277064603 5.9175190633 5.8645747951 5.8633526179 5.8890590019 5.9175190633 5.8653031859 5.8656315214 5.9054842843 5.9175190634 5.8911061657 5.8713444805 5.9084961159 5.9830337159 5.9181694522 5.8979599319 5.9235136378 6.0091058217 6.0534931763 6.0931739596 6.0938592253 6.2519275763 6.1791390409 6.1774301761 6.1858211201 6.2519275763 6.2337143294 6.1881636360 6.2212384982 6.8924958846 6.3389841121 6.2351621742 6.2495647495 6.8924958846 6.9243611459 6.9932249981 6.9754385177 6.9213088464 6.9478541781 7.0049857553 7.0578514218 89 6.9213088464 6.9627188788 7.0113097799 7.0667600306 6.9225375397 6.9732352338 7.0171827744 7.0771959063 6.9225375397 7.0277766801 7.0726157582 7.1177505661 6.9225375397 7.0326373762 7.0984752386 7.1382460342 6.9225375397 7.0414064919 7.1002906049 7.1692912232 7.1376783841 7.1304294353 7.1635004673 7.2300655754 7.1376783841 7.2593717677 7.3075405352 7.3485746049 7.2101746031 7.2785222444 7.3296364329 7.3850685048 7.2101746031 7.2948243438 7.3391840530 7.4408961375 7.4366871322 7.4966575697 7.4803058175 7.4471025377 7.4366871322 7.5022780666 7.6391675023 7.7002881441 7.5067784085 7.5621167240 7.6652735609 7.7456766665 7.5242789143 7.5677855984 7.6778112328 7.7506284653 7.5242789143 7.5863742825 7.6960010300 7.8407001019 7.7712060987 7.7679570089 7.8067920063 7.9029343777 7.7712060987 7.8004697585 7.9111215816 8.0339522213 7.8550877591 7.8544553311 7.9442611587 8.0535089126 7.8550877591 7.8863558781 7.9854364162 8.0653895545 7.9625178645 7.9282034032 8.0096528576 8.1252804322 7.9625178645 7.9430591403 8.0502600153 8.1576778965 7.9625178645 8.0325896841 8.1510977279 8.2067461996 7.9625178645 8.0333250115 8.1517251323 8.2163900886 8.0344087534 8.1047538259 8.1718879709 8.2437285913 8.0344087534 8.1336841137 8.2018998271 8.2796621546 8.0344087534 8.1410480305 8.2136934145 8.3141413336 8.0344087534 8.1665140264 8.2424595730 8.3773820196 8.0980634827 8.1670305420 8.2639130095 8.3919921828 8.0980634827 8.1740532043 8.2883639660 8.3949605780 8.0980634827 8.2032169403 8.2935124381 8.4363503800 8.0980634827 8.2457668994 8.3135624156 8.4415559750 8.1804233071 8.2716458303 8.3819254791 8.4779701310 8.1804233071 8.3025681814 8.4032552051 8.5030876161 8.2383022487 8.3127257165 8.4338602514 8.5233415136 8.2383022487 8.3529668955 8.4345483047 8.5663089010 8.2781957797 8.3616083432 8.4767387046 8.6071633768 8.4295193671 8.4109512429 8.4993443325 8.6521502164 8.5258898518 8.6309654149 8.7352656326 8.8397049654 8.5893873240 8.6658454523 8.7506276711 8.8585177522 12.5849948567 12.8924184433 13.0851997159 12.6663967322 13.2774591943 13.4717726398 13.5251432206 12.7167465050 13.2774591943 13.5346814207 13.6353801893 12.8114173422 13.3888975708 13.6083651614 13.7496233185 13.1700353507 13.4697322777 13.6891889931 13.8992833430 13.4174482707 13.4697322777 13.8052319022 13.9315951122 13.5070328522 13.8085075131 13.9452679670 13.9784815697 13.6440276125 13.8120291754 13.9959339204 14.0753042536 13.6907702133 13.8226704723 14.0362183872 14.1984714263 13.8550134811 13.8322826773 14.0920372880 14.2503053529 13.8948707184 13.8896438342 14.1594399491 14.2829778442 13.9289603839 13.8896438342 14.1639448279 14.3375520336 14.0616449434 13.9643066353 14.1969736481 14.4190516704 14.1980362314 13.9643066353 14.2101281166 14.4215731772 14.3658661491 13.9643066353 14.2775005304 14.4399991796 14.4145848823 13.9643066353 14.3558316367 14.4655810873 14.4211456983 90 14.2476613467 14.4373376483 14.6209254074 14.4646826153 14.2529648624 14.5090586342 14.6406889405 14.5060835311 14.2891165448 14.5314036749 14.6459554703 14.5359696300 14.2891165448 14.5790760662 14.7418509672 14.5786539442 14.3077225363 14.6073879967 14.7645551172 14.6231591075 14.3661648070 14.6223242420 14.7769857954 14.7300935432 14.4209478378 14.6644368624 14.8502944755 14.8010332273 14.4209478378 14.6858593658 14.8716919647 14.8974444518 14.4209478378 14.6969297058 14.8932567920 14.9183400331 14.4209478378 14.7114390802 14.9620226891 14.9489978865 14.4286849181 14.7294437088 14.9831432424 14.9638395519 14.4286849181 14.7349215494 14.9902494172 15.0695246556 14.4491658314 14.7434561593 15.0004270862 15.0796982075 14.4491658314 14.7711784989 15.0255371494 15.1084120892 14.4539040097 14.7790705724 15.0522863892 15.1189290417 14.4539040097 14.8070987744 15.0597791593 15.1560995794 14.4554011939 14.8296729763 15.0621294355 15.1842474525 14.4554011939 14.8430604264 15.0659706048 15.1991780723 14.5146585650 14.8667068679 15.1044008899 15.2009486819 14.5146585650 14.8723992583 15.1200287018 15.2403167990 14.5195294841 14.8838910071 15.1244946962 15.2600686692 14.5195294841 14.9003053395 15.1518494367 15.2698063089 14.6444163801 14.9179429609 15.1562985100 15.3496676980 14.6444163801 14.9404123949 15.2072963402 15.3618683240 14.6444163801 14.9546742807 15.2201556234 15.3874157696 14.6444163801 14.9588487047 15.2311065409 15.4116933347 14.6521569067 14.9760200629 15.2676075914 15.4510072561 14.6521569067 14.9807640201 15.2710528137 15.4723430978 14.6521569067 14.9881655226 15.2786475594 15.4897569858 14.6521569067 15.0260546824 15.3056824270 15.5170834072 14.7110971586 15.0560729051 15.3273751159 15.5265015393 14.7110971586 15.0883944708 15.3822934769 15.5579341260 14.7176622955 15.0922272457 15.4156433035 15.6137224286 14.7176622955 15.1349637946 15.4204090073 15.6338440654 14.8263256706 15.1501888231 15.4443371248 15.6450875029 14.8270682874 15.2061412143 15.4788232654 15.6713145586 14.9641778186 15.3058801532 15.5472156020 15.7225889017 14.9641778186 15.3074962830 15.5530326461 15.7387749585 14.9671386250 15.3292113290 15.5683707875 15.7524795298 14.9671386250 15.3452279088 15.6217677652 15.8006125336 14.9794962086 15.3657732643 15.6306414505 15.8161164928 14.9794962086 15.3923953307 15.6740105876 15.8464019263 14.9951538804 15.4194369643 15.6953982413 15.8755205923 15.0218646177 15.4294491516 15.7108575602 15.9068660024 15.1348212931 15.4390614210 15.7192493619 15.9675639965 15.1348212932 15.4580506400 15.7417425900 16.0136193949 15.1364752129 15.4613034353 15.7956045922 16.0336391712 15.1364752131 15.4881491814 15.7962143196 16.0447089521 15.1450241683 15.5016885170 15.8169055666 16.1283589612 15.1450241683 15.5470895590 15.8936630314 16.1456524685 15.1450241683 15.5854969031 15.9022352626 16.2247899289 15.1450241683 15.6051208026 15.9121778296 16.2302092149 15.1489557074 15.6381713183 15.9543738940 16.3201555995 15.1489557074 15.6560224958 16.0007255091 16.3712202144 91 15.1489557074 15.6715034582 16.0244694932 16.4294837703 15.1489557074 15.6969770654 16.0327663541 16.4428630273 15.2678401049 15.8102300612 16.1138201393 16.4515093277 15.2678401049 15.9523221984 16.1723809293 16.5437908256 15.5548240833 15.9574114069 16.2367547722 16.5519850493 15.5548240833 15.9746706491 16.2689586360 16.5664578942 15.5548240833 15.9898569816 16.2890634086 16.6794619042 15.5548240833 16.0499924512 16.3536798937 16.7420546694 15.6211985593 16.0541721552 16.3806915569 16.7676902820 15.7577574356 16.1526971306 16.5161320555 16.9257747080 15.7577574357 16.1980593098 16.5319661187 16.9432147301 16.0042578711 16.3070800836 16.5633273450 16.9588895963 16.1124180540 16.4247649424 16.7229724816 17.0081200134 16.1124180540 16.5243067803 16.7893989602 17.0318547925 16.2720838265 16.5268875810 16.8367982239 17.2065970785 16.2720838265 16.5771728899 16.8462779087 17.2592419005 16.3194272263 16.5861018853 16.8738816767 17.3030052252 16.3194272266 16.6016702702 16.9323365227 17.3133277190 16.3194272266 16.7103299860 16.9936385890 17.3870077954 16.3194272266 16.7638347752 17.1428031221 17.4082886499 16.4777677675 16.7867793199 17.1570640316 17.4877038092 16.4777677675 16.8401110646 17.1903607283 17.4963757363 16.4909888675 16.9171439889 17.2637906018 17.5110428215 16.4909888675 16.9322487245 17.2716873427 17.5469898919 16.8160991295 17.1202518777 17.3003088225 17.5560547122 16.8160991297 17.1495606766 17.3235295264 17.6108379924 16.8390232515 17.1595220141 17.3658084170 17.6378429488 16.8390232515 17.1784148076 17.3822142475 17.6814730473 16.8919687621 17.2112054153 17.4445374587 17.8203870633 16.8919687623 17.2443906820 17.4702749055 17.8812449002 16.9950855684 17.2799929066 17.4768035489 17.8861445768 16.9950855685 17.2992674997 17.4869370845 17.9150093718 16.9950855685 17.3272470698 17.5732110087 17.9409583780 16.9950855685 17.3965497862 17.6137102755 18.0225000460 17.0755372282 17.4797191297 17.6944610239 18.0773680240 17.0755372282 17.5212522235 17.7071873477 18.0839745526 17.1431734402 17.5682081871 17.7600121236 18.1161336363 17.1431734404 17.5831660601 17.7885564721 18.1775738538 17.2562240505 17.6009203050 17.8115876147 18.2445784541 17.2562240505 17.6171122083 17.8965131290 18.2684680273 17.2720381498 17.7464298000 17.9339959111 18.2797006731 17.2720381498 17.8269233631 17.9898351111 18.2854823833 17.4240362801 17.8704236265 18.0565663175 18.3403197266 17.7343366574 18.0221379860 18.0858021789 18.3822900726 17.8305356484 18.0918555200 18.1769840939 18.4102265354 17.8467760819 18.1039231407 18.1964644963 18.4257465881 17.8467760819 18.1048891602 18.2081859476 18.4698094421 17.8467760819 18.1439277823 18.2322444381 18.5164939194 17.8467760819 18.1576285025 18.2677711692 18.5376151559 17.8574539165 18.1751726085 18.3119211514 18.5831366546 17.8602996225 18.1818882050 18.3566168233 18.6048781190 17.8602996225 18.1939674842 18.3674035754 18.6189545961 17.8602996225 18.2042162440 18.3971446012 18.6831354235 17.8602996225 18.2239249946 18.4192535559 18.7034847830 92 17.8690516795 18.2347727914 18.4204455999 18.7463713971 17.8912899292 18.2448842174 18.4340733339 18.8755474102 17.8912899292 18.2493732840 18.4694756123 18.9101665766 17.9081742080 18.2727778466 18.4845372985 18.9463876291 17.9081742080 18.2900266295 18.5070228690 18.9734694914 17.9719243230 18.2934659086 18.5554992437 19.0676261947 17.9719243230 18.3473085641 18.5585096055 19.1538279497 17.9824828704 18.3627895148 18.5928956698 19.2725460886 17.9824828704 18.3811448223 18.6411853711 19.6617841972 18.0561757830 18.3997562234 18.7014592935 19.7548570623 18.0561757831 18.4111635099 18.7257030974 19.8082870335 18.0561757831 18.4269742175 18.7362562835 19.8884749133 18.0561757831 18.6061634480 18.8097065999 19.9329974238 18.0912730117 18.6378922948 18.8386169446 20.0651943401 18.1005457434 18.7237695755 18.9055978467 20.1323919924 18.3094022041 18.7794013927 18.9423216323 20.1694749464 18.3094022042 19.1113092709 19.1607488696 20.3320747537 18.3986245718 18.4489079543 18.4489079543 Appendices

APPENDIX D DATA OF THERMAL PROPERTIES 93 USING LDA+U Pristine ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 96 num_modes: 147456 num_integrated_modes: 147352 zero_point_energy: 599.4031657 high_T_entropy: 7990.7900681 thermal_properties: - temperature: 0.0000000 free_energy: 599.4031657 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 599.4031657

- temperature: 10.0000000 free_energy: 599.3983452 entropy: 1.5519662 heat_capacity: 3.6875714 energy: 599.4138649

- temperature: 20.0000000 free_energy: 599.3516808 entropy: 9.4444211 heat_capacity: 27.5867280 energy: 599.5405693

- temperature: 30.0000000 free_energy: 599.1588451 entropy: 32.3074206 heat_capacity: 98.0557181 energy: 600.1280677

- temperature: 40.0000000 free_energy: 598.6400484 entropy: 74.5125746 heat_capacity: 203.9150052 energy: 601.6205514

- temperature: 50.0000000 free_energy: 597.6161833 entropy: 132.4045891 heat_capacity: 319.2854638 energy: 604.2364128 94 Zero ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 95 num_modes: 145920 num_integrated_modes: 145920 zero_point_energy: 596.9697582 high_T_entropy: 7931.1452413 thermal_properties: - temperature: 0.0000000 free_energy: 596.9697582 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 596.9697582

- temperature: 10.0000000 free_energy: 596.9642186 entropy: 1.8518054 heat_capacity: 4.5502244 energy: 596.9827366

- temperature: 20.0000000 free_energy: 596.9076304 entropy: 11.4334492 heat_capacity: 32.8905855 energy: 597.1362994

- temperature: 30.0000000 free_energy: 596.6788996 entropy: 37.7079136 heat_capacity: 109.7722428 energy: 597.8101370

- temperature: 40.0000000 free_energy: 596.0863751 entropy: 83.8317708 heat_capacity: 218.9855160 energy: 599.4396459

- temperature: 50.0000000 free_energy: 594.9515464 entropy: 145.1608387 heat_capacity: 334.6457935 energy: 602.2095883 95 One ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 95 num_modes: 145920 num_integrated_modes: 145916 zero_point_energy: 602.4530219 high_T_entropy: 7933.1240164 thermal_properties: - temperature: 0.0000000 free_energy: 602.4530219 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 602.4530219

- temperature: 10.0000000 free_energy: 602.4481880 entropy: 1.8700509 heat_capacity: 5.3028730 energy: 602.4668885

- temperature: 20.0000000 free_energy: 602.3840673 entropy: 13.2817526 heat_capacity: 38.6168196 energy: 602.6497024

- temperature: 30.0000000 free_energy: 602.1204535 entropy: 42.9677036 heat_capacity: 120.7616527 energy: 603.4094846

- temperature: 40.0000000 free_energy: 601.4574660 entropy: 92.5865395 heat_capacity: 231.8321000 energy: 605.1609275

- temperature: 50.0000000 free_energy: 600.2202979 entropy: 156.7441873 heat_capacity: 346.8404145 energy: 608.0575073 96 Two ZnO

# Thermal properties / unit cell (natom)

unit: temperature: K free_energy: kJ/mol entropy: J/K/mol heat_capacity: J/K/mol

natom: 95 num_modes: 145920 num_integrated_modes: 145912 zero_point_energy: 609.6228800 high_T_entropy: 7917.7646916 thermal_properties: - temperature: 0.0000000 free_energy: 609.6228800 entropy: 0.0000000 heat_capacity: 0.0000000 energy: 609.6228800

- temperature: 10.0000000 free_energy: 609.6187398 entropy: 1.7173060 heat_capacity: 5.1195001 energy: 609.6359129

- temperature: 20.0000000 free_energy: 609.5563090 entropy: 13.1980537 heat_capacity: 39.3998787 energy: 609.8202701

- temperature: 30.0000000 free_energy: 609.2907730 entropy: 43.4907738 heat_capacity: 122.9294397 energy: 610.5954962

- temperature: 40.0000000 free_energy: 608.6190152 entropy: 93.7979266 heat_capacity: 234.2762718 energy: 612.3709323

- temperature: 50.0000000 free_energy: 607.3670587 entropy: 158.4512271 heat_capacity: 348.7284991 energy: 615.2896200