Computational and Quantum Mechanical Investigations of ...

Computational and Quantum Mechanical

Investigations of Oxide and Halide Perovskites

using First-principles Study

A dissertation Submitted for Partial Fulfilment of

The Requirements for the degree of

Doctor of Philosophy (Ph.D.)

In

Physics

By

NAZIA ERUM

Roll No. Ph.D.-1303

Under the Kind Supervision of

Prof. Dr. Muhammad Azhar Iqbal

Department of Physics

University of the Punjab,

Lahore, Pakistan.

August 2018

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CERTIFICATE

It is certified that the research work contained in this thesis has been performed by Miss Nazia

Erum, Ph.D. scholar in the subject of Physics, Roll No. Ph.D.-1303, enrolled in Fall 2013 in the

Department of Physics, University of the Punjab, Lahore, hereby declare that the matter printed in

this thesis entitled “Computational and Quantum Mechanical Investigations of Oxide and

Halide Perovskites using First-principles Study” is the result of my own original investigation,

no part of this thesis falls under plagiarism and has not been submitted as a whole or in part for

any degree or diploma at this or any other university. If found otherwise, I will be responsible for

the consequences.

Nazia Erum

Supervisor Prof. ® Dr. Muhammad Azhar Iqbal

Department of Physics,

University of the Punjab, Lahore, ______________________

Pakistan.

Chairman Dr. Mahmood-ul-Hassan

Department of Physics,

University of the Punjab, Lahore, ______________________

Pakistan.

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DEDICATIONS

I dedicate this thesis to my spiritual father, great saint HAZRAT ALLAMA ASAD NIZAMI

CHISTI SULEMANI Rehmatullahi A'laih spiritual son of Sheikh-ul-Islam wal-Muslimeen

Hazrat Baba Fareed-ud-din Masood Ganjsaker Chisti Farooqi Rehmatullahi A'laih and to my

parents who have supported me all the way since the beginning of my studies. Finally, this thesis

is dedicated to all those who believe in the richness of learning.

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Title

Computational and Quantum Mechanical

Investigations of Oxide and Halide Perovskites

using First-principles Study

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ABSTRACT

Fluoride and oxide perovskite structures are attracting huge interest in recent years due to their

special functionalities. In this thesis, the theoretical investigation on wide range of useful

compounds from perovskite family have been studied thoroughly for their possible

technological applications. Within the framework of Density Functional Theory (DFT),

structural, elastic, mechanical, electronic, optical, magnetic and thermodynamic properties are

studied by employing Full Potential-Linearized Augmented Plane Wave (FP-LAPW) method.

For the said investigation, the WIEN2k package is utilized.

The investigations on fluorine based strontium series of perovskites SrMF3 (M = Li, Na, K,

Rb) reveals that in these mechanically stable fluoroperovskites, brittleness and ionic behavior

are dominated which decreases from SrLiF3 to SrRbF3. Calculated energy band profiles

confirm wide and direct (Γ-Γ) bandgap. A predominant characteristic associated with cation

replacement shows that Li by Na, Na by K, and K by Rb significantly reduces the direct

bandgap in SrMF3 (M = Li, Na, K, Rb) compounds. This crucial variation is responsible for

working in different Ultra-Violet regions of the spectrum. Furthermore, from application point

of view, they could preferably be used in lens materials because they would not tolerate

birefringence that would make design of lenses difficult but also can be used in the confinement

of light for Light Emitting Devices.

The optimizations of structural parameters for rubidium based fluoroperovskite, RbHgF3 is

done with variety of approximations, which validates through comparison with available

experimental data. Energy band profile authenticates that inspected material is a narrow and

indirect energy bandgap (M–Γ) semiconductor while contour maps of electron density verifies,

mixed covalent-ionic behavior. In addition to it, optical responses show wide range of

absorption and reflection in high frequency regions.

Several elastic and mechanical parameters, reveals that protactinium based oxide series of

perovskites XPaO3 (X = K, Rb) are mechanically stable and possesses weak resistance to shear

deformation as compared with resistance to unidirectional compression while flexible and

covalent behaviors are dominated in them. The analysis of band profile through Tran–Blaha

modified Becke–Johnson (TB-mBJ) potential highlights the underestimation of bandgap with

traditional Density Functional Theory (DFT) approximation. Specific contribution of

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electronic states are investigated by means of total and partial density of states and it can be

evaluated that both compounds are direct bandgap (Γ–Γ) semiconductors. The study on BaMO3

(M= Pa,U) explores, type of chemical bonding with the help of variations in electron density

difference distribution that is induced due to changes of second cation. The results of electronic

properties illustrate direct bandgap (Γ-Γ) semi-conductive nature with the bandgap of 4.20 eV

and 4.01 eV for BaPaO3 and BaUO3 compounds respectively. The band gap dependent optical

properties such as complex dielectric function Ԑ (ω), optical conductivity σ (ω), refractive

index n (ω), reflectivity R (ω), and effective number of electrons (neff) via sum rules are

reported for the first time.

The investigations on KXF3 (X = V, Fe, Co, Ni) authenticates that this class of

fluoroperovskites are elastically as well as mechanically stable and anisotropic while KCoF3

is harder than rest of the compounds. The calculated spin dependent magnetoelectronic

properties in these compounds shows that exchange splitting is dominated by N-3d orbital. The

stable magnetic phase optimizations verify the experimental observations at low temperature.

The present methodology represents an influential approach to calculate the whole set of

mechanical and magneto-opto-electronic parameters, which would support to understand

various physical phenomena and empower device engineers for implementing these materials

in spintronic applications.

The pressure induced structural, elastic, mechanical, electronic, optical and thermodynamic

properties of SrLiF3, SrNaF3, SrKF3, SrRbF3, and CaLiF3 are computationally calculated for

their possible technological outcomes. All elastic and mechanical parameters are linearly

dependent on applied pressure and an increase in pressure improves tensile strength and

stiffness, on the other hand, reduces brittleness and compressibility of these cubic

fluoroperovskites. It is observed that an increase in pressure considerably improves the wide

and direct (Γ-Γ) electronic bandgap. The optical parameters of SrLiF3 and SrNaF3 shows that

all optical responses shift towards higher energy ranges which divulges that both are more

suitable for optoelectronic devices at higher pressure ranges. Consequently, our theoretical

work has been benchmarked various quantum mechanical effects, which will motivate research

scholars to done theoretical as well as experimental investigations on fluoride and oxide

perovskites that must be considered to understand and utilize these materials in fabricating

practical devices for optoelectronic, microelectronic, spintronic and piezoelectric applications.

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ACKNOWLEDGEMENTS

“A man can only attain knowledge with

the help of those who possess it.

This must be understood from the very beginning.

One must learn from him who knows.”

George Gordjieft.

First and foremost, all praise to Almighty ALLAH, the most Merciful, the Compassionate, the

creator and sustainer of all the universe, who is the origin of all knowledge and wisdom, who

gave me the courage and power to accomplish this research work. Without his grace and mercy,

this work would have not been accomplished. In this universe for the guidance of human being

almighty ALLAH bestow Anbiya-Karam. By these holy souls God remove the nastiest billows

of infidelity & incredulity and illustrate his creation towards the precise way of persistent

religious conviction and last but not the least accomplished Islam by sending Hazoor Sarwar-

kainat Fakhar-mojodaat Khatam-ul-anbiya Hazrat Muhammad Mustafa (Sallala ho tala aleh

Wasalam), Subhan Allah! After Prophets, the same vocation was handed over to Saints, for

preaching & persuading Islam, for propagating Toheed & Risalaat and in spreading civilization

& social intercourse these consecrated characters completely follow Hazrat Muhammad

Mustafa (Sallala ho tala aleh Wasalam).

I offer my humblest words of thanks to Holy Prophet (Sallala ho tala aleh Wasalam), the

source of unbounded knowledge, who is forever a torch of guidance and who has guided his

“Ummah” to seek knowledge from cradle to grave, whose holy teaching inspired me to

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accomplish this research work in time. I would further like to convey my sincere gratitude to

Sheikh-ul-islam wal-Muslimeen Hazrat Baba Fareed-ud-din Masood Ganjsaker Chisti

Farooqi Rehmatullahi A'laih spiritual son of Khwaja-e-Khwajgan Hazrat Khwaja

Ghareeb Nawaz Moinuddin Chisti Ajmeri Rehmatullahi A'laih, whom trustworthy high

ranked & superb personality endow with precious services for scattering the luminosity of

Islam in the subcontinent, to point up off track Muslims the right alleyway and to incline new

muslins towards Islam. Without his special favor, this work would have not been

accomplished.

To the casual observer, a thesis may appear to be solitary work. However, to complete a report

of this magnitude requires a network of support, and I am deeply indebted to many people.

In the first place, I would like to record my gratitude to my respectable supervisor, Prof. Dr.

Muhammad Azhar Iqbal, for his dynamic supervision, advice, support, generosity from the

very early stage of this work. Above all and the most needed, he provided me unflinching

encouragement and support in various ways. His truly scientific intuition has made him as a

constant oasis of ideas in major areas of theoretical as well as experimental physics, which

exceptionally not only inspire but enrich my growth as a student, a researcher and a scientist

want to be. I am indebted to him more than he knows. I acknowledge the unwavering support

received from the Dr. Mahmood-ul-Hassan, Chairman Department of Physics, University of

Punjab, Lahore, who supported and helped me a lot in resolving many issues. This is his

dedicated support that made this work to its final end.

I especially thanks to Prof. Dr. Peter Blaha, Prof. Dr. Karlheinz Schwarz, Dr. Fabien Tran,

Dr. Andreas Troster, Leila Kalantari, Jan Doumont from Vienna University of Technology,

Institute of Materials Chemistry Austria, they directly or indirectly helped me a lot in

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software issues during this thesis. Next, I would also like to recognize Dr. Abed Breidi, School

of Metallurgy and Materials, University of Birmingham, United Kingdom and Dr. Songke

Feng Northwestern Polytechnical University, China for innovating my concepts in various

ways. I also thanks to Professors A. Savin, R. Dronskowski, A. J. Maeland, and G. Kresse, for

fruitful scientific communications.

The accomplishments in this work cannot be fulfilled without valuable comments provided by

various journal reviewers from Computational Material Science, Physica B, Communications

in Computational Physics, Materials Research Express, Solid State Communications, and

Chinese Physics B, they not only make me eligible to do number of effective publications but

helped me to improve thesis quality a lot.

During my years pursuing the Ph.D. at Department of Physics, University of Punjab, I have

had the pleasure of meeting many intellectual people. I am thankful to all of them who helped

me in their own way all this time. I extend sincere felicitations to all the learned staff members,

technical and non-technical staff for their courteous cooperation. I am also thankful to library

staff members, especially chief librarian for giving me opportunity to utilize digital resources

effectively. I also want to express my truthful thankfulness to members of Doctoral

Programme Coordination Committee (DPCC), University of Punjab for their assistance and

useful guidance.

Where would I be without my family? My parents deserve special mention for their inseparable

support and prayers. This thesis cannot be completed without constant cooperation and

encouragement of my lovely husband Engr. Rao Muhammad Abdullah Asadi, who

sincerely raised my intellectual pursuit, his involvement with originality has nourished my

academic maturity. His dedication, care, love and persistent confidence in me, has taken the

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load off my shoulders. Last but not least, I express my very special thanks to my mother-in-

law, for my brothers and sisters. Hopefully this is not the end but the end of a new beginning

Insha-Allah!

Nazia Erum

Table of contents

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TABLE OF CONTENTS Page

TITLE PAGE Ι

CERTIFICATE ΙΙ

DEDICATIONS ΙΙΙ

ABSTRACT V

ACKNOWLEDGMENTS VΙΙ

TABLE OF CONTENTS XΙ

LIST OF TABLES XIX

LIST OF FIGURES XXΙV

LIST OF SYMBOLS AND ABBREVIATIONS XXXVΙΙ

Chapter 1 1

Introduction 1

1.1 Overview 1

1.2 The driving forces 2

1.3 Scope and objective 4

1.3.1 Quantum mechanical investigation 4

1.3.2 First principles studies 6

1.3.3 Computational analysis 7

1.4 Perovskites 8

1.5 Crystallographic details of perovskite structures 8

1.5.1 Tolerance factor criteria for perovskites 11

1.5.2 Types of perovskites 12

1.6 Aim of the research 15

1.7 Outline of thesis 16

Chapter 2 18

Perovskite materials: From synthesis to applications 18

2.1 Overview 18

2.2 Synthesis methods for perovskite materials 19

Table of contents

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2.2.1 Conventional inorganic solid-state synthesis 20

2.2.2 Solution-based synthesis methods 20

2.3 Prediction methods for structural properties of perovskites 21

2.4 Physical characteristics of perovskites 23

2.5 Band chemistry of perovskites 24

2.6 From insulating to superconducting perovskites 25

2.7 Magnetism and electronic correlations in perovskites 29

2.8 Thermodynamic valence stability in transition metal based perovskites 33

2.9 Properties of perovskites 34

2.9.1 Property based tentative classification of perovskites 36

2.9.2 Opto-electronic properties 36

2.9.3 Dielectric properties 38

2.9.4 Piezoelectricity 40

2.9.5 Multiferroicity 42

2.9.6 Electronic conductivity 45

2.9.7 The Seebeck coefficient 48

2.9.8 Polarons 48

2.9.9 Thermal expansion 49

2.10 Application of perovskites 50

Chapter 3 55

Literature Review 55

3.1 Overview 55

3.2 Background of materials 55

3.3 Structural properties-Previous research 57

3.4 Optoelectronic properties-Previous research 59

3.5 Elastic and mechanical properties-Previous research 62

3.6 Magnetic properties-Previous research 64

3.7 Thermodynamic properties-Previous research 70

3.8 Conclusion 72

Chapter 4 73

Theory and Computational details 73

Table of contents

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4.1 Introduction 73

4.2 Many body problems and Schrodinger wave equation 74

4.3 The Basic Methods of Electronic Structure 77

4.4 The Density Functional Theory (DFT) 80

4.5 Hohenberg-Kohn Theorems and Kohn Sham Equations 83

4.6 The Exchange and Correlation approximations 88

4.6.1 The Local Density approximation (LDA) 88

4.6.2 The Generalized Gradient approximation (GGA) 90

4.6.3 The modified Becke–Johnson (mBJ) potential 93

4.7 Methods for solution of Kohn Sham Equations 95

4.8 Full-Potential Linearized Augmented Plane Wave Method (FP-LAPW) 96

4.9 Simulation techniques 98

4.9.1 The WIEN2k Package 100

4.10 Applications of Density functional theory (DFT) 102

Chapter 5: Results and discussion Ι; 106

Elastic, and optoelectronic investigation of SrMF3 (M = Li, Na, K, Rb) and RbHgF3

fluoroperovskites 106

5.1 Introduction 106

5.2 Structural, elastic and mechanical properties of SrMF3 (M = Li, Na, K, Rb) 106

5.2.1 Structural properties 107

5.2.2 Elastic properties 109

5.2.3 Mechanical behavior 110

5.2.3.1 Elastic moduli calculations 110

5.2.3.2 Cauchy’s pressure and shear constant calculations 111

5.2.3.3 Poisson’s ratio and elastic anisotropy calculations 112

5.2.3.4 Melting temperature Tm and Kleinman’s parameter calculations 113

5.2.3.5 Lame’s constant calculations 114

5.3 Opto-electronic investigation of SrMF3 (M = Li, Na, K, Rb) 120

5.3.1 Electronic properties 120

5.3.1.1 Band structure calculations 120

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5.3.1.2 Density of States (DOS) calculations 121

5.3.1.3 Electron density contour calculations 122

5.3.2 Optical parameters 122

5.3.2.1 Complex dielectric constant calculations 123

5.3.2.2 Optical conductivity and energy loss calculations 124

5.3.2.3 Sum rules calculation via neff 125

5.4 Opto-electronic investigation of RbHgF3 for low birefringent lens materials 149


5.4.2 Electronic Properties 150




5.4.3 Optical properties 152


5.4.3.2 Absorption coefficient calculations 153

5.4.3.3 Optical reflectivity calculations 153

5.5 Conclusion 166

Chapter 6: Results and discussion ΙΙ; 168

Investigation of mechanical and optoelectronic behavior of actinoid based oxide

Perovskites 168


6.2 Mechanical and optoelectronic study of XPaO3 (X= K, Rb) 169


6.2.2 Elastic constant calculations 171

6.2.3 Mechanical parameters 172

6.2.3.1 Elastic moduli calculations 172

6.2.3.2 Cauchy’s pressure and Poisson’s ratio calculations 173

6.2.3.3 Shear constant and elastic anisotropy calculations 173

6.2.3.4 Lame’s constant calculations 174

6.2.3.5 Melting temperature calculations 174

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6.2.4 Electronic behavior 175




6.2.5 Optical characteristics 176


6.2.5.2 Optical conductivity and energy loss calculations 178

6.2.5.3 Refractive index and reflectivity calculations 178

6.2.5.4 Absorption coefficient calculations 179


6.3 Ab initio study of high dielectric constant BaMO3 (M=Pa, U) oxide perovskite 206

6.3.1 Structural parameters 206

6.3.2 Electronic behavior 208



6.3.2.3 Electron density Calculations 209

6.3.3 Optical characteristics 209


6.3.3.2 Optical conductivity calculations 210

6.3.3.3 Refractive index and reflectivity calculations 211


6.4 Conclusion 232

Chapter 7: Results and discussion ΙΙΙ; 234

Band profiles and magneto-optic properties of KXF3 (X= V,Fe,Co,Ni) 234


7.2 Structural stability 235

7.2.1 Analytical calculations of lattice constants 236

7.2.2 Tolerance factor calculations 237

7.3 Elastic properties 237

7.3.1 Calculation of elastic constants 237

Table of contents

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7.4 Mechanical properties 238

7.4.1 Calculation of elastic moduli 239

7.4.2 Calculation of Cauchy’s pressure, B/G and Poisson’s ratio 240

7.4.3 Calculation of shear constant and elastic anisotropy 240

7.4.4 Calculation of Kleinman’s parameter and Lame’s constant 241

7.5 Thermal properties (Calculation of the Debye temperature) 242

7.6 Electronic and magnetic properties 243

7.6.1 Spin-dependent band structure calculations 244

7.6.2 Spin-dependent Density of States (DOS) calculations 244

7.6.3 Spin-dependent electron density calculations 245

7.6.4 Calculation of magnetic properties 245

7.7 Optical properties 246

7.7.1 Calculation of complex dielectric function 246

7.7.2 Calculation of energy loss function 248

7.7.3 Calculation optical conductivity 248

7.7.4 Calculation of absorption coefficient 248

7.7.5 Calculation of reflectivity 249

7.7.6 Calculation of refractive index 249

7.7.7 Calculation of sum rule via neff 250

7.8 Conclusion 277

Chapter 8: Results and discussion ΙV; 278

Effect of pressure variation on strontium and calcium based fluoroperovskites 278


8.2 Background of investigation 279

8.3 Pressure variation on physical properties of SrLiF3 281

8.3.1 Pressure variation on structural properties 282

8.3.2 Pressure variation on electronic properties 283

8.3.3 Pressure variation on elastic properties 285

8.3.4 Pressure variation on mechanical properties 287

8.3.5 Thermodynamic properties 289

Table of contents

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8.3.5.1 The Quasi-harmonic Debye model 289

8.3.5.2 Pressure and temperature variation on thermodynamic properties 291

8.3.6 Pressure variation on optical properties 292

8.4 Effect of pressure variation on physical properties of SrNaF3 318



8.4.3 Pressure variation on elastic and mechanical properties 321

8.4.4 Pressure and temperature variation on thermodynamic properties 324

8.4.5 Effect of pressure variation on optical properties 326

8.5 Pressure variation on physical properties of SrKF3 352





8.5.5 Pressure variation on Debye temperature (θD) 357

8.5.6 Pressure and temperature variations on thermodynamic properties 359

8.6 Pressure variation on physical properties of SrRbF3 377






8.7 Pressure variation on physical properties of CaLiF3 396






8.8 Conclusion 413

Chapter 9 417

Conclusions and future work 417

Table of contents

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9.1 Conclusions 417

9.2 Mechanical and opto-electronic properties of Perovskites 418

9.3 Magneto-opto-electronic properties of fluoroperovskites 420

9.4 Pressure and temperature dependent physical aspects of fluoroperovskites 421

9.5 Future work plan 424

REFERENCES 426

LIST OF PUBLICATIONS 462

List of Tables

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LIST OF TABLES

Page

Table 1.1: Some perovskite, their tolerance factor and structures. 14

Table 2.1: Applications of perovskites along with respective properties. 54

Table 5.1: Comparison of calculated equilibrium lattice constants (ao), ground state

energies (Eo) and bulk modulus (Bo) with experimental and other theoretical

values of SrMF3 (X = Li, Na, K and Rb) compounds. 117

Table 5.2: Bond-lengths of SrMF3 (X= Li, Na, K, Rb) compounds. 117

Table 5.3: Calculated values of elastic constants C11, C12, C44, for SrMF3 (X = Li, Na, K

and Rb) compounds. 118

Table 5.4: Calculated values of Bulk modulus B0, Voigt’s shear modulus GV, Reuss’s

shear modulus GR, Hill’s shear modulus GH, Young’s modulus Y, and Pugh’s

index of ductility Bo/GH. 118

Table 5.5: Calculated values of Shear constant(𝐶′), Cauchy pressure (𝐶′′), Poisson’s

ratio (ѵ), Anisotropy constant (A), Kleinman parameter (ξ), Lame’s

coefficients (λ and μ), and Melting temperature (Tm). 119

Table 5.6: Band gap of SrMF3 (M = Li, Na, K, Rb) at different symmetry points

compared with experimental and other theoretical results. 148

Table 5.5: Comparison of Present calculation with previous experimental and theoretical

values for lattice constants (ao), ground state energies (Eo), bulk modulus (Bo)

and its pressure derivative (Bp) of RbHgF3 compound. 165

Table 6.1: Comparisons of calculated values of bond length, equilibrium lattice constant

(ao in Ǻ), ground state energy (Eo in RY), bulk modulus (Boin GPa) and its

pressure derivative (BP) with experimental and other theoretical results for

XPaO3 (X = K, Rb) compounds. 201

Table 6.2: Calculated values of tolerance factor for XPaO3 (X = K, Rb). 202

Table 6.3: Calculated values of elastic constants C11, C12, C44, for XPaO3 (X = K, Rb)

compounds. 202

Table 6.4: Calculated values of Bulk modulus B0, Reuss’s shear modulus GR, Voigt’s

shear modulus GV, Hill’s shear modulus GH, Young’s modulus Y and Pugh’s

index of ductility Bo/GH. 203

List of Tables

XX

Table 6.5: Calculated values of Shear constant (C′), Cauchy pressure (C′′), Lame’s

coefficients (λ and μ), Anisotropy constant (A in GPa) and Poisson’s ratio

(ѵ in GPa) and the melting temperature (Tm in K) for XPaO3 (X= K, Rb)

compounds. 204

Table 6.6: Band gap comparison of XPaO3 (X = K, Rb) at different symmetry points. 205

Table 6.7: Comparison of calculated equilibrium lattice constants ao (in Ǻ), ground state

energies Eo (in Ry), bulk modulus Bo (in GPa), its pressure derivative BP (in

GPa), and bond lengths with experimental and other theoretical values of

BaXO3 (X = Pa, U) compounds. 230

Table 6.8: Calculated tolerance factor for BaXO3 (X = Pa, U). 231

Table 6.9: Band gap comparison of BaXO3 (X = Pa, U) at different symmetry points. 231

Table 7.1: Comparison of experimental and calculated values of equilibrium lattice

constants (ao in Ǻ), ground state energies (Eo in Ry), bulk modulus (Bo in GPa)

and its pressure derivative (BP), and bond lengths of KXF3 (X = V,Fe,Co,Ni)

compounds. 271

Table 7.2: Calculated tolerance factor for KXF3 (X = V,Fe,Co,Ni) compounds. 272

Table 7.3: Calculated values of elastic constants (C11, C12 and C44 in GPa), for KXF3

(X = V,Fe,Co,Ni) compounds. 272

Table 7.4: Calculated values of Bulk modulus (B0 in GPa), Young’s modulus (Y in GPa),

Voigt’s shear modulus (GV in GPa), Reuss’s shear modulus (GR in GPa), and

Hill’s shear modulus (GH in GPa) for KXF3 (X = V,Fe,Co,Ni) compounds. 273

Table 7.5: Calculated values of B/G ratio, Shear constant (C’), Cauchy pressure (C’’),

Lame’s coefficients (λ and μ), Kleinman parameter (ξ in GPa), Anisotropy

constant (A in GPa) and Poisson’s ratio (ѵ in GPa) for KXF3

(X = V,Fe,Co,Ni) compounds. 274

Table 7.6: Comparison of experimental and calculated values of longitudinal (υl in Km/s),

transverse (υt in Km/s), average sound velocity (υm in Km/s), Debye

temperature (θD in K) and the melting temperature (TMelt in K) for KXF3 (X =

V,Fe,Co,Ni) compounds. 275

Table 7.7: Comparison of calculated interstitial (minst), local and total magnetic moment

(MT) in μB of KXF3 (X= V,Fe,Co,Ni) compounds with available

experimental and other theoretical data. 276

List of Tables

XXİ

Table 8.1: Comparison of the present calculation with the previous experimental and

theoretical values for the lattice constants (ao), ground state energies (Eo), bulk

modulus (Bo) and its pressure derivative (B′) at ambient pressure of SrLiF3

compound. 315

Table 8.2: Comparison of previous and calculated values of Pressure (P in GPa),

Energies (E in Ry), Volume of unit cell (V in (a.u.)3), Energy Gap (eV), and

Bond length (dSr-F, dLi-F). 315

Table 8.3: Calculated values of elastic constants (C11, C12, C44) of SrLiF3 at pressure

from 0-50 GPa. 316

Table 8.4: Derived elastic constants characterizing mechanical stability (Equations

8.1-8.3) of SrLiF3 at pressure from 0-50 GPa. 316

Table 8.5: Calculated values of elastic moduli Bulk modulus (B0), Voigt’s shear

modulus (GV), Reuss’s shear modulus (GR) and Hill’s shear modulus

(GH), and Young’s modulus (Y) of SrLiF3 at pressure from 0-50 GPa. 317

Table 8.6: Calculated values of Shear constant (C’), Cauchy pressure (C’’), Poisson’s

ratio (ѵ) Anisotropy constant (A), Kleinman parameter (ξ), and melting

temperature (Tm) of SrLiF3 at pressure from 0-50 GPa. 317


theoretical values for the lattice constants (ao), ground state energies (Eo),

bulk modulus (Bo) and its pressure derivative (B′) at ambient pressure of

SrNaF3 compound. 349



Bond length (dSr-F, dNa-F). 349

Table 8.9: Calculated values of elastic constants (C11, C12, C44), of SrNaF3 at pressure

from 0-25 GPa. 350

Table 8.10: Calculated values of Bulk modulus (B0), Voigt’s shear modulus (GV),

Reuss’s shear modulus (GR) and Hill’s shear modulus (GH), Young’s

modulus (Y), and B/G ratio, of SrNaF3 at pressure from 0-25 GPa. 350

Table 8.11: Calculated values of Poisson’s ratio (ѵ), Cauchy pressure (C’’), Anisotropy

constant (A), Kleinman parameter (ξ), and melting temperature (Tm) of

SrNaF3 at pressure from 0-25 GPa. 351

Table 8.12: Derived elastic constants characterizing mechanical stability (Equations

8.33-8.35) of SrNaF3 at pressure from 0-25 GPa. 351

List of Tables

XXİİ




SrKF3 compound. 373



Bond length (dSr-F, dK-F). 373

Table 8.15: Calculated values of elastic constants (C11, C12, C44) of SrKF3 at pressure

from 0-25 GPa. 374


Reuss’s shear modulus (GR) and Hill’s shear modulus (GH), Young’s

modulus (Y), and B/G ratio of SrKF3 at pressure from 0-25 GPa. 374


constant (A), Kleinman parameter (ξ), melting temperature (Tm),

longitudinal (υl in m/s), transverse (υt in m/s), average sound velocity (υm in

m/s), and Debye temperature (θD in K) of SrKF3 at pressure from 0-25 GPa. 375

Table 8.18: Derived elastic constants characterizing mechanical stability (equation 8.36-

8.38) of SrKF3 at pressure from 0-25 GPa. 376



bulk modulus (Bo) and its pressure derivative (B′) at ambient pressure for

SrRbF3 compound. 393


Energies (E in Ry), Volume of unit cell (V in (a.u.)3) and Bond length

(dSr-F, dRb-F) of SrRbF3 compound. 393

Table 8.21: Calculated values of elastic constants (C11, C12, C44) of SrRbF3 at pressure

from 0-25 GPa. 394

Table 8.22: Calculated values of derived elastic constants characterizing mechanical

stability of SrRbF3 at pressure from 0-25 GPa. 394


Reuss’s shear modulus (GR) Hill’s shear modulus (GH), Young’s modulus

(Y), and B/G ratio of SrRbF3 at pressure from 0-25 GPa. 395

List of Tables

XXİİİ


constant (A), Kleinman parameter (ξ), melting temperature (Tm),

longitudinal (υl in m/s), transverse (υt in m/s), average sound velocity (υm in

m/s), and Debye temperature (θD in K) of SrRbF3 at pressure from 0-25 GPa. 395




CaLiF3 compound. 410


Energies (E in Ry), Volume of unit cell (V in (a.u.)3) and Bond length

(dCa-F, dLi-F) of CaLiF3 compound. 410

Table 8.27: Calculated values of elastic constants (C11, C12, C44) of CaLiF3 at pressure

from 0-50 GPa. 411

Table 8.28: Calculated values of derived elastic constants characterizing mechanical

stability of CaLiF3 at pressure from 0-50 GPa. 411


Reuss’s shear modulus (GR), Hill’s shear modulus (GH), Young’s

modulus (Y), and B/G ratio, of CaLiF3 at pressure from 0-50 GPa. 412

Table 8.30: Calculated values of Poisson’s ratio (ѵ), Anisotropy constant (A), Kleinman

parameter (ξ), melting temperature (Tm) longitudinal (υl in m/s), transverse

(υt in m/s), average sound velocity (υm in m/s), and Debye temperature

(θD in K) of CaLiF3 at pressure from 0-50 GPa. 412

List of Figures

XXİꓦ

LIST OF FIGURES Page

Figure 1.1: Schematic diagram showing correlation between First principles study,Quantum

mechanical investigation and computational analysis. 9

Figure 1.2: Comparison between fundamental quantum mechanics and technology of the

classical world. 9

Figure 1.3: Schematic representation showing journey of quantum mechanical

Investigation for evaluating physical properties of material. 10

Figure 1.4: A generic perovskite structure of the form ABX3. Note however that the two

structures are equivalent – the left-hand structure is drawn so that atom B is at

the <0,0,0> position while the right-hand structure is drawn so that atom

(or molecule) A is at the <0,0,0> position. Also note that the lines are a guide

to represent crystal orientation rather than bonding patterns. 10

Figure 1.5: The ideal ABX3 perovskite structure showing the octahedral and icosahedral

(12-fold) coordination of the B and A-site cations, respectively. 13

Figure 1.6: Illustration of the simple cubic perovskite unit along one of the main unit cell

axes in (a) an ideal cubic perovskite with a larger A-site cation and (b) a

smaller A site cation. 14

Figure 2.1: Perovskite mineral species (CaTiO3) along with Lev Aleksevich von

Perovski. 19

Figure 2.2: Structure and morphology of perovskite mineral. 23

Figure 2.3: Schematic illustration of the band gap in solid materials. 27

Figure 2.4: A band gap diagram showing the approximate band energies in ABO3 that

from the density of states (DOS) in a perovskite. 28

Figure 2.5: A band gap diagram showing the different sizes of band gaps for conductors,

semiconductors, and insulators. 28

Figure 2.6: Block diagram breakdown of chemical and physical properties of matter. 35

Figure 2.7: Schematic illustration for the phenomenon of piezoelectric effect. 42

Figure 2.8: Multiferroics combine the properties of ferroelectrics and ferromagnets. 46

Figure 2.9: Block diagram illustration of perovskites multiferroics 46

Figure 2.10: The multiferroics totem; illustrating the three main ferroic orders with their

respective fields and crossed interactions. 47

List of Figures

XXꓦ

Figure 2.11: Conditions required for ferroelectricity (polarization) and ferromagnetism

(unpaired electron spin motion). 47

Figure 2.12: Various applications of perovskites quantum dot, nanowire and nanosheet. 53

Figure 3.1: Illustration of the different orbitals that overlap with a) strong eg-pσ and b)

t2g-pπ overlaps between two transition metals with dn configuration and an

oxygen, i.e., the M-O-M bonds. 68

Figure 3.2: Emergence of the novel interface magnetic state at the heterointerfaces of

LSMO/BFO. (a) Novel interfacial magnetic state in the LSMO/BFO

heterostructure (b) Evolution of the interface magnetism and exchange bias

coupling with temperature. The vertical guiding line indicates the blocking

temperature of the exchange bias coupling and the magnetic transition

temperature of the interface magnetic state. 69

Figure 3.3: A general schematic illustration for calculating thermodynamic properties. 71

Figure 4.1: Block diagram representation of various theoretical methods. 75

Figure 4.2: Schematic chemistry of atoms and molecules in solids. 78

Figure 4.3: The evolution and classification of quantum mechanical methods. 81

Figure 4.4: Schematic presentation of Quantum methods. 82

Figure 4.5: A schematic representation of the relationship between the "real" many body

system (left hand side) and the non-interacting system of Kohn Sham density

functional theory (right hand side). 82

Figure 4.6: Schematic description of the SCF cyclic procedure in solving the

Kohn-Sham equations. 87

Figure 4.7: Partitioning of the unit cell into atomic spheres (I) and an interstitial

region (II). 99

Figure 4.8: The unit cell divided into muffin-tin region and interstitial region. 99

Figure 4.9: Flow chart of WIEN2k code SCF cycle in single mode and in parallel

Mode. 102

Figure 5.1: Crystal structures of SrMF3, Where M = Li, Na, K, and Rb (Sr+2: Blue, M+1:

Green, F-1 : Red). 115

List of Figures

XXꓦİ

Figure 5.2: Lattice constants versus change in bond lengths between M and F of SrMF3

(M = Li, Na, K, Rb). 116

Figure 5.3: Melting temperature Tm (K) Vs Klienmann parameter ξ (GPa). 116

Figure 5.4: The mBJ-electronic band dispersion curves for SrLiF3. 126

Figure 5.5: The mBJ-electronic band dispersion curves for SrNaF3. 127

Figure 5.6: The mBJ-electronic band dispersion curves for SrKF3. 128

Figure 5.7: The mBJ-electronic band dispersion curves for SrRbF3. 129

Figure 5.8: The Density of States for SrLiF3 by mBJ potential. 130

Figure 5.9: The Density of States for SrNaF3 by mBJ potential. 131

Figure 5.10: The Density of States for SrKF3 by mBJ potential. 132

Figure 5.11: The Density of States for SrRbF3 by mBJ potential. 133

Figure 5.12 (a): Calculated mBJ total two and three-dimensional electronic charge

densities for SrLiF3 in (100) plane. 134

Figure 5.12 (b): Calculated mBJ total two and three-dimensional electronic charge

densities for SrNaF3 in (100) plane. 135

Figure 5.12 (c): Calculated mBJ total two and three-dimensional electronic charge

densities for SrKF3 in (100) plane. 136

Figure 5.12 (d): Calculated mBJ total two and three-dimensional electronic charge

densities for SrRbF3 in (100) plane. 137


densities for SrLiF3 in (110) plane. 138


densities for SrNaF3 in (110) plane. 139

Figure 5.13 (c): Calculated mBJ total two and three-dimensional electronic charge

densities for SrKF3 in (110) plane. 140

Figure 5.13 (d): Calculated mBJ total two and three-dimensional electronic charge

densities for SrRbF3 in (110) plane. 141

List of Figures

XXꓦİİ

Figure 5.14: Total two-dimensional electron density plots in (110) plane for (a) SrLiF3

, (b) SrNaF3, (c) SrKF3, (d) SrRbF3. 142

Figure 5.15 (a): Calculated imaginary part Ԑ2 (ω) of the dielectric function for the SrMF3

(M=Li, Na, K, Rb) compounds. 143

Figure 5.15 (b): Calculated real part Ԑ1 (ω) of the dielectric function for the SrMF3

(M=Li, Na, K, Rb) compounds. 144

Figure 5.15 (c): Calculated energy loss function L (ω) for SrMF3 (M=Li,Na,K,Rb

compounds. 145

Figure 5.15 (d): Calculated conductivity σ (ω) for SrMF3 (M= Li, Na, K, Rb)

compounds. 146

Figure 5.15 (e): Calculated sum rule for SrMF3 (Li,Na,K,Rb) compounds. 147

Figure 5.16: Cubic crystal structure of RbHgF3. 154

Figure 5.17: Variation of total energy as a function of unit cell volume for RbHgF3. 155

Figure 5.18: Comparison of band structures in high symmetry directions with mBJ and

PBE-GGA schemes for RbHgF3. 156

Figure 5.19: The Density of States for RbHgF3 by mBJ potential. 157


densities in (100) plane for RbHgF3. 158


densities in (110) plane for RbHgF3. 159

Figure 5.21 (a): Total two-dimensional electron density plots in the (100) plane for

RbHgF3. 160

Figure 5.21 (b): Total two-dimensional electron density plots in the (110) plane for

RbHgF3. 160

Figure 5.22 (a): Calculated imaginary part Ԑ2(ω) of the dielectric function for RbHgF3

compound. 161

Figure 5.22 (b): Calculated real part Ԑ1(ω) of the dielectric function for RbHgF3

compound. 162

List of Figures

XXꓦİİİ

Figure 5.22 (c): Calculated absorption coefficient α (ω) of dielectric function for

RbHgF3 compound. 163

Figure 5.22 (d): Reflectivity R (ω) as a function of energy for RbHgF3 compound. 164

Figure 6.1 (a): Cubic crystal structure of KPaO3 181

Figure 6.1 (b): Cubic crystal structure of RbPaO3 182

Figure 6.2 (a): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u)3)

for KPaO3. 183

Figure 6.2 (b): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u)3)

for RbPaO3. 184

Figure 6.3: Electronic energy dispersion curves for (a) KPaO3 and (b) RbPaO3 along

some high symmetry directions in the Brillouin zone (BZ) within modified

Becke-Johnson (mBJ) Potential. 185

Figure 6.4 (a): The Density of States for KPaO3 by mBJ potential. 186

Figure 6.4 (b): The Density of States for RbPaO3 by mBJ potential. 187


densities for KPaO3 in (100) plane. 188


densities for RbPaO3 in (100) plane. 189


densities for KPaO3 in (110) plane. 190


densities for RbPaO3 in (110) plane. 191

Figure 6.7: Total two-dimensional electron density plots in (110) plane for (a) KPaO3,

(b) RbPaO3. 192


(b) RbPaO3. 192

Figure 6.9 (a): Calculated imaginary part Ԑ2 (ω) of the dielectric function for XPaO3

(K, Rb) compounds. 193

Figure 6.9 (b): Calculated real part Ԑ1 (ω) of the dielectric function for XPaO3 (K, Rb)

compounds. 194

Figure 6.9 (c): Calculated conductivity σ (ω) for XPaO3 (K, Rb) compounds. 195

List of Figures

XXİX

Figure 6.9 (d): Calculated energy loss function L (ω) for XPaO3 (K,Rb) compounds. 196

Figure 6.9 (e): Refractive index n (ω) as a function of energy for XPaO3 (X=K, Rb)

compounds. 197

Figure 6.9 (f): Reflectivity R (ω) as a function of energy for XPaO3 (X=K, Rb)

compounds. 198

Figure 6.9 (g): Absorption coefficient α (ω) as a function of energy for XPaO3

(X=K, Rb) compounds. 199

Figure 6.9 (h): Calculated sum rule (Neff) for XPaO3 (K, Rb) compounds. 200

Figure 6.10 (a): Cubic crystal structure of BaPaO3. 212

Figure 6.10 (b): Cubic crystal structure of BaUO3. 213


for BaPaO3. 214


for BaUO3. 215

Figure 6.12: Electronic energy dispersion curves for (a) BaPaO3 and (b) BaUO3 along

some high symmetry directions in the Brillouin zone (BZ) within WC-GGA.216

Figure 6.13 (a): The Density of States for BaPaO3 by WC-GGA approximation. 217

Figure 6.13 (b): The Density of States for BaUO3 by WC-GGA approximation. 218

Figure 6.14 (a): Calculated total two and three-dimensional electronic charge densities

for BaPaO3 in (100) plane. 219

Figure 6.14 (b): Calculated total two and three-dimensional electronic charge densities

for BaUO3 in (100) plane. 220

Figure 6.15 (a): Calculated total two and three-dimensional electronic charge densities

for BaPaO3 in (110) plane. 221

Figure 6.15 (b): Calculated total two and three-dimensional electronic charge densities

for BaUO3 in (110) plane. 222

Figure 6.16: Total two-dimensional electron density plots in (100) plane for (a) BaPaO3

, (b) BaUO3. 223

Figure 6.17: Total two-dimensional electron density plots in (110) plane for (a) BaPaO3,

(b) BaUO3. 223

List of Figures

XXX

Figure 6.18 (a): Calculated imaginary part Ԑ2 (ω) of the dielectric function for BaXO3

(Pa, U) compounds. 224

Figure 6.18 (b): Calculated real part Ԑ1 (ω) of the dielectric function for BaXO3 (Pa, U)

compounds. 225

Figure 6.18 (c): Calculated conductivity σ (ω) for BaXO3 (X=Pa, U) compounds. 226

Figure 6.18 (d): Refractive index n (ω) as a function of energy for BaXO3 (X=Pa, U)

compounds. 227

Figure 6.18 (e): Reflectivity R (ω) as a function of energy for BaXO3 (Pa,U)

compounds. 228

Figure 6.18 (f): Calculated sum rule (Neff) for BaXO3 (Pa,U) compounds. 229

Figure 7.1 (a): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u) 3)

for KVF3. 251

Figure 7.1 (b): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u) 3)

for KFeF3. 252

Figure 7.1 (c): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u) 3)

for KCoF3. 253

Figure 7.1 (d): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u) 3)

for KNiF3. 254

Figure 7.2 (a): The LSDA (Spin up)-electronic band dispersion curves for KXF3 (X=

V,Fe,Co,Ni). 255

Figure 7.2 (b): The GGA (Spin up)-electronic band dispersion curves for KXF3 (X=

V,Fe,Co,Ni). 256

Figure 7.2 (c): The mBJ (Spin up)-electronic band dispersion curves for KXF3 (X=

V,Fe,Co,Ni). 257

Figure 7.2 (d): The LSDA (Spin down)-electronic band dispersion curves for KXF3 (X=

V,Fe,Co,Ni). 258

Figure 7.2 (e): The GGA (Spin down)-electronic band dispersion curves for KXF3 (X=

V,Fe,Co,Ni). 259

Figure 7.2 (f): The mBj (Spin down)-electronic band dispersion curves for KXF3 (X=

V,Fe,Co,Ni). 260

List of Figures

XXXİ

Figure 7.3: Spin-dependent total and partial density of states for (a) KVF3, (b) KFeF3,

(c) KCoF3 and (d) KNiF3. 261

Figure 7.4: Spin-dependent electron charge densities in (110) planes for KXF3

(X= V, Fe, Co and Ni). 262

Figure 7.5: The calculated imaginary part Ԑ2 (ω) of the dielectric function for KXF3

(X= V,Fe,Co,Ni) compounds. 263

Figure 7.6: The Calculated real part Ԑ1 (ω) of the dielectric function for KXF3 (X=

V,Fe,Co,Ni) compounds. 264

Figure 7.7: Calculated energy loss function L (ω) of the dielectric function for KXF3


Figure 7.8: Calculated conductivity σ (ω) of the dielectric function for KXF3


Figure 7.9: Calculated absorption coefficient α (ω) of the dielectric function for KXF3


Figure 7.10: Calculated reflectivity R (ω) of the dielectric function for KXF3


Figure 7.11: Refractive index n (ω) of the dielectric function for KXF3 (X= V,Fe,Co,Ni)

compounds. 269

Figure 7.12: Calculated sum rule (Neff) of the dielectric function for KXF3


Figure 8.1: The Pressure variation of Lattice Constant (a) GGA (b) LDA. 297

Figure 8.2: The Pressure variation of Bonds length (a) Sr-F (b) Li-F. 297

Figure 8.3: The Pressure dependence of Band Gap (a) GGA (b) mBj. 298

Figure 8.4: The electronic band structures of SrLiF3 under the application of pressure

(0, 10, 20, 30, 40 and 50 GPa) calculated using GGA Approximation. 299

Figure 8.5: The Total and Partial Density of states (TDOS & PDOS) of SrLiF3 at 0 GPa

using GGA Approximation. 300

Figure 8.6: Stability criteria for cubic SrLiF3 compound as a function of pressure. 301

Figure 8.7: Calculated pressure dependence of elastic constant/moduli (a) C11 (b) C12

for SrLiF3 compound. 301

List of Figures

XXXİİ

Figure 8.8: Calculated pressure dependence of (a) Elastic constant/moduli (C44) (b) Bulk

modulus (B) for SrLiF3 compound. 302

Figure 8.9: Calculated pressure dependence of Kleinman parameter (ξ), and Melting

temperature (Tm) for SrLiF3 compound. 302

Figure 8.10 (a): Variation of the specific heat capacities (Cp) versus temperature at

different pressures for SrLiF3 compound. 303

Figure 8.10 (b): Variation of the heat capacities (CV) versus temperature at different

pressures for SrLiF3 compound. 304

Figure 8.10 (c): Temperature dependence of the volume expansion coefficient α (T) at


Figure 8.10 (d): Variation of the Debye temperature (θD) as a function of temperature at


Figure 8.11 (a): Calculated Imaginary part Ԑ2 (ω) of the dielectric function as a function

of pressure for SrLiF3 compound. 307

Figure 8.11 (b): Calculated Real part Ԑ1 (ω) of the dielectric function as a function of

pressure for SrLiF3 compound. 308

Figure 8.11 (c): Calculated Refractive index n (ω) as a function of pressure for SrLiF3

compound. 309

Figure 8.11 (d): Calculated Reflectivity R (ω) as a function of pressure for SrLiF3

compound. 310

Figure 8.11 (e): Calculated Conductivity σ (ω) as a function of pressure for SrLiF3

compound. 311

Figure 8.11 (f): Calculated Absorption coefficient α (w) as a function of pressure for

SrLiF3 compound. 312

Figure 8.11 (g): Calculated Energy loss function L (ω) as a function of pressure for

SrLiF3 compound. 313

Figure 8.11 (h): Calculated Sum rule as a function of pressure for SrLiF3 compound. 314


Figure 8.13: The Pressure variation of Bond lengths (a) Sr-F (b) Na-F. 330

Figure 8.14: The Pressure dependence of Band Gap (a) GGA (b) mBj. 331

List of Figures

XXXİİİ

Figure 8.15: The electronic band structures of SrNaF3 under the application of pressure

(0, 5, 10, 15, 20 and 25 GPa) calculated using GGA Approximation. 332

Figure 8.16: The Total and Partial Density of states (TDOS & PDOS) of SrNaF3 at 0

and 25 GPa using GGA Approximation. 333

Figure 8.17: Calculated pressure dependence of elastic constants/moduli (a) C11

(b) C12 (c) C44 (d) Bulk modulus, B for SrNaF3 compound. 334

Figure 8.18: Stability criteria for cubic SrNaF3 compound as a function of pressure. 334

Figure 8.19 (a): Variation of the Lattice constant as a function of temperature at

different pressures for SrNaF3 compound. 335

Figure 8.19 (b): Variation of the unit cell volume as a function of temperature at

different pressures for SrNaF3 compound. 336

Figure 8.19 (c): Variation of the Bulk modulus as a function of temperature at different

pressures for SrNaF3 compound. 337

Figure 8.19 (d): Variation of the Debye temperature (θD) as a function of temperature

at different pressures for SrNaF3 compound. 338

Figure 8.19 (e): Variation of the specific heat capacities of Cv as a function of

temperature at different pressures for SrNaF3 compound. 339

Figure 8.19 (f): Variation of the specific heat capacities of Cp as a function of

temperature at different pressures for SrNaF3 compound. 340

Figure 8.20 (a): Calculated Imaginary part Ԑ2 (ω) of the dielectric function as a function

of pressure for SrNaF3 compound. 341


pressure for SrNaF3 compound. 342

Figure 8.20 (c): Calculated Refractive index n (ω) as a function of pressure for SrNaF3

compound. 343

Figure 8.20 (d): Calculated Reflectivity R (ω) as a function of pressure for SrNaF3

compound. 344

Figure 8.20 (e): Calculated Conductivity σ (ω) as a function of pressure for SrNaF3

compound. 345

Figure 8.20 (f): Calculated Absorption coefficient α (w) as a function of pressure for


List of Figures

XXXİꓦ

Figure 8.20 (g): Calculated Energy loss function L (ω) as a function of pressure for


Figure 8.20 (h): Calculated Sum rule as a function of pressure for SrNaF3 compound. 348


Figure 8.22: The Pressure variation of Bond lengths (a) Sr-F (b) K-F. 362

Figure 8.23: The Pressure dependence of Band Gap (a) GGA (b) mBj 363

Figure 8.24: The electronic band structures of SrKF3 under the application of

pressure (0, 5, 10, 15, 20 and 25 GPa) calculated using GGA

Approximation. 364

Figure 8.25: The Total and Partial Density of states (TDOS & PDOS) of SrKF3 at

0 and 25 GPa using GGA Approximation. 365

Figure 8.26: Calculated pressure dependence of elastic constants/moduli

(a) C11 (b) C12 (c) C44 (d) Bulk modulus, B for SrKF3 compound. 366

Figure 8.27: Stability criteria for cubic SrKF3 compound as a function of

pressure. 366


different pressures for SrKF3 compound. 367


different pressures for SrKF3 compound. 368


pressures for SrKF3 compound. 369


at different pressures for SrKF3 compound. 370


temperature at different pressures for SrKF3 compound. 371


temperature at different pressures for SrKF3 compound. 372


Figure 8.30: The Pressure variation of Bond lengths (a) Sr-F (b) Rb-F. 383

Figure 8.31: Calculated pressure dependence of elastic constants (a) C11 (b) C12 (c) C44

for SrRbF3 compound. 384

List of Figures

XXXꓦ

Figure 8.32: Stability criteria for cubic SrRbF3 compound as a function of pressure. 384

Figure 8.33: Calculated pressure dependence of elastic parameters (a) Bulk modulus (B)

(b) Shear modulus (G) (c) Young’s modulus (Y) (d) B/G Ratio for SrRbF3

compound. 385

Figure 8.34 (a): Calculated pressure dependence of elastic wave velocities (a) υl (b) υt

(c) υm for SrRbF3 compound. 386

Figure 8.34 (b): Calculated pressure dependence of Debye temperature (θD) for SrRbF3

compound. 386


different pressures for SrRbF3 compound. 387


different pressures for SrRbF3 compound. 388


pressures for SrRbF3 compound. 389


at different pressures for SrRbF3 compound. 390


temperature at different pressures for SrRbF3 compound. 391


temperature at different pressures for SrRbF3 compound. 392

Figure 8.36: The Pressure variation of Lattice Constant (a) LDA (b) GGA. 402

Figure 8.37: The Pressure variation of Bond lengths (a) Ca-F (b) Li-F. 402


for CaLiF3 compound. 403

Figure 8.39: Stability criteria for cubic CaLiF3 compound as a function of pressure. 403

Figure 8.40: Calculated pressure dependence of isotropic elastic parameters (a) Bulk

modulus (B) (b) Shear modulus (G) (c) Young’s modulus (Y) (d) B/G ratio

for CaLiF3 compound. 404

Figure 8.41 (a): Calculated pressure dependence of elastic wave velocities (a) υl (b) υt

(c) υm for CaLiF3 compound. 405

Figure 8.41 (b): Calculated pressure dependence of Debye temperature (θD) for CaLiF3

compound. 405

List of Figures

XXXꓦİ

Figure 8.42 (a): Variation of the specific heat capacities of Cv as a function of

temperature at different pressures for CaLiF3 compound. 406

Figure 8.42 (b): Variation of the specific heat capacities of Cp as a function of

temperature at different pressures for CaLiF3 compound. 407

Figure 8.42 (c): Temperature dependence of the volume expansion coefficient α (T) at

different pressures for CaLiF3 compound. 408


at different pressures for CaLiF3 compound. 409

List of Symbols and Abbreviations

XXXꓦİİ

LIST OF SYMBOLS AND ABBREVIATIONS

Å Angstrom

G Shear modulus

α Absorption coefficient

Eg Band gap energy

eV Electron volt

UV Ultra-Violet

Bp Pressure derivative of bulk modulus

CP Heat capacity at constant pressure

CV Heat capacity at constant volume

G* Gibbs function

γ Gruneisen parameter

BS Adiabatic bulk modulus

n Number of atoms per chemical formula

Y Young’s modulus

𝐂′ Shear constant

𝐂′′ Cauchy’s pressure

A Elastic anisotropy parameter

θD Debye temperature

Tm Melting temperature

ξ Kleinman parameter

υl Longitudinal sound velocity

υt Transverse sound velocity


XXXꓦİİİ

υm Average sound velocity

minst Interstitial magnetic moment

MT Total magnetic moment

rav Average ionic radii

VBM Valence band maxima

CBM Conduction band minima

DOS Density of States

MO Molecular orbital

AFM Antiferromagnetic

FM Ferromagnetic

CMR Colossal magnetoresistance

MOSFET Metal-oxide semiconductor field effect transistor

GDM Giant dielectric constant materials

MEMS Microelectromechanical system

pH Potential of Hydrogen

LED Light Emitting Diodes

VUV Vacuum Ultra-Violet

VUVLED Vacuum-Ultraviolet Light Emitting Diodes

SOFC Solid Oxide Fuel Cell

HTSC High-temperature superconductor

MCSCF Multi-Configurations Self Consistent Field

HF Hartree-Fock

SCF Self-consistent field


XXXİX

QMC Quantum Monte Carlo

LDA Local Density Approximation

LSDA Local Spin Density Approximation

GGA Generalized Gradient Approximation

TB-mBJ Tran-Blaha modified Becke–Johnson

LAPW Linearized Augmented Plane Wave

MTOs Muffin tin orbitals

FP-LAPW Full-Potential Linearized Augmented Plane Wave Method

PPW Pseudopotential plane wave

APW+lo Augmented Plane Wave Plus Local Orbitals

SIC Self-interaction correction

NMR Nuclear Magnetic Resonance

BZ Brillouin Zone

SrLiF3 Strontium Lithium Trifluoride

SrNaF3 Strontium Sodium Trifluoride

SrKF3 Strontium Potassium Trifluoride

SrRbF3 Strontium Rubidium Trifluoride

KVF3 Potassium Vanadium Trifluoride

KFeF3 Potassium Iron Trifluoride

KCoF3 Potassium Cobalt Trifluoride

KNiF3 Potassium Nickel Trifluoride

BaPaO3 Barium Protactinium Trioxide

BaUO3 Barium Uranium Trioxide


XL

KPaO3 Potassium Protactinium Trioxide

RbPaO3 Rubidium Protactinium Trioxide

Chapter 1 Introduction

Page | 1

Chapter 1

Introduction

“There is nothing more difficult to take in hand,

more perilous to conduct, or more uncertain in its success,

than to take the lead in the

introduction of a new order of things.”

Niccolo Machiavelli

The most entertaining part of a dissertation is its motivational introduction for its writer to

work on because here he or she is allowed to be little bit frivolous, while in rest of the work

individual have to be bound around modest language of the scientific writing in favor of the

relevant field of study.

1.1 Overview

Material science serves the mankind to understand the world. Pure and applied material

science have resolved mysteries of many worldly problems. To look into nature, one has to

establish new methods. In this regard, the scientists especially material scientists are working

hard to promote the resources of the natural world and to govern over its applications. The

theoretical physics of material science is perhaps a field where the principle of reducing raw

information of a material can be clearly observed in terms of theoretical calculations,

graphical interpretation, logical evaluation, mathematical expressions and the results of these

approaches are not only successful but may well be called as remarkable. The primary

concern of material science is to search about fundamental understanding of structure,

internal properties as well as processing of materials. In an innovative industrial sector, the


Page | 2

growth can be prompted by continuous development of new materials whose products greatly

transformed the relationship between humans and their environment (Snaith 2013).

The goals of this introductory chapter are to provide the general motivation to perform this

work, followed by focus of the thesis in terms of evaluating a correlation between

computational analysis, first principles study and quantum mechanical investigation. Then

the chapter includes crystallographic details of various perovskite structures. At the end, aim

of the research and outline of the thesis are discussed.

1.2 The driving forces

Imagine a world fifty year from now… a world where cars are driven by hydrogen produced

from solar energy and water. A world where the air is clean from particulate matter and toxic

fumes from vehicles. A world where all the energy and the materials used are produced and

recycled in a sustainable and clean way. This utopic picture of the future is probably several

decades away for the developed countries and even further away for the rest of the world. It

is therefore necessary to improve today’s technology with discoveries that take the science

and technology a big step forward in order to accelerate the process. Historically, many

discoveries such as the superconducting pnictides (Norman 2008), Teflon (Bellis 2013),

vulcanized rubber and the cuprate superconductors (Sleight 1988) have all been found

primarily by chance. However, as example of an accidental discovery, the new blue pigments

in the YIn1-xMnxO3 (Smith 2009) system were found by investigating the solid solution for

interesting electronic properties. The discovery was made without the anticipation to discover

the first, new and highly stable inorganic blue pigment in more than 200 years since the

discovery of cobalt blue. The knowledge from the discovery was used to find other

compositions with the same structure type and produce green, yellow, orange and red colors


Page | 3

through careful control of the composition (Smith et al., 2011). The new pigments have

become a big success and are now being investigated further by several large companies in

order to commercialize them. The discovery is therefore a very good example of why the

systematic studies of different chemical systems are necessary in order to find correlations

between composition, structure and properties that can be transferred to different areas of

usage.

Although numerous binary AO2, ternary ABO3, ABF3 oxide and halide systems where A and

B are two different cations have been studied throughout the years, plenty of work still

remains, both to find new applications for old materials and to find new materials for current

and new technologies based on more complex interplay between the composition, structure

and properties in quaternary and higher order systems. Especially for complex systems where

simplified theories for magnetic interactions, conductivity, catalysis and ionic conductivity

stop working, additional investigations are necessary. At this point in time, predictions

through computational simulations are quite cheap, accurate and fast enough to predict the

structure and properties from only an initial input of the composition for complex systems.

Thus, systematic investigations of compositional variations in, perovskite-related systems, to

understand the more complex compositions, are an essential first step. One of the goals will

be to improve the computational models in order to find new materials with desired

properties more efficiently in the future. The computational models of new materials based

on the initial research goals are complemented by the measurement of the properties in order

to improve the synthesis that are used to predict new materials. However, apart from the

exploratory research that is driven by curiosity to explain different phenomena or to find new

ones, most research is often application driven and that type of research is mainly driven by


Page | 4

the need to solve specific problems such as to increase efficiency, lower the price and

decrease the impact on the environment.

1.3 Scope and objective

The important part of thesis is its theme of underlying research work. Because for

investigating in a right direction, it is necessary to know, is our destination clear, are we

reaching towards one write point and be very vigilant about what you are get into but in fact,

research is a vista that have no bounds. There are three important phenomena which are

carried out to conduct this investigation, as shown in schematic representation via Figure 1.1,

which are quantum mechanical investigation, computational analysis, and first principles

study. All of them are correlated in this work. This correlation opens up the opportunity to

investigate various physical phenomenon of oxide and fluoride perovskites within reasonable

ease. In next few sections let’s correlate them by exploring few lines of thought.

1.3.1 Quantum mechanical investigation

This section is dedicated to answer the significant question that what is the actual concept of

quantum mechanical investigation and how to utilize it in present thesis? So, let’s explain

that how applications of quantum mechanics can be utilized in terms of material’s properties

by few lines of thought.

Whenever we see a material, we observe that nature solve some fundamental equation of

physics in order to arrange the atoms. Material scientists try to study complex behavior of

any material at atomic level in a very literal way. Quantum mechanics explores the behavior

of electrons within the atoms. It is in fact, the physics of very small. At Quantum level

electron nature turns to be at wave nature which can be described by a wave function. The


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wave function is a physical object that can be described in terms of mathematical formalism

by using well-known time independent Schrodinger wave equation (Schrodinger 1926).

Many body wave functions can be used to apply for evaluating physical properties of any

crystalline material but to solve many particle Schrodinger’s wave equation Density

Functional Theory (DFT) should be applied to solve system of N electrons via electron

charge density instead of electron wave function (Lany and Zunger 2009). Hence the

corresponding first principles quantum mechanical investigations are mainly done with DFT,

according to which many-body problem of interacting electrons and nuclei is aligned into

series of one electron equation, well known as Kohn-Sham equation (Hohenberg and Kohn

1964) (The detailed description of Density Functional Theory (DFT) is given in chapter 3).

As a result, required material properties can be calculated. The properties of the material can

be of various types in which two of them are important namely physical or chemical

properties. In this thesis, our main concern is with physical properties due to an aspect of

matter that can be measurable without changing it and whose value describes a state of the

physical system. Schematic representation showing journey of quantum mechanical

investigation for evaluating physical properties of material is shown in Figure 1.1.

The importance of quantum mechanical investigation lies in the fact that it governs the

electronic structure of the material at atomic scale or in another way at Angstrom level.

Quantum mechanical investigation delivers complete information regarding to relative

stability, chemical bonding, phase transitions, atomic relaxation, mechanical, electrical,

optical, vibrational and magnetic behavior at atomic scale while the determination criteria of

these parameters depends upon several factors like structure, composition, disorder,

temperature, pressure and so on (Born 1927). The properties of solid composites especially


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crystalline solids are of great potential interest to the material scientists and delivers many

technological benefits. The relationships between fundamental quantum mechanics and the

technology of the classical world is shown in Figure 1.2.

1.3.2 First principles studies

In simulation field there are various techniques for the analysis of a given problem at

molecular or atomic level. Among them Monte-carlo statistical analysis (Hastings 1970),

molecular dynamic simulations (Alder 1959), and first principles calculations (Irwin 1988)

are worthy to mention according to current scope of the thesis. The main concern of all these

techniques, is to cover the phenomenon of length scale because the dominant concept to

investigate the properties of corresponding material can be changed from meters (m) down to

micrometers (µm) in classical mechanics and continuum models, while depending on the

criteria of the application they can be governed by various length and time scales.

First principles calculations or ab-initio study is a new-fangled third pillar of investigation

which opens up the possibility to study a complex system by performing computer

simulations. The method involved in these simulations can be achieved with variety of ways

ranging from classical to quantum mechanical approaches. It is one of the best theoretical

tool of choice for predicting new material. It holds fully quantum mechanical treatment of

electrons. The dependence of these calculations is hidden in Density Functional Theory

(DFT) which can well be described by famous Schrodinger’s equation in non-relativistic case

and Dirac’s equation in relativistic case. The purpose to solve Schrodinger’s equation via

Density Functional Theory (DFT) approach is to calculate properties of given material. In


Page | 7

addition to it, these calculations do not require any experimental knowledge to carry out such

investigations (Martin 2004).

The main motivation to do ab-initio study lies in the fact that it directly starts at an

established level of condensed matter physics and does not make assumptions such as

parameter fitting and empirical modelling. The success is hidden in the fact that the only

powerful probe to investigate the physical or chemical properties of material relies on atomic

constants as input parameters in order to solve the Schrodinger’s equation. Further, it needs

no experimental information to envisage the behavior of a material ahead of its synthesis, for

instance, in an ab-initio Density Functional Theory (DFT) approach electronic structure

calculations can be done by using Schrodinger’s equation that do not require fitting the

model to the experimental data. The method involved in these simulations can be performed

with the variety of ways ranging from classical to quantum mechanical approaches. To date

thousands of material properties are being calculated by using these methods. These valuable

procedures evolved into different varieties for ease of applications and are used by material

scientists, biochemists, geologists, drug designers, and even by astrophysicists as well.

1.3.3 Computational analysis

The important resource to explore here is computational analysis. In fact, we live in the era of

technology and there are many effective ways to speed up this technology, among them the

one which saves time as well as money is computational procedures which allows to analyze

and interpret physical properties of a given material by solving number crunching

calculations in small span of time. This sort of analysis allows to plan future experiments

instead to go through all kinds of experimental procedures and allows one to narrow the


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design space so in order to study a complex system, need is to search out best practical

computational resources. However, it is also true that modelling and simulations of material

have attracted much more consideration during the last few decades because of substantial

improvement and growth in processing speed and algorithms.

1.4 Perovskites

In the market of research solids are available with various crystal structures, for example

ionic solids, metallic solids, network atomic solids, atomic solids, molecular solids, and

amorphous solids as well. Among all the different structure types, the perovskite structure,

named after the Russian mineralogist Lev A. Perovski (Tilley 2016), has, since its discovery

in 1839 by Gustav Rose (Marc and McHenry 2007), been found to be one of the most

versatile structures for the development of technologically very important applications, for

example catalysts (Lombardo and Ulla 1998), batteries (Yang et al., 2012), thermoelectrics

(Robert et al., 2007 & Raveau 2005), dielectrics (Kim and Woodward 2007),

superconductors (Torardi 1988) and colossal magnetoresistance (CMR) (Raveau et al., 1998)

devices. The unique ability of perovskite-type structures to accommodate most of the

elements in the periodic table makes them useful for many different types of studies

(Woodward 1997). These are investigations into how different properties vary with

composition. The commonly investigated parameters include structural changes, magnetic

properties, electronic and ionic transport, in most cases to further develop today’s advanced

technology.

1.5 Crystallographic details of perovskite structures

The ideal ABX3 perovskite structure, as shown in Figure 1.4, with oxide and halide as anion,

is composed of cubic close packed layers of AX3, as shown in Figure 1.5, along the cubic


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Figure 1.1: Schematic diagram showing correlation between First principles

study,Quantum mechanical investigation and computational analysis.

Figure 1.2: Comparison between fundamental quantum mechanics and technology of the

classical world (Weinberg 2013).

Quantum mechanical

investigation

Computational analysis

First principles

study


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Figure 1.3: Schematic representation showing journey of quantum mechanical

investigation for evaluating physical properties of material.

Figure 1.4: A generic perovskite structure of the form ABX3. Note however that the two

structures are equivalent – the left-hand structure is drawn so that atom B is at the <0,0,0>

position while the right-hand structure is drawn so that atom (or molecule) A is at the

<0,0,0> position. Also note that the lines are a guide to represent crystal orientation rather

than bonding patterns (Snaith 2013).

Quantum mechanical investigation

Problem in solution of many body wave function

Propose Density functional theory

Khon-Sham equation

Need for approximate solutions

Evaluation of physical properties


Page | 11

unit cell. The larger A cation (usually a lanthanoid, alkaline earth, or alkaline earth element)

is coordinated by 12 X ions. The smaller B cation (usually a transition metal, or s/p-block

element) is positioned in the octahedral interstices formed by the X-ions (Snaith 2013 &

Navrotsky 1998).

The large flexibility of the perovskite structure makes it ideal for systematic substitutions of

the different ions in the structure. The most common substitutions are those of the A- and/or

B-site cations by other cations of different charge, size and electronic structure. The ideal

cubic perovskite structure (Pm3m) can therefore be tuned relatively easy if the difference in

the ionic radius between the ions on the same site are kept below ~10-15% (Zhang et al.,

2007). Structural changes upon substitution on the A-site, assuming rigid BO6 octahedra, can

be illustrated by the substitution of Sr2+ ions in SrTiO3 with the smaller Ca2+ ions as shown in

Figure 1.6 respectively. The example illustrates how the perovskite structure adapts changes

in the ratio between the ionic radii of the A and B cations. The resulting tilt of the BO6

octahedra is mainly the result of a minimization of electrostatic ion-ion interactions. By

treating the A and B site ions as hard spheres with different ionic radii, it is possible to

estimate the distortion of the perovskite structure from the ideal cubic Pm-3m symmetry

(Tilley 2016).

1.5.1 Tolerance factor criteria for perovskites

Tolerance factor or Goldschmidt's tolerance factor is an indicator to determine the stability

and distortion of crystal structures. Goldschmidt recognized that if the B-O distance is twice

the unit cell edge, and if twice the A-O distance equals the face diagonal of the unit cell,

perfect cubic close packing would be obtained. The relationship was quantified in the form of

(Goldschmidt 1926):


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𝑡 =0.707<𝐴−𝑋>

<𝐵−𝑋> (1.1)

where t is now commonly known as the Goldschmidt tolerance factor. For more complex

compositions with several different A/B cations on each site, the weighted cationic radii are

used. It has been found that the perovskite structure is formed for compounds with 0.78 < t <

1.05 (Woodward 1997). The tolerance factor can therefore be used to estimate the expected

amount of octahedral tilt. For an ideal cubic perovskite, the tolerance factor should be equal

to 1. However, the cubic symmetry is also found for perovskites with t-values slightly

deviating from the ideal one. Nevertheless, larger deviations of t from 1 are followed by

changes in symmetry (Travis et al., 2016; Bhalla et al., 2000 & Mitzi 1999). The details of

some perovskites and their tolerance factors are mentioned in Table 1.1.

1.5.2 Types of perovskites

The structures of perovskites are determined by short range attractive forces and repulsive

forces between nearby ions, along with long range electrostatic interactions between unit

cells. Crystallography determines the balance of these forces, and therefore the structure.

Which, in turn, contributes to the properties and performance of the material. Depending on

composition and chemistry of the constituent elements, perovskites have five different types

like simple perovskites, inverse perovskites, double perovskites, antiperovskites, and double

antiperovskites. From the available literature it is evident that above types can be found in

five different structures, including cubic, tetragonal, orthorhombic, hexagonal, and

rhombohedral (Smith 2015). The recent advancements in identifying crystallographic

technology have made it feasible to accurately determine structure of many perovskite

compounds for subsequent modeling.


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Figure 1.5: The ideal ABX3 perovskite structure showing the octahedral and icosahedral (12-

fold) coordination of the B and A-site cations, respectively (Navrotsky 1998).


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Figure 1.6: Illustration of the simple cubic perovskite unit along one of the main unit cell

axes in (a) an ideal cubic perovskite with a larger A-site cation and (b) a smaller A site

cation (Zhang et al., 2007).

Table 1.1: Some perovskite, their tolerance factor and structures.

a) (Travis et al., 2016), b) (Mitzi 1999), c) (Bhalla et al., 2000)

Tolerance

factor Structure Explanation Example

>1 Hexagonal A ion too big or B

ion too small BaNiO3

a

0.93-1.02 Cubic A and B ions have

ideal size SrTiO3

b, BaTiO3a

0.71-0.9 Orthorhombic/Rhombohedral

A ions too small

to fit into B ion

interstices

GdFeO3c,

CaTiO3a

<0.71 Different structures A ions and B have

similar ionic radii FeTiO3

a


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1.6 Aim of the research

In today’s modern society where technology plays a major role in our lives, it is necessary to

continue the development further through discoveries of new material and/or improve

existing ones. For technological needs, the importance of the ability to control the properties

is crucial in order to maintain high efficiency while avoiding material degradation at various

temperatures.

The main motivation for this thesis has therefore been to investigate the relations between the

composition, structure and properties of some perovskite-related materials of interest for

potential applications. The first three parts of the thesis involves the investigation of the

structural, optoelectronic, elastic, mechanical, thermodynamic and magnetic properties of

SrLiF3, CaLiF3, SrNaF3, SrKF3, SrRbF3, KVF3, KFeF3, KCoF3, KNiF3, KPaO3, RbPaO3,

BaUO3 and BaPaO3 oxide and halide perovskites employing the first principles density-

functional calculations. Accurate information of the following proposed compounds and their

structural trends could, however provide valuable background information for resolving

physical relationship in their scientifically more important analogues.

The next part involves the study to correlate the existing theoretical and previous

experimental works by extending pressure induced structural, elastic, mechanical, electronic,

optical and thermodynamic properties of SrLiF3, CaLiF3, SrNaF3, SrKF3, SrRbF3 fluoride

perovskites. The main emphasize of each property is to evaluate its application and ways to

implement in practical device fabrication.

The final overall motivation of this thesis is to investigate the interrelation between structure,

composition, and structural, elastic, mechanical, electronic, magnetic, optical, thermal,


Page | 16

thermodynamic properties of some selected perovskite materials for improving lens

fabrication technology, opto-electronic, spintronic and piezoelectric devices. Systematic

studies on this kind of perovskite-related materials are especially important in these strongly

correlated systems where the collective behavior of individual contributions can lead to

unexpected properties. Hopefully this study is an attempt to compensate lack of information

on theoretical and experimental data of aforementioned properties of these materials and to

add some new converged physics and innovative investigation in them. As far as, for a one-

man project, it would be too ambitious to solve all the current challenges but hopefully the

present work, does add some valuable advances in the field of ab-initio quantum mechanical

investigation.

1.7 Outline of thesis

The thesis is presented in nine chapters followed by the references. Following the

introductory chapter, chapter one, the second chapter serves to explain perovskite materials:

from synthesis to applications. The aim of Chapter 3, is to gather the general overview of

available literature on oxide and halide perovskites. Chapter 4, is reserved for the detailed

description of theory and computational details of underlying methodology, followed by brief

introduction of Density Functional Theory (DFT) and to acknowledge various simulation

techniques.

The results and discussion are discussed in four chapters (Part of the text from the published

papers is included in the respective discussion sections). Chapter 5, is devoted to investigate

elastic, and optoelectronic properties of alkali and alkaline earth fluoroperovskites

(Publication 1 to 3). Chapter 6, outlines mechanical and optoelectronic behavior of actinoid


Page | 17

based oxide perovskites (publication 4 and 5). The comparison of electronic band profiles

and magneto-optic properties of transition metal based fluoroperovskites are presented in

chapter 7 (publication 6) and chapter 8 (publication 7 to 11), distinguishes effect of pressure

variation on strontium and calcium based fluoroperovskites. The general conclusions, an

outlook and suggestions for further work are covered in chapter 9.

Chapter 2 Perovskite materials

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Chapter 2

Perovskite materials: From synthesis to applications

“I am neither clever

nor especially gifted. I am only

very, very curious”

Albert Einstein

2.1 Overview

Perovskites obtain their name from calcium titanium oxide (CaTiO3) structure. The term

“perovskite” was initially reserved for CaTiO3, but it was later applied to synthetic

compounds with a similar stoichiometry and crystal structure to CaTiO3 (Marc and McHenry

2007). Figure 2.1 shows naturally occurring compound CaTiO3 species. Goldschmidt

(Goldschmidt 1926) extensively studied the first synthetic perovskite and pioneered many

principles that are even today, remains applicable to the structure. Recently, perovskite

structured ceramics have become one of the worldwide materials due to their peculiar

properties via ferroelectric (Choi et al., 2004), thermo-electric (Obara et al., 2004),

pyroelectric (Chan et al., 2005), dielectric (Arlt and Hennings 1985) and optical properties

(erum and Iqbal, November 2017). Depending on these peculiar properties perovskite

ceramics have several extraordinary applications such as in, random-access memories

(Kingon et al., 1996), tunable microwave devices (Nenasheva et al., 2004), capacitors

(Dimos and Mueller 1998), displays (Protesescu et al., 2015), piezoelectric devices (Uchino

et al., 1998), actuators (Muralt et al., 2009), sensors (Obayashi et al.,1976), and wireless

communications (Sebastian 2010 & Uchino et al., 1998). Perovskites can be prepared in

various forms like nanocrystalline, bulk, thin films and rods, depending on their applications.


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This chapter provides a brief overview on synthesis methods for perovskite materials,

physical characteristics, band chemistry, significant properties and at the end the chapter

highlights the eventual technological applications.

Figure 2.1: Perovskite mineral species (CaTiO3) along with Lev Aleksevich von Perovski

(Jana 2008).

2.2 Synthesis methods for perovskite materials

Historically, inorganic solid-state physics has been a very central branch of physics

industrially. Especially for the development of different technologically important materials

such as functional glasses, photocatalysts, batteries, lasers, magnetic materials, metals and

alloys, it has been and will continue to be important. The conventional “shake and bake”

solid state synthesis method (Anthony 2014) is usually enough for standard characterization

methods such as electron microscopy, although it is sometimes more beneficial to use other

routes to produce the same material, but with different morphological characteristics to


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change the size/shape dependent properties. These can be, for example, electronic and

magnetic properties in low dimensional materials or conductivity/catalytic properties from

particle size or surface mediated effects.

The following sections will briefly describe the commonly used synthesis procedures for the

unconversant reader in this field.

2.2.1 Conventional inorganic solid-state synthesis

The traditional synthesis route in inorganic solid-state chemistry has been through grinding

of the reactants, usually the oxides, carbonates or nitrates of the desired elements in

stoichiometrically right proportions in a mortar with a pestle. The corresponding mixture is

then pressed with a hydraulic press into a pellet in a dye. The pellet is then placed in a

furnace with the desired temperature programmed, in order to form the desired product (Rao

and Gopalakrishnan 1997). Due to the long diffusion lengths in the particles of the starting

materials (if oxides are used) >1μm, it is usually necessary to regrind and press a new pellet

with subsequent heating until a single-phase material is observed by powder x-ray diffraction

(Cullity 1977).

2.2.2 Solution-based synthesis methods

In some cases, it is more desirable to use a solution-based route to produce the products, for

example, when there are difficulties in obtaining single-phase materials due to unreactive

starting materials, large diffusion distances with slow diffusion rates at the desired annealing

temperature, and the need to have a high surface area and nano-sized particles of the material

for applications. Typically, the salts, e.g., nitrates, oxalates, carbonates or oxides are

dissolved in a water solution together with acid citric acid in the desired ratio to the cation

content at 70-90°C. If necessary, for example, when alkaline oxides are used, nitric acid


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needs to be used in order to dissolve the starting materials. The next step is to raise the

Potential of Hydrogen (pH) of the solution with ammonia in order to deprotonate the citrate

molecules in order to chelate - bind to the metal-ions in the solution. If any acidic oxides are

used, these will dissolve at this step.

In the next step the solution is kept at a higher temperature ~300°C to increase the solubility,

while most of the water and ammonia is boiled off. This step will also speed up the gelation

process while avoiding precipitation of any metal salts as the pH decreases when the

ammonia evaporates. The final step is to combust the citrate at a temperature above ~350°C

in order to form a nano-powder of the product. The resulting voluminous powder can then be

again grinded, pressed and annealed at higher temperatures (McHale 1995).

2.3 Prediction methods for structural properties of perovskites

To find electronic structure of any material first ingredient is its structural properties. The

electronic structure of any material is important because the properties concerned to the

electrical, magnetic, optical and thermoelectric behaviors are dependent on it (Erum and

Iqbal 2016 & Wang and Kang 1998). Structural properties of perovskites can be evaluated by

experimental as well as theoretical methods. Experimentally basic structural parameters can

be calculated by different diffraction methods such as X-ray and neutron powder diffraction

methods, powder diffraction and structure refinement methods (Cullity 1977). However

according to scope of this thesis focus is paid to theoretical methods in detail. In theoretical

fields there are several ways to determine structural properties ranging from analytical

models to computational methodology. In this lieu, the first attempt was made by Mooser and

Perason, (Mooser and Pearson 1959) they introduced structural map technology to study the

average principle quantum number by evaluating difference between anion as well as cation.


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This well-known method is called as structural map method. Kumar and their fellows

(Kumar 2008) suggested to plot “structural map” for cubic perovskites. The same method

was followed by Muller and Roy (Muller and Roy 1974).

An important feature to identify crystalline materials and their structural properties is the

lattice constant. To reduce experimental temporal cost and time, recently researchers are

aiming to develop lattice constant prediction models to accurately investigate concerned

lattice parameters. This includes seminal work based on linear regression techniques by Jiang

and their fellows (Jiang et al., 2006), Moreira and Dias (Moreira and Dias 2007), as well as

Ubic (Ubic 2007) but due to linear regression these models are incapable to attain

nonlinearity involved in correlating lattice constant with atomic parameters. As a result,

appreciably, error cannot be able to minimize. In another work Lufaso with his fellows

(Lufaso and Woodward 2001) had established mathematical model for material scientists

built on SPuDS program, that delivers structural aspects for synthesized perovskites. In 2008

another model based on number of valence electrons and average ionic radii (rav) have been

proposed by Verma and their co-fellows (Verma et al., 2008) for cubic perovskites. All these

analytical models are contributing to explore structural trends of perovskites and delivers

many useful data to academia community but there are some known limitations in these

models with comparison to computational and experimental work due to contributing factors

of equations involved in these models (Erum and Iqbal, February 2017).

In theoretical framework another way is computational methodology which includes ab-initio

or first principles investigations (Lany and Zunger 2009). In this method of calculation, the

structural parameters are predicted by volume optimization process, through reducing the


Page | 23

total energy of the unit cell using Murnaghan’s equation of state (Murnaghan 1944). The

resultant ground state structural properties include various lattice parameters.

2.4 Physical characteristics of perovskites

To distinguish perovskites from other class of compounds it is necessary to have information

about its appearance and physical characteristics governing them. These compounds exhibit

variable color from gray, brown, black, and orange to yellow. On surface white to gray

streaks are observed as shown in Figure 2.2. In lustrous form it is submetallic to adamantine,

waxy or greasy. In crystalline form they are transparent and their crystals are opaque with

orthorhombic or pseudo cubic crystal symmetry. General values for hardness and specific

gravity is 5.5 and 4.0 respectively (Tilley 2016). Their associated minerals include andradite,

chlorite, leucite, melilite, serpentine, talc, nepheline, and sphene (Wenk and Bulakh 2004 &

Chakhmouradian and Mitchell 1998).

Figure 2.2: Structure and morphology of perovskite mineral (Wenk and Bulakh 2004).


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2.5 Band chemistry of perovskites

To understand material properties, based on the electronic configuration, the knowledge of

how the energy levels of the orbital in a crystalline solid change with structure and

composition is required (Martin 2004). Simple schematic view of bandgap in solids can be

illustrated from Figure 2.3. In principle, it is possible to qualitatively estimate the energy

levels for the orbitals that formulate the valence and conduction bands for an ideal cubic

perovskite as shown in Figure 2.4. The information needed for this type of drawing of the

orbital energy levels is mainly due to knowledge of the composition and crystal structure for

the compound of interest as well as ionization potentials and electronegativities of the

elements. The Density of States (DOS) for a solid phase can often be tracked back to the

molecular orbital (MO) scheme for a molecule with the same polyhedron; the difference

comes from the extension of the units in three dimensions for a solid. For a generic cubic

perovskite, the band structure is based on the original molecular energies created primarily

from the ligand field theory of an isolated MO6-octahedron (Muller and Roy 1974). The

energy levels of the bands vary throughout the crystal, extended in three dimensions

according to the topology and strength of the bonds. The band structure of a compound is

therefore the link between the structure, bonding and physical properties such as the

electronic conductivity, catalytic activity, and magnetic as well as optical properties in a

material.

The number of available energy levels between an energy E and dE (DOS) depends on how

the bands run in the crystal. These can be either narrow or broad depending on the degree of

orbital overlaps in the different directions of the crystal. The bands that forms the DOS are


Page | 25

thus affected by chemical and structural changes. In general, some simple ways to

manipulate the bands in the perovskite structure can be used and include (Hoffmann 1987):

• Isovalent or aliovalent A/B-site substitution for ions of different size and charge →

tilting of the octahedras → decrease/increase of the M-O-M orbital overlap →

narrowing/broadening of the corresponding bands.

• Substitution for more/less polarizing (electronegative) ions → more/less orbital

overlap → broader/narrower bands.

To get a more realistic picture of the energy levels of a compound it is necessary to use more

advance modeling, for example, DFT calculations (Knížek et al., 2006). Even then it is very

difficult to estimate the exact electronic structure for more complex systems. The difficulties

arise from the influences of defects, local distortions, impurities, strains, spin-orbit couplings,

mixtures of spin-states and so on.

2.6 From insulating to superconducting perovskites

Based on electronic bandgap configuration, behavior of perovskites can be classified into

insulators, semiconductors, conductors, and superconductors as shown in Figure 2.5. The

absence or abundance of free electrons play a vital role in distinguishing among the various

perovskite materials. Perovskites like SrTiO3 (Obara et al., 2004), SrLiF3, SrNaF3 (Erum and

Iqbal, March 2017), BaLiF3 (Mousa and Mamoud 2013), reveals insulating behavior. Since

they do not have any electrons in conduction band. Other perovskites like LaCuO3, KNbO3,

LaCoO3, SrRbF3, RbPbF3 (Murtaza et al., 2013 & Bringley et al., 1993) shows

semiconducting behavior. Some perovskites like LaNiO3, LaCoO3, LaMnO3, and LaCuO3

(He and Franchini 2012) manifests conductive behavior. The electrical conductivity can be


Page | 26

increased by enhancing the total number of mobile charge carriers. However, the major

source of distinction between properties is electronic configuration of the B-ion. Such as do

Ti+4 is responsible for insulating character in SrTiO3, meanwhile d7 Ni3+ and d8 Cu3+ are

responsible for conductive behavior both in LaNiO3 and LaCuO3 respectively. The

corresponding d6 and d8 configuration results a splitting between the filled t2g and eg orbital

in conduction band which can give rise to insulating ground state in the corresponding

compounds.

In addition to their diverse electrical properties, perovskites have been extensively examined

for superconductivity. BaPb1-xBixO3 (0.05 < x < 0.31) discovered in 1975, is a super

conducting oxide with a critical temperature (Tc) of 13 K (Sleight et al., 1975). Since then, a

number of superconductors with Tc > 77 K including the well-known YBa2Cu3O7-δ have

been discovered (Knizhnik 2003). Now-a-days an interesting property of perovskite material

is superconductivity at high temperature. This is unique property of perovskite however the

critical temperature for superconducting transition to occur is about 130-155 K for

HgBa2Ca2Cu3O8+δ. However, the increase value of Tc is achievable by adding more number

of Cu-O layers (Chu et al., 1993). The complex compounds at a certain critical temperature

Tc includes complex cuprate perovskite oxides and their sub-units. The presence of copper

ion at B-sites is mandatory to achieve superconductivity. However, La2-xBaxCuO4 was the

first perovskite related material. Another example of such compound is BaPb1-xBixO3

(Bednorz and Muller 1986).


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Figure 2.3: Schematic illustration of the band gap in solid materials (Holgate 2009).


Page | 28

Figure 2.4: A band gap diagram showing the approximate band energies in ABO3, that form

the density of states (DOS) in a perovskite (Hoffmann 1987).

Figure 2.5: A band gap diagram showing the different sizes of band gaps for conductors,

semiconductors, and insulators (Wikimedia commons 2015).


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2.7 Magnetism and electronic correlations in perovskites

When a perovskite material is put in a magnetic field, H, it gains a molar magnetization, M,

that together can be described as the magnetic induction, B, in equation (2.1) using the

centimeter-gram-system with electromagnetic units (c.g.s. e.m.u.):

𝐵 = 𝐻 + 4𝜋𝑀 (2.1)

The magnetic susceptibility can be expressed as:

𝜒𝑚 = (

𝑀

𝐻 (𝑒𝑚𝑢)).𝑀𝑊 (

𝑔𝑟𝑎𝑚

𝑚𝑜𝑙𝑒)

𝑚 (𝑔𝑟𝑎𝑚). 𝑓𝑖𝑒𝑙𝑑 (𝑂𝑒) 𝐶

𝑇 (2.2)

Here χm is the molar magnetic susceptibility in the linear response region, i.e., where the

magnetization changes linearly with the field, m is the mass in grams and Mw the formula

weight per magnetic ion. This type of linear behavior is also referred to as the Curie law,

where C is the Curie constant.

From the Curie constant it is possible to deduce the average magnetic moment per magnetic

ion through the following expression:

µ𝑒𝑓𝑓 = √3𝑘𝐵𝑇𝜒𝑚

𝑁𝐴µ𝐵2 = √8𝜒𝑚𝑇 (2.3)

where kB is Boltzmann’s constant, T the actual temperature, NA Avogadro’s constant, χm the

molar magnetic susceptibility and μB the Bohr magneton. All materials interact with a

magnetic field by responding with a force opposite to the applied field. This is called

diamagnetism and is an inherent additive property caused by the interaction of the magnetic

field with the motion of the electrons in their orbits. The effect is independent of the applied

field and temperature. The diamagnetic susceptibility is approximately in the order 10-6 emu

and negative. The opposite effect with an attractive force from the magnetization of unpaired

electrons is called paramagnetism and is several magnitudes larger, e.g., 10-4-0.1 emu and


Page | 30

overrides the diamagnetic signal. The magnetic susceptibility is therefore mainly based on

two components:

𝜒𝑡𝑜𝑡 = 𝜒𝐷 + 𝜒𝑃 (2.4)

where χD and χP are the diamagnetic and paramagnetic contributions, respectively.

As a consequence of the lower thermal energy, kBT with lower temperature, an increase of

the magnetic susceptibility is observed for a paramagnetic material as more and more

electrons align in the direction of the field. The type of paramagnetism that contains some

type of cooperative orientation of the moments is better described by the Curie-Weiss law:

𝜒𝑚 = 𝑐

(𝑇−𝛳) (2.5)

Here ϴ is the Weiss constant and describes the strength of the magnetic interactions, which is

either negative (anti-parallel interactions) or positive (parallel interactions). In some cases,

the magnetic interactions between the electrons in a material can interact strongly and couple

below a critical temperature. The interactions/couplings between the magnetic centras

(unpaired electrons and orbital angular moments) are mainly classified into two major

groups, those where the majority of the electrons are coupled antiparallel (antiferromagnetic

(AFM) interactions/ordering) and those with the majority of the electrons coupled in parallel

(ferromagnetic (FM) interaction/ coupling). In between these two groups many different

types of magnetic effects/orderings can exist.

In general, the type of interactions between electrons can be described by the exchange

energy, Hex between atoms i and j separated with a distance rij with the total spins Si and Sj:

𝐻𝑒𝑥 = − ∑ 𝐽(𝑟𝑖𝑗)𝑆𝑖𝑆𝑗 (2.6)

In this expression the effective exchange parameter J(rij) describes the type of interaction

between the atoms, as a function of the distance between them. A positive value indicates


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that parallel alignment of the moments is favored and a negative value that an antiparallel

alignment is favorable. The exchange coupling can be divided into two different classes. The

direct exchange, where inter-atomic exchanges of electrons occur through atoms that are

close enough for their orbitals to overlap significantly, and indirect exchange, which couples

the magnetic moments over larger distances through intermediate ions such as oxygen ions in

oxides. The latter is commonly referred to as super-exchange. It follows that a material in

which the magnetic moments are spontaneously aligned in the field direction below a defined

temperature, i.e., the Curie temperature, TC, is a FM. In this type of material, the

magnetization increases rapidly and saturates quickly with M as the larger magnetic domains

grows at the expense of the smaller.

As opposed to FMs with one magnetic cell, the magnetic moments of two identical sub-

lattices can align oppositely with identical magnetic moments below a certain temperature,

i.e., the Neel temperature, TN. The material is referred to as an AFM. Hence, it is not possible

to have an AFM ordering in non-crystalline solids. A special case of AFM ordering called

ferrimagnetism occur when the two sublattices are chemically different although

ferromagnetically ordered in each sub-lattice, that results in a residual net magnetic moment.

Hence, ferrimagnetic ordering can exist in amorphous materials as the oppositely aligned

magnetic moments are not required to be of the exact same size.

Due to local variations in a material that cause anisotropical variations in the crystal field

strength, meta-magnetism can occur, which is typically an AFM below the TN but undergoes

a magnetic transition at high field strengths that causes the magnetization to increase. This

can be observed as a change in the slope in the M vs. H curve at fields exceeding the

saturation of the magnetization.


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Another type of magnetic behavior of solids is the so-called speromagnetism, which involves

randomly ordered localized magnetic moments with no significant net magnetization or

ordering beyond the nearest neighbors, except for randomly coupled spins locally. These

systems are usually referred to as spin glasses and behave mainly as a paramagnet above the

“freezing” temperature TSG.

Above a certain concentration of magnetic ions, clustering can occur, where the material is

classified as a mictomagnet which essentially is a “cluster glass” (Mattis 2006). Spin glass

behavior can be characterized by using A.C. magnetic measurement instead of D.C. that only

measures the equilibrium value of the magnetization. The A.C. susceptibility measures the

susceptibility as a function of the variation of the field, H and frequency ω that varies

sinusoidally. This will provide a high sensitivity to the change of magnetization at a given

time and frequency. For spin glasses the freezing temperature will show a shift in the

temperature of the “cusp” with changing frequency of the field due to relaxations and

irreversible ordering at non-equilibrium conditions (Stein and Newman 2012).

In transition metal compounds the measured effective magnetic moment, ueff can often

deviate significantly from the calculated value given by the expression (Handley 2000):

µ = [𝐿(𝐿 + 1) + 4𝑆(𝑆 + 1)]1

2 µ𝐵 (2.7)

where L is the maximum value of the orbital quantum number, S the total spin quantum

number, for the electrons outside closed shells and µB bohr magneton. The total angular

momentum J = |L + S| is conserved and the degeneracy of the energy levels for a specific J

value will be either (2S + 1) or (2L + 1), depending on which one has the smallest value.

With a “spin-orbit” coupling new quantum states are created for each J with a degeneracy of

2J + 1. Adjacent energy levels, e.g., J and J + 1 will then be separated by an energy


Page | 33

corresponding to (J + 1) λ, for which λ is the spin-orbit coupling constant with the unit of cm-

1. For strong spin orbit couplings, the energy separation will be relatively large and can thus

be seen clearly by spectroscopic methods, e.g., XANES.

For many of the 3d transition metal compounds, the observed magnetic moment frequently

deviates from the values that includes spin-orbit coupling, and are therefore closer to the

spin-only value of the effective magnetic moment:

µ𝑆.𝑂 = [4𝑆(𝑆 + 1)]1

2 = [𝑛(𝑛 + 2)]

1

2 µ𝐵 (2.8)

where n is the number of unpaired electrons responsible for the magnetic moment. In these

cases, the orbital angular momentum is either very small or negligible; hence the orbital

contribution is “quenched” (Mabbs and Machin 2008).

2.8 Thermodynamic valence stability in transition metal based

perovskites

The properties of the perovskites primarily depend on the electronic structure of the

constituting ions and their interactions through bonds. However, properties such as the

electronic or ionic conductivity can deliberately be changed by taking advantage of the

thermodynamic stability of different transition metal ions in their different oxidation states.

In general, the thermodynamic driving force for a reaction is to minimize the Gibbs free

energy, ΔrG towards zero to reach equilibrium. For a spontaneous reaction the ΔrG is lower

than zero, that is, ΔrG < 0 and ΔrG > 0 for a non-spontaneous reaction via following

equation:

𝛥𝑟𝐺 = 𝛥𝑟𝐻 − 𝑇𝛥𝑟𝑆 (2.9)

Here ΔrG is the Gibbs free energy of reaction, ΔrH the enthalpy of reaction, ΔrS the entropy

change of reaction and T the actual temperature. It is clear that metals with a large negative


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ΔrG for the reaction with oxygen to form an oxide; also form the oxide much more easily.

Hence, oxides with the least negative ΔrG, are reduced more easily. In general, the trend

among the 3d transition-metal elements can roughly be approximated to the following order,

where the oxide form of the first element is the easiest to reduce to metal at 1000°C: Ni > Co

> Fe > Mn > V > Cr > Ti (Kawada and Yokokawa 1997). However, although this trend is

true for the binary BxOy-oxides, it is important to emphasize that the same trend is also

observed for the corresponding ABO3 perovskites.

As an additional difference between the perovskite oxides and the simple binary oxides, the

higher valence for transition metal ions of varying oxidation states in perovskites are further

stabilized through the lanthanide ion; therefore, an additional term called the stabilization

energy δ(ABO3) needs to be reduced. However, it is observed that the stabilization of higher

oxidation states in the LaMO3 decreases with a decrease in the Goldschmidt tolerance factor,

t (Goldschmidt 1926). This implies that a higher oxidation state of the transition metal ion is

better stabilized for structures with high symmetry.

2.9 Properties of perovskites

A property is an attribute which describes the features of any system. The properties can be

classified in terms of physical and/or chemical as shown in Figure 2.6. A physical property is

an aspect of matter that can be measurable without changing it and whose value describes

state of the physical system. Some examples of physical properties are molecular weight,

volume and color. Chemical properties are contrasted with physical properties which use to

determine the way a material behaves in a chemical reaction and it can only be observed by

changing the chemical identity of a substance. According to the need of this thesis attention

is only paid to physical properties of perovskites.


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Many examples of perovskite compositions with various interesting properties can be found

in the literature. For example, some compounds exhibiting colossal magnetoresistance

(CMR) are the perovskite related La1-xAxMnO3+δ materials where A2+ is Ca, Sr, Ba, or Pb

(Van et al., 1993). Compounds exhibiting CMR have potential uses in data storage

technologies, such as computer hard drives and floppy disks. Different perovskite

compositions also result in a wide variety of electrical behaviors. For example, SrFeO3 is a

metallic conductor at room temperature, BaBiO3 is a semiconductor and stoichiometric

LaMnO3 is an insulator (Muller and Roy 1974). Due to these tunable properties of perovskite

materials, very large variety of applications can be obtained and many more are still to be

found in the future.

Figure 2.6: Block diagram breakdown of chemical and physical properties of matter.


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2.9.1 Property based tentative classification of perovskites

Perovskite materials possess various versatile properties. Based on these properties one can

tentatively classify perovskites into the following major groups such as semiconducting,

conducting, and superconducting perovskites, dielectric perovskites, piezoelectric

perovskites, pyroelectric perovskites, ferroelectric perovskites, ferro-magnetic and anti-

ferromagnetic perovskites, colossal magnetoresistive perovskites, photovoltaic perovskites,

and catalytically active perovskites. Based on these properties perovskite materials can

classify accordingly.

2.9.2 Opto-electronic properties

Perovskites have emerged as a revolutionary class of materials having excellent optical and

photoluminescence properties. Several areas where the light interacts with the matter are

obviously of practical interest. Studies of the optical properties of solids have been proven as

a strong tool for electronic and atomic structure of the desired materials. The optoelectronic

properties of a material are important to understand the optical nature over a wide spectral

range to acknowledge the application of that material in photonics and optoelectronic

devices. The major requirements for these devices are wide and direct bandgap

semiconductors (Fox 2001). It helps to identify the internal character of that material

(Wooten 1972). Perovskites have gained huge hype in opto-electronic industry due to direct

wide bandgaps and high thermal stability. SrThO3, SrZrO3, and SrRbF3 have direct wide

bandgap semiconductors, (Shein et al., 2007) which makes these oxide and halide

perovskites very favorable for usage in optoelectronic applications. The optical density of

CaTiO3 was reported by Linz and Herrington (Linz and Herrington 1958). The absorption


Page | 37

characteristics are quite similar to those of SrTiO3 crystals with the exception that the

absorptions are shifted to shorter wavelengths. BaTiO3 and SrTiO3 have been considered for

high temperature infrared windows. The electro-optic properties of KTaO3, K(Ta0.65Nb0.35)

O3, BaTiO3 and SrTiO3 in the paraelectric phase were measured by Geusic and their fellows

(Geusic et al., 1964). The electro-optic coefficients of these perovskites are nearly constant

with temperature and can vary from material to material when the distortions of the optical

indicators are expressed in terms of the induced polarization. Many ab-initio or first

principles investigation have successfully explored the opto-electronic properties of these

materials which are in reasonable agreement with the experimental studies.

Now-a-days, there has been considerable attention in resources to be used for laser

applications. Using of perovskite laser host materials is a great deal. The ion Nd3+ appears to

be the most popular for introduction into relatively large crystallographic sites. However,

except when LaF3 is used as a host, compensating ions are required in these substitutions.

Divalent Tm2+ and Dy2+ can be substituted in CaF2 without compensating ions but these

divalent rare earth ions relatively unstable. For crystallographic sites for the Al3+, Cr3+ proved

to be ideal for substitution.

In photovoltaic industry, originally the perovskites were viewed as a curious replacement to

dye molecules in mesoscopic type of sensitized solar cells, due to their high absorption

coefficient and broad sense of absorption spectrum. However, it was soon realized that

perovskite materials are unique semiconductors, different from dye molecules or other

organic absorbers, and very suitable for the inorganic semiconductors for photovoltaic

applications, for example Si or GaAs. The perovskites materials have long carrier diffusion

length and remarkable performance in planar heterojunction architectures. The conversion


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efficiency of perovskite solar cells has reached up to 16% in its current version. The fast-

paced improvement, coupled with inexpensive materials and preparation methods indicates

that perovskite solar cells can break the prevailing paradigm and can be achieved in low cost

and excellent performance. In addition, multi-junction hybrid solar cells based on perovskites

are a very promising route to deliver a higher efficiency, cost-effective solar technology that

will compete favorably with today’s technologies, and we believe this application is likely to

be the first commercial appearance of the perovskite solar cells. Due to hysteresis free, such

cells inherit the advantages of organic photovoltaics. These benefits will lead these

perovskites go beyond the crystalline silicon (Chen et al., 2015).

Bandgap tailoring is another technique to design new materials for application in

optoelectronic industry. There are many theoretical and experimental techniques to vary

bandgap of materials to make them applicable for optoelectronic applications. The perovskite

photovoltaics can be processed on a variety of substrates via either solution or vapor phase

processing and have already delivered very high efficiency on a flexible format. In particular

fun illustration of the future technologies may enable extremely lightweight high-power

perovskite photovoltaics (Holgate 2009).

2.9.3 Dielectric properties

The detailed study on dielectric properties of perovskites have been done in the recent past

because some perovskites have revealed interesting unusual high value of the dielectric

constant, in which BaTiO3, PbTiO3 (Chou and Chen 1998), BaUO3, BaPaO3 (Erum and

Iqbal, February 2017), are worthy to mention. This feature of any compound is highly

desirable in micro as well as nano electronic devices, especially in their capacitive and

thermistive components. In oxide perovskites, such as PbTiO3, BaTiO3, and BaUO3 the


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collective polar displacement of metal ion on B site of perovskite, in accordance with oxygen

sub-lattice is responsible for large value of dielectric constant which can be associated with

their ferroelectric behavior (Charmola et al., 2011). The complex form of perovskite solid

solution attains a growing interest in last few decades because of numerous unexpected

properties. Relaxer ferroelectric is a class of perovskite compounds which retains pronounced

frequency dispersion and large value of dielectric constant and variation in the value of

dielectric constant as a function of temperature, which is of potential interest in

manufacturing practical devices. It is a well-known fact that dielectric property, response to

external excitations. The solid solutions of these ferroelectric materials are disordered by

variation in doping concentration and their dielectric behavior can be controlled by

controlling concentration of doping in them. This concerns several groups of materials

termed as mixed metal perovskites “super Q” solid solution such as SrxCa1-xTiO3, BaxSr1-

xTiO3, PbZn1/3Nb2/3O3-xPbTiO3, PbMg1/3Nb2/3O3-xPbTiO3 (Wu and Davies 2006). There are

many other useful techniques to control dielectric response of perovskites such as application

of pressure, temperature and so on. These properties widen the significance of these

materials while provoking future efforts to study such material under different conditions.

According to the value of dielectric constant and associated parameters perovskite

compounds exhibit many intriguing applications. Among them an important application is in

metal-oxide semiconductor field effect transistor (MOSFET). The perovskite oxide

CaCu3Ti4O12 is known to display largest value of dielectric constant between 104-105

(Alonso et al., 2003 & Sinclair et al., 2002). This class of compound is classified as giant

dielectric constant materials (GDM). The reason behind this value is still under investigation

however one possible reason is reported, regarding to barrier layer capacitor model. The


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major utilization of dielectric constant lies in the fact that it can eventually determines the

miniaturization capacity of electronic component. Now-a-days material scientist are

struggling hard to explore the ties between structural and dielectric properties, in order to

utilize them in future applications and each of above mentioned factors needs to be a subject

of special investigation (Homes 2001).

2.9.4 Piezoelectricity

The root of piezoelectric properties lies in between two crucial words piezo and electrics.

Hence the combined meaning of the word implied the concept of pressure-electricity. The

materials which retains this phenomenon conversely possesses geometric strain or

deformation in proportion to an applied electric field as shown in Figure 2.7. The

piezoelectric effect was first discovered in 1880 by curie brothers in single crystal quartz

(Katzir 2003). The perovskite material which can well be known as the dawn of piezoelectric

material is the ceramics of Barium titanate (BaTiO3). However original discovery of BaTiO3

was not concerned with piezoelectric properties rather it was related to high capacitance

(Ogawa 1947). Now-a-days the widely commercialized perovskite ceramics are primary

piezoelectric materials (Jaffe and Cook 1971). These piezoelectric type of perovskite

materials are applied to various devices ranging from sensors to actuators. There are five

essential key ingredients in the development of piezo-materials which includes performance

to reliability, macro to nano, hard to soft, homo to hetero, and single to multi-functional.

There are many perovskites including simple to complex one which exhibits piezoelectric

property. For example Ba (Mg1/3Ta2/3)O3, Pb(Mg1/2W1/2)O3, and Ba(Mg1/3Ta2/3)O3. Other

examples of lead free piezoelectric includes three major group of compounds namely (Bi,

Na)TiO3, (Na,K)NbO3, and tungsten bronze based perovskites (Uchino 2014). From


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crystallographic point of view depending on B-ion ordered/disordered arrangement some of

these structures show simple/complex perovskite symmetry. The significant difference in

physical properties of these compounds is observed corresponding to ionic ordering because

on increasing the ionic valence difference, tendency of short-range ordering increases. High

K-piezoelectric perovskite are applied to study medical acoustic. Another astonishing

application of piezoelectric Pb(Zr,Ti)O3 perovskite is its ability of passive mechanical

damping. Furthermore, it can also be utilized in piezoelectric microelectromechanical system

(MEMS) (Roberts 1947). However, from technological point of view some emphasize is

required to utilize piezoelectric property of perovskite in new technologies such as ultrasonic

disposal technology, in reduction of gas contamination, in generating new energy-harvesting

systems and in developing economic medical instruments. In disaster prevention application

piezoelectric perovskite can play a vital role. These applications include nuclear power plant

safety systems, earthquake monitoring as well. In actuator materials the primary (linear) and

secondary (quadratic) phenomenon are piezoelectric and electrostatic effects respectively.

Certain perovskite oxides, for example solid solutions of PbZrO3 and PbTiO3, exhibit

piezoelectric properties and are used for numerous applications including microphones,

stereo speaker fuses (for lighters) and ultrasonic cleaners (Jaffe and Cook 1971).


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Figure 2.7: Schematic illustration for the phenomenon of piezoelectric effect (Uchino 2010).

2.9.5 Multiferroicity

“Materials should exist, which can be polarized by a magnetic field and magnetized via

an electric field.” (Curie 1894)

Multiferroics symbolize an extraordinary class of materials exhibiting simultaneous

ferromagnetic, ferroelectric and ferroelastic ordering. Figure 2.8-2.10 depicts the multiferroic

materials possessing respective properties. The distinctiveness of these materials lies on the

possibility of simultaneous utilization of both their magnetization and polarization states, a

massive potential which would make them outstanding candidates for new generation

memory devices and sensors (Scott 2007 & Ramesh and Spaldin 2007).

Multiferroic perovskites can be classified into different groups which exhibits in two coupled

ferroic order (vice-versa antiferroic order) distinguishing between two major classes of

multiferroic materials. These two groups are responsible for two major properties, namely

ferroelectricity and ferromagnetism. Some special perovskites retain both ferroelectric and


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ferromagnetic behavior at the same time such as BiFeO3, BiCrO3, and BiMnO3, are the

materials which shows multiferroicity even at room temperature (Bordet et al., 2007).

Figure 2.8 depicts perovskite materials with regard to their ferroelectric, ferromagnetic and

multiferroic character. Among the various explored multiferroics, BiFeO3 oxide perovskite,

is receiving nonstop attentions since it possesses both ferroelectric order and anti-

ferromagnetic order for a widespread temperature range which is greatly above room

temperature (Ederer and Spaldin 2005). Another example includes BiMnO3 contain Mn ion

which is associated with mixed perovskite state with d0 and dn ions simultaneously in

dissimilar ionic states. The origin of multiferroic compound starts from the discovery of first

ferromagnetic material which was complex combination of several multiferroic boracite

compounds known as nickel iodine boracite Ni3B7O13I (Ascher et al., 1966). The coupling

factor is very weak and rare in these perovskites.

There are several sources of ferromagnetism and ferroelectricity in multiferroic materials.

However, these two phenomena try to exclude one another. Multiferroic materials are single

phase materials which is combination of two or more forms of ferroic order including

ferromagnetism, ferroelectricity, ferroelasticity, and ferrotoroidicty (Scott 2007). Block

diagram illustration of perovskite multiferroics are presented in Figure 2.9. However, these

materials are rare in nature because the d-transition metal ion (which is essential for

magnetism), generally reduces the capacity of off-centering ferroelectric distortion. In order

to simultaneous co-existence of ferroelectricity and ferromagnetism an additional driving

force must be present. Some multiferroic materials have strong response towards both

electric and magnetic field at room temperature that is why very rare of them are used in


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practical applications. A ferroelectric material must be an insulator and most of the

ferromagnetic materials are generally metals (Ascher et al., 1966).

“You may say anything you like but, we all are made up of ferroelectrics”

(Matthias 1949)

Hence, the absence of ferromagnetic insulators limits the concurrent manifestation of

ferromagnetic and ferroelectric ordering. Figure 2.10 reveals crossed interaction between

ferroelectricity and ferromagnetism. However schematic illustration of both phenomenon is

presented in Figure 2.11. Magnetic ordering of any kind takes place due to the presence of

unpaired d electrons, whereas ferroelectric materials such as common perovskite oxides

(ABO3) have a d0 configuration on the small B cation. Magnetoelectric multiferroic materials

should have some distortion in the crystal structure with some unpaired electrons in the d

orbitals. Recently, it has been found that even in the absence of any structural distortion,

magnetic spin ordering can produce ferroelectricity. This greatly expands the number of

potential ferroic materials (Cora and Catlow 1999).

Now-a-days modern research is focused on the materials which have the ability to bear

ferroelectricity with ferro or anti-ferro magnetic state simultaneously. This type of modern

multi multiferroic materials have potential applications in hybrid memory and spintronic

devices because of advantage of no time reversal and spatial symmetry. Advantages of these

multiferroic materials includes highly sensitive magneto-sensors, multistate memory

elements, and sensitive detection of magnetic field. However, there are some challenges that

BiFeO3 type of multiferroic materials have to face such as the value of leakage current is

very large in them, retaining small remnant polarization with commonly high value of

coercive field. However, the current challenges with BiFeO3 type of perovskite compounds


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can be overcome by applying electric field by introduction of manganese and titanium in

BiFeO3, in order to reduce the amount of leakage current greatly.

2.9.6 Electronic conductivity

Electronic conduction in perovskite-related materials is one of the main properties that

governs the feasibility of the material for many applications, for example, catalysts (Li et al.,

2009), thermoelectrics (Kobayashi et al., 2001), spintronic devices (Serrate et al., 2007) and

in various electrodes (Molenda et al., 2007) as well. In 3d, 4d or 5d transition metal-based

perovskites, different types of bonding to the oxygens should be expected. In general, the

nuclear charge is more efficiently screened by the closed shells when moving down in the

periodic table. It is therefore expected that the outer electrons are more loosely bonded for 5d

elements relative to 3d elements. Therefore, the hybridization of the d-orbitals with the

oxygen 2p orbitals should be larger for 5d-elements. The orbital overlap of the metal-oxygen

(M-O) or metal-halide (M-H) bond governs a large part of the localization of the electrons.

Using a band picture this can be described with the large overlap of the M-O/M-H orbitals

that lead to broader bands, and hence higher electronic conductivity. The behavior of the

electrons can be generalized into two types, itinerant and localized. The itinerant picture

approximates the electron as a “cloud” and hence electrons that travel through the material

without any major resistance. This type of behavior is typical for metallic oxides such as

CaVO3, SrCoO3 and SrMoO3 (Goodenough 1967). For the localized picture, it is assumed

that the orbital overlap is not large enough for the electrons to move freely, i.e., the inter-

atomic distances are larger than the itinerant electron case. This results in particle-like

electrons “jumping” from one cation to another, and hence the conductivity will be activated


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Figure 2.8: Multiferroics combine the properties of ferroelectrics and ferromagnets.

Figure 2.9: Block diagram illustration of perovskites multiferroics


Page | 47

Figure 2.10: The multiferroics totem; illustrating the three main ferroic orders with their

respective fields and crossed interactions (Wang et al., 2015).

Figure 2.11: Conditions required for ferroelectricity (polarization) and ferromagnetism

(unpaired electron spin motion) (Mabbs and Machin 2008)


Page | 48

by thermal energy, which increases the probability for a successful “jump” as the thermal

vibrations increases. These types of materials behave as semiconductors for which the

conductivity increases with increasing temperature. In fact, conductivity does not give any

information about the majority type of carrier, i.e., electrons/holes and will therefore need to

be complemented by measurements of the Seebeck coefficient.

2.9.7 The Seebeck coefficient

When a part of material is exposed to heat, charge carriers, i.e., electrons and holes will start

to diffuse to the cold part. The accumulation of one type of charge carriers at the colder part

will give a difference in chemical potential that corresponds to an electric potential difference

ΔV, i.e., the Seebeck voltage. The created potential difference will depend on the material’s

ability to separate the two different “heat carriers,” mainly the negative electrons, e, and the

positive holes, h. Hence, the build-up of the Seebeck voltage will depend on the mobility of

the two types of carriers and will thus be different from one material to another. This effect

arises from the entropy change induced per charge carrier, e.g., spin, mixing and vibrational

contributions. The calculation of the Seebeck coefficient can be defined in different ways

depending on type of material (Kobayashi et al., 2001).

2.9.8 Polarons

Many of the perovskite materials exhibits the phenomenon of lattice vibrations. When the

lattice phonons (lattice vibrations) are small enough (appropriate wavelength) to match the

local deformations caused by the electron when moving through the lattice, a potential well is

formed. The electron is therefore trapped by the lattice deformation caused by the local

dielectric polarization of the lattice. The electron and its lattice distortion behave as one unit,


Page | 49

which is most often classified as a polaron. When the wave function and the corresponding

lattice distortions cover many lattice sites, a so-called large polaron is formed. The large

polaron moves as a free electron through the lattice and is not entirely trapped, hence the

metallic behavior of transition metal oxides when electronic conduction is dominated by

large polarons (Cox 1992). In cases where the interaction of the electron with the lattice is

large enough, i.e., polaron-binding energy is larger than half the band width of the electron,

the electron will be classified as a small polaron (Goodenough 1971). The conduction

through small polaron-hopping is thermally activated and will often lead to a semi-

conductive behavior of the transition metal oxide and fluorides.

2.9.9 Thermal expansion

All perovskite materials are vibrating at all temperatures, even at absolute zero the atoms still

vibrate inside a material. As the temperature increases, the thermal energy of the system

increases, which in turn gives the atoms more kinetic energy to increase their vibrating

motion anharmonically and thus occupy a larger volume. The thermal energy not only

increases the volume of the atoms but can also induce more drastic changes in the atomic

volume in the crystal. This includes the reduction of the transition metal ion through the

formation of anion or cation vacancies or changes in the electronic structure from thermal

excitation of one spin-state to a higher spin-state as in Co3+ (Shannon 1976). In other cases,

the thermal energy will at some point induce a phase transition that can be either a change in

the state, e.g., solid to liquid to gas phase or a structural change within the same, e.g., solid to

solid state. If such transition occurs, it will, if the change in the unit cell volume is large

enough, be seen in the thermal expansion of the material. For solids, it is common to measure


Page | 50

the average linear expansion of a pressed and sintered powder along one direction by a

dilatometer. Typically, the sintered cylindrical pellet that is to be measured is polished to a

size close to the sapphire reference used for calibrating the instrument. The linear thermal

expansion is measured as a function of temperature in one dimension and can be roughly

calculated over a temperature range.

2.10 Application of perovskites

Perovskite materials exhibit intriguing and extraordinary physical properties that have been

extensively studied for both theoretical modeling and practical applications. These solids are

currently gaining considerable importance in the field of electronics, geophysics,

astrophysics, nuclear, optics, medical, and environment as well (Housecroft and Sharpe

2008). Oxide and halide based perovskites are fascinating nanomaterials for wide

applications due to its structural stability, wide bandgaps, variety of available compounds,

large number of intriguing properties, that eventually lead to many real-world applications.

The ABX3 type halides and oxides with direct band gaps (> 3 eV) have strong stability

against high temperature and strong radiation. Thus, the wide band gap ABX3 type halides

and oxides are potential candidates for next generation UV photodetectors, vacuum

ultraviolet transparent lens martials, vacuum-ultraviolet light emitting Diodes (VUVLEDs)

(Erum and Iqbal, March 2017 & Roth 1961). These materials demonstrate diversified

physical phenomenon such as piezoelectric, pyroelectric, ferroelectric, dielectric,

superconductive, multiferroic, Colossal Magnetoresistance (CMR), and Giant

Magnetoresistance (GMR) etc.

The strontium and barium based perovskites (BaZrO3 and SrZrO3) are currently being

developed as the electrolyte material for Solid Oxide Fuel Cells (SOFCs) (Gemmen and Liu


Page | 51

2010). Perovskites have an unusually high tolerance for oxygen ion vacancies, which makes

it an ideal electrolyte or cathode material for fuel cells. Oxygen vacancies can be created in

the material by doping with rare earth elements such as Y(III) or Yb(III), which replace the B

atom, and create one oxygen vacancy in order to maintain charge neutrality. These randomly

distributed oxygen vacancies can significantly change the conductivity of the material and

make it a very efficient electrolyte (Winter and Brodd 2004). Another current area of

research involving perovskites is high-temperature superconductors (HTSCs). The cuprate-

perovskite type is a specific kind of superconducting material. Cuprate perovskites are

related to CaTiO3 type perovskites but differ in that the B atoms have eight-fold coordination

with oxygen atoms. Starting from the CaTiO3 prototype and going to YBa2Cu3O7, Ba and Y

substitute for Ca, while Cu substitutes for Ti (Smith et al., 2011). Some cuprate-perovskite

ceramic HTSCs currently have an operating temperature of 90 K, which is significantly

higher than most other superconducting materials. Furthermore, various applications of

perovskites such as solar cells, Light Emitting Diodes (LED), Photodetectors, waveguides

and nano lasers at different length scales can be illustrated from Figure 2.12. Table 2.1 gives

some more important applications of different perovskite structured materials along with the

respective properties.

Depending on these distinct properties perovskites are useful for numerous applications for

example (Das 2017; Alvarez 2016; Dogan 2015; Brittman 2015; Benedek and Fennie 2013;

Raveau 2005 & Shaw 2000):

• Thin film capacitors

• Non-volatile memories

• Photo electrochemical cells


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• Recording applications

• Read heads in hard disks,

• Spintronics devices

• Laser applications

• For windows to protect from high temperature infrared radiations.

• High temperature heating applications (Thermal barrier coatings)

• Frequency filters for wireless communications

• Non-volatile memories

• Sensors, actuators and transducers,

• Drug delivery

• Catalysts in modern chemical industry

• Ultra-sonic imaging, ultrasonics & underwater devices


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Figure 2.12: Various applications of perovskites quantum dot, nanowire and nanosheet

(Wang and Kang 1998).


Page | 54

Table 2.1: Applications of perovskites along with respective properties (Das 2017; Alvarez

2016; Dogan 2015; Brittman 2015; Benedek and Fennie 2013; Raveau 2005 & Shaw 2000)

Reference

compound

Properties Possible/ Actual

applications BaTiO3 Dielectric

Ferroelectric

Multilayer ceramic capacitors

(MLCCs),

sensor, PTCR resistors,

embedded capacitance

PbTiO3 Pyroelectric

piezoelectric

Transducer, pyrodetector, under

water devices

(BaSr)TiO3 Non-linear dielectric

properties,

pyroelectric

Tunable microwave devices,

pyrodetector

Pb(ZrTi)O3 Dielectric,

Pyroelectric,

Piezoelectric,

Electro-optic

Nonvolatile memory,

ferroelectric memories,

Surface wave acoustic devices,

pyrodetector,

substrate wave guide devices

Bi4Ti3O12,

high Tc cuprate

compounds

Ferroelectric with high

Curie temperature

superconductivity

High-temperature actuators,

FeRAMs

BaCeO3, BaZrO3 Proton conduction Electrolyte in protonic solid

oxide fuel cells (PSOFCs)

LaNiO3 chemical catalysts

(La,Sr)MnO3 Colossal Magnetoresistance Spintronics devices

Pb(Mg1/3Nb2/3) O3,

CaCu3Ti4O12,

Pg3MgNb2O9

dielectric Memory, capacitor

Resonators,

K(TaMb)O3 Pyroelectric,

Electro-optic

Waveguide device, frequency

doubler

BiFeO3 Magnetoelectric coupling,

high Curie temperature

Magnetic field detectors,

memories

(La,Sr)BO3

(B = Mn, Fe, Co)

Mixed conduction, catalyst Cathode material, oxygen

separation

membranes, membrane

reactors, controlled

oxidation of hydrocarbons

BaTiO3,

(K0.5Na0.5) NbO3,

Na0.5Bi0.5TiO3

Ferroelectricity,

piezoelectricity

Computer Memory, Lead-free

piezoceramics, Ultrasounds

SeCuO3 multiferroicity Memory Devices

LaAlO3

YAlO3

Host materials for rare-earth

luminescent ions

Lasers Substrates for epitaxial

film deposition

SrTiO3, SrLiF3 Insulators Microelectronics

Ba2MgTa2O9 Highest Melting Point Space Craft

Chapter 3 Literature Review

Page | 55

Chapter 3

Literature Review

“If you have an apple and I have an apple and we exchange apples

then you and I still have one apple.

but if you have an idea and I have an idea and

we exchange ideas,

then each of us will have two ideas”

George Bernard Shaw

3.1 Overview

To understand science, it is necessary to know its history. This section of the work is

contextual area of the research to justify and ensure existing body of knowledge and

illustrates the vision that, in which extent subject has been studied previously while

highlighting the flaws and gaps in the previously studied work. Further it enhances concern

of the work, add understanding and knowledge in the field of study, establish a theoretical

framework between methodology and the material. In this chapter, the previous research

work carried out on oxide and halide perovskites are reviewed briefly. In general, this chapter

provides background on physical properties of perovskites with broad-line of interest.

3.2 Background of materials

In perovskites, metallic and non-metallic elements are combined together to form an ideal

cubic solid structure. They equip properties of ceramic materials and are abundantly found in

earth’s crust. The perovskites are composed of calcium titanium oxide mineral which is

based upon calcium titanate. It contains the chemical formula ABX3 having three


Page | 56

compositional variables A, B, and X. Here A is usually large cation like Rb, Na, K, Ca, Sr,

and Ba; B is supposed to be smaller cation like Ti, Nb, Mn, Co, Fe, and Zr; and X anion can

be halides, hydrides and oxides such as F, Cl, I, Br, H, O) (Brik 2011). (The detailed

description of perovskite and its structure can be found in Chapter 1 and 2).

Configuration of different ions in perovskite family of compound provide information about

its basic structure however perovskite structure can be deviated from idealized crystal

structure due to several reasons so knowledge about different properties of idealized & non-

idealized compounds are extremely important. The rapid improvement in physical properties

such as structural properties, mechanical properties, opto-electronic properties, magnetic,

electromagnetic and magneto-opto-electronic properties, thermal, thermoplastic,

thermoelectric, thermomechanical, and thermodynamic properties, vibrational and dynamical

properties of perovskites has gained huge interest from academic community. In fact, they

are rising star of the exploration world. These exceptional physical properties exhibit many

exotic applications, revealing many intriguing features like:

Ferroelectricity as in BiFeO3 (Arnold et al., 2009), Paraelectricity as in SrTiO3 (Salehi 2011).

Thermoelectricity as in LaCoO3 (Anzai et al., 2011 & Androulakis et al., 2004), in

CH3NH3AI3 (A = Pb and Sn) (He et al., 2014), in Ni-doped perovskite-type YCo1-xNixO3 (Yi

et al., 2013), in HoMnO3 (Khan et al., 2015), in SrTiO3 (Muta et al., 2004 & Ohta et al.,

2005) and in CaMnO3 (Kobayashi et al., 1991 & Ohtaki et al., 1995). Piezoelectricity as in

PbTiO3 (Duan et al., 2004; Eitel et al., 2001 & Zhang et al., 2003). Superconductivity as in

La0.9Sr0.1CuO3, YBa2Cu3O7, HgBa2Ca2Cu2O8 (Maeno et al., 1994 & Ishihara 2009). Electric

conductivity & catalytic activity as in LaCoO3, and LaMnO3 (Spinicci 2003). Ferrimagnetism

as in Titanium based perovskite Oxides, RTiO3: R = Lanthanoids (Turner et al., 1980).


Page | 57

Colossal magnetoresistance and half-metallicity as in manganites La1−x SrxMnO3 (LSMO)

(Sawada et al., 2009). These solids have fascinating importance in field of astrophysics,

geophysics, high energy physics, nuclear physics, photon physics, phonon physics and in

various areas of electronics as well (Scullin et al., 2008; Woerner et al., 2009 & Yamada et

al., 2009). Based on versatility they have widespread application in different fields such as

gas sensors (Fergus et al., 2007), random access memory units (Mehonic et al., 2012), lasers

and photoelectrolysis (Luo et al., 2014), high-density capacitors (Lu et al., 2003), ceramic

materials (Moskvin et al., 2010), high performance solar cells (Jeon et al., 2015), efficient

nano generators (Park et al., 2013) and in pyroelectric nanotubes (Zhu et al., 2010). So, the

boundary of their captivating applications not only delivers experimental achievements but

theoretical outcomes as well.

3.3 Structural properties-Previous research

First part of this work is to calculate structural properties of oxide and halide specially

fluoroperovskites at fixed and varying pressure ranges which is very useful in planning &

development of innovative materials. The perovskite structure with cubic symmetry having

space group Pm-3m (No. 221) is attaining huge interest from academia and industry.

Depending on structural chemistry of the constituent compounds perovskites have five

different types like simple perovskites, inverse perovskites, double perovskites,

antiperovskites, and double antiperovskites. The above types can be found in five different

structures including cubic, tetragonal, orthorhombic, hexagonal and rhombohedral (Moskvin

et al., 2010 & Weeks et al., 2010).


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Ionic radius plays a crucial role in finding crystal structure of ionic compounds. In this

succession, a model is constructed for cubic perovskites which is based upon average ionic

radii. The purpose is to investigate correlation between ionic charge and lattice constants.

Findings suggest that tolerance factor play an important part in structural distortion and

analysis of structural distortion is important to design new buffer materials (Verma et al.,

2008). The lattice constants of CsSnCl3, CsSnBr3, and CsSnI3 are also successfully predicted

by using the ionic radii method (Koferstein et al., 2014).

Lattice constant is another significant tool for identifying structural properties of material.

Through this lattice mismatch problem in substrate materials can be resolved. For this

purpose, many structural analytical techniques are used which include linear regression

technique (Jiang et al., 2006; Moreira et al., 2007 & Ubic et al., 2007), common neighbor

analysis (CNA) technique (Stukowski et al., 2012) and Voronoi tessellation analysis are

renowned one (Zhang et al., 2012). However, these models have some drawbacks so it is

difficult to predict precise lattice constants for developmental stage compounds. To

accomplish this need software makers, invent some new software and bring it into the market

of research. These software products are purely based on mathematical modeling among

them famous one is SPuDs program (Lufaso et al., 2013).

In 2015 Slassi (Slassi 2015) use semiclassical Boltzmann equation in Density Functional

Theory (DFT) to find structural distortion in doped Barium Tin Oxide with some elements of

Lanthanoid series. For this purpose, he used 2×2×2 cubic perovskite supercell and observed

that Lanthanum doping initiates donor states at shallow level to enhance the electrical

conductivity and optical transparency which can be useful for their application regarding to

transparent conducting oxide (TCO) based devices.


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3.4 Optoelectronic properties-Previous research

Next part of the work is to calculate optoelectronic properties of oxide and halide perovskites

at fixed and varying pressure ranges while focusing on fluoroperovskites which helps in

understanding internal character as well as electronic and atomic structure of different

materials. Electronic charge distribution is very concerned phenomenon in opto-electronics

which provides information about its chemical bonding, detailed character of density of

states and its opto-magnetic traits. Ultimate need of optically active materials in opto-

electronic devices have motivated researchers to find complex dielectric function, refractive

index, absorption spectrum and many other parameters (Fox 2001). High temperature

perovskites are another field of wide interest. Based on electronic configuration they show

useful conducting, semiconducting and superconducting properties. Some compounds like

LaNiO3, LaCuO3, KNbO3, SrTiO3, LaCoO3 is semi-conductors but with different ionic

distribution they can change their behavior and convert into insulators, conductors or

superconductors (Roy and Vanderbilt 2010).

Strontium series of fluoride as well as oxide perovskites are of huge attention due to their

multidirectional aspects. Shein and their fellows (Shein et al., 2008) studied polycrystalline

SrMO3 (M = Ti, V, Zr, and Nb) in comparison with SrSnO3 compound. They through their

work try to provide a starting picture of these functional material to study further about the

effects of their structural distortion in relation with phase transition. In another study opto-

electronic trend of compounds SrLiF3, SrKF3, SrNaF3 & SrRbF3 are investigated briefly

(Mubarak 2014). In this investigation limited optical properties are discussed in terms of

electronic nature. The calculations reveal that the ratio of dielectric constant is high due to

collective polar displacements while dielectric constant increase with decrease in bandgap


Page | 60

which is demanding for microelectronic devices. Hence some of these materials can be

utilized in opto-electronic as well as microelectronic devices.

In another study Mousa and its fellows (Mousa et al., 2013) shed light on optoelectronic

parameters of XLiF3 (X= Ca, Sr, Ba) fluoroperovskites and predict direct band gap with

mixed covalent and ionic bonding nature by performing an ab-initio Density functional

theory (DFT) study but their investigation is limited to opto-electronic response of the

corresponding materials. Another comparative experimental study on Lithium based

fluoroperovskite SrLiF3 and BaLiF3 is also done by (Nishimatsu et al., 2002). The authors

explore that SrLiF3 possesses wider and direct bandgap than BaLiF3. However, this

experimental investigation is limited due to complexity in their synthesis and volatile nature.

Lang with his coworkers (Lang et al., 2014) studied the behavior of band structures, density

of states and analyze chemical trends of conduction band as well as valence bands in cubic

ABX3 halide perovskites and concluded that when B cation changes from Pb to Sn and when

X anion changes from Cl to I, the bandgap will decrease. This happening is due to variant

behavior of symmetry distribution. They also estimated influence of spin-orbit coupling

effect on electronic properties of materials and prove that all materials have direct bandgap at

R point. Hence, they can be good materials for opto-electronic industry.

In 2015, another investigation (Ahmad et al., 2015) find correlation between bandgap and

dielectric constant in terms of Penn model that proves dielectric constant is in accordance

with incident photons (Penn 1962). Optoelectronic properties of CsSnM3 (M = Cl, Br, I)

(Hayatullah et al., 2013) and CsMCl3 (M=Zn, Cd) (Hayatullah et al., 2013) are verified by

different researchers. Furthermore, the optoelectronic response of compounds like KCaF3 and

KCaCl3, exists in the cubic phase, have been well studied by Mousa (Mousa 2014). He


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explores in his study that these materials are extremely transparent in infrared, visible as well

as in low frequency ultraviolet section hence possibly utilized in transparent optical coatings

in these regions, still there is no through theoretical or experimental work on their elastic,

mechanical and thermal behavior. In a subsequent experimental study, KCaF3 have been

successfully explored by using single crystal neutron diffraction experiment which

determines the nature of high conductivity in this compound (Demetriou 2005). However, in

another experimental study, temperature dependence of KCaF3 is demonstrated by

phenomenon of Raman scattering and concludes high temperature structural instabilities in

these compounds. While Flocken with his coworkers (Flocken et al., 1986) analyzed the

reason of relative instability in KCaF3 is due to rotation of the CaF6 octahedra. So, with

reference to optoelectronic properties the target of this thesis is to improve previous analysis

and add some more converged physics on oxide as well as fluoride class of composite

perovskites.

Another part of this study is the effect of hydrostatic pressure on opto-electronic response of

strontium and calcium based fluoroperovskites because pressure imparts a significant impact

to tune electronic properies, complex dielectric coefficients, refractive index, reflectivity as

well. For example, the high pressure structural stability of some group Ι-ΙΙ inverse

fluoroperovskites has been investigated so far (Yalcin et al., 2016; Vaitheeswaran et al.,

2010 & Besnalah et al., 2003). In which Korba with his fellows (Korba et al., 2009) have

calculated opto-electronic properties of BaLiF3 under the influence of pressure and found that

the valence bandwidth increases monotonically with the pressure. The chemistry of structural

stability for SrLiF3 is expected to be in similar accordance with BaLiF3. Since these

perovskites are made up of the network of corner linked polyhedral, tilt or distortion of


Page | 62

polyhedral upon application of temperature or pressure plays a crucial role in their stability.

In fact, the electron transfer from s to p band under pressure, the so-called s-p transition, is

not only the driving force behind many structural and opto-electronic transitions in many

alkali and alkaline earth based fluoroperovskite but it also imparts an important role in the

stability of the crystal structure (Erum and Iqbal, March 2017).

3.5 Elastic and mechanical properties-Previous research

Third main focus of this work is to calculate elastic and mechanical properties at fixed and

varying pressure ranges. Elastic properties of solid can play a significant role in explaining

the valuable information about the structural stability and the binding characteristics (Sadd

2005). Using data of single crystal elastic properties, one can calculate various poly

crystalline elastic properties (Erum and Iqbal 2016). The major importance of elastic

constants is hidden in its response towards an applied macroscopic stress. Generally, the

basic idea that used to calculate elastic coefficient for cubic crystals, (C11, C12, C44) is the

application of homogenous deformations within finite value by using first-principles

investigation (Hill 1952), by using Charpin method (Charpin 2001).

In this rapidly growing era researchers are continuously working for giving ease in handling

problems through formulation of new software. Recently very efficient open source software

is being developed by Jamal and his fellows (Jamal et al., 2014), for solving elastic constants

for the systems of cubic crystals. Before this development these elastic constants have to

calculate manually which was very time consuming.

Nowadays, material scientists are paying attention to fabricating faster, efficient, and flexible

opto-electronic devices. Among the perovskite-type oxides, KPaO3, and RbPaO3 which has


Page | 63

been reported as an ideal cubic perovskite-type structure, (Keller 1965) have received

considerable attention because of their mechanical flexibility, good optical quality, high

catalytic activity, and high melting point. These XPaO3 (X = K, Rb) compounds can possibly

be used for fabricating flexible perovskite solar cells for optical pathways, (Bayindir 2004) as

high-performance broadband photodetectors for color-sensitive photodiodes, (Campbell

1991) as flexible neural recording devices for medical diagnoses (Kim 2010), as well as light

emitting diodes (LED) for integrated devices (Xiao 2017) but unfortunately neither

experimental nor theoretical effort have been paid to investigate elastic and mechanical

properties of XPaO3 (X = K, Rb) compounds.

Most of the ternary fluoroperovskite compounds such as CsCaF3 and CsCdF3 (Salmankurt

2016), RbZnF3, RbCdF3 and RbHgF3 (Murtaza 2013) are characterized by stimulated effects

of photo and thermo luminescence, tunable laser actions, electron-phonon interactions, and

capability of phase transition. Among them wide-band gap alkali earth based strontium series

of fluoroperovskite, has gained prominent interest because it is a prospective candidate for

vacuum ultraviolet-transparent lens materials in optical lithography and anti-reflective

coatings. It can also use effectively as a dose during radiation therapy (Nishimatsu et al.,

2002).

The effect of pressure variation on elastic and mechanical properties of fluoroperovskites is

another part of this investigation because pressure is an important entity to tune physical

properties, such as lattice parameters, elastic constants, elastic moduli, stiffness coefficients,

Debye temperature, melting temperature, and so on. Generally, fluoroperovskites are made

up of the network of corner linked polyhedral, tilt or distortion of polyhedral upon

application of temperature or pressure plays a crucial role in their stability because change in


Page | 64

pressure transfer electron from s state to p state imparts an important role in the stability of

crystal structure and eventually on elastic and mechanical parameters. So, pressure induced

investigation of fluoroperovskites on mechanical parameters is highly desirable and needs an

intensive investigation.

3.6 Magnetic properties-Previous research

Virtually every investigation is incomplete without magnetic properties because all materials

interact with a magnetic field by responding with a force opposite to the applied field. This is

called diamagnetism and is an inherent additive property caused by the interaction of the

magnetic field with the motion of the electrons in their orbits. The effect is independent of

the applied field and temperature. The diamagnetic susceptibility is approximately in the

order 10-6 emu and negative. The opposite effect with an attractive force from the

magnetization of unpaired electrons is called paramagnetism and is several magnitudes

larger, overrides than diamagnetic signal. An important emphasize is to search out

ferroelectricity, and ferromagnetism in different materials.

To define and understand magnetism in perovskites, there are number of proposed models,

such as the concept of itinerant magnetism, is explained by stoner model and the Jahn-Teller

effect is used to explain spin and orbital ordering mechanism (Mabbs and Machin 2008). The

phenomenon of magnetism in oxide, halide and hydride class of perovskites is due to

occurrence of localized spin of corresponding d-state. The perovskite materials are classified

as multifunctional materials due to its wide range of magnetic properties. These magnetic

phenomenon lead to many diversified mechanisms in terms of application (Sagar et al.,


Page | 65

2011). The different magnetic interactions for the 180° M-O-M perovskite cases, as shown in

Figure 3.1 can be summarized in three main situations.

For the first situation the interactions between two d5 cations e.g. Fe in LaFeO3 with oxygen

in between is considered. In this situation the Fe3+ ion is assumed to be high spin with a half-

filled eg orbital directed towards the oxygen pσ orbital that connects to the adjacent Fe3+. The

virtual transfer of electrons will therefore move electrons from one of the d5 ions to the other

and back. Because of the Pauli exclusion principle, the electrons have to align antiparallel

during the exchange. This will result in a strong AFM interaction between the eg-pσ bonds

and a weak AFM interaction between the t2g and pπ orbitals. However, the t2g orbitals interact

less with the pπ orbitals and are not contributing significantly to the overall result. In total,

these interactions will lead to a strong AFM interaction.

In the second situation a d3 metal ion, e.g., Cr3+ in LaCrO3 with empty eg orbitals interacts

with another adjacent d3 element through the intermediate oxygen. In this case the eg orbitals

are empty and contribute very little. The main contributions to the magnetic interactions

come from half-filled t2g orbitals, where the electrons are interacting weakly with the

electrons in the adjacent t2g orbitals antiferromagnetically. The overall magnetic interaction

from this type of configuration will lead to a relatively weak AFM interaction (Kanamori

1959).

The third situation involves the interactions between an element with half-filled t2g and eg

orbitals, d5, and one with empty eg orbitals, d3, as in Fe3+ and Cr3+ for a hypothetical

compound of LaCr0.5Fe0.5O3. In this case the virtual electron transfer of half-filled eg orbitals

to empty eg orbitals will maintain the parallel alignment of the electrons and will therefore

give a FM interaction which is stronger and overrides the AFM interaction between the half-


Page | 66

filled t2g orbitals of the d5 and d3 ions. The overall result will be a dominating FM interaction

(Goodenough 1963). A schematic overview of FM/AFM transition in magnetic state at

heterointerfaces of La0.7Sr0.3MnO3/BiFeO3 (Yu et al., 2010) is presented in Figure 3.2.

To meet with the requirements of progressive technological needs, half-metallic compounds

are worthy to mention. These are cheap and efficient compounds which retains only one spin

direction at around Fermi level. Generally, the half-metallic investigation in perovskite

materials are concerned with their possible applications in the field of magnetoresistive

sensors, magnetoresistive memories and in spintronic devices such as spin valve and

magnetic storage systems (Ali et al., 2015; Narayan and Ramaseshan 1978 & Pisarev et al.,

1969).

Abbes and Noura (Abbes and Noura 2015) pay attention on highly correlated d-band

electrons of SrRuO3, BaRuO3 and CaRuO3. In this study it is evaluated that strong coupling

is due to manifestation of magnetic phases which allows understanding of spin effects in this

class of material. In technical field magnetic recording devices are in great demand. Linear

Muffin-Tin Orbitals (LMTO) calculation by Hocine and their fellows (Hocine et al., 2014)

shows that magnetic moment increases with RuFe3N and it decrease with OsFe3N which

imply them in high density magnetic recording devices. For non-magnetic and ferromagnetic

compounds, metallic character retains and magnetic phase stability can be determined from

total energy calculation in both spin states.

The complex perovskites fluorides have general formula ABF3 here A and B are cations

while F is monovalent fluorine anion. Among them KXF3 (X= V,Fe,Co,Ni) perovskite

structures are subject of many unique properties such as half-metallicity, colossal magneto

resistivty, high temperature superconductivity, phase separation, ferroelectricity,


Page | 67

piezoelectricity, semiconductivity, catalytic activity, thermoelectricity, phenomenon of

metal-insulator transition and photoluminescence (Bhalla et al., 2000). Based on above

mentioned properties these materials have attained great attention in different areas of

spintronics which is highly concerned with spin polarized materials to improve tunneling

magneto resistance (TMR) in magnetic tunnel junctions (MJTs) (Okazaki and Suemune

1961). In general, KXF3 crystals crystallizes in the ideal cubic perovskite structure which

have been confirmed experimentally by Lee and their fellows (Lee et al., 2003) and

Manivannan and their fellows (Manivannan et al., 2008). In another study, Ito and Morimoto

(Ito and Morimoto 1977) observe magnetic phase transition in KFeF3 between 4.2 K and 300

K. Furthermore, they propose spin arrangement below Curie temperature (TC). Shafer with

his coresearchers (Shafer et al., 1967) report the novel ferrimagnetic compositions in

RbMgF3-RbCoF3 system where only Co2+ is the magnetic transition metal ion. Theoretically,

Punkkinen (Punkkinen 1999) investigates d-states correlation phenomenon in the potassium

based perovskites KFeF3 and KCoF3. From the previous studies, it can be expected that

KXF3 compounds have beneficial electronic and magnetic properties. In this dissertation, we

will contribute to search thermal and magneto-opto-electronic parameters of these ternary

fluorides, which are still under cover.


Page | 68

Figure 3.1: Illustration of the different orbitals that overlap with a) strong eg-pσ and b) t2g-pπ

overlaps between two transition metals with dn configuration and an oxygen, i.e., the M-O-M

bonds (Handley 2000).


Page | 69

“Figure 3.2: Emergence of the novel interface magnetic state at the heterointerfaces of

LSMO/BFO. (a) Novel interfacial magnetic state in the LSMO/BFO heterostructure (b)

Evolution of the interface magnetism and exchange bias coupling with temperature. The

vertical guiding line indicates the blocking temperature of the exchange bias coupling and the

magnetic transition temperature of the interface magnetic state. Adapted from (Yu et al.,

2010). Copyright 2010, American Physics Society”.


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3.7 Thermodynamic properties-Previous research

Thermodynamic properties of matter encompass wide range of aspects in their calculations,

as shown in Figure 3.3. The most significant parameter to calculate in ab-initio

thermodynamic aspect is Debye temperature. The Debye temperature (θD) or Debye cut-off

frequency is a significant form of temperature, which used to quantify several

thermodynamic properties in the solid. It is basically a measure of the vibrational response of

the crystal. In actual, it is the temperature above which the crystal behaves classically.

Significance of Debye temperature (θD) lies in the fact that it can use to quantify several

thermodynamic properties. There are two main methods to calculate Debye temperature (θD)

including elastic constant method and specific heat measurement method (Anderson 1963).

Effect of pressure variation on thermodynamic aspects of material is another phenomenon of

interest because both temperature and pressure have inverse relation with lattice parameters

and bulk modulus because at a given pressure with increasing temperature, lattice parameters

such as lattice constants and volume expansion increase and bulk modulus decreases. So,

effect of pressure and temperature are inversely proportional to each other. In this lieu, Gu

and his fellows (Gu et al. 2014) reveals under pressure investigation of Alkaline-Earth

Barium Hafnates due to its good thermodynamic stability, chemical resistance, fine optical

quality, high melting point and good temperature specifications. The special part of this work

is to use Quasi-harmonic Debye model to analyze elastic and thermodynamic effects above

zero kelvins which overcome limitation of the program WIEN2k. By varying temperature

and pressure, it came to know that temperature and pressure are inversely proportional which

lead to mechanical stability of BaHfO3 compounds.


Page | 71

If material’s bandgap is within 1-2 eV then material’s thermal aspects are dominated by

thermoelectric transport properties are dominated such as Banaras and their fellows (Banaras

et al., 2015) investigates thermoelectric transport properties of pure and doped Holmium

Manganese Oxide HoMnO3. In this study effect of chemical potential is examined on all

parameters. The suitable value of seebeck coefficient, thermal conductivity and electric

conductivity makes them ideal for their use in alternative energy devices.

As it can be observed from literature survey, some researchers have intended to explore

several thermal aspects of fluoride and oxide perovskites but unfortunately still there is lack

of investigation on pressure induced thermodynamic response of strontium and calcium

based group ΙA and ΙΙA perovskites.

Figure 3.3: A general schematic illustration for calculating thermodynamic properties


Page | 72

3.8 Conclusion

The review of literature revealed that various perovskites have been widely investigated for

their structural, mechanical, opto-electronic and magnetic properties by number of

researchers but in spite of successful cases reported earlier, detailed physical aspects of many

oxide and halide perovskite are still under cover. Therefore, it is considered very important to

explore them for their possible applicability in various applications. This study could be

useful to provide new information regarding to behavior of alkali and alkaline based

fluoroperovskites, under different pressure and temperature conditions. Furthermore, another

purpose of this research is to develop an insight about significant properties of actinoid based

oxide perovskites for their possible technological benefits. The corresponding results are

presented in chapter 5 to 8, while the conclusion of the entire study is drawn in chapter 9.

Hopefully the present investigation will contribute towards scientific information on fluoride

and oxide perovskites for their structural, elastic, mechanical, electronic, optical, magnetic

and thermodynamic aspects under constant and variable temperature as well as pressure

conditions which would explore opportunities for material scientists to implement them in

numerous applications.

Chapter 4 Theory and Computational details

Page | 73

Chapter 4

Theory and Computational details

“All the mathematical sciences founded on relations

between physical laws and laws of numbers, so that the aim

of exact science is to reduce the problems of the nature to the

determination of quantities by operations with numbers”

James Clerk Maxwell

4.1 Introduction

To reinforce the trust of potential investors, technological development should be speed up

by involvement of researchers in competition of higher efficiencies. There are several ways

to investigate solution of a given research problem. Two major routes are theoretical work

and observational or experimental work. In the previous century a novel, third pillar of

research has been emerged which is an ab-initio or first principles investigation. The block

diagram representation of various theoretical methods is shown in Figure 4.1. The

requirement of ab-initio study lies to perform computational simulation because number-

crunching calculational power of computers have reached a critical mass so theoretically, the

only possibility to study the complex crystalline structure containing several atoms is to

perform computer simulations. Computational science and computational resources steadily

grew over the last few decades. Now-a-days simulation techniques are able to undertake

detailed and accurate calculations on an increasingly wide and complex range of materials

because microscopic description of physical and chemical properties is a complex problem.

They involve collection of interacting atoms which may be stimulated by external field.

These particles may be in gas to condensed phase, molecules or clusters, solids, surfaces,


Page | 74

wires or they can be molecules in solution, and adsorbates on surfaces (but in all cases, we

have to describe a system in terms of number of nuclei and electrons which are interacting

through electrostatic forces (Lejaeghere et al., 2013).

Computer simulations can be performed with variety of ways ranging from classical to

quantum mechanical approaches. The classical mechanical approach is based upon semi-

empirical scheme, which requires extensive input parameters in order to attain nearby

experimental results. While first-principles method based on quantum mechanical theories

allow the treatment of much smaller unit cell and do not require any experimental knowledge

as input to carry out such calculations. For the past 30 years, the quantum mechanical

simulation of the periodic systems has been dominated by density functional theory (Lany

and Zunger 2009).

In this chapter our focus is to describe the theoretical methods and approximate solutions

used in calculating physical properties of fluoride and oxide perovskites. The main purpose

of this chapter is to discuss background theories of computational methods used throughout

in this research work, covers bibliographic details of different exchange and correlation

schemes, which will briefly sketch computational elegance and simple conceptual framework

of ab-initio DFT studies.

4.2 Many body problems and Schrodinger wave equation

The microscopic description of the chemical and physical properties of the particles is a

complex problem. In general, ensemble of particles can be in variety of phases from gas

phase (clusters and molecules) to condensed phase (surface, solids, liquids, and wires) as

shown in Figure 4.2. In general, one of the major problems in condensed matter solid-state

physics is to determine ground state properties with a collection of N interacting atoms,


Page | 75

Figure 4.1: Block diagram representation of various theoretical methods (Weinberg 2013).

which may also be affected by external local potential provided by the nuclei. In quantum

mechanics, solution of such problems can be found in ground state wave function

|Ѱ(𝑟1, … , 𝑟𝑁)⟩ that holds all of the physical information about a system by solving time

independent Schrodinger equation (Schrodinger 1926) given as:

H|Ѱ⟩ = 𝐸 |Ѱ⟩ (4.1)

The expression for average total energy E is given by expectation value of Hamiltonian Ĥ

that is:

𝐸 = ⟨Ѱ|H|Ѱ⟩ (4.2)


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Here the notation [Ѱ] corresponds that energy is a functional of wavefunction because it

takes function as input and gives number as output.

In equation 4.1 Ĥ represents Hamiltonian operator describing the N electrons of a system,

can be given by applying of well-known Born-Oppenheimer approximation (Born and

Oppenheimer 1927).

The purpose of this approximation is to decouple the entire system between two parts,

electronic part, and the nuclear part because mass of the nuclei is much greater than mass of

the electrons. For this reason, another name of Born-Oppenheimer approximation is

clamped-nuclei approximation. Hence, the complete Hamiltonian of many-body system can

be defined by sum of five terms given as:

H = Te + Tn +Ven +Vee +Vnn (4.3)

Here Te represents sum of kinetic energy operator of the electrons, Tn represents Kinetic

energy of nuclei, Ven represents electron-nuclei attractive interaction, Vee = electron-electron

repulsive interaction, Vnn = nuclei-nuclei repulsive interaction.

Detailed form of equation 4.3 can be written as:

H = =

−eN

em 1

22

2

=

−

nN

k k

k

m1

22

2

= = −

−e nN N

k k

k

Rr

eZ

1 1 0

2

||4 +

= −

e eN N

rr

e

1 0

2

||4 +

= −

n nN

j

N

jk kj

jk

RR

eZZ

1 0

2

||4

(4.4)

While i and j denote the N electrons in the system.

In general, the time independent Schrodinger wave equation 4.1 can be solved to attain

physical properties of a many body systems. However, practically it is not so easier because

within a given system each electron is influenced by potential of all other electrons, which


Page | 77

prohibit to separate this equation into the set of independent equations. This problem arises in

all many body systems with the only exception of one-electron system. Therefore, it is

impossible to solve the Schrodinger equation and further simplification is required, in order

to achieve a set of single particle wave function. To resolve this problem Schrodinger

equation is solved for all electrons by different approximate methods. The main motivation to

introduce approximation techniques in Schrodinger equation lies in the fact to calculate

electronic wave function and electronic energy, which leads to attain quantum mechanical

properties of many body systems (Messiah 1961). For solving such problems several basic

and quantum mechanical, ab initio approximations are employed as mentioned in the

incoming sections.

4.3 The Basic Methods of Electronic Structure

The primary goal of quantum mechanical approaches is to control the electronic structure. As

mentioned in the previous section, that electronic structure can be analyzed by solving the

Schrödinger equation. The evolution and classification of quantum mechanical methods are

shown in Figure 4.3. The most popular classes of ab initio electronic structure methods

include Hartree-Fock Generalized Valence Bond, Moller-Plesset perturbation theory, Multi-

Configurations Self Consistent Field (MCSCF), Configuration interaction, Multi-Reference

Configuration Interaction, Coupled cluster, Quantum Monte Carlo, Reduced density matrice

approaches, and the finally the most popular one which is ab-initio Density functional theory

(DFT) (Kutzelnigg 2006). The schematic presentation of quantum methods is displayed in

Figure 4.4.


Page | 78

Figure 4.2: Schematic chemistry of atoms and molecules in solids (Mitzi 1999).


Page | 79

Among them one of the most common type of an ab initio electronic structure approach is

called the Hartree-Fock (HF) method. D. R. Hartree and V. Fock proposed this

approximation scheme in 1930 (Hartree 1928 & Fock 1930). They attempted to solve many

body systems by using self-consistent field (SCF) approximation also known as Hartree-Fock

approximation which postulates that many-electron wave function can be written as a product

of one-electron wave function. The electronic wave function of N electron is approximated

by slater determinant, which fulfill the requirement that electrons satisfy Fermi-Dirac

statistics and Pauli Exclusion Principle. The exact form of approximation can be expressed as

Slater determinant matrix (Slater 1930):

Ѱ(𝑥1, 𝑥2, … 𝑥𝑁 ) = det (Ѱ1(𝑥1) … Ѱ1(𝑥𝑛). . .Ѱ𝑛(𝑥1) … Ѱ𝑛(𝑥𝑛)

) (4.5)

In equation 4.5 variational principle is applied to obtain minimum ground state energy. In

fact, it provides the way of choice to chemists for calculating the accurate atomic shell

structure with good description of inter-atomic bonding.

On a practical point of view, there are some major limitations in this theory because the

Hartree-Fock interaction neglects the electron correlation effect, which result to increase

repulsion energy of electrons, overestimating the ionic character and further influence on

band gap width and positioning of Fermi level especially in metals. It translates discrepancy

in some properties with respect to experimental measurements and for accurate results some

post-Hartree-Fock methods are required but they are computationally cost (Csavinszky 1968).

Many types of calculations such as Moller-Plesset perturbation theory (MP) and Coupled

Cluster (CC) begin with HF calculation and subsequently correct for the missing electronic


Page | 80

correlation. Another method which avoids making the variational overestimation in its

diffusion, variational, and Green's functions flavors in its first place is Quantum Monte Carlo

(QMC) (Onida et al. 2002). But these calculations can be very time consuming because these

methods work with an explicitly correlated wavefunction and evaluate integrals numerically

using a Monte Carlo integration. In order to overcome limitations in basic methods to

determine electronic structure, an alternative theory well known as Density Functional

Theory (DFT) is introduced which has been applied extensively in this thesis work.

4.4 The Density Functional Theory (DFT)

“Is simplicity best,

Or simply the easiest?

Martin L.Gore

Density Functional Theory (DFT) is a quantum mechanical ground state approach in which

emphasis is on the use of electrons charge density instead of electronic wave function. DFT

has proved to be highly successful tool for the calculation of many physical properties like

structural, elastic, electronic, mechanical, magnetic optical, thermal, thermodynamic, and

thermoelectric properties of metals, semi-metals, half-metals, semiconductors, conductors,

superconductors and insulators covering all types of bulk substances and nanostructures.

DFT is extensively used in various research areas of chemistry and physics because of its

computational simplicity. In fact, DFT is an operationally independent-particle theory

(Kohanoff and Gidopoulos 2003).


Page | 81

Figure 4.3: The evolution and classification of quantum mechanical methods (Griffiths

2004).


Page | 82

Figure 4.4: Schematic presentation of Quantum methods.

Figure 4.5: A schematic representation of the relationship between the "real" many body

system (left hand side) and the non-interacting system of Kohn Sham density functional

theory (right hand side) (Lany and Zunger 2009).

QUANTUM METHODS

Wavefunctions

Basic methods

Electron density

Modern DFT


Page | 83

Although DFT has its theoretical roots in the Thomas-Fermi model but it was put on a solid

hypothetical foundation by Walter Kohn (who worked out the theory) and John Pople (who

implemented the theory) they combinedly proposed the concept of modern DFT. In

recognition to importance of this theory, both scientists were awarded with the chemistry

Noble Prize in 1998. The main idea behind DFT is to describe a fermion interacting system

via its charge density instead of its many body wave function which reduces 3N variables of

N particles to three variables of density, ρ (x, y, z,) (Suryanarayana et al., 2010). This theory

can be best described within framework of Hohenberg-Kohn theorems and Kohn-Sham

equations.

4.5 Hohenberg-Kohn Theorems and Kohn Sham Equations

The basics of DFT lie in two fundamental theorems proposed by Hohenberg and Kohn in

1964 (Hohenberg and Kohn 1964). They gave mathematical proof of the concept that the

energy of a system can only be defined in terms its electron density. This long mathematical

proof was explained in terms of two theorems discussed in this section and schematic

representation is shown in Figure 4.5, in which left hand side represents the relationship

between the real many body system and right-hand side represents the non-interacting system

of Kohn Sham density functional theory.

The Hamiltonian of the system can be best described by following equation:

�� = �� + �� + �� (4.6)

In equation 4.6, the first term is due kinetic energy, the second term represents electron-

electron interaction, and the third term introduces electron interaction with nuclei and

external potential.


Page | 84

According to the first theorem (Martin 2004) for any system of interacting particles, the

external potential V(ṟ) is uniquely determined, except for a constant, by the electronic density

n(r). Then the Hamiltonian operator uniquely determined corresponding ground state

stationary wave function |Ѱ[𝑛]⟩ and all underlying properties of the many body systems. In

this way electron-electron, interaction energy, and kinetic energy can be expressed by the use

of the density. However, this theorem gives no information how one can calculate the density

of a system, which is proposed in the second theorem.

The second theorem (Kohanoff and Gidopoulos 2003) shows that in case of any particular

external potential V(ṟ), the accurate electron density is global minimum value of the ground

state energy functional, in short density obeys variational principle. The contribution of

expectation value can be expressed as functionals of 𝜌(ṟ). So, we therefore have:

𝐸𝑔[𝜌(ṟ)] = ∫ 𝑉(ṟ)𝜌(ṟ)𝑑ṟ + 𝐹[𝜌(ṟ)] (4.7)

Where 𝐹[𝜌(ṟ)] represents kinetic energy of electrons and mutual interaction between them,

V is the potential and 𝜌 is the density of electrons.

𝐹[𝜌(ṟ)] can be broken down into two parts:

𝐹[𝜌(ṟ)] = 𝑇[𝜌(ṟ)] + 𝑈[𝜌(ṟ)] (4.8)

In equation 4.8 𝑇[𝜌(ṟ)] represents influence of kinetic energy of a non-interacting electronic

system whose distribution is 𝜌(ṟ) and electron-electron interaction energy of a system is

given by 𝑈[𝜌(ṟ)], where𝐹[𝜌(ṟ)], denotes universal functional, for this functional a

variational principle hold, which is valid for any external potential with finite number of

particles. For a known 𝐹[𝜌(ṟ)], within a given potential, the evaluation of ground state


Page | 85

energy and density will be easy. Therefore, the expression for total energy whose distribution

is 𝜌(ṟ) can written as:

𝐸𝑡𝑜𝑡[𝜌(ṟ)] = ∫ 𝑉(ṟ)𝜌(ṟ)𝑑ṟ + 𝑇[𝜌(ṟ)] + 𝑈[𝜌(ṟ)] (4.9)

As mentioned previously that 𝑇[𝜌(ṟ)] is the kinetic energy, it cannot be minimized with

reference to 𝜌. So Kohn-Sham proposed an indirect method for total energy minimization by

using the Lagrange multiplier to constrain the number of electrons given as:

𝛿𝐸𝑡𝑜𝑡

𝛿𝜌(ṟ)= 𝑉(ṟ) +

𝛿𝑇[𝜌(ṟ)]

𝛿𝜌(ṟ)+

𝛿𝑈[𝜌(ṟ)]

𝛿𝜌(ṟ)= 𝜇 (4.10)

Kohn and Sham demonstrated that how the properties of a homogenous gas can be applied in

the theoretical investigation of inhomogeneous system.

The potentials in equation 4.10 can be collected by introducing an effective potential 𝑣𝑒𝑓𝑓 :

𝑣𝑒𝑓𝑓 = 𝑉(ṟ) +𝛿𝑈[𝜌(ṟ)]

𝛿𝜌(ṟ) (4.11)

By using above equation, the equation 4.10 can be written as:

𝛿𝑇[𝜌(ṟ)]

𝛿𝜌(ṟ)+ 𝑣𝑒𝑓𝑓(ṟ) = 𝜇 (4.12)

Where 𝜇 is a constant, which represents chemical potential. So, for a system of non-

interacting electrons the expression for total energy can be written as:

𝐸𝑡𝑜𝑡[𝜌(ṟ)] = ∫ 𝑉(ṟ)𝜌(ṟ)𝑑ṟ + 𝑇[𝜌(ṟ)] +𝑒2

8лє0∬

𝜌(ṟ)𝜌(ṟ)

|ṟ−ṟ|𝑑ṟ𝑑ṟ + 𝐸𝑋𝐶[𝜌(ṟ)] (4.13)

In equation 4.13 the last term 𝐸𝑋𝐶[𝜌(ṟ)] denotes exchange correlation energy, functionally

depending on the entire density distribution 𝜌(ṟ) which contains all terms whom exact


Page | 86

solution is unknown. (In next section, it will discuss in detail). The method developed by

Khon-Sham, proposed a practical solution of DFT equation by introducing a one-electron

Hamiltonian like Schrodinger equation, in the following manner:

−ħ2

2𝑚𝛻2Ѱ𝑛(ṟ) + 𝑣𝑒𝑓𝑓(ṟ)Ѱ𝑛(ṟ) = є𝑛Ѱ𝑛(ṟ) (4.14)

Where −ħ2

2𝑚𝛻2 (non-relativistic approximation) denotes Kinetic energy operator and the

resulting total electron density is given by:

𝜌(ṟ) = ∑ |Ѱ𝑛(ṟ)|2𝑛 (4.15)

In summary, 𝑣𝑒𝑓𝑓 can be calculated from an initial guess of density, which can further put

into equation 4.14 to attain є𝑛Ѱ𝑛(ṟ) and used to calculate a new density. The contribution of

DFT is to provide a precise prescription to determine effective potential (𝑣𝑒𝑓𝑓) for

calculating the total ground state energy.

The mathematical framework for finding ground state density is taken from Kohn-Sham

equations, which utilizes standard independent particle methods. As shown schematically in

Figure 4.6 that these equations can be solved self-consistently. It consists of following steps:

1. An initial guess of electron density 𝜌(ṟ), is generated.

2. Then it is used to calculate effective potential𝑣𝑒𝑓𝑓(ṟ).

3. Which can further put into equation 4.14 to attain є𝑛Ѱ𝑛(ṟ) and used to calculate a

new density 𝜌𝑜𝑢𝑡.

4. This process is repeated until self-convergence is achieved, within a chosen numerical

accuracy i.e. 𝜌𝑜𝑢𝑡(ṟ) = 𝜌𝑖𝑛(ṟ) (Dreizler et al., 1990).


Page | 87

Figure 4.6: Schematic description of the SCF cyclic procedure in solving the Kohn-Sham

equations (Blaha et al., 2008).


Page | 88

4.6 The Exchange and Correlation approximations

The main challenge of modern DFT is to achieve the balance between accuracy of practical

calculation and computational cost. For this purpose, several approximations have to be

introduced because in Kohn-Sham equations, exchange-correlation energy 𝐸𝑋𝐶[𝜌(ṟ)]

functional is the only unique term. It is the minimum energy for all possible many-body wave

functions within the given density. The phenomenon of exchange-correlation energy 𝐸𝑋𝐶 or

exchange-correlation potential describes the fact of Coulomb potential and Pauli Exclusion

Principle beyond the pure electrostatic interaction because electrons being fermions, require

their wave functions to be anti-symmetric when two electrons are interchanged (Parr and

Yang 1989). The generation of Exchange and Correlation approximations have emerged

novel innovations in the field of DFT research.

4.6.1 The Local Density approximation (LDA)

The first and simplest approximation, has been proposed in the seminal paper, by Kohn and

Sham, is known as Local Density Approximation (LDA). For a long time, the LDA has been

the most widely used approximation for the exchange-correlation energy in density

functional theory (DFT) (Pople et al., 1992). The LDA method uses exchange correlation

energy density calculated from uniform electron gas to measure the exchange energy. This

scheme has been widely used to portray a varied range of exchange-correlation interactions

in covalently bonded systems but these functionals cannot identified accurately and they

must be calculated approximately. LDA the simplest approximation is known to be local in

the sense that at any point in the space corresponding electron exchange and the correlation

energy is a function of electron density at that point only. Within this approximation, the


Page | 89

expression for total exchange- correlation energy is approximated as integral of all

contributions. Thus:

𝐸𝑋𝐶𝐿𝐷𝐴[𝜌(ṟ)] = ∫ є𝑋𝐶

𝐿𝐷𝐴[𝜌(ṟ)] 𝜌(ṟ)𝑑ṟ (4.16)

Where є𝑋𝐶𝐿𝐷𝐴[𝜌(ṟ)] is the exchange-correlation energy density of the real non-uniform

electron gas.

In case of magnetic systems, where the spin also plays a vital role while involving open

electronic shells in the properties of a material; better approximations to exchange-

correlation functional can be obtained by introducing the effect of up and down spin densities

in LDA approximation and named as Local Spin Density Approximation (LSDA). In LSDA,

the exchange and correlation contribution are separated as (Wu and Cohen 2006):

𝐸𝑋𝐶𝐿𝐷𝐴[𝜌 ↑ (ṟ), 𝜌 ↓ (ṟ) ] 𝜌(ṟ) = ∫ є𝑋𝐶

𝐿𝐷𝐴[𝜌 ↑ (ṟ), 𝜌 ↓ (ṟ) ] 𝜌(ṟ)𝑑ṟ (4.17)

Where є𝑋𝐶𝐿𝐷𝐴[𝜌 ↑ (ṟ), 𝜌 ↓ (ṟ) ] is the exchange-correlation spin-polarized energy density of the

real non-uniform electron gas.

In fact, in the LDA scheme 𝐸𝑋𝐶𝐿𝐷𝐴[𝜌(ṟ)] is function of the local density, which can be easily

separated into exchange and correlation parts such as:

𝐸𝑋𝐶𝐿𝐷𝐴[𝜌(ṟ)] = 𝐸𝑋

𝐿𝐷𝐴[𝜌(ṟ)] + 𝐸𝐶𝐿𝐷𝐴[𝜌(ṟ)] (4.18)

Similarly, for LSDA:

𝐸𝑋𝐶𝐿𝐷𝐴[𝜌 ↑ (ṟ), 𝜌 ↓ (ṟ) ] = 𝐸𝑋

𝐿𝐷𝐴[𝜌 ↑ (ṟ), 𝜌 ↓ (ṟ) ] + 𝐸𝐶𝐿𝐷𝐴[𝜌 ↑ (ṟ), 𝜌 ↓ (ṟ) ] (4.19)

The significant point that proves this system easier to solve is that it reduces the energy

functional to a local function of the density. Hence no more complication added in solving


Page | 90

Schrodinger’s equation. The properties such as phase stability, vibrational frequencies,

charge moments, and elastic moduli can calculate with accuracy.

Despite of its accuracy LDA as well as LSDA have some shortcomings. Typical error ratio

with respect to experimental value is about 1% on atomic positions and of about 5% for

phonon frequencies. In LDA and LSDA schemes exchange-correlation energy density

depends only on local potential while it should be non-local, which means it should depends

on the system at every point in the space. LDA retains problems for highly localized and

strongly correlated electrons like compound of rare-earth elements and transition metal

oxides. One of the serious limitation of LDA include that it cannot be able to provide

estimation for long-ranged exchange interaction such as Van der Waal’s interaction. This

type of interaction is long ranged (electronic interaction) which mainly add in primary stage

of material development and in reactions such as crystal growth, chemical stability and

physical absorption. LDA have another serious limitation that in atoms bond energy

calculations are not accurate where the electrons are quite localized and the electron densities

are poor. In this way, ground state energies of atoms are underestimated and binding energies

are overestimated. In addition, the local structure of the energy expression does not justify

electronic arrangement in bonds that excludes chemistry from the functional expression

(Ceperley and Alder 1980).

4.6.2 The Generalized Gradient approximation (GGA)

For certain systems LDA works well but it fails for other materials due to several reasons,

one of them is that the distribution of electrons within a molecule is not uniform. The next

generation of functional have aimed for treating inhomogeneous electron system including


Page | 91

the gradient of density into locally expanded functional. This type of functional has attracted

much attention due to its abstract simplicity and moderate computational workloads. In

general, this relation can be expressed as (Wu and Cohen 2006):

𝐸𝑋𝐶𝐺𝐺𝐴[𝜌(ṟ)] = ∫ є𝑋𝐶

𝐺𝐺𝐴[𝜌(ṟ)] 𝜌(ṟ)𝐹𝑋𝐶[𝜌(ṟ), 𝛻𝜌(ṟ), 𝛻2𝜌(ṟ), … . ]𝑑ṟ (4.20)

Here 𝐹𝑋𝐶 is the factor, which use to modify the LDA functional in order to consider the

variation of 𝜌(ṟ) at the reference point and which is asked to satisfy a number of formal

conditions for the exchange-correlation hole, long range decay, sum rule and so on. The

approximation of this type is known as Generalized Gradient Approximation (GGA).

GGA type of functionals are often referred to as functional zoo because a very large number

of functionals have been proposed. What is needed for the functional is a from that mimics a

re-summation to infinite order, however for GGA there is not unique recipe. A through

comparison of different GGA functionals are done by Filippi and his fellows (Filippi et al.,

1994). These functionals may vary from parameter to parameter. Naturally, not all the formal

properties can be enforced at the same time, and differentiates one functional from other such

as while the functionals that contain second derivative of charge density (to improve

accuracy of kinetic energy functional) are called meta-GGA. Some of the renowned GGA

functionals includes Lee, Yang and Parr’s correlation functional (LYP) (Miehlich et al.,

1989), Becke’s exchange function (B88) (Becke 1988), Perdew, Burke and Ernzerhof’s

exchange-correlation functional (PBE) (Perdew 1996), Perdew and Wang’s (PW91)

correlation functional (Perdew and Wang 1992), Wu and Cohen (WC) density gradient

functional (Wu and Cohen 2006) are worthy to mention.


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In GGA functionals higher order expansion in equation 4.20 have been developed but it will

lead to violate some suitable conditions for exchange and correlation as showed by Perdew

(Perdew 1986) that originally did not yield any notable improvement in the exchange

energies and analytical equation for the potentials become highly complicated. Despite of the

fact, these functionals may not necessarily base on new physical ideas but GGA functionals

yields better results than the LDA because different LDAs modeling the electron gas yield

nearly identical results, different GGAs can yield very different energetic and structural

results because of the freedom in the parameters of the gradient expansion.

In general, GGA methods can improve the computed binding energies, electron affinities,

atomic energies and give better representation for non-uniform densities over the LDA.

According to report of Perdew (Perdew 1986) GGA exhibits an error of 1 % while LDA has

14% error in the calculation of exchange correlation energies. Many calculations assessing

the accuracy of GGA demonstrates that it substantially corrects the LDA error in the

cohesive energies of solids and molecules.

Despite of its accuracy GGA have some shortcomings. The lattice parameters by GGA

always rise in comparison with LDA, however a close agreement with experimental data is

observed for alkali 3d and 4d metals. For molecules thermos-chemistry GGA schemes also

delivers poor results. Similarly, for opto-electronic calculations GGA underestimates these

results, for instance the theoretical bandgap of AlN in zinc blende structure is 4.2 eV

(Litimein et al., 2002) and 5.40 eV (Amin et al., 2011) by LDA and GGA schemes

respectively while the calculated value of experimental bandgap is of 6.2 eV (Martinez et al.,

2001). The same problem arises for almost all kind of compounds. Unlike LDA, GGA

functionals are not unique. These functionals cannot be classified as purely ab-initio


Page | 93

calculation because some experimental data is required to solve parameters in GGA

functional. However, performance may vary from type of material under investigation. In

comparison with experimental results GGA scheme has some limitations. Hence for attaining

good results researchers in this field are aiming to get their results close to experimental

values so they use different flavors of LDA and GGA for strongly correlated systems like

LDA+U and GGA+U (Anisimov et al., 1997).

4.6.3 The modified Becke–Johnson (mBJ) potential

In majority solid state calculations, determination of electronic structure is done with Kohn-

sham equations by using various approximations such as local density and generalized

gradient approximations for exchange correlation energy as well as for potential. But in many

circumstances these local or semi local approximations cannot be able to interpret accurate or

enough experimental data. The calculated values of bandgap for LDA and GGA schemes

underestimates the bandgap because of the fact that GGA as well as LDA schemes

undervalues bandgaps in semiconductor as well as insulators. The wrong interpretation of the

true unoccupied states with respect to corresponding Khon–Sham DFT states, results this

underestimation (Grabo et al. 1997).

Tran and Blaha in 2009 (Tran and Blaha 2009) tested and verified modified Becke and

Johnson (mBJ) potential for accurate bandgap calculations of sp-semiconductors, wide

bandgap insulators and for strongly correlated 3d transition-metal oxides. This newly

developed scheme minimizes the conflict between the theoretical and experimental

investigations. It is an exchange potential which used to handle exchange and correlation


Page | 94

energy for strongly correlated electron systems (Koller et al., 2011). It was proposed by

following expression:

𝑈𝑋,𝜎𝑀𝐵𝐽(𝒓) = 𝐶𝑈𝑋,𝜎

𝐵𝑅 (𝒓) + (3𝐶 − 2)1

𝜋√

5

12 √

2𝑡𝜎(𝒓)

𝜌𝜎(𝒓) (4.21)

where 𝑡𝜎 =1

2∑ (𝛻Ѱ𝑖,𝜎

∗ ). (𝛻Ѱ𝑖,𝜎)𝑁𝜎𝑖=1 is the kinetic energy density, 𝜌𝜎 =

1

2∑ |Ѱ𝑖,𝜎|

2𝑁𝜎𝑖=1 is the

density of the electron, while the Becke-Russel potential is given by (Becke and Johnson

2006):

𝑈𝑋,𝜎𝐵𝑅 (𝒓) = −

1

𝑏𝜎(𝒓)(1 − 𝑒−𝑥𝜎(𝒓) −

1

2𝑥𝜎(𝒓)𝑒−𝑥𝜎(𝒓)) (4.22)

and C in equation 1 can be elaborated by:

𝑐 = 𝑎 + 𝑏√1

𝑉𝑐𝑒𝑙𝑙∬

|𝛻𝜌(𝒓)|

(𝜌(𝒓))𝑐𝑒𝑙𝑙𝑑3𝑟 (4.23)

In the above equation Vcell is the unit cell volume where a and b are two independent

parameters, with the values of -0.012 (dimensionless) and 1.023 Bohr1/2, respectively.

In fact, mBj is an exchange correlation, not exchange correlation functional so first one has

to use modern LDA or GGA approximation for structural properties and then to use

mBJLDA or mBJGGA potential for the proper calculations of the corresponding band

structure. In fact, the Becke and Johnson (BJ) potential can either be used in combination

with LDA or GGA (Perdew and Wang 1992), and that can be termed as mBJLDA or

mBJGGA. These potential leads to more sophisticated results with more demanding

computational orders of magnitude. As a result, a very accurate bandgap specially of wide

bandgap semiconductors and insulators can be obtained. This is a type of semi local potential

which can compete in accuracy with other expensive hybrid methods. So, it can be efficiently

applied to large systems which was certainly not possible with other hybrid methods.


Page | 95

The Tran-Blaha functional have attracted a lot of interest recently due to the accurate

predictions of the band gaps in semiconductors and insulators and the low-computational

cost. However, this functional has also some drawbacks which should be emphasized e.g. the

TB-mBJ functional is not a functional derivative and cannot be used to calculations of forces,

lattice constants, phase stability etc (Koller et al., 2011). For cubic perovskites without

strongly localized f- or d-states the TB-mBJ functional gives similar results to the

computationally more expensive hybrid HSE functional (Kaczkowski and Jezierski 2013).

However, for localized states better agreement with experimental results is obtained within

LDA+U approach. There are some limitations and some benefits in these theoretical schemes

but their selection can be based upon selection of system under consideration.

4.7 Methods for solution of Kohn Sham Equations

There are several techniques to solve DFT Kohn Sham equations, once they are described

into functional terms. Nowadays different codes are available in the simulation market that

can solve these equations with ease but they can differentiate in terms of basis sets. The

different basis sets used are in actual the linear combination of atomic orbitals (abbreviated

as LCAO), Slater type or Gaussian orbitals (abbreviated as STOs, GTOs), plane wave

(abbreviated as PW) with or without augmentations, Linearized Augmented Plane Wave

(abbreviated as LAPW) with or without Local Orbitals, and muffin tin orbitals (abbreviated

as MTOs). The wave functions, which represent these basis sets, can be all-electron wave

functions or pseudo-wave functions (Patterson et al., 2010). However, calculations in this

thesis have been based upon Full-Potential Linearized Augmented Plane Wave Method (FP-

LAPW) which is integrated into Wien2k Package of simulation software (Schwarz et al.,

2010).


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4.8 Full-Potential Linearized Augmented Plane Wave Method

(FP-LAPW)

To solve one-electron Kohn-Sham equation in terms of Khon-Sham orbitals, one of the most

reliable methods is the Full-Potential Linearized Augmented Plane Wave Method (FP-

LAPW) which allows most precise calculation of the electronic structure and other physical

quantities of crystals and surfaces. The use of this technique depends upon binding strength

of electrons in solids which leads to maximize its applicability in atomic forces. The muffin

tin Augmented Plane Wave (APW) and Linearized Augmented Plane Wave (LAPW)

approaches were frequently used in the era of 1970s. In these methods, potential is assumed

constant in the interstitial region and symmetrically spherical in the muffin-tin region. These

approximations are highly useful in order to explain the metallic systems but not very

efficient in the calculation of structure and opto-electronic parameters of open structured as

well as covalently bonded solids. A comprehensive description of three types of schemes

APW, LAPW, APW+lo is explained in detail by Schwarz (Schwarz et al., 2010). To

overcome these limitations and to obtain better predictions for these properties, the non-

muffin tin approximation (no shape approximation) were applied which is named Full-

Potential Linearized Augmented Plane Wave Method (abbreviated as FP-LAPW) but indeed

FP-LAPW computation need considerable higher computational effort in comparison with

the pseudopotential plane wave (PPW) based methods. The FP-LAPW method combines the

basis of LAPW with full potential treatment to solve the Kohn-Sham equations for the

ground state total density, ground state total energy, and eigenvalues of many body electron

system by introducing finite set of basis. In this approach unit cell potential, and charge

density are expanded by linear combinations of lattice harmonics inside the atomic sphere.


Page | 97

This alteration is achieved by partitioning the unit cell as shown in Figure 4.7 into (Ι) non-

overlapping atomic circles; centered at atomic sites and (ΙΙ) outside the sphere an interstitial

region; region between two spaces (Madsen et al., 2001). Another pictorial representation of

unit cell division is displayed in Fig. (3.7). The FP-LAPW method extends the potential

inside and outside the sphere in the following from:

𝑉( 𝑟) = ∑ 𝑉𝐿𝑀𝐿𝑀 (𝑟)𝑌𝐿𝑀(𝑟) Inside sphere (4.24)

𝑉( 𝑟) = ∑ 𝑉𝐾 𝐿𝑀 𝑒𝑖��.�� Outside sphere (Interstitial region) (4.25)

where the equation 4.24 is for inside the sphere and equation 4.25 is for outside or interstitial

region of the atomic sphere. These non-muffin tin corrections do not affect the choice of

basis functions in the interstitial regions but by using true crystal potential, the radial wave

functions are evaluated. So, the benefit of FP-LAPW method is that it is free in selection of

sphere radii as compared to old APW as well as LAPW methods. The effects for relativistic

valence states can be either incorporated in scalar relativistic handling or with the second

dissimilarity technique including spin-orbit coupling. Furthermore, in FP-LAPW method the

core states are computed self-consistently by using the spherical part of the crystal potential

within Muffin Tin (MT) spheres. Potential is spherically symmetric inside the sphere while it

is constant outside the sphere. However, eigenstates and eigenvalues treatment of core

electrons is fully relativistic and it is semi-relativistic for the valence electrons. The union of

basis set in FP-LAPW method is controlled by a disconnect parameter RMT х Kmax, where

RMT is the smallest atomic sphere radius in the unit cell and Kmax is the magnitude of the

corresponding largest K vector.


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Recently FP-LAPW method has showed important progress. Now-a-days researchers are

working out on several nuclear and magnetic quantities using FP-LAPW method such as

isomer shifts, electric field gradients, hyperfine fields, and core level shifts. The recent

optimization has significantly reduced the CPU time of these calculations. However, FP-

LAPW implementations are fairly suitable for complicated systems, because of its

computational expense and memory requirements. One successful implementation of this

technique is simulation program package of WIEN2K discussed in the coming section.

4.9 Simulation techniques

First principle calculations have been the foremost and modern tool to achieve theoretical

understanding and prediction of physical properties, kinetic and thermally driven

phenomenon which otherwise are beyond the reach of several other computational

techniques. In this field variety of computer codes are available such as VASP (Parlinski and

Kawazoe 2000), CRYSTAL code (Piskunov et al., 2004), SIESTA code (Coulaud et al.,

2013), Quantum Espresso (Giannozzi et al., 2009), ABINIT code package (Roy et al., 2010),

CASTEP code (Segall et al., 2002), FEFF (Rehr et al., 2013), Lmtart computer code

(Savrasov et al., 2004), CP2K (Carignano et al., 2014) but in this thesis attention is paid to

WIEN2k code (Schwarz et al., 2010 & Erum and Iqbal 2016) due to its various features such

as graphical user interface, low convergence time, accuracy, efficiency, user friendliness,

robustness & portability.


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Figure 4.7: Partitioning of the unit cell into atomic spheres (I) and an interstitial region (II)

(Blaha et al., 2002).

Figure 4.8: The unit cell divided into muffin-tin region and interstitial region

(Schwarz et al., 2010).


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4.9.1 The WIEN2k Package

Almost all computational work presented in this thesis on oxide and halide perovskite

compounds has been performed by using WIEN2K code (Blaha et al., 2008). This code is

embedded in the framework of density functional theory. Its development brings computation

of various material properties such as structural, chemical, opto-electronic, mechanical,

thermal, and magnetic by structure generation of periodically arranged unit cells at absolute

zero. The calculation of aforementioned properties can be done by applying different

physical laws within framework of density functional theory.

Over a period of more than twenty years, a full-potential LAPW method has been developed

for crystalline solids but its copyrighted first version was called named as WIEN (Blaha et

al., 1990). In succeeding years significantly, improved UNIX based versions of WIEN were

WIEN93, WIEN95 and WIEN97 but based on alternative basis set newly updated version is

WIEN2k (WIEN2k_16.1) which is LINUX based and written in many independent

FORTRAN 90 programs. The sequences of individual modules are linked together through

C-shell scripts alongwith F90 compiler. The WIEN package contains several sub-programs.

The WIEN developers have the strategy to provide a general code with the help of these sub

programs.

Major steps to evaluate crystalline properties includes generation of structure followed by

space group selection of the chosen material with suitable lattice coordinates. In the next

step, initialization of the code is done through step by step process which ultimately detects

minimum separation energy required to stabilize the unit cell structure. Then finally SCF

calculations are performed. The flow chart of WIEN2k code SCF cycle in single mode and in


Page | 101

parallel mode is shown in Figure 4.9. The two major parts in program flow of WIEN2K are

initialization and the self-consistent field (SCF) cycle. The purpose of initialization step is to

generate the unit cell and set up the initial density. The construction of the effective potential

is done with the help of LAPW0 program. Then the program LAPW1 solves the Khon-Sham

equation for valence electrons. The construction of new electron density is done with the help

of LAPW2. The LCORE program treats the core electrons. Finally, the input density for the

next iteration is done with LMIXER program. In current platform of WIEN2k core states has

flexibility to describe atomic-like states near the nuclei and electron-like states in the

interstitial.

The all electron nature of WIEN2K means it calculates all electron states including explicitly

the tightly bound, atomic like deep core states (Laskowski et al., 2004). It is an Augmented

Plane Wave Plus Local Orbitals (APW+lo) Program for calculating crystal properties and

allows major upgrading particularly related to speed, universality, and user accessibility

(Blaha et al., 2002).WIEN2k can treat all atoms of periodic table in similar manner such as

from heavy atoms like U to lighter atoms like C in similar manner and this balance is due to

mixed basis set of plane waves and atomic functions while solving radial Dirac equation

numerically for each SCF iteration. This allows them to expand or contract according to their

potential. Besides all these specifications, WIEN2k code is one of the fastest and consistent

simulation codes among to calculate crystalline properties at atomic level. Almost all

computational work presented in this thesis on oxide and halide perovskites are done using

WIEN2k code which is embedded in the framework of Density functional theory. In fact,

there are some limitations in this code and to improve that several bugs are adding on daily


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basis in this code and hope can be generated that future WIEN2K will be much better than

present WIEN2K.

Figure 4.9: Flow chart of WIEN2k code SCF cycle in single mode and in parallel mode

(Schwarz et al., 2010).

4.10 Applications of Density functional theory (DFT)

Density functional theory (DFT) has made an unparalleled impact on the application of

quantum mechanics and challenging problems in condensed matter physics. It is a subtle,

provocative, and seductive business in the field of theoretical research. It can drive one mad

due to its basic premise that all the intricate motions and pair correlations in many electron

systems are contained by total electron density alone. This theory provides an unbiased tool

to compute the ground state energy in relatively realistic models of bulk materials and

surfaces. The reliability of such calculations is based upon accurate approximations to


Page | 103

exchange-correlation functional. It can be applicable to simple as well as complex systems

such as atoms, molecules, 3d solids, 2d surfaces and interfaces. Density functional theory

(DFT) has been immensely successful in its ability to predict physical properties, and, in

particular, structures of condensed matter systems. It can investigate atomic and molecular

structures, the understanding and design of catalytic, ionization potentials, vibrational

spectra, chemical reactions in biomolecules, nature of active sites in catalysts, phase

transition in solids, bond length, bond strength, liquid metals, and properties of magnetic

materials, processes in enzymes and zeolites, electron transport, solar energy, harvesting and

conversion, drug design in medicine, as well as many other problems in science and

technology. The story behind the success of DFT over Hartree-Fock (H-F) is hidden in its

electron correlation energy, which is generated by the interaction of pair for electrons. The

consideration of correlation energy in HF method is required by Pauli repulsion energy

(Martin 2004).

One of the advantage of DFT lies in the phenomenon of electron correlation which is the

major reason for localization and delocalization of electrons in a system. For example, for

solids LDA approximation turns out to be computationally much more successful than HF

due to true exchange potentials and slightly simpler Slater’s local exchange approximation.

Another significant advantage of DFT is due to its reduced cost over HF method (Zieger

1991) because DFT calculations scales to N3 while HF calculations scales to N6 where N

represents number of electrons in a system. It means that larger simulation cells can be

simulated using DFT for the same computational cost (Gill 1992).

An interesting aspect of DFT is that even the simplest systems can show details and

challenges reflecting those of much larger and complex systems. One example of this is the


Page | 104

understanding of the extensively used term, “strong correlation.” In literature, strong

correlation is meant to mention the breakdown of the single-particle picture, which is based

on a determinant of single particle Kohn Sham orbitals. The Strongly correlated systems

offer significant and new challenges for the functional. The challenge to demonstrate strong

correlation for density functionals can help to realize the enormous potential of DFT (Lany

and Zunger 2009).

Despite the applications and successes of DFT in many branches of science and engineering,

there are some future challenges and known issues for DFT. DFT detractors complain about

the prerequisite of KS-DFT, that the exchange-correlation functional should be exact, then

quantum mechanical nature of matter can be described correctly by DFT. In fact, it is the

approximate nature of the exchange correlation functional that is the reason both for the

success as well as for the failure of DFT applications because exchange-correlation

functional and its underlying hole will never be expressible in closed analytical form, which

made DFT heaven probably, be unattainable. This leads to major limitation of standard DFT

in describing strongly correlated systems. DFT computations in all known implementations

are found to qualitatively break down into certain strongly correlated electron systems,

sometimes it predicts a compound to be a metal while experimentally it is an insulator. This

problem is known as self-interaction error. In real system, each electron is under influence of

every other electron other than it-self while in DFT all electrons are described by coulomb

interaction with all others and with themselves through Hartree term. Therefore, the term

“self-interaction” referred as Coulomb interaction of one electron with its own electron

density. When electrons interact with it-self, they affect the modeling of charge localization,

magnetic materials, and superconductivity. To remove this limitation, there are several ways


Page | 105

to correct this error, with the commonly used self-interaction correction (SIC) or through

using DFT+U (Szabo and Ostlund 1982). So new and deeper theoretical insights are needed

to aid the development of new functionals. Another problem in DFT is incorrect description

of cohesive energy of a system. The DFT functionals especially GGA functional

underestimates cohesive energy of the systems, which results in lattice parameters larger than

experimental one. This is due to the fact that atoms are not bonded to each other as strongly

as they are in nature and therefore the separation distance increases between the atoms. On

the other hand, LDA results lattice parameters smaller than the experimental results. This is

because the cohesive energy is over predicted by using this exchange-correlation assumption.

Beyond all these limitations, users are willing to pay the price due to simplicity, efficacy, and

speed of DFT calculations. The crucial development in KS-DFT finds itself under increasing

pressure to deliver higher accuracy to adapt problems that are more challenging. Therefore,

the next fifty years in this research field is as interesting as the first. However, these

calculations will preserve the Kohn-Sham philosophical, theoretical, and computational

framework.

Chapter 5 Results and discussion Ι

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Chapter 5: Results and discussion Ι;

Elastic, and optoelectronic investigation of SrMF3 (M = Li,

Na, K, Rb) and RbHgF3 fluoroperovskites

“This problem, too, will look simple

After it is solved.”

Charles Kettering

5.1 Introduction

Now-a-days highly efficient materials can be obtained by involvement of researchers in

competition for achieving highly efficient and resourceful materials. In continuation with

this, fluorine based perovskites are fortified ingredients for manufacturing transparent lenses

in Vacuum Ultra-Violet (VUV) region of electromagnetic (EM) spectrum (Green et al.,

2014). The purpose of this chapter is to investigate detailed information about electronic

structure, mechanical stability and opto-electronic trend of alkali and alkaline earth based

fluoroperovskites. This chapter comprises of three major sections. The first two sections

contain structural, elastic, mechanical, and opto-electronic parameters of SrLiF3, SrNaF3,

SrKF3, and SrRbF3 while third section is built upon electronic structure calculations and

opto-electronic response of RbHgF3 compound.

5.2 Structural, elastic and mechanical properties of SrMF3

(M = Li, Na, K, Rb)

In a previous experimental research Düvel and their fellows (Düvel et al., 2011) employed

Nuclear Magnetic Resonance (NMR) spectroscopy to examine bond chemistry in BaLiF3 and

SrLiF3 fluoroperovskites. In a subsequent theoretical study Mousa and their fellows (Mousa

et al., 2013) explored direct bandgap XLiF3 (X= Ca, Sr, Ba) fluoroperovskites by opto-


Page | 107

electronic properties and their bonding character. But as per best of our information no one

have explored elastic and mechanical behavior of these compounds. Hence detailed

theoretical investigation on these fundamental but crucial parameters are required in detail,

for their eventual technological applications. The upcoming subsections are dedicated to

reconnoiter results and discussion about electronic structure, elastic and mechanical behavior

of SrMF3 (Li, Na, K, Rb) fluoroperovskite and all results are compared by previously

available work where data is available.

5.2.1 Structural properties

To compute structural properties of SrMF3, the total ground state energy is determined at

various unit cell volumes. In actual, it is an energy minimization process. The ultimate lattice

parameters are calculated by employing first and third order Birch Murnaghan’s equation of

state, (Murnaghan 1944) as shown in Table 5.1, to produce energy versus volume curve. The

cubic unit cell crystal structure is displayed in Figure 5.1. It can be observed from figure the

that SrMF3 fluoroperovskites possesses one molecule per cubic unit cell containing space

group classification of type Pm-3m (no. 221). In an elementary cell, the atomic positions of

respective Sr, M and F ions are located at Wyckoff coordinates of (1a,1b,3c) at (0,0,0),

(0.5,0.5,0.5), and (0,0.5,0.5) respectively. According to crystallographic positioning of

SrMF3, Sr lies at each corner, M at body centered position, and fluorine ions retain their

positions at face centers of cubic unit cell. As the element traversed from Li to Rb, an

increasing trend of lattice constant is observed, because M (M= Li, Na, K, Rb) atoms contain

larger atomic radii. These fluoroperovskites show higher Sr-F bond length then M-F bond

distance, although for the bond length of CsMCl3 (M= Zn, Cd) compounds, similar behavior

can be observed (Hayatullah et al., 2013).


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The analytical calculations of lattice constants are also performed in this work by using two

famous methods. The first method employed ionic radii of the constituent atoms to calculate

lattice constant values. This ionic radius method uses following formula (Clementi et al.,

1963):

𝑎0 = 𝛼 + 𝛽 (𝑟𝑆𝑟 + 𝑟𝐹) + Ɣ(𝑟𝑀 + 𝑟𝐹) (5.1)

The α, β and γ constants in above equation have values of 0.0674, 0.4905 and 1.2921

respectively. And the values for ionic radii are rSr = 1.44 Å, rLi = 1.61 Å, rNa = 1.39 Å, rK =

1.64 Å, rRb = 1.72 Å, rF = 1.285 Å respectively (Erum and Iqbal 2016). The second method

proposed by Verma and Jindal depends on average ionic radii rav, valence electrons number

in Sr, M, and F and some cubic constants, like K (2.45) and S (0.09). The subsequent relation

is as follows (Verma et al., 2008):

𝑎0 = 𝐾(𝑉𝑆𝑟𝑉𝑀𝑉𝐹)𝑠𝑟𝑎𝑣 (5.2)

It can be noticed from Table 5.1 that DFT and analytical calculation of lattice constants

reveal some deviation within 3-4%. This deviation is due to several reasons: Firstly, the

empirical relation for calculating lattice constant by V.J method depends on average ionic

radii. Secondly, the empirical relation for calculating lattice constant by V.J method depends

on number of valence electrons of each atom and thirdly, the error might be due to constants

K (2.45) and S (0.09) which are involved in this empirical relation. Therefore, it can be

concluded that empirical relation given by V.J method needs a lot of improvement to attain

lattice constant values near to experimental one.

To evaluate bonding nature, an important parameter is bond length which depicts an average

distance in between the center point of two bonded atoms because chemical bonding helps to


Page | 110

traditional mechanical and cubic stability condition at P = 0 GPa, which can be mentioned by

the following relation C11- C12 > 0, C11 > 0, C44 > 0, C11 + 2C12 > 0 (mechanical stability

condition), and C12 < B < C11 (cubic stability condition). In general, these relations indicate

mechanical stability criteria and cubic stability criteria respectively. Table 5.3 reveals

proportion for elasticity in length C11 is highest for lithium based fluoroperovskite SrLiF3

while it is lowest for rubidium based fluoroperovskite SrRbF3. Meziani and Belkhir (Meziani

and Belkhir 2012) observed the similar results for unidirectional compression along the

principle crystallographic direction. The elasticity in shape can be well explored by elastic

constant of C44. Hence the present calculations reveal that SrMF3 retains more resistance for

shear deformation C44 in comparison with unidirectional compression C11 because the value

of C11 is 605.98%, 486.13%, 649.22%, 306.87%, than C44 for SrRbF3, SrKF3, SrNaF3, SrLiF3

correspondingly.

5.2.3 Mechanical behavior

Many real world versatile mechanical parameters can be evaluated by using elastic constants.

The thermo-elastic stress and strain (internal) are renowned industrial applications of

mechanical properties. The mechanical properties are calculated by GGA approximations as

shown in Table 5.4 and 5.5 respectively.

5.2.3.1 Elastic moduli calculations

The standard description of bulk and shear modulus can be best described by Voigt-Reuss-

Hill (VRH) method (Reuss and Angew 1929). The measure of resistance to reversible

deformation can be best explained by shear modulus G, from following relation (Shafiq et

al., 2011):


Page | 111

𝑮𝑽 = 𝟏

𝟓(𝑪𝟏𝟏 − 𝑪𝟏𝟐 + 𝟑𝑪𝟒𝟒) (5.3)

𝑮𝑹 =𝟓𝑪𝟒𝟒(𝑪𝟏𝟏−𝑪𝟏𝟐)

𝟒𝑪𝟒𝟒+𝟑(𝑪𝟏𝟏−𝑪𝟏𝟐) (5.4)

𝑮 =𝑮𝑽+𝑮𝑹

𝟐 (5.5)

However, expression of bulk modulus B are mention from following equation (Kittel 2005):

𝑩 =𝟏

𝟑 (𝑪𝟏𝟏 + 𝟐𝑪𝟏𝟐) (5.6)

It can be observed from Table 5.4 that SrLiF3 has largest value (74.48 GPa, 52.41GPa),

SrRbF3 contains lowest value (33.05GPa, 21.55GPa), of the bulk as well as shear modulus

respectively. In conclusion, SrLiF3 is the hardest material and rest of the compounds are less

harder as compared to SrLiF3 fluoroperovsktie. The response of a material towards linear

strain can be well defined by young’s modulus Y via following relation (Jenkins & Khanna

2005):

𝒚 =𝟗𝑩𝑮

(𝟑𝑩+𝑮) (5.7)

The highest value of B, Y, and G for SrLiF3, implies that there is more tendency of charge

transfer in SrLiF3, rather than SrRbF3, SrKF3 and SrNaF3 respectively. Another significant

ratio which is related to resistance for plastic deformation namely Pugh’s index of ductility or

B/G ratio. For B/G < 1.75 and B/G >1.75 distinguishes compound as brittle or ductile

respectively (Pugh 1954). The SrMF3 series of compounds reveals slightly brittle character

because they satisfied former criteria of Pugh’s index of ductility.

5.2.3.2 Cauchy’s pressure and shear constant calculations

Another important parameter which used to describe angular character in atomic bonding is

Cauchy’s pressure. It can be well defined as follows (Brik 2011):


Page | 112

𝑪′′ = 𝑪𝟏𝟐 − 𝑪𝟒𝟒 (5.8)

Negative values of Cauchy’s pressure (𝐶′′) indicate high angular characteristics in bonding

while compound with positive Cauchy’s pressure tend to form metallic bond in nature. For

B/G less than 1.75 and Cauchy’s pressure less than zero implies that SrMF3 compounds are

not ductile in nature and retain less tendency of covalent bonding. As a result, compounds

have dominant ionic behavior. To further distinguish between, ionic or covalent behavior the

present analysis is extended to evaluate shear constant. However low values of shear

constants depict ionic behavior and vice versa. It can be expressed as:

𝑪′ =𝟏

𝟐(𝑪𝟏𝟏 − 𝑪𝟏𝟐) (5.9)

From Table 5.5 it is evident that SrMF3 have low values of shear constant so tend towards

ionic in behavior.

5.2.3.3 Poisson’s ratio and elastic anisotropy calculations

Next, ratio of compression to relative expansion can be expressed in terms of Poisson’s ratio,

as follows (Pettifor 1992):

ѵ =(𝟑𝑩−𝟐𝑮)

𝟐(𝟑𝑩+𝑮) (5.10)

As per Gu and his fellows (Gu et al., 2014) for central force solids respective upper and

lower limits are 0.25 and 0.5. In another investigation Haines with his co-researchers (Haines

et al., 2001) suggested that for ionic material it is less than 0.1. Hence it can be clearly

demonstrated from Table 5.5 that SrMF3 compounds resist more for ionic character and least

partial covalent character and from several mechanical parameters it can be well concluded

that SrLiF3 show more brittle behavior than rest of the compound.

In manufacturing disciplines, elastic anisotropy parameter (A) plays an imperative character.

The relation can be well defined as follows (Jamal et al., 2016):


Page | 113

𝑨 =𝟐𝑪𝟒𝟒

(𝑪𝟏𝟏−𝑪𝟏𝟐) (5.11)

If A is equal to unity, then the crystal can be completely characterized as isotropic. However,

when micro-cracks are introduced within the material than the value of A, deviates from

unity and the degree of deviation measures amount of elastic anisotropy. However, value of

A of SrMF3 is less than unity that clearly shows anisotropic behavior of corresponding

materials. In fact, anisotropy decreases on addition of the cation with less atomic size and

vice versa.

5.2.3.4 Melting temperature Tm and Kleinman’s parameter calculations

To calculate melting tendency of SrMF3 compounds, the next task is to explore melting

temperature, above which material changes from its solid phase to its liquid phase (Fine et

al., 1984):

Tm = 607 + 9.3B + 555 (5.12)

The GGA calculation of melting temperature are basically consistent with aforementioned

behavior of the compounds. The pictorial representation of melting temperature from Figure

5.3 shows that melting temperature increases as compounds traverses from SrRbF3 to SrLiF3

respectively. Another significant parameter which was introduced by Kleinman, used to

quantify material’s behavior towards bond stretching or bond bending; if minimum bond

stretching then Kleinman parameter (ξ), ξ=1 but If compound possess minutest value for

bond bending then ξ=0 (Kleinman 1962), as:

𝝃 =𝑪𝟏𝟏+𝟖𝑪𝟏𝟐

𝟕𝑪𝟏𝟏−𝟐𝑪𝟏𝟐 (5.13)

The range of value lie between 0.22-0.42 as compound changes from SrRbF3 to SrLiF3 which

shows that in SrRbF3 bond bending is prevalent while bond stretching is dominant in SrLiF3

fluoroperovskite.


Page | 114

5.2.3.5 Lame’s constant calculations

At the end, the study is related with stress to strain by acknowledging two important

constants namely first λ and second μ Lame’s constant. The expression for these constants

can be derived from various mechanical parameters in the following form (Alouani 1991):

𝝀 = 𝒀ѵ

(𝟏+ѵ)(𝟏−𝟐ѵ) (5.14)

𝝁 =𝒀

𝟐(𝟏+ѵ) (5.15)

Above equations reveals that these constants are in direct relation with the value of Y. Our

calculated values not fulfill the specific criteria 𝜆 = 𝐶12 𝑎𝑛𝑑 𝜇 = 𝐶′ for isotropic material.

So SrMF3 is a class of anisotropic compounds which is in accordance with the calculated

value of anisotropy parameters. In conclusion, an increase in anisotropy is observed in the

following fashion SrLiF3→ SrNaF3→ SrKF3→ SrRbF3. Hence these compounds can be used

in manufacturing low birefringence lens materials. As a result, through this study various

quantum mechanical effects have benchmarked which are very beneficial to utilize and

understand in manufacturing practical devices.


Page | 115

Figure 5.1: Crystal structures of SrMF3, Where M = Li, Na, K, and Rb (Sr+2: Blue, M+1:

Green, F-1 : Red)


Page | 116

Figure 5.2: Lattice constants versus change in bond lengths between M and F of SrMF3

(M = Li, Na, K, Rb).

Figure 5.3: Melting temperature Tm (K) Vs Klienmann parameter ξ (GPa).

SrLiF3

SrNaF3

SrKF3

SrRbF3

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.2

3.21

3.22

4 4.2 4.4 4.6 4.8 5

Bo

nd

-len

gth

s M

-F (

Å)

Lattice constant (Å)

SrLiF3SrNaF3

SrKF3 SrRbF3

0

500

1,000

1,500

2,000

2,500

3,000

0.42 0.23 0.24 0.22

Tm

(K)

ξ(GPa)


Page | 117

Table 5.1: Comparison of calculated equilibrium lattice constants (ao), ground state energies

(Eo) and bulk modulus (Bo) with experimental and other theoretical values of SrMF3 (X = Li,

Na, K and Rb) compounds.

a (Mousa et al., 2013), b (Mubarak 2014), c (Düvel et al., 2011), d (Yamanoi et al., 2014), e (Ouenzerfi

2004) (Other theoretical work) f (Castro 2002), g (Mishra 2011) (Experimental Work)

Table 5.2: Bond-lengths of SrMF3 (X= Li, Na, K, Rb) compounds.

Compound Present

─────────

GGA

Work

─────────

LDA

─────────

I.R method

─────────

V.J method

Theoretical

work

Experimental

work

SrLiF3

ao (Å)

Eo (Ry)

Bo (GPa)

4.11

-6974.884

73.482

4.08

-6974.901

4.81

4.79

3.88a, b,

3.87c

3.76d, 3.75e

72.87a,71.5b

4.45f

72.07g

SrNaF3

ao (Å)

Eo (Ry)

Bo (GPa

4.23

-7284.587

56.59

4.16

-7284.601

4.71

4.45

4.18a

55.81a

4.44f

SrKF3

ao (Å)

Eo (Ry)

Bo (GPa)

4.43

-8163.855

37.62, 47.52

4.41

-8163.728

4.88

4.71

4.38a

37.69a

4.49f

29.64f

SrRbF3

ao (Å)

Eo (Ry)

Bo (GPa)

4.49

-12922.447

32.24

4.46

-12922.386

4.98

5.01

4.55a

31.62a

4.47f

30.6f

SrLiF3

Bond

length

(Å)

SrNaF3

Bond

length

(Å)

SrKF3

Bond

length

(Å)

SrRbF3

Bond

length

(Å)

Sr-F (Å) 2.52 Sr-F (Å) 2.55 Sr-F (Å) 2.58 Sr-F (Å) 2.60

Li-F (Å) 1.85 Na-F (Å) 2.23 K-F (Å) 2.60 Rb-F (Å) 2.74

Sr-Li (Å) 3.23 Sr-Na (Å) 3.61 Sr-K (Å) 3.98 Sr-Rb (Å) 4.15


Page | 118

Table 5.3: Calculated values of elastic constants C11, C12, C44, for SrMF3 (X = Li, Na, K and

Rb) compounds.

Table 5.4: Calculated values of Bulk modulus B0, Voigt’s shear modulus GV, Reuss’s shear

modulus GR, Hill’s shear modulus GH, Young’s modulus Y, and Pugh’s index of ductility

Bo/GH.

Sr.No. Parameters SrLiF3 SrNaF3 SrKF3 SrRbF3

1 C11 (GPa) 151.741 145.490 98.101 85.390

2 C12 (GPa) 37.353 12.140 9.021 6.881

3 C44 (GPa) 49.447 22.410 20.180 14.091


1 Bo(GPa) 74.481 56.590 38.710 33.051

2 Gv(GPa) 52.541 40.122 29.924 24.156

3 GR(GPa) 52.279 30.511 25.833 18.954

4 GH(GPa) 52.414 35.315 27.872 21.551

5 Y (GPa) 127.361 87.703 67.427 53.107

6 Bo/GH (GPa) 1.421 1.605 1.385 1.532


Page | 119

Table 5.5: Calculated values of Shear constant (𝐶′), Cauchy pressure (𝐶′′), Poisson’s ratio

(ѵ), Anisotropy constant (A), Kleinman parameter (ξ), Lame’s coefficients (λ and μ), and

Melting temperature (Tm).


1 𝐶′ 57.19 66.67 44.54 39.25

2 𝐶′′ -12.09 -10.27 -11.16 -7.21

3 Ѵ (GPa) 0.21 0.24 0.22 0.23

4 A (GPa) 0.86 0.34 0.45 0.36

5 ξ(GPa) 0.42 0.23 0.24 0.22

6 λ 39.53 32.64 20.17 18.56

7 μ 52.40 35.36 27.86 21.58

8 Tm(K) 1854 1688 1522 1469


Page | 120

5.3 Opto-electronic investigation of SrMF3 (M = Li, Na, K, Rb)

5.3.1 Electronic properties

The electronic properties of SrLiF3, SrNaF3 SrKF3 SrRbF3 fluoroperovskites are analyzed by

electronic band dispersion curves, detailed states for density while chemical nature of

bonding is interpreted by electron density contour maps.

5.3.1.1 Band structure calculations

Dispersion curves of computed bandgap are displayed in Figure 5.4 - 5.7 respectively. The

bandgap results are taken in first brillouin zone. Here to get reliable results, DFT bandgap is

treated with five distinct exchange and correlation schemes. Detailed theoretical chemistry

for band structure are itemized in Table 5.6. The current calculations from LDA as well as

PBE-GGA approximations are in good agreement with previously available work. Though,

these approximations underestimate the bandgap in semiconductors as well as in insulators

due to wrong interpretation of the true unoccupied states with respect to consistent Khon–

Sham DFT states (Wu and cohen 2006). To acquire nearby experimental values and to

overcome band gap underestimation, we graphically only present results by modified Becke-

Johnson potential (mBJ).

Calculated outcomes depict SrMF3 contains both maxima as well as minima of valence and

conduction bands at (Γ- Γ) symmetry points as a result provides minimum direct band gap

and it can be analyzed from Table 5.6 that bandgap is going to decrease from SrLiF3 to

SrRbF3 and vice versa because minima of conduction band move closer to Fermi level (EF).

The similar results for direct (Γ-Γ) gap are also observed for BaLiF3, CaLiF3 and CsSrF3

(Mousa et al., 2013 & babu et al., 2012) however different behavior is observed for RbCdF3,


Page | 121

RbZnF3, as well as RbHgF3 (Murtaza et al., 2013) having indirect bandgap at (M-Γ) which

depicts that nature of electronic properties for fluoroperovskites varies as of compound to

compound. Additionally, through mBJ potential a bandgap near to experimental values is

obtained like Yalcin and their fellows (Yalcin et al., 2016), calculated 8.2 eV bandgap for

mBJ potential, alike to BaLiF3 experimental bandgap of 8.41 eV. Consequently, just 1.78%

deviation has been observed. Analogous findings are observed for SrMF3 compounds.

5.3.1.2 Density of States (DOS) calculations

The investigated outcomes of density of states (DOS) indicate wide dispersion of electronic

bands as shown in Figure 5.8 (DOS for SrLiF3), Figure 5.9 (DOS for SrNaF3), Figure 5.10

(DOS for SrKF3), and Figure 5.11 (DOS for SrRbF3) respectively. In conduction band Sr-3d

peaks appears nearby 5.9, 6.6, 8.8, 9.2 eV for SrRbF3, SrKF3, SrNaF3, SrLiF3 correspondingly.

Though, SrRbF3 and SrKF3 shows crossed crests of Rb-3d and K-3d within range of 5-10 eV.

The valence band resides within energy range as of 0 to -30 eV. The evolution of fluorine 2p

governs valence band region till -2.5 eV, together with little influence of Sr and M states.

Likewise, hybridized crests are observed in -20 to -23 eV for F-2s as well as M states

respectively. Furthermore, triplet degenerate energy levels at Γ symmetry point are observed

for lower conduction and upper valence gap regions. At -13.8 and -15.1 eV, peaks in SrLiF3

and SrNaF3 are due to Sr-4p states individually. In valence band region, for SrKF3 and

SrRbF3, K-3p and Rb-4p states are observed at -8.5 eV and -6.5 eV. In SrMF3 different peaks

at different energy levels are appeared for example, Na-2p state at -17.2 eV, K-4s states at -

25.8 eV, Rb-5s state at -22.6 eV for SrNaF3, SrKF3, SrRbF3 respectively.


Page | 122

5.3.1.3 Electron density contour calculations

In crystalline solids nature of chemical bonding, can be determined via electron density,

contour maps (Hoffman 1988). Electron density distribution curves are presented in 2D and

3D, along (100) and (110) planes. It can be observed from Figure 5.12 (a-d) that, in (100)

plane dispersal of charge for SrLiF3 is not clearly spherical within Sr and F cation as well as

anion respectively. However, it is perfectly spherical for SrRbF3, SrNaF3, and SrKF3 that is

due to predominant ionic bonding nature. Whereas as of view from (110) plane, as shown in

Figure 5.13 (a-d) and 5.14, it can be analyzed that, charge is transferred between M cation as

well as F anion because of huge electronegativity variance where transference of charge in

octahedral differs as per subsequent relation: LiF6 > NaF6 > KF6 > RbF6 in addition to it,

nature of covalent bonding corresponds accordingly: SrRbF3 > SrKF3 > SrNaF3 > SrLiF3. In

fact, atomic magnitude becomes larger when moving down in a group and corresponding

electrons in bigger atoms are not capable to stay strongly in comparison with smaller ones

which reduces extent of electronegativity. Hence increase in ionic behavior can be observed

as compound moves from SrLiF3 to SrRbF3 While this investigation matches well by

consequences of DOS where maximum p-d hybridization is obtained in SrLiF3. Furthermore,

parallel trend of outcomes was also observed in the work done by Harmel and their co-

fellows (Harmel et al., 2015 & Harmel et al., 2012).

5.3.2 Optical parameters

In order to calculate fundamental and derived optical responses of SrMF3, exchange

correlation potential of modified Becke–Johnson (mBJ) with GGA is applied. The details of

evaluated parameters are mentioned in few upcoming headings.


Page | 123

5.3.2.1 Complex dielectric constant calculations

The Ԑ(ω) can be defined as follows (Fox 2001):

Ԑ(𝜔) = Ԑ1(𝜔) + 𝑖Ԑ2(𝜔) (5.16)

In equation 5.16 Ԑ(ω), Ԑ1(ω) and Ԑ2(ω) represents complex, real as well as imaginary

dielectric function respectively. Generally, Ԑ1(ω) can influences in two ways such as

intraband as well as interband transitions. For metals, the contribution of intraband transition

is exclusively significant. While further classification of interband transitions can be done in

terms of indirect and direct transitions (Dressel 2001). However, the present calculations are

performed by considering results of direct (interband) transitions. Ԑ2(ω) can be given as:

Ԑ2(𝜔) = (4𝜋2𝑒2

𝑚2𝜔2) ∑ ∫ ⟨𝑖|𝑀|𝑗⟩2𝑓𝑖(1 − 𝑓𝑖)𝛿(𝐸ј,𝑘 − 𝐸𝑖,𝑘 − 𝜔)𝑘𝑖,𝑗 𝑑3𝑘 (5.17)

In equation 5.17 initial and final states are denoted by ὶ and ј respectively, the dipole matrix

is represented by M, ω is the resonance frequency. In spectra of Ԑ2(ω), as shown in Figure

5.15 (a) wide-ranging characteristic peaks are observed which can be directly linked to DOS

of the respective compounds for further clarification. First critical point or threshold energy

for SrLiF3, SrNaF3, SrKF3 and SrRbF3 are occurred approximately at 9.3 eV, 8.9 eV, 7.2 eV

and 5.8 eV respectively. For direct optical transitions, this threshold energy gives

corresponding fundamental gap at equilibrium, which is also recognized as fundamental

absorption edge. It ensues due to splitting between highest states of valence band at (Γv) to

lowest state of conduction band at (Γc). In fact, the materials with band gaps greater than 3.1

eV can work well in the Ultra-Violet region so all compounds can work well in ultraviolet

region of electromagnetic spectrum (Wooten 1972). It can be observed that fundamental


Page | 124

absorption edge shift towards lower energy as we move towards Li to Rb. After the incent of

critical point, Ԑ2(ω) dispersion curves increases abruptly due to rapid contribution of

electronic states. As a result, the principle peak is situated at 12.2 eV for SrLiF3, 10.9 eV for

SrNaF3, 8.3 eV for SrKF3 and 6.9 eV SrRbF3, correspondingly which is in accordance with

the trends of DOS and band-structure. These peaks are attributed to transitions of F-2p state

alongwith minor influences of Sr-3d and M-states positioned just below zero energy Fermi

level (EF). However, using the knowledge of optical matrix element SrKF3 and SrRbF3

compounds reveals hybridized peaks around 5 to 10 eV of K-3d and Rb-3d respectively. The

real part of dielectric function Ԑ1(ω) is given by the well-known Kramers-Kronig relation via

corresponding equation (Erum and Iqbal, November 2017):

Ԑ1(𝜔) = 1 +2

𝜋𝑃 ∫

ὠԐ2(ὠ)𝑑ὠ

ὠ2−ὠ2

ⱷ

0 (5.18)

It describes Ԑ1(ω) defines electric polarizability and absorptive behavior of the material. The

dielectric function (static part), as shown in Figure 5.15 (b), Ԑ1(0) at zero frequency limit is

calculated at about 1.61 eV to 1.71 eV for SrMF3 compounds, correspondingly. Similar trend

of static refractive index is observed. The curves of Ԑ1(ω) starts increasing from the (zero)

limit of frequency and attains maxima value to expose the principle peak which is at about

6.6 eV, 7.9 eV, 9.5 eV and 10.3 eV for SrRbF3, SrKF3, SrNaF3 and SrLiF3, exclusively.

5.3.2.2 Optical conductivity and energy loss calculations

The calculated values of complex dielectric function Ԑ(ω) can allow to determine key optical

function for the electromagnetic spectrum such as optical conductivity σ(ω), and electron

energy loss spectrum L(ω) which can be determined by following expressions (Fox 2001):


Page | 125

𝐿(𝜔) = 𝐼𝑚 (−1

Ԑ(𝜔)) (5.19)

𝜎(𝜔) = 2𝑊𝐶𝑉ℏ𝜔

𝐸02 (5.20)

In equation 5.20 WCV is the transition probability between conduction and valence band. The

calculated spectrum of L(ω) is figured in Figure 5.15(c). It relates microscopic as well as

macroscopic responses of material and elucidates characteristic contribution associated with

plasma resonance frequency (ωp) (Waghmare 2001). The peaks in L (w) spectra narrate

energy loss of electron that is traversing rapidly within the material. The region of distinct

sharp peak covers range within 7 eV to 14 eV for M compounds respectively. The calculated

optical conductivity σ(ω) for SrMF3 fluoroperovskites are shown in Figure 5.15(d). The drift

of figure reveals that the phenomenon of σ(ω) optical conductivity begins at approximately 4

eV. The journey of optical conductivity traverses from small and large peaks and finally

reveals steady behavior in high-energy ranges. It is evident from above analysis that from

cation Rb to Li, the spectrum of σ(ω) shift towards low energy ranges.

5.3.2.3 Sum rules calculation via neff

In the end, the sum rule is evaluated to consider the number of effective valence electrons via

corresponding formula (Abelès 1972):

𝑛𝑒𝑓𝑓(𝜔) = ∫ 𝜎(ὠ)𝜔

0ὠ 𝑑ὠ (5.21)

From Figure 5.15 (e) trends of sum rule can be analyzed, which originates the inter-band

transition of electrons at about 5.5 eV. Then there is slow increase in trendline, however

advent of (sharp) peak reveals abrupt increment of electron which saturates in the range at


Page | 126

about 14-16 eV and onwards. The results of neff are in similar accordance with the above

calculated optical parameters.

Figure 5.4: The mBJ-electronic band dispersion curves for SrLiF3

En

ergy

(eV

)


Page | 127

En

erg

y (

eV)

Figure 5.5: The mBJ-electronic band dispersion curves for SrNaF3


Page | 128

En

ergy

(eV

)

Figure 5.6: The mBJ-electronic band dispersion curves for SrKF3


Page | 129

Figure 5.7: The mBJ-electronic band dispersion curves for SrRbF3

En

ergy

(eV

)


Page | 130

Figure 5.8: The Density of States for SrLiF3 by mBJ potential

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 131

Energy (eV)

DO

S (

Sta

tes/e

V)

Figure 5.9: The Density of States for SrNaF3 by mBJ potential


Page | 132

Energy (eV)

DO

S (

Sta

tes/e

V)

Figure 5.10: The Density of States for SrKF3 by mBJ potential


Page | 133

DO

S (

Sta

tes/e

V)

Energy (eV)

Figure 5.11: The Density of States for SrRbF3 by mBJ potential


Page | 134

Figure 5.12 (a): Calculated mBJ total two and three-dimensional electronic charge densities

for SrLiF3 in (100) plane.


Page | 135

Figure 5.12 (b): Calculated mBJ total two and three-dimensional electronic charge densities

for SrNaF3 in (100) plane.


Page | 136

Figure 5.12 (c): Calculated mBJ total two and three-dimensional electronic charge densities

for SrKF3 in (100) plane.


Page | 137

Figure 5.12 (d): Calculated mBJ total two and three-dimensional electronic charge densities

for SrRbF3 in (100) plane.


Page | 138


for SrLiF3 in (110) plane.


Page | 139


for SrNaF3 in (110) plane.


Page | 140

Figure 5.13 (c): Calculated mBJ total two and three-dimensional electronic charge densities

for SrKF3 in (110) plane.


Page | 141

Figure 5.13 (d): Calculated mBJ total two and three-dimensional electronic charge densities

for SrRbF3 in (110) plane.


Page | 142

Figure 5.14: Total two-dimensional electron density plots in (110) plane for (a) SrLiF3, (b)

SrNaF3, (c) SrKF3, (d) SrRbF3.


Page | 143

Figure 5.15 (a): Calculated imaginary part Ԑ2 (ω) of the dielectric function for the SrMF3

(M=Li, Na, K, Rb) compounds.


Page | 144

Figure 5.15 (b): Calculated real part Ԑ1(ω) of the dielectric function for the SrMF3

(M=Li, Na, K, Rb) compounds.


Page | 145

Figure 5.15(c): Calculated energy loss function L (ω) for SrMF3 (M=Li,Na,K,Rb)

compounds.


Page | 146

Figure 5.15(d): Calculated conductivity σ(ω) for SrMF3 (M= Li, Na, K, Rb) compounds.


Page | 147

4.3. Opto-electronic investigation of Rubidium based Fluoro-Perovskite for

. Figure 5.15(e): Calculated sum rule for SrMF3 (Li,Na,K,Rb) compounds.


Page | 148

Table 5.6: Band gap of SrMF3 (M = Li, Na, K, Rb) at different symmetry points compared

with experimental and other theoretical results.

Compound Symmetry

Point

Bandgap

Type

𝐸𝑔𝑀𝐵𝐽

(eV) 𝐸𝑔𝐿𝐷𝐴(eV) 𝐸𝑔

𝑊𝐶−𝐺𝐺𝐴(eV) 𝐸𝑔𝑃𝐵𝐸−𝐺𝐺𝐴(eV) 𝐸𝑔

𝑃𝐵𝐸𝑠𝑜𝑙−𝐺𝐺𝐴(eV) Other

work

PBE-GGA

(LDA)

SrLiF3 Γ-Γ Direct 9.20 7.01 7.17 7.28 7.16 7.21a,7.30b

(7.19c)

R-R Direct 10.10 8.30 8.80 8.65 8.70

M-M Direct 10.20 9.00 9.40 9.10 9.20

X-X Direct 10.00 7.90 8.20 8.10 8.00

SrNaF3 Γ-Γ Direct 8.30 5.53 5.47 5.61 5.35 5.58a,

(5.94c)

R-R Direct 9.10 7.40 7.80 7.70 7.75

M-M Direct 9.00 7.60 7.20 7.30 7.32

X-X Direct 8.70 6.60 6.90 7.00 7.18

SrKF3 Γ-Γ Direct 6.80 3.20 3.80 3.31 3.50 3.27a

R-R Direct 6.50 4.30 4.80 5.00 5.20

M-M Direct 6.30 4.40 5.20 4.80 5.10

X-X Direct 5.90 3.20 3.80 3.70 3.90

SrRbF3 Γ-Γ Direct 5.60 2.21 2.28 2.30 2.32 2.29a

R-R Direct 5.10 4.63 4.69 4.70 4.71

M-M Direct 5.50 3.84 3.86 3.90 4.10

X-X Direct 5.30 2.31 2.38 2.40 2.50

a) (Mubarak 2014), b) (Mousa et al., 2013), c) (Nishimatsu et al., 2002).


Page | 149

5.4 Opto-electronic investigation of RbHgF3 for low birefringent

lens materials

Physical properties of material have played principle role in implementation of them in

specific area of device fabrication. To fulfill this curiosity material scientists are vigilant to

inquire opto-electronic response of different perovskite compounds. Now-a-days rubidium

series of mercury fluoroperovskite is renowned to own several technological benefits. One of

the important application of RbHgF3 is in fabricating lens materials with low birefringence

because high birefringence can make design of lenses difficult. Further it conceives wide

range of opto-electronic and photonic applications such as in optical pathways, UV detectors,

transparent optical coatings and Light Emitting diodes (LED) (Lang et al., 2014 &

Vaitheeswaran et al., 2007).


In order to investigate ground state structural aspects of RbHgF3, the total energy is

determined at various volumes. The structural properties are calculated by using energy

minimization process as described in section 5.1.1 in detail. It can be observed from Figure

5.16 that RbHgF3 fluoroperovskite compound crystallized it-self in cubic structure while sites

of Wyckoff coordinates are situated at 1a, 1b, and 3c for respective cations and anions of Rb,

Hg and F. These structural findings are in similar accordance with comprehensive

experimental investigation done by Dotzler and their fellows (Dotzler et al., 2008).

The variation of total energy as a function of unit cell volume is displayed in Figure 5.17 and

calculated lattice parameters such as ground state energy (Eo), equilibrium lattice constant

(ao), bulk modulus (Bo) and its pressure derivative (Bp) are tabulated in Table 5.5 with four


Page | 150

different schemes (LDA, PbE-GGA, WC-GGA and PBEsol-GGA). The slightly overvalued

DFT calculated lattice constant as compared to experimental one can be attributed to the use

of traditional DFT approximations. Furthermore, good crystal rigidity is evident from the

value of bulk modulus but on the other hand, a contrary relation between value of bulk

modulus and lattice constant can be observed from Table 5.5, in similar accordance with the

pattern of another halide perovskite (Ghebouli et al., 2012; Brik 2011& Rose et al., 1993).

5.4.2 Electronic Properties

In this section, calculations of electronic behavior of RbHgF3 have been done in terms of

band structure, Density of states (DOS), (Total as well as Partial) and nature of bonding is

explained in terms of electron density plots.


In this study the computation of bandgap is done with the help of GGA plus Tran-Blaha

modified Becke–Johnson (TB-mBJ) potential (Tran and Blaha 2009). The comparison of

energy band structures at high symmetry direction from PBE-GGA and mBJ schemes are

shown in Figure 5.18, which proves bandgap underestimation by PBE-GGA approximation.

It can be noted that trend of overall dispersion in a band structure curves are almost same and

lower edge of conduction and higher edge of valence band lies at M and Γ symmetry points

along the Brillouin Zone (BZ), resulting an indirect (M-Γ) bandgap of about 0.7 eV from

PBE-GGA and 3.1 eV from mBJ schemes respectively. However, in order to make a

reasonable comparison, there is lack of experimental data of bandgap. An important

application of this material is in Ultra-Violet region of electromagnetic spectrum because it


Page | 151

has bandgap larger than 3.1 eV so work efficiently for UV-based practical devices (Lang

2014).


The detailed analysis of partial and total density of states are done by analyzing various states

of corresponding energy density distribution, as shown in Figure 5.19. The significant peaks

of DOS are found within -10 to15 eV. At -10 eV, a fine peak can be notified because of Rb-

4p states. Next, in valence band region from fermi-level to -6.1 eV an overlapping is

observed between F-2p and Hg-3d states and above the fermi level, the upper and lower part

of conduction band is filled by Hg-4s as well as Rb-4d states respectively.


In crystalline materials, nature of chemical bonding can be communicated via map of

electron density plots (Gelatt 1983). The 2D and 3D contour maps along (100) and (110)

planes are shown in Figure 5.20 (a, b) and 5.21 (a, b) respectively. In contour maps of Rb

cation and F- anion due to large electronegativity difference, transfer of charge occurred.

However, the perfectly spherical charge distribution confirms strong ionic character in Rb-F

bond. Furthermore, uniform distribution of charge between Hg cation and F anion validates

covalent character of HgF3 type octahedra. The bonding nature by these plots are in exact

accordance with the plots of Density of states as shown in Figure 5.19, where in between Hg-

3d and F-2p states, the p-d hybridization is maximum. Hence confirms mixed covalent and

ionic bonding character in rubidium based fluoroperovskite. In a similar study, Harmel and

their fellows (Harmel et al., 2015) observed the same results for cesium based

fluoroperovskites.


Page | 152

5.4.3 Optical properties

The tool of optical analysis is employed to expose internal behavior of RbHgF3 compound.

Imaginary and real part of dielectric function are two fundamental optical responses.

However, in concern with, practical device applications analysis is extended to plot spectra

of reflectivity and optical absorption.


The complex dielectric function consists of two parts, namely imaginary and real part (Fox

2001):

Ԑ(𝜔) = Ԑ1(𝜔) + 𝑖Ԑ2(𝜔) (5.22)

Imaginary part of dielectric function can be denoted in terms of Ԑ2(ω) while Ԑ1(ω) depicts

about real part of dielectric function. The comprehensive response of a material due to

applied electromagnetic radiation in terms of Ԑ2(ω) are shown in Figure 5.22(a). In this

optical study focus is paid to direct interband transition while taking into account appropriate

element of transition dipole matrix (Brik 2011). The band structure and DOS of the

investigated compound follow the pattern of widespread peaks of Ԑ2(ω). The major peak in

Ԑ2(ω) are located at approximately 22.1 eV and threshold energy point occurs at 4.4 eV

approximately. The occurrence of these diversified peaks are observed due to hybridized

states of Rb-4d with some p states of mercury and fluorine. The real part of dielectric

function Ԑ1(ω) depicts the absorptive behavior as figured out in Figure 5.22(b). The dielectric

function, static part is notified at 1.88 eV. The ascending peaks in Ԑ1(ω) attains maximum


Page | 153

position around 6 eV, which confirms an overall narrow bandgap semi-conductive nature,

while minimum attains at 20 eV.

5.4.3.2 Absorption coefficient calculations

The absorption of electromagnetic radiation in optical absorption spectra of RbHgF3 starts at

about 5.05 eV as shown in Figure 5.22 (c). This particular energy point is in exact

accordance with the trend of bandgap. After some ascending peaks, RbHgF3 starts absorbing

effectively, delivers prominent peaks at around 21.5 eV. After trivial variations, the peaks

again going to decrease. The absorption spectra analysis concludes application of RbHgF3 for

wide absorption purposes at about 21.5 eV in Ultra-Violet region of electromagnetic

spectrum.

5.4.3.3 Optical reflectivity calculations

Figure 5.22 (d), depicts phenomenon of reflectivity as a function of energy. It can be

observed that up to 19 eV the phenomenon of reflectivity stays below 9-10%. However, at 22

eV material attains high value of reflectivity in high energy region. As a result, it can be

concluded that RbHgF3 is one of the prospective material for efficient lenses as well as

transparent coating devices because in infrared (IR) as well as visible regions these materials

remain highly transparent.


Page | 154

Figure 5.16: Cubic crystal structure of RbHgF3


Page | 155

Figure 5.17: Variation of total energy as a function of unit cell volume for RbHgF3


Page | 156

Figure 5.18: Comparison of band structures in high symmetry directions with mBJ and

PBE-GGA schemes for RbHgF3

En

erg

y (

eV

)


Page | 157

Energy (eV)

DO

S (

Sta

tes/e

V)

Figure 5.19: The Density of States for RbHgF3 by mBJ potential


Page | 158


in (100) plane for RbHgF3

.


Page | 159


in (110) plane for RbHgF3.

.


Page | 160

Figure 5.21 (a): Total two-dimensional electron density plots in the (100) plane for RbHgF3.

Figure 5.21 (b): Total two-dimensional electron density plots in the (110) plane for RbHgF3.


Page | 161

Figure 5.22 (a): Calculated imaginary part Ԑ2(ω) of the dielectric function for RbHgF3

compound.


Page | 162

Figure 5.22 (b): Calculated real part Ԑ1(ω) of the dielectric function for RbHgF3 compound.


Page | 163

Figure 5.22 (c): Calculated absorption coefficient α (ω) of dielectric function for

RbHgF3 compound.


Page | 164

Figure 5.22(d): Reflectivity R(ω) as a function of energy for RbHgF3 compound.


Page | 165

Table 5.5: Comparison of Present calculation with previous experimental and theoretical

values for lattice constants (ao), ground state energies (Eo), bulk modulus (Bo) and its

pressure derivative (Bp) of RbHgF3 compound.

a) (Muller & Roy 1974), b) (Moreira & Dias 2007)

Compound

RbHgF3

Present

work

————

PBE-GGA

Present

work

————

WC-GGA

Present

work

————

PBEsol-

GGA

Present

work

————

LDA

Experimental

work

Other

theoretical

work

ao (Å)

4.60 4.57 4.53 4.49 4.47a 4.46b

Eo (Ry) -45854.51 -45854.43 -45854.40 -45854.39

Bo (GPa) 48.84 49.01 49.39 49.81 48.32a

BP(GPa) 5.61 5.58 5.53 5.51


Page | 166

5.5 Conclusion

In this chapter, systematic first principles calculation of five fluoroperovskites (SrLiF3,

SrNaF3, SrKF3, SrRbF3 and RbHgF3) have been carried out successfully.

Comprehensive results of structural, elastic and mechanical properties of strontium based

group 1A compounds, (section 5.2), reveals that value of lattice constants increases, as cation

shift from Lithium to Rubidium, while value of bulk modulus decreases, that can be

attributed to higher extent of atomic radii of Rubidium. Furthermore, these elastically and

mechanically stable compounds have dominant brittle and ionic behavior.

Section 5.3 delivers unique theoretical strategy to calculate detailed opto-electronic trends of

strontium based group 1A fluoroperovskites, via various exchange and correlation schemes,

provides accurate description of band profiles, which permits to investigate reliable

predictions of electronic charge density and density of states. These calculations argue

against the existence of low bandgap values that have been studied previously with less

reliable Local Density Approximation (LDA) and Generalized Gradient Approximation

(GGA) schemes but there is lack of experimental data so in description we compare Tran-

Blaha modified Becke–Johnson (TB-mBJ) band gap results that are generally similar to

experimental band profile of BaLiF3 compounds. On the basis of above exploration, it can be

concluded that SrMF3 (M= Li, Na, K, Rb) are ionic, wide, and direct bandgap

fluoroperovskites and need an extensive experimental research for their possible utilization in

Ultra-Violet (UV) transparent lens material and in advanced lithographic technology.

In section 5.4, structural, and opto-electronic properties of RbHgF3 have been discussed, with

four different exchange-correlation approximations. The analysis of energy band profiles


Page | 167

authenticates indirect narrow energy bandgap (M–Γ) semi-conductive nature. While

contribution of different bands ensures mixed covalent, and ionic behavior. The

advantageous optical responses explore wide range of absorption and reflection in high

frequency regions. Hence RbHgF3 can be efficiently applied for manufacturing high class

lens material with low birefringence and by suitable doping can be utilized in photovoltaic

applications as well.

Chapter 6 Results and discussion ΙΙ

Page | 168

Chapter 6: Results and discussion ΙΙ;

Investigation of mechanical and optoelectronic behavior of

actinoid based oxide perovskites

“Prediction is very difficult,

Especially about the future.”

Niels Bohr

6.1 Introduction

Now-a-days faster, flexible and efficient devices are attaining huge attention. In this regard,

material scientists are struggling hard for reconnoitering new materials. In the last few

decades, much interest has been given to perovskite oxides which have large value of static

dielectric constants. Materials possessing greater value than that of silicon, (for silicon Ԑ0 = 7)

can be classified in terms of high dielectric constant materials. In fact, Ԑ0 have the ability to

decide about miniaturization extent of any material and this idea can be ultimately utilized in

integrated device synthesis.

In section 6.2 potassium and rubidium based protactinium oxide perovskites are investigated

to explore their possible aspects. Technically XPaO3 (X = K, Rb) compounds are sound but

due to radio-active nature and high cost of protactinium, little systematic investigation has

been stated on them. So, this theoretical work motivated us to inquire XPaO3 in detail. In the

present work, to address significant aspects various physical properties of XPaO3 are

explored for their possible technological applications.

The section 6.3 of this chapter is dedicated to inquire Ba based oxide perovskites BaPaO3 and

BaUO3 that fall in the class of high dielectric constant materials (Erum and Iqbal, Februrary

2017) but experimental as well as theoretical information are unavailable due to radioactive


Page | 169

nature of these compounds. So, in this study aim is to contribute on scientific information of

these compounds which are equally important to investigate about possible properties such as

electronic structure, elastic behavior, mechanical stability, opto-electronic and dielectric

properties on them.

6.2 Mechanical and optoelectronic study of XPaO3 (X= K, Rb)

The crystal structure of both KPaO3 and RbPaO3 have been reported as an ideal cubic

perovskites-type structure. The first experimental report about these compounds was the

work of Keller (Keller 1965). He reported that these systems have cubic perovskite structure

with the lattice constants 4.341 Å and 4.368 Å for KPaO3 and RbPaO3, respectively. Next

was the work of Iyer and Smith (Iyer and Smith 1971). However, they probably do not obtain

KPaO3, but product which was richer in Pa2O5 rather than KPaO3. It seems that, the work of

Keller (Keller 1965) is the only experimental report so far in which cubic KPaO3 and

RbPaO3 have been obtained.


As mentioned in literature survey that ternary oxide perovskites KPaO3 and RbPaO3, have

space group Pm-3m (no. 221) crystallizes in simple cubic structure. The cubic unit cell

crystal structure is displayed in Figure 6.1 (a) and 6.1 (b) for KPaO3 and RbPaO3,

respectively. The total energy is determined at various volumes as illustrated by Figure 6.2

(a) and 6.2 (b) for KPaO3 and RbPaO3, correspondingly. The XPaO3 structural aspects are

calculated by same energy minimizing process as mentioned in section 5.2 of chapter 5

(Murnaghan 1944). In which corresponding equilibrium unit cell volume and total energy are

evaluated by corresponding set of various lattice parameters. The details of resultant lattice


Page | 170

parameters calculated from GGA and LDA approximations are shown in Table 6.1. That

reveals value of lattice constant for RbPaO3 is larger than KPaO3 and vice versa, and it can

be clarified in terms of following relation R(K) = 2.43 Å is smaller than that of R(Rb) =

2.65 Å.

Furthermore, to clarify this chemistry two analytical methods are also employed to calculate

lattice constants. Both ionic radii and Verma Jindal calculation of lattice constants are based

on equations 6.1 and 6.2 respectively,

a0 = α + β (rX + rO) + Ɣ(rPa + rO) (6.1)

a0 = K(VXVPaVO)srav (6.2)

using ionic radii of rK = 1.64 Å, rRb = 1.72 Å, rPa = 0.90 Å, and ro = 1.35 Å, for K, Rb, Pa, and

O correspondingly, with valence electrons of 1, 2 and 6 for X, Pa, and O. Due to the

dependence of equations 6.2 on the number of valence electrons, the calculated lattice

constants via V.J method illustrates deviation within 7-8 % while for I.R methods this

deviation is within 2-3% with respect to DFT method but experimental versus DFT

calculated outcomes are in reasonable similarity by each other. Crystal rigidity of XPaO3 are

estimated with the help of bulk modulus value. It can be manipulated from Table 6.1 that

RbPaO3 have smaller value of bulk modulus than KPaO3 in contrary relation with value of

lattice constant. The same contrary relation between bulk modulus and lattice constant have

been found for another perovskite reported previously by Gheloubi with his fellows

(Gheloubi et al., 2012), as well as by Erum and Iqbal (Erum and Iqbal, March 2017).


Page | 171

The interpretation of chemical bonding is performed with chemical trends. Further this

structural chemistry is used to estimate tolerance factor, where the length or distance between

the two bonded atoms is well known as bond length. It can be noticed from Table 6.1 as

KPaO3 approaches to RbPaO3, there is increment between bond lengths of X and O atoms.

Furthermore, for X, Pa, and O, similar trend is observed. Next, we evaluate the criteria of

tolerance factor by using bond length for XPaO3 compounds (Goldschmidt 1926).

𝑡 =0.707<𝑋−0>

<𝑃𝑎−𝑂> (6.3)

In above equation <Pa-O> and <X-O> depicts X, O and Pa respectively, which satisfies good

tolerance factor criteria within 0.93-1.02 as presented in Table 6.2.

6.2.2 Elastic constant calculations

Elastic properties give reliable information regarding to stability of structure and binary

chemistry (Sadd 2005). In Table 6.3, three major elastic constants are summarized which are

C11, C12 and C44 respectively. All these calculations are carried out by using Charpin’s

method (Charpin 2001). As this is first theoretical approach for calculating elastic constants

of XPaO3 compounds, so this work can be a comparative approach for other scientists

working on the same direction. To ensure cubic as well as mechanical stability, it is a good

finding that all elastic constants satisfy condition of traditional mechanical stability at

pressure of 0 GPa. The details of traditional mechanical stability condition can be found in

our recent paper authored by Erum and Iqbal (Erum and Iqbal 2016). For KPaO3 the

resistance for unidirectional compression is high (432.182 GPa) along the (principle)

crystallographic direction while it possesses a lower value (311.413 GPa) for RbPaO3, which

validates the weak unidirectional resistance for KPaO3 compound. Meziani and Belkhir

(Meziani and Belkhir 2012) observed the similar results for C11 elastic constant. The results


Page | 172

of C44 which is shear or volumetric deformation reveals that both compounds retains greater

resistance for compression rather than shear deformation.

6.2.3 Mechanical parameters

The purpose of this section is to compute polycrystalline mechanical aspects by utilizing data

information from elastic constants. The evaluated parameters include detailed elastic moduli,

Poisson’s ratio, coefficients for elastic stiffness, and melting temperature.

6.2.3.1 Elastic moduli calculations

The hardness of material is an important entity which can use to estimate rigidity of any

crystalline structure. For this purpose, shear and bulk modulus are equally employed which

can conveniently be explored by Voigt Reuss-Hill (VRH) approximation (Reuss and Angew

1929; Shafiq et al., 2015):

𝑮𝑽 = 𝟏

𝟓(𝑪𝟏𝟏 − 𝑪𝟏𝟐 + 𝟑𝑪𝟒𝟒), (6.4)

𝑮𝑹 =𝟓𝑪𝟒𝟒(𝑪𝟏𝟏−𝑪𝟏𝟐)

𝟒𝑪𝟒𝟒+𝟑(𝑪𝟏𝟏−𝑪𝟏𝟐) , (6.5)


𝟐 , (6.6)

𝑩 =𝟏

𝟑 (𝑪𝟏𝟏 + 𝟐𝑪𝟏𝟐) (6.7)

For shear and bulk modulus, it can be noticed from Table 6.4 that KPaO3 (70.25 GPa, 203.78

GPa) have larger value than that RbPaO3 (69.48GPa, 146.98GPa) respectively. The

contribution of material for linear strain is estimated through Young’s modulus by (Kittel

2005):

𝒚 =𝟗𝑩𝑮

(𝟑𝑩+𝑮). (6.8)


Page | 173

RbPaO3 has lower values for shear, bulk and Young’s modulus rather than KPaO3 which

depicts lower tendency of charge transfer among cation and anion respectively. Next to

identify the correct information regarding to brittle/ flexible or ductile character of XPaO3,

the criteria for B/G ratio is evaluated. Typical values of B/G less than 1.75 and greater than

1.75 refers material to be brittle or ductile/flexible (Pugh 1954). The calculated B/G values

(2.901 GPa, 2.120 GPa) for KPaO3, and RbPaO3 confirms strong flexibility of respective

compounds.

6.2.3.2 Cauchy’s pressure and Poisson’s ratio calculations

Next, the Cauchy’s pressure is utilized to confirm flexible nature by taking difference

between C12 and C44 elastic constant respectively. If the value of this particular pressure is

negative/ positive, then material tend towards brittle/flexible in nature (Brik 2011), as shown

in Table 6.5:

𝑪′′ = 𝑪𝟏𝟐 − 𝑪𝟒𝟒 . (6.9)

Meanwhile according to Frantsevich and his fellows (Frantsevich et al., 1983), if material

retains Poisson’s ratio v > 0.26 than material will dominate with flexible characteristics, as

shown in Table 6.5:

ѵ =(𝟑𝑩−𝟐𝑮)

𝟐(𝟑𝑩+𝑮) (6.10)

At this point, it can be observed that for both compounds 𝐶′′ > 0, v > 0.26 and B/G > 1.75

indicates that XPaO3 contains high directional bonding and is flexible in nature.

6.2.3.3 Shear constant and elastic anisotropy calculations

Next the shear constant is calculated to differentiate bonding characteristic (Nakamura 1995),

as shown in Table 6.5:

𝑪′ =𝟏

𝟐(𝑪𝟏𝟏 − 𝑪𝟏𝟐) (6.11)


Page | 174

High value of shear constant reveal that XPaO3 contains dominant covalent behavior. The

isotropic behavior of any crystal in manufacturing disciplines can be estimated through

elastic anisotropy parameter “A” (Jamal et al., 2016), as shown in Table 6.5:

𝑨 =𝟐𝑪𝟒𝟒

(𝑪𝟏𝟏−𝑪𝟏𝟐). (6.12)

Cubic crystals can be completely categorized by means of anisotropic factor (A), if A is

equivalent to unity, then material is isotropic but present calculations notified that it deviates

from 1 so XPaO3 can be completely characterized as anisotropic class of compounds because

of deviation of its value from unity.

6.2.3.4 Lame’s constant calculations

Next, the study is related with stress to strain by acknowledging two important constants

namely first λ and second μ Lame’s constant. The expression for these constants can be

derived from various mechanical parameters in the following form (Alouani 1991):

𝝀 = 𝒀ѵ

(𝟏+ѵ)(𝟏−𝟐ѵ) (6.13)

𝝁 =𝒀

𝟐(𝟏+ѵ) (6.14)

Above equations reveals that these constants are in direct relation with the value of Y. Our

calculated values not fulfill the specific criteria λ = C12 and μ = C′ for isotropic material.

So XPaO3 is a class of anisotropic compounds which is in accordance with the calculated

value of anisotropy parameters as well.

6.2.3.5 Melting temperature calculations

Another important factor is melting temperature (Tm) that elaborates the extent of melting for

a specified material via following relation (Fine et al., 1984):

Tm = 607 + 9.3B + 555 (6.15)


Page | 175

Table 6.5 shows that RbPaO3 have lower tendency of melting rather than KPaO3.

6.2.4 Electronic behavior

In this section electronic behavior of XPaO3 are evaluated by electronic band structure, by

evaluating density of states (DOS) as well as bonding nature is calculated via contour maps

of electron density from Tran-Blaha modified Becke–Johnson (TB-mBJ) potential (Tran and

Blaha 2009).


At this point of investigation, band structure comparison of XPaO3 at different symmetry

points with various approximations are presented in Table 6.6. It can be observed that at all

symmetry points except (Γ-Γ), gap value is high. Though pictorial demonstration of band

structure with (TB-mBJ) at Brillouin Zone are displayed in Figure 6.3. The XPaO3

compounds without any external pressure reveals (Γ-Γ) symmetry direct bandgap of about

3.60 eV and 3.14 eV for KPaO3 as well as RbPaO3 correspondingly. These materials have

larger bandgap than 3.1 eV, so work well for ultraviolet region of (electromagnetic) spectrum

(Lang 2014).


The computation of partial and total density of states is complementary to calculate

accessible number of positions to occupy. From Figure 6.4 (a) and 6.4 (b), it can be observed

that total interval of energy encompasses region of fermi level EF from (EF – 20eV) up to (EF

+ 15eV). However, appearance of sharp peak due to O 2s state is observed at (EF – 20eV).

Then 5p states of Pa, dominate the energy interval at (EF - 15eV). After fermi level to -4.6 eV

(the upper valence band region) dominates via O-2p states whereas Pa-4f states occupies


Page | 176

conduction band till 6 eV approximately. As a result, formation of conduction band is

because of mixed of X: 3d and Pa: d states.


In crystalline solids, the nature of bonding can be analyzed by contour maps of electron

density (Gelatt 1983). From (110) plane charge densities between X, Pa, O atoms can be

observed as shown in Figure 6.6 and 6.7, for two as well as three dimensions respectively.

However, from view of (100) plane contour plots of just corresponding X and O ions can be

seen as displayed in Figure 6.5 and 6.8, for two as well as three dimensions respectively. It

can be observed that between X and O ions the nature of bonding is ionic, since of the fact

that very low hybridization occurs between X-O ions. However, there is large sharing

between Pa and O ion, which depicts strong covalent character and it is well known reality

that hybridization between cations and anions causes covalent nature of respective

compounds (Erum and Iqbal, February 2017).

6.2.5 Optical characteristics

This part of the chapter is dedicated to compute, the optical properties of XPaO3 compounds

by means of Trans-Blaha modified Becke–Johnson (TB-mBJ) potential. The basic optical

parameter covers imaginary part of dielectric function Ԑ2(ω), real part of dielectric function

Ԑ1(ω), extinction coefficient k(w), absorption coefficient α(ω), reflectivity R (ω), optical

conductivity σ(ω), energy loss function L(ω), and effective number of electrons neff via sum

rules.


Here complex part of dielectric function can be written as (Fox 2001):


Page | 177

Ԑ(𝜔) = Ԑ1(𝜔) + 𝑖Ԑ2(𝜔) (6.16)

First part of equation denotes real part Ԑ1(ω) while second part of above equation represents

imaginary part Ԑ2(ω). The Ԑ1(ω) can be used to express comprehensive response of a

compound via following equation (Wooten 1972):

Ԑ2(𝜔) = (4𝜋2𝑒2

𝑚2𝜔2) ∑ ∫ ⟨𝑖|𝑀|𝑗⟩2𝑓𝑖(1 − 𝑓𝑖)𝛿(𝐸ј,𝑘 − 𝐸𝑖,𝑘 − 𝜔)

𝑘𝑖,𝑗 𝑑3𝑘. (6.17)

In above equation M, 𝑓𝑖, 𝐸𝑖,𝑘 and 𝐸ј,𝑘 are diploe matrix, i-th state fermi distribution function,

and i-th as well as j-th state energy of electron correspondingly. Ԑ2(ω) peaks represents

perfect outline of band structure and DOS respectively. As shown in Figure 6.9 (a) critical

point of threshold occurs at about 3.99 eV and 3.81 eV for RbPaO3 and KPaO3 accordingly.

However, occurrence of principle peak occurs due to Pa-4f at about 6 eV for XPaO3, which

refers towards transition between states of valence band (occupied) to conduction band

(unoccupied). After that till 15 eV diversified peaks are observed due to overlapping of states

between X-3d and few d plus p states of Pa and O respectively.

The polarization phenomenon can be observed by Ԑ1(ω) through subsequent relation (Brik

2011):

Ԑ1(𝜔) = 1 +2

𝜋𝑃 ∫

ὠԐ2(ὠ)𝑑ὠ

ὠ2−ὠ2

ⱷ

0 , (6.18)

Here principle value of corresponding integral is denoted by P. It can be observed from

Figure 6.9 (b) that the dielectric function Ԑ1(0), the static part, noticed at about 4.5 eV for

both compounds. The Ԑ1(ω) curves starts increasing and achieves their extreme rate nearby

3.7 eV for XPaO3.


Page | 178

6.2.5.2 Optical conductivity and energy loss calculations


function for the electromagnetic spectrum such as optical conductivity σ(ω), and electron

energy loss spectrum L(ω) which can be determined by following expressions (Fox 2001):


𝐸02 (6.19)

𝐿(𝜔) = 𝐼𝑚 (−1

Ԑ(𝜔)) (6.20)

The journey of optical conductivity σ(ω) reveals that it starts at approximately 2 eV with

minor mounting peaks, then ultimately attain highest peak at about 6 eV for XPaO3 as

displayed in Figure 6.9 (c). A prominent fact can be observed from σ(ω) that it shifts towards

high energy region as compound traversed from K to Rb due to increase in band width of the

corresponding compound. The characteristic contribution of plasma resonance frequency ωp

can be depicted with the help of energy loss function L(ω) (Murtaza and Ahmad 2012).

Figure 6.9 (d) reveals for both compounds a prominent crest is detected at about 10 eV.

6.2.5.3 Refractive index and reflectivity calculations

The key optical parameters such as refractive index n(ω), and reflectivity R can be

determined by following expressions (Wooten 1972):

𝑛(𝜔) = 1

√2[√Є1(𝜔)2 + Є2(𝜔)2 + Є1(𝜔) ]

1

2 (6.21)

𝑅 = |𝑛−1

𝑛+1|

2

(6.22)


Page | 179

In Figure 6.9 (e) and 6.9 (f), the calculated spectrum of refractive index as well as reflectivity

are shown. The refractive index (static parts) n (0) for KPaO3 and RbPaO3 are at value of

2.07 and 2.09 respectively. The maxima in values of refractive index attains at 4 eV (2.81)

and 4.1eV (2.82) for KPaO3 and RbPaO3 respectively. However, Figure 6.9 (f) depicts that

XPaO3 initiates high reflection and acquires maxima in 23-27 eV range. So, in this specific

range, materials show high transparency.

6.2.5.4 Absorption coefficient calculations

The absorption coefficient α (ω) can be calculated via following relations (Harmel et al.,

2015):

𝛼(𝜔) =4𝜋ƙ(𝜔)

𝜆 (6.23)

From the plot of Figure 6.9 (g) (plot of absorption coefficient α (ω)), it can be observed that

XPaO3 starts absorption phenomenon at about 4.25 eV. The particular point of threshold is in

similar resemblance with the behavior of bandgap trends, along with effective absorption

occurs in 21-25 eV range. After the incent of highest peak diversified small and large peaks

are observed till 30 eV. The wide range of absorption suggests use of XPaO3 in absorption

purpose applications, characteristically nearby 23 eV.





0ὠ 𝑑ὠ (6.24)


Page | 180

Finally, sum rule is evaluated to estimate effective valence electron, which are available to

utilize inter and intra band transitions, appear at about 3.5 eV as figured in Figure 6.9 (h).

The trend-line follow slow, then linear rise in effective number of electrons. Then there

occurs a sharp peak in a region of 12-14 eV, showing saturation of electrons.


Page | 181

Figure 6.1(a): Cubic crystal structure of KPaO3


Page | 182

Figure 6.1(b): Cubic crystal structure of RbPaO3


Page | 183


For KPaO3.

Volume (a.u)3

To

tal

ener

gy

(R

y)


Page | 184


for RbPaO3.

Volume (a.u)3

To

tal

ener

gy

(R

y)


Page | 185

Figure 6.3: Electronic energy dispersion curves for (a) KPaO3 and (b) RbPaO3 along some

high symmetry directions in the Brillouin zone (BZ) within modified Becke-Johnson (mBJ)

Potential.

En

ergy

(eV

)


Page | 186

Figure 6.4(a): The Density of States for KPaO3 by mBJ potential

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 187

Figure 6.4(b): The Density of States for RbPaO3 by mBJ potential

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 188


for KPaO3 in (100) plane.


Page | 189


for RbPaO3 in (100) plane.


Page | 190


for KPaO3 in (110) plane.


Page | 191


for RbPaO3 in (110) plane.


Page | 192


(b) RbPaO3.


(b) RbPaO3.


Page | 193

Figure 6.9(a): Calculated imaginary part Ԑ2(ω) of the dielectric function for

XPaO3 (K, Rb) compounds.


Page | 194

Figure 6.9 (b): Calculated real part Ԑ1(ω) of the dielectric function for

XPaO3 (K, Rb) compounds.


Page | 195

Figure 6.9 (c): Calculated conductivity σ(ω) for XPaO3 (K, Rb) compounds.


Page | 196

Figure 6.9 (d): Calculated energy loss function L (ω) for XPaO3 (K,Rb) compounds.


Page | 197

Figure 6.9 (e): Refractive index n (ω) as a function of energy for XPaO3 (X=K, Rb)

compounds.


Page | 198

Figure 6.9 (f): Reflectivity R (ω) as a function of energy for XPaO3 (X=K, Rb)

compounds.


Page | 199

Figure 6.9 (g): Absorption coefficient α (ω) as a function of energy for XPaO3 (X=K, Rb)

compounds


Page | 200

Figure 6.9 (h): Calculated sum rule (Neff) for XPaO3 (K, Rb) compounds.


Page | 201

Table 6.1: Comparisons of calculated values of bond length, equilibrium lattice constant (ao

in Ǻ), ground state energy (Eo in RY), bulk modulus (Boin GPa) and its pressure derivative

(BP) with experimental and other theoretical results for XPaO3 (X = K, Rb) compounds.

a) (Muller & Roy 1974) b) (Morss et al., 2010) c) (Jain et al., 2013) (Experimental Work), d) (Majid and Lee

2010), e) (Verma et al.,2008), f) (Jiang 2006), g) (Moreira and Dias 2006) (Other theoretical work)

Compound Present work

——————— GGA

Present work

—————

——

LDA

Present Analytical

work

———————

I.R method

Present Analytical

work

———————

V.J method

Experimental

work

Other

theoretical

work

KPaO3

ao (Å)

4.37 4.32 4.29 3.99 4.34a,4.36b,

4.38c 4.34d,4.32d, 4.18e,4.24f,

4.23g

Eo (Ry) -56260.653 -56260.713

Bo (GPa) 203.51 203.63

BP(GPa) 3.99 3.98

Bond-

lengths

K-O(Å) 2.51

K-Pa(Å) 3.23

Pa-O(Å) 1.81

RbPaO3

ao (Å)

4.42 4.35 4.32 4.05 4.37a,4.40b,

4.39c

4.36d,4.37d,

4.27e,4.26f,

4.26g

Eo (Ry) -60982.210 -60982.279

Bo (GPa) 146.58 146.65

BP(GPa) 4.122 4.291

Bond-

lengths

Rb-O(Å) 2.61

Rb-Pa(Å) 3.81

Pa-O(Å) 1.82


Page | 202

Table 6.2: Calculated values of tolerance factor for XPaO3 (X = K, Rb).

a (Goldschmidt 1926)

Table 6.3: Calculated values of elastic constants C11, C12, C44, for XPaO3 (X = K, Rb)

compounds.

Tolerance Factor Present work

——————————

Bond length formula

Present work

—————————

Goldschmidt’s formula

Other work

KPaO3 0.985 0.992 0.993a

RbPaO3 1.014 1.018 1.019a

Sr.No. Parameters KPaO3 RbPaO3

1 C11 (GPa) 432.182 311.413

2 C12 (GPa) 89.586 64.767

3 C44 (GPa) 34.816 46.377


Page | 203

Table 6.4: Calculated values of Bulk modulus B0, Reuss’s shear modulus GR, Voigt’s shear

modulus GV, Hill’s shear modulus GH, Young’s modulus Y and Pugh’s index of ductility

Bo/GH.


1 Bo(GPa) 203.785 146.982

2 GR(GPa) 51.101 61.802

3 Gv(GPa) 89.408 77.155

4 GH(GPa) 70.254 69.478

5 Y (GPa) 189.038 180.06

6 Bo/GH (GPa) 2.901 2.120


Page | 204

Table 6.5: Calculated values of Shear constant (C′), Cauchy pressure (C′′), Lame’s

coefficients (λ and μ), Anisotropy constant (A in GPa) and Poisson’s ratio (ѵ in GPa) and the

melting temperature (Tm in K) for XPaO3 (X= K, Rb) compounds.


1 C′ 171.29 123.32

2 C′′ 54.77 18.39

3 Ѵ (GPa) 0.35 0.30

4 A (GPa) 0.20 0.38

5 λ 156.77 100.04

6 μ 70.27 69.52

7 Tm(K) 3057 2529


Page | 205

Table 6.6: Band gap comparison of XPaO3 (X = K, Rb) at different symmetry points.

Compound Symmetry

Point

Bandgap

Type 𝐄𝐠

𝐌𝐁𝐉(eV) 𝐄𝐠

𝐋𝐃𝐀(eV) 𝐄𝐠𝐖𝐂−𝐆𝐆𝐀(eV) 𝐄𝐠

𝐏𝐁𝐄−𝐆𝐆𝐀(eV) 𝐄𝐠𝐏𝐁𝐄𝐬𝐨𝐥−𝐆𝐆𝐀(eV)

KPaO3 Γ-Γ Direct 3.60 3.41 3.51 3.49 3.45

R-R Direct 4.41 4.22 4.33 4.31 4.27

M-M Direct 4.45 4.32 4.42 4.39 4.36

X-X Direct 4.39 4.21 4.31 4.29 4.26

RbPaO3 Γ-Γ Direct 3.14 2.93 3.04 3.01 2.97

R-R Direct 3.38 3.27 3.34 3.31 3.29

M-M Direct 3.42 3.21 3.39 3.37 3.23

X-X Direct 3.21 2.98 3.18 3.12 3.10


Page | 206

6.3 Ab initio study of high dielectric constant BaMO3 (M=Pa, U)

oxide perovskite

In some earlier experimental studies BaUO3 has been quite successfully explored, in which

enthalpies of formation and calculation of molar Gibbs energy by solution calorimetric

analysis are evaluated (Williams et al. 1984 & Deacon-Smith 2015). In another, study

thermochemical and thermoelectric properties have been investigated successfully (Chen et

al., 1999 & Kurosaki 2001).

6.3.1 Structural parameters

The crystals of BaMO3 (M=Pa, U) perovskite oxides crystallizes in Pm-3m (no. 221) space

group having ideal cubic structure as shown in Figure 6.10 (a)-6.10 (b) for BaPaO3 and

BaUO3 respectively. The sites of Wyckoff coordinates are positioned at (0 0 0), (1/2,

1/2,1/2), (1/2, 1/2, 0) for Ba, X= (Pa, U) and O atoms respectively. In order to achieve,

optimum volume for BaMO3, total energy is determined at various volumes by utilizing

equation of state proposed by Murnaghan (Murnaghan 1944). The optimization plots of each

compound are displayed Figure 6.11 (a) and 6.11 (b) respectively. The calculated optimized

ground state structural parameters lattice constants (ao), bulk moduli (Bo), and ground state

energies (Eo) are tabulated in Table 6.7. The optimized lattice parameters are also compared

with experimental and previous theoretical results.

In this section, lattice constants are also evaluated by two well-known analytical methods

namely ionic radii and Verma Jindal method. The formula for ionic radii method can be

interpreted as (Clementi et al., 1963):

𝑎0 = 𝛼 + 𝛽 (𝑟𝐵𝑎 + 𝑟𝑂) + Ɣ(𝑟𝑋 + 𝑟𝑂) (6.25)


Page | 207

Where rBa, rX (x= Pa,U) and rO are ionic radii for Ba (1.61 A0), Pa (0.90 A0), U (0.89 A0) and O

(1.35 A0) respectively. Verma and Jindal model depends upon the average ionic radii and

number of valence electrons as follows (Verma et al., 2008):

𝑎0 = 𝐾(𝑉𝑋𝑉𝑃𝑎𝑉𝑂)𝑠𝑟𝑎𝑣 (6.26)

Where average ionic radii are denoted by rav and 2, 2, while 6 are number of valence

electrons for Ba (VBa), X (VX) as well as O (VO) correspondingly. While S (0.09) and K

(2.45) are constants for equation. It can be noticed from Table 6.7 that Density Functional

Theory (DFT) and analytical (ionic radii as well as Verma and Jindal method) calculation of

lattice constant reveals some deviation within 2-3 % as compared to experimental results.

Next, the rigidity of crystal structure can be determined in terms of bulk modulus which

depicts, BaPaO3 possesses lower bulk modulus than BaUO3. In addition to it, inverse relation

between bulk modulus as well as lattice constant can be observed which is in similar

accordance with KPaO3 and RbPaO3 perovskite compounds respectively (Erum and Iqbal,

February 2017).

Next part of this section is to determine extent of activation energy for oxygen migration

which can be clearly illustrated by calculating critical radius. In doping selection procedure

critical radius illustrates an imperative part by using following formula (Marezio et al., 1970

& Blundell 2001):

𝑟𝑐 =1.4141𝑟𝑀 −𝑟𝑜 −3.414𝑟𝐵𝑎+5.828(𝑟𝐵𝑎− 𝑟𝑜)2

2𝑟𝐵𝑎+0.828 𝑟𝑀+2.828𝑟𝑜 (6.27)

Table 6.7 explores that BaPaO3 have larger value of critical radius than BaPaO3. So, it can be

concluded that BaUO3 have larger migration energy. The information of important chemical

trends can be estimated by acknowledging bond lengths which are shown in Table 6.7.


Page | 208

Further tolerance factor is calculated to interpret structural symmetry in BaMO3 compounds

as follows (Goldschmidt 1926):

𝑡 =0.707<𝐵𝑎−𝑂>

<𝑋−𝑂> (6.28)

Where X, Ba, and O are average lengths of corresponding bonds as shown in Table 6.8.

Both BaPaO3 and BaUO3 satisfy good criteria of tolerance factor for cubic crystal which lies

within 0.93-1.02 range.

6.3.2 Electronic behavior

In this sub-section electronic properties of BaMO3 are investigated with the aid of electronic

band structure, Density of states (DOS) total as well as partial, and electronic charge density

distribution.


The bandgap results can be observed from Table 6.9 that are computed with four various

types of exchange and correlation schemes of ab-initio study. However, for this thesis

graphical attention is only paid to bandgap by WC-GGA approximation, keeping in mind its

accuracy for metals and semiconductors, as shown in Figure 6.12. It can be observed that

minima of conduction band minima (CBM) and maxima of valence band (VBM) occupies

(Γ-Γ) symmetry point revealing direct bandwidth of 4.05 eV for BaPaO3 and 3.98 eV for

BaUO3, however the trend of overall dispersion curves for bands remains identical. It can be

closely examined, that for both compounds conduction band traverses from fermi level while

valence band are well below the fermi level, which verifies metallic nature in BaXO3.

Furthermore, these materials have larger bandgap than 3.1 eV, so work well for ultraviolet

region of (electromagnetic) spectrum (Lang 2014).


Page | 209


The states of density distribution comprises on four different regions from (EF – 20 eV) to

(EF + 15 eV) as shown in Figure 6.13 (a) and 6.13 (b) for BaXO3. The core states are

dominated by X-6p states which occupies a region between -20 to -14.4 eV. After that a

distinct peak is observed at -15 eV due to Ba-4p state, however maxima of valence band due

to O-2p state occurs within the interval of -7 to 4.2 eV respectively. Then 5f states of X plays

a dominant role in overall physical properties of conduction band, which is further hybridized

with some of Ba (F and d states) respectively.

6.3.2.3 Electron density Calculations

A very impact full tool for explaining bonding nature of various atoms in crystalline solid is

through contour maps of charge density (Gelatt 1983). From view of (100) plane contour

plots of just corresponding Ba and O ions can be seen as displayed in Figure 6.14 (a-b) and

6.16, for two as well as three dimensions correspondingly. However, from (110) plane charge

densities between Ba, X, O atoms can be observed as shown in Figure 6.15 (a-b) and 6.17,

for two as well as three dimensions respectively. The spherical charge distribution is

observed for Ba and O atoms, without any overlapping, which helps to justify ionic nature

bonding in BaXO3. Similar ionic nature is observed for Ba and O ions due to low extent of

hybridization between Ba as well as O ions is observed. This similar bonding nature for other

perovskites are also observed by Murtaza and his fellows (Murtaza et al., 2011).

6.3.3 Optical characteristics

In this subsection, fundamental and derived optical responses for BaMO3 (X= Pa, U) are

calculated by WC-GGA approximation.


Page | 210


The complex part of dielectric function Ԑ (ω), provides basis for the calculation of all

fundamental optical responses. It can be depicted from Figure 6.18 (a) of Ԑ2(ω) that threshold

point occurs at about origin. Next the transition from unoccupied conduction band states to

occupied valence band states causes a major peak for both compounds which is located

approximately at 0.3-0.4 eV. The existence of these peaks is due to 5f states of X (X= Pa, U)

and these peaks imparts a crucial role in overall internal response of BaMO3. Then till 14 eV,

the steady peaks are observed. From Ԑ2(ω), another important phenomenon of strong

absorption (within 0 to 2 eV) can be observed, which might be due to induced electric field,

that results in large collective excitation of effective mass in the interfaces.

The real part of dielectric function Ԑ1(ω) helps to analyze polarizability of a given material

(Brik 2011). It can be analyzed from Figure 5.18 (b) that the peak for Ԑ1(0) (static dielectric

constant) is 40 and 72 for BaPaO3 and BaUO3 respectively.

The corresponding high value of static dielectric constant in both compounds classifies their

high degree of miniaturization in them. After that a sharp decrease in Ԑ1(ω) curve of BaXO3

is observed, which eventually attains a lowest value approximately at 0.3 - 0.4 eV. As a

whole, till 14 eV a narrow bandgap semi-conductive nature is observed.

6.3.3.2 Optical conductivity calculations

The conduction phenomenon can be interpreted from the plot of optical conductivity σ(ω) as

shown in Figure 6.18 (c) that the phenomenon of optical conduction initiates from origin and

attains maximum position approximately at 0.3-0.4 eV, followed by certain decrease in

oscillations.


Page | 211

6.3.3.3 Refractive index and reflectivity calculations

Next the plot of refractive index n(ω) and reflectivity R are calculated. Further, it can also be

revealed from the curve of refractive index n (ω) that it possesses a close relationship with

the trend of Ԑ1(ω). It can be observed from Figure 6.18 (d), n (0) lies at 6.3 for BaPaO3 and

9.0 for BaUO3 respectively. In high energy region curve of n (ω) starts vanishing which

ultimately reflects that studied materials loss transparency beyond certain energy limit and

absorbs photon in high energy region. The reflectivity spectrum, R(ω) demonstrates that in

accordance with Ԑ2(ω), BaXO3 starts reflecting highly from the origin which is less for

BaPaO3 than BaUO3 as displayed in Figure 6.18 (e). The R(ω) spectrum curve suffers trivial

variations followed by optimum reflectivity peaks within region of 7.5 to 8.0 eV respectively.

As per particular reflecting properties of both materials they can be classified as highly

transparent materials in infrared region of electromagnetic spectrum.


At the end, sum rule is assessed in terms of effective number of valence electrons per unit

cell (Fox 2001). The curves of sum rules deliver distinguished peak at about 2.0 eV as shown

in Figure 6.18 (f). Then the interband transition of electrons increases slowly that ultimately

saturates at approximately 12-14 eV range respectively.


Page | 212

Figure 6.10 (a): Cubic crystal structure of BaPaO3


Page | 213

Figure 6.10 (b): Cubic crystal structure of BaUO3


Page | 214

Figure 5.3: Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u)3) for

KPaO3.

Volume (a.u.)3

En

ergy

(R

y)

Figure 6.11(a): Variations of total energy (E, in Ry) with unit cell volume (V, in

(a.u)3) for BaPaO3.


Page | 215

Volume (a.u.)3

En

ergy

(R

y)

Figure 6.11(b): Variations of total energy (E, in Ry) with unit cell volume (V, in

(a.u)3) for BaUO3


Page | 216

En

ergy

(eV

)

Figure 6.12: Electronic energy dispersion curves for (a) BaPaO3 and (b) BaUO3 along

some high symmetry directions in the Brillouin zone (BZ) within WC-GGA

approximation.


Page | 217

Figure 6.13 (a): The Density of States for BaPaO3 by WC-GGA approximation.

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 218

Figure 6.13 (b): The Density of States for BaUO3 by WC-GGA approximation.

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 219

Figure 6.14 (a): Calculated total two and three-dimensional electronic charge densities for

BaPaO3 in (100) plane.


Page | 220

Figure 6.14 (b): Calculated total two and three-dimensional electronic charge densities for

BaUO3 in (100) plane.


Page | 221

Figure 6.15 (a): Calculated total two and three-dimensional electronic charge densities for

BaPaO3 in (110) plane.


Page | 222

Figure 6.15 (b): Calculated total two and three-dimensional electronic charge densities for

BaUO3 in (110) plane.


Page | 223


(b) BaUO3.


(b) BaUO3.


Page | 224

Figure 6.18 (a): Calculated imaginary part Ԑ2 (ω) of the dielectric function for

BaXO3 (Pa, U) compounds.


Page | 225

Figure 6.18 (b): Calculated real part Ԑ1 (ω) of the dielectric function for

BaXO3 (Pa, U) compounds.


Page | 226

Figure 6.18 (c): Calculated conductivity σ (ω) for BaXO3 (X=Pa, U) compounds.


Page | 227

Figure 6.18 (d): Refractive index n (ω) as a function of energy for BaXO3 (X=Pa, U)

compounds.


Page | 228

Figure 6.18 (e): Reflectivity R (ω) as a function of energy for BaXO3 (Pa,U) compounds.


Page | 229

Figure 6.18 (f): Calculated sum rule (Neff) for BaXO3 (Pa,U) compounds.


Page | 230

Table 6.7: Comparison of calculated equilibrium lattice constants ao (in Ǻ), ground state

energies Eo (in Ry), bulk modulus Bo (in GPa), its pressure derivative BP (in GPa), and bond

lengths with experimental and other theoretical values of BaXO3 (X = Pa, U) compounds.

a (Morss 2010), b (Radiochemie and Hochschule 1965), c (Majid and Lee 2010) (Experimental Work); d (Verma 2008), e (Jiang 2006), f (Moreira and Dias 2007), g (Muller and Roy 1974) h (Yamanaka 1999),

I (Hinatsu 1993) (Other theoretical work)


———————

GGA

Present work

――—————

LDA

Present

Analytical work

———————

I.R method

Present

Analytical work

———————

V.J method

Experimental

work Other

theoretical

work

BaPaO3

ao (Å)

4.47 4.44 4.43 4.21 4.45a, 4.49b 4.45c,4.42c

4.47d,4.39e

4.38f

Eo (Ry) -71329.561 -71329.585

Bo (GPa) 126.91 126.57

BP(GPa) 4.33 4.31

rc (Å)

0.963

Bond-

lengths

Ba-O(Å) 2.89

Ba-Pa(Å) 3.65

Pa-O(Å) 1.98

BaUO3

ao (Å)

4.39 4.37 4.41 4.02 4.38g,

4.40h, 4.41I

4.41c,4.40c,

4.45d,

4.38e

4.36f

Eo (Ry) -72853.611 -72853.634

Bo (GPa) 140.61 140.49 138.11f

BP(GPa) 3.94 3.87

rc (Å)

0.950

Bond-

lengths

Ba-O(Å) 3.09

Ba-U(Å) 3.77

U-O(Å) 2.18 2.19I


Page | 231

Table 6.8: Calculated tolerance factor for BaXO3 (X = Pa, U).

Tolerance

Factor Present work

—————————— Bond length formula

Present work

—————————— Goldschmidt’s formula

Other work

BaPaO3 0.932 0.928 0.930a

BaUO3 0.951 0.962 0.934a

a) (Goldschmidt 1926)

Table 6.9: Band gap comparison of BaXO3 (X = Pa, U) at different symmetry points.

Compound Symmetry

Point

Bandgap

Type 𝐄𝐠

𝐖𝐂−𝐆𝐆𝐀(eV) 𝐄𝐠𝐋𝐃𝐀(eV) 𝐄𝐠

𝐏𝐁𝐄−𝐆𝐆𝐀(eV) 𝐄𝐠𝐏𝐛𝐄𝐬𝐨𝐥−𝐆𝐆𝐀(eV)

BaPaO3 Γ-Γ Direct 4.20 4.15 4.18 4.17

R-R Direct 4.40 4.31 4.37 4.34

M-M Direct 4.51 4.42 4.48 4.46

X-X Direct 4.37 4.26 4.33 4.31

BaUO3 Γ-Γ Direct 4.01 3.91 3.97 3.94

R-R Direct 4.38 4.29 4.33 4.31

M-M Direct 4.49 4.41 4.46 4.44

X-X Direct 4.35 4.26 4.30 4.29


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6.4 Conclusion

In this chapter, systematic first principles calculation of four actinoid based oxide perovskites

(KPaO3, RbPaO3, BaPaO3 and BaUO3) have been carried out successfully.

Comprehensive results of structural, elastic, mechanical and opto-electronic properties of

protactinium based group 1A compounds (section 6.2), reveals that value of lattice constants

increases, as cation shift from potassium to Rubidium, while value of bulk modulus

decreases, that can be attributed to higher extent of atomic radii of Rubidium. These

elastically and mechanically stable compounds own less resistance for shear distortion in

comparison with resistance to unidirectional compression, whereas flexible and covalent

behaviors are dominated. Furthermore, explicit influence of electronic states and band

dispersion curves reveals that both compounds are direct bandgap (Γ-Γ) semiconductors. The

fundamental optical aspects in high frequency regions, reveals extensive extent of absorption

and reflection. So, by shielding radioactivity, these beneficial features can make these

compounds suitable for implementing them in flexible opto-electronic applications.

Section 6.3 delivers unique theoretical strategy to calculate detailed opto-electronic trends of

Barium based actinoid perovskite oxides (BaPaO3 and BaUO3), via various exchange and

correlation schemes. Electronic aspects authenticate metallic nature with mixed ionic and

covalent bonding. However optical particulars such as prominent value of static dielectric

constant, recommends significant role of these materials in implementing them in micro as

well as nano-scale devices.

In summary, these actinoid based oxide perovskites, have valuable features in one aspect or

another so by extensive experimental research via properly handling their radioactive nature,


Page | 233

versatile outcomes can be achieved for their possible technological benefits. Furthermore,

this investigation can be upgraded if the two materials can be doped with another magnetic

semiconductor element to make BaPaO3 and BaUO3 semiconductor compounds.

Chapter 7 Results and discussion ΙΙΙ

Page | 234

Chapter 7: Results and discussion ΙΙΙ;

Band profiles and magneto-optic properties of KXF3 (X=

V,Fe,Co,Ni)

“In physics you don’t have to go

around making trouble for yourself

nature does it for you”

Frank Wilczek

7.1 Introduction

Half metallic compounds are equally important to meet with the needs of modern technology.

They are cheap and efficient alternatives which at fermi level retain their one spin direction.

The general interest area of investigation is concerned with their utilization in the field of

magnetoresistive sensors, and magnetoresistive memory devices (Ali et al., 2015; Narayan

and Ramaseshan 1978 & Pisarev et al., 1969). The dependence of spintronic mechanism

entirely based on charge of an electron and its spin, as a result they have tendency to deliver

gigabit memory devices (Ohno et al., 1996 & Rao and Raveau 1995) The perovskite

symmetry composed of KXF3 where X = V, Fe, Co, Ni are in focus for versatile aspects like

colossal magneto resistivity, half-metallicity, high temperature superconductivity,

ferroelectricity, semi-conductivity, piezoelectricity, thermoelectricity, phase separation,

catalytic activity, photoluminescence, and phenomenon of metal-insulator transition.

Experimental studies (Manivannan et al., 2008 & Lee et al., 2003) confirm the ideal cubic

crystal structure of KXF3. However experimental investigation on KFeF3 by Ito and

Morimoto (Ito and Morimoto 1977) confirms the magnet phase transitions between 4.2 and

300 K while suggesting spin alignment lower than the Curie temperature. In another


Page | 235

experimental investigation Shafer and their fellows (Shafer et al., 1967) confirms that Co 2+

is the only magnetic ion of transition metal for rubidium based ferrimagnetic system

RbMgF3- RbCoF3. In a subsequent theoretical investigation on KFeF3 and KCoF3 (Punkkinen

1999), explores d-states correlation phenomenon but still there is lack of detailed theoretical

investigation on these fluoroperovskites.

Hence to attain eventual technological application, this section of the thesis is dedicated to

cover lack of previous studies on structural analysis, thermal stability, and magneto-opto-

electronic properties of KXF3. The significant findings of study at ambient pressure are

summarized in terms of electronic structure, magnetic, mechanical, thermal, and optical

properties of KXF3.

7.2 Structural stability

The crystallization of KXF3 occurs in cubic structure. The position of K, X, and F ions are

located at Wyckoff coordinates of (1a,1b,3c) at (0,0,0), (0.5,0.5,0.5), and (0,0.5,0.5)

respectively. To compute structural properties, the total ground state energy is determined at

various unit cell volumes. The ultimate lattice parameters are calculated by employing first

and third order equation of state (Murnaghan 1944), to produce energy versus volume curve

as shown in Figure 7.1 (a-d). As a result, spin polarized structural parameters (ground state)

for example corresponding lattice parameters with their derivatives are tabulated in Table

7.1, with exchange correlation LSDA and GGA approximation. All above mentioned

parameters agreed well with previous theoretical and available experimental results.

Furthermore, decrease in lattice constant is observed as transition metal in KXF3 changes

from V to Ni, in accordance with the decrease in atomic size.


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7.2.1 Analytical calculations of lattice constants

To validate the results calculated by ab-initio study, two analytical methods are also

employed namely ionic radii, Verma and Jindal (V.J) method. Ionic radii method is given by

(Clementi et al., 1963):

a0 = α + β (rK + rF) + Ɣ(rX + rF) (7.1)

Where α, β, and Ɣ are the equation constants having values 0.06741, 0.4905, and 1.2921

respectively while ionic radii of K, X (X= V,Fe,Co,Ni) and F are (1.38 Ǻ), (0.59 Ǻ, 0.66 Ǻ,

0.75 Ǻ, 0.69 Ǻ) and (1.33 Ǻ) respectively (Ubic 2007). The subsequent relation for Verma

and Jindal model entirely based upon ionic radii and number of valence electrons (Verma et

al., 2008):

𝑎0 = 𝐾(𝑉𝐾𝑉𝑋𝑉𝐹)𝑠𝑟𝑎𝑣 (7.2)

Here K and S are equation constants for cubic system with values of 2.45 and 0.09

respectively and average ionic radii is denoted by rav. The lattice constants calculated by

equation 6.1 and 6.2 possesses reasonable discrepancy as compared to ab-initio calculation of

lattice constants. This deviation is due to several reasons: Firstly, these empirical relations

depend upon average ionic radii. Secondly, the empirical relation for calculating lattice

constant by V.J method depends on number of valence electrons of each atom and thirdly,

the error might be due to constants α, β, Ɣ, K (2.45) and S (0.09) which are involved in these

empirical relations. Therefore, it can be concluded that in these empirical relations a lot of

improvement is required, to attain lattice constant values near to experimental one. However,

for this study, DFT versus experimental results are in reasonable agreement with each other.


Page | 237

7.2.2 Tolerance factor calculations

The nature of chemical bonding can be explored by means of bond lengths. It can be

observed from Table 7.1, between cation-anion bond lengths K versus X decreases from

Vanadium, V to Nickel, Ni because of reduced atomic size of Ni. Similar behavior is also

observed for X and F accordingly. Next, we evaluate the criteria of tolerance factor by using

bond length for KXF3 compounds (Goldschmidt 1926).

𝑡 =0.707<𝐾−𝐹>

<𝑋−𝐹> (7.3)

Here average bond length between K, X and F is denoted by <K-F> and <X-F> respectively.

All compounds fulfill good tolerance factor criteria between 0.95 and 1.04 for cubic

perovskites.

7.3 Elastic properties

Elastic properties give reliable information regarding to mechanical behavior of crystalline

solids. In solids, the valuable parameters concerning stability of structure as well as binding

characteristics can be obtained with the help of elastic constants. Another important

contribution of these parameter is to differentiate phenomenon of elasticity from the

mechanism of plasticity.

7.3.1 Calculation of elastic constants

There are three independents elastic constants for cubic systems, denoted by C11, C12, and

C44. Several mechanical parameters can be evaluated from these elastic constants which

includes reuss’s modulus, hill’s modulus, young’s modulus, bulk modulus, voigt’s modulus,

shear modulus, poisson’s ratio, melting temperature and elastic stiffness coefficients. The

details of calculated elastic constants are tabulated in Table 7.3. All these calculations are


Page | 238

carried out by using Charpin’s method (Charpin 2001). In continuation with this, the

calculated elastic constant of KMF3 obey traditional mechanical and cubic stability condition

at P = 0 GPa, which can be mentioned by the following relation C11- C12 > 0, C11 > 0, C44 >

0, C11 + 2C12 > 0 (mechanical stability condition), and C12 < B < C11 (cubic stability

condition) respectively (Erum and Iqbal 2016). The detailed explanation of C11 elastic

constant can be found in chapter 5, section 5.2, that is lowest for KNiF3 and it is highest for

KVF3, which validates strong resistance of KNiF3 towards unidirectional compression. The

elasticity in shape can be well explored by elastic constant of C44.The present calculations

reveal that KMF3 retains more resistance for shear deformation C44 in comparison with

unidirectional compression of C11 because the value of C11 is approximately 65.04%,

66.29%, 69.32%, and 88.93% than C44 for KNiF3, KCoF3, KFeF3, and KVF3 respectively.

Meziani and Belkhir (Meziani and Belkhir 2012) receive similar trends for elastic constants

C11 and C44. Likewise, these values are also compared with existing experiment results

(Dovesi et al., 1997 & Aleksiejuk et al., 1975). In general, thermal expansion have a tendency

to lessen values of elastic constants at finite temperature, as confirmed by experimental as

well as theoretical investigation of some perovskites for example KMgF3, KZnF3, and

CsCdF3 (Patel et al., 1976 & Sugano 1970).

7.4 Mechanical properties

The purpose of this section is to compute polycrystalline mechanical aspects by utilizing data

information from elastic constants. The evaluated parameters include detailed elastic moduli,

Poisson’s ratio, coefficients for elastic stiffness, as well as extent of melting are calculated in

accordance with some proposed formulas as stated earlier in section 5.2 of chapter 5

respectively.


Page | 239

7.4.1 Calculation of elastic moduli

Hardness of material can be determined through value of both bulk and shear modulus. The

expression of bulk modulus B are mention from following equation (Kittel 2005):

𝑩 =𝟏

𝟑 (𝑪𝟏𝟏 + 𝟐𝑪𝟏𝟐) (7.4)

Another important mechanical aspect which is shear modulus G, can be calculated by using

following expressions (Shafiq et al., 2015):

𝑮𝑽 = 𝟏

𝟓(𝑪𝟏𝟏 − 𝑪𝟏𝟐 + 𝟑𝑪𝟒𝟒) (7.5)

𝑮𝑹 =𝟓𝑪𝟒𝟒(𝑪𝟏𝟏−𝑪𝟏𝟐)

𝟒𝑪𝟒𝟒+𝟑(𝑪𝟏𝟏−𝑪𝟏𝟐) (7.6)


𝟐 (7.7)

And the response of a material towards linear strain can be well defined by Young’s modulus

via following relation (Jenkins & Khanna 2005):

𝒚 =𝟗𝑩𝑮

(𝟑𝑩+𝑮) (7.8)

In this study bulk modulus is calculated by two methods one is from elastic constant and

another by Equation of State by Murnaghan (Murnaghan 1944). These values are in

reasonable agreement with each other, which depicts validity of both methods. From the

trend of Young’s modulus (Y), Bulk modulus (B0), Reuss’s shear modulus (GR), Voigt’s

shear modulus (GV), and Hill’s shear modulus (GH), it can be inferred that KCoF3 is stiffer

and have more tendency of charge transfer as compared to KVF3, KFeF3, and KNiF3 hence

stiffer as compared to rest of the compounds which is tabulated in Table 7.4.


Page | 240

7.4.2 Calculation of Cauchy’s pressure, B/G and Poisson’s ratio

Another important parameter which used to describe angular characteristics in atomic

bonding is Cauchy’s pressure. It can be well defined as follows (Brik 2011):

𝑪′′ = 𝑪𝟏𝟐 − 𝑪𝟒𝟒 (7.9)

Negative values of Cauchy’s pressure indicate high angular characteristics in bonding

whereas compound with positive Cauchy’s pressure tend to form metallic bond in nature. In

addition to it, the ratio of compression to relative expansion can be expressed in terms of

Poisson’s ratio as follows (Pettifor 1992):

ѵ =(𝟑𝑩−𝟐𝑮)

𝟐(𝟑𝑩+𝑮) (7.10)

Haines and their fellows (Haines et al., 2001) suggested that for ionic material it is less than

0.1. Hence it can be inferred from Table 7.5 that 𝐶′′ > 0, B/G > 1.75, and ѵ > 0.26 implies

that except KFeF3, the rest of the compound, contains high directional bonding and are

ductile.

7.4.3 Calculation of shear constant and elastic anisotropy

To further distinguish between, ionic or covalent behavior the present analysis is extended to

evaluate shear constant (Nakamura 1995), as shown in Table 7.5. It can be expressed as:

𝑪′ =𝟏

𝟐(𝑪𝟏𝟏 − 𝑪𝟏𝟐) (7.11)

High value of shear constant reveals that KXF3 contains dominant covalent behavior. The

isotropic behavior of any crystal in manufacturing disciplines can be estimated through

elastic anisotropy parameter A. The relation can be well defined as follows (Jamal et al.,

2016):


Page | 241

𝑨 =𝟐𝑪𝟒𝟒

(𝑪𝟏𝟏−𝑪𝟏𝟐) (7.12)

The crystals can be completely categorized as isotropic if the value of elastic anisotropic

parameter A, equals to unity and any deviation from this value reveals extent of elastic

anisotropy in the given material (Jenkins & Khanna 2005). It can be observed from Table

7.5, that the values of A for KMF3 is less than or greater than unity, which clearly indicates

anisotropic behavior of these compounds.

7.4.4 Calculation of Kleinman’s parameter and Lame’s constant

Another significant parameter which was introduced by Kleinman, used to quantify

material’s behavior towards bond stretching or bond bending; if minimum bond stretching

then Kleinman parameter ξ=1 but If compound possess minutest value for bond bending then

ξ=0 as (Kleinman 1962):

𝝃 =𝑪𝟏𝟏+𝟖𝑪𝟏𝟐

𝟕𝑪𝟏𝟏−𝟐𝑪𝟏𝟐 (7.13)

The range of value lie between 0.37-0.64 as compound changes from KVF3 to KNiF3 which

shows that in KVF3 bond bending is prevalent while bond stretching is dominant in KNiF3

fluoroperovskite.

Next, the study is related with stress to strain by acknowledging two important constants

namely first λ and second μ Lame’s constant. The expression for these constants can be

derived from various mechanical parameters in the following form (Alouani 1991):

𝝀 = 𝒀ѵ

(𝟏+ѵ)(𝟏−𝟐ѵ) (7.14)

𝝁 =𝒀

𝟐(𝟏+ѵ) (7.15)

These constants are in direct relation with the value of Y. The calculated values not fulfill the

specific criteria 𝜆 = 𝐶12 𝑎𝑛𝑑 𝜇 = 𝐶′ for isotropic material. So KMF3 is a class of anisotropic


Page | 242

compounds which is in reasonable agreement with the calculated value of anisotropy

parameters. Furthermore, both of these constants confirm the extent of shear stiffness that is

in accordance with previous work of RbNiF3 (Mubarak and Saleh 2015).

7.5 Thermal properties (Calculation of the Debye temperature)

Several thermodynamic parameters like heat capacities, thermal conductivity, as well as

melting temperature can be quantified with the help of an important quantity well known as

Debye temperature (θD) or Debye cut-off frequency. In general, θD can be calculated by two

easy to access methods namely specific heat measurement method and elastic constant

method. In this study, elastic constant method is employed to extract θD and their related

quantities. The standard method for calculating Debye temperature (θD) and associated

parameters from the elastic constants is derived by Anderson (Anderson 1963), which

expresses the link between θD and the mean elastic wave velocity (Wachter et al., 2001) as:

𝛳𝐷 = ℎ

𝑘𝐵[

3𝑛

4𝜋𝑉𝑎]

1

3ѵ𝑚 (7.16)

where h is Planck’s constant, kB is Boltzmann’s constant, Va is the atomic volume, and n is

the number of atoms per unit volume while the average propagation velocity of the acoustic

wave is given by (Anderson 1963):

ѵ𝑚 = [1

3(

2

ѵ𝑡3 +

1

ѵ𝑙3)]

−1

3 (7.17)

Furthermore, the propagation velocities of the transverse and longitudinal acoustic waves of

a polycrystalline material can be obtained by the following relations (Schreiber and Anderson

1973):


Page | 243

ѵ𝑙 = (3𝐵+4𝐺

3𝜌)

1

2 (7.18)

ѵ𝑡 = (𝐺

𝜌)

1

2 (7.19)

Where B is the bulk modulus, G is the shear modulus and ρ is the density of the material. It

can be observed from Table 7.6 that resultant calculated quantities are in reasonable

agreement with previously available theoretical results on RbFeF3 and RbNiF3 respectively

but there is some deviation within present and experimentally calculated previous results, due

to the fact that these calculations are done at 0 K while experimental investigation are carried

out at finite temperature. Furthermore, according to Sakho and their fellows (Sakho et al.,

2006), at 0 K density of compound is low and possesses inverse relation with regard to θD. As

far as, the difference in melting temperature (± 300 K) is concerned, it endorses due to

various schemes of exchange-correlation potential, it will eventually lead to miscalculations

between total energies of atoms. The calculations regarding to longitudinal and transverse

sound (υt and υl) velocities reveals that with respect to rest of compound (KFeF3, KCoF3, and

KNiF3) the fluoroperovskite KVF3 possesses higher values for sound velocity. In

continuation with this, KVF3 have highest value of θD, because θD have direct relation with

average sound velocities. The results of melting temperature are in similar accordance with

above conclusions.

7.6 Electronic and magnetic properties

In this subsection, detailed magneto-electronic aspects have been explored for KXF3. The

electronic properties of KVF3, KFeF3, KCoF3, and KNiF3 fluoroperovskites are analyzed by

electronic band dispersion curves, detailed states for density while chemical nature of


Page | 244

bonding is interpreted by electron density contour maps. Furthermore, the magnetic

properties of KVF3, KFeF3, KCoF3, and KNiF3 are calculated by calculating various

magnetic moments.

7.6.1 Spin-dependent band structure calculations

The various form of LSDA, GGA approximations with mBJ (Tran and Blaha 2009),

potential is employed to calculate spin dependent electronic band structures. It can be

observed from Figure 7.2 (a-f), that by different exchange and correlation schemes, structures

of energy band profiles are almost similar with minor difference in gap details. It is evident

from plots that KVF3 as well as KCoF3 both fluoroperovskites behaves as narrow gap

semiconductor and narrow bandgap insulator for corresponding up and down spin channels

respectively. While KFeF3 as well as KNiF3 reveals full spin polarization, with half metallic

nature about fermi level, that is in similar accordance thru earlier work done by Naraya,

Manivannan and their fellows (Manivannan et al., 2008 & Narayan and Ramaseshan 1978).

Additionally, when cation traversed from V to Ni, decrease in hybridization is observed

between X and 2p states of F, which ultimately increase bandgap of the corresponding

compounds.

7.6.2 Spin-dependent Density of States (DOS) calculations

During this study we testify entire network of densities through various schemes such as

LSDA, GGA, and mBJ but there is small difference in detail so for the sake of precision and

to avoid repetition, only the pictorial outcomes with GGA are presented in Figure 7.3 (a-d).

The calculated DOS are determined within energy interval (EF – 10eV) to (EF + 10eV). The

principle contributing states in DOS includes K: 3s, 3p, 3d, F: 2s, 2p and X: 3s, 3p, 3d. The


Page | 245

fluorine F-2p states, induces some narrow peaks within -5 to -10 eV in valence band region.

The 3d states of transition metals occupies energy interval from (EF - 5eV) to (EF + 5eV).

However, in region above fermi level, the hybridized states K: 3d, 3p and 3s states are

responsible for the formation of conduction band. In general, as a whole, coulomb’s

repulsion between states X-3d as well as F-2p states is responsible to generate crystal fields,

as a result splitting will occur in terms of 3d as t2g and eg non-degenerate states (Zener 1951).

7.6.3 Spin-dependent electron density calculations

The contours of electronic charge density help to explore nature of bonding in solids with

crystalline characteristics (Hoffman 1988). The plots for (110) direction are displayed in

Figure 7.4 (a-d), for corresponding channels of up and down spin symmetries. For majority

spin channel, the X states are almost spherical and it can be associated with partially filled 3d

states of transition metal. Furthermore, ionic nature is observed for ions of K and F due to

reduced hybridization in between them. While in down spin channel modifications for X

states, alters its shape from spherical to dumbly, that can validate ionic illustration for 2p

states of F. In summary, it can be evaluated that covalent bonding is dominant between X-F

ions because of large charge sharing between them and this covalent nature depends upon pd-

hybridization among cation and anions.

7.6.4 Calculation of magnetic properties

The origin of magnetism can be attributed to existence of partly occupied shells of electrons

(Blundell 2001). In this section, the concept of magnetism is calculated in terms of total,

local, and interstitial magnetic moment. It can be observed from Table 7.7 that DFT

calculated results remains good accordance with available experimental as well as earlier


Page | 246

theoretical findings. For Potassium, values of mK are 0.00259 for KVF3, -0.0017 for KFeF3, -

0.0016 for KCoF3 and -0.0089 for KNiF3 correspondingly. In general, total extended orbital

polarization is responsible for the origin of magnetic moment. In fact, negative sign in mK

for KFeF3, KCoF3, as well as KNiF3, reveals anti-parallel K-atoms corresponding X atoms,

as a result net magnitude of magnetic moments reduces. However, at interstitial sites and F

atoms, the positive value of magnetic moment reveals that KXF3 possesses parallel magnetic

moments. Finally, for transition metal, the total value of magnetic moment becomes 3, 4, 3,

and 2 for KXF3 respectively. These prominent variations in value of magnetic moments can

be attributed to transfer of electrons from partially filled X atoms to F atoms accordingly.

However lowest magnetic behavior is accessed for KVF3 because of low overlapping in F-2p

as well as X-3d states while strongest magnetization is determined in KFeF3 because in Fe

atom value of mF is highest. The overall integer characteristic in KXF3 magnetic moments

follows Slater-Pauling rule (Slater 1936).

7.7 Optical properties

To quantify the internal behavior of any material optical properties are employed. This part

of the chapter is dedicated to compute fundamental and derived optical responses by mBJ

potential. The details of evaluated parameters with their mathematical formulas are

mentioned in few upcoming headings.

7.7.1 Calculation of complex dielectric function

The root of optical properties lies in complex dielectric function, well represented as Ԑ (ω)

via subsequent relation (Fox 2001):


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Ԑ(𝜔) = Ԑ1(𝜔) + 𝑖Ԑ2(𝜔) (7.20)

In equation (7.20) Ԑ1(ω) and Ԑ2(ω) represents real and imaginary part of dielectric function

respectively. The imaginary part of dielectric function Ԑ2(ω) can be given as:

Ԑ2(𝜔) = (4𝜋2𝑒2

𝑚2𝜔2) ∑ ∫ ⟨𝑖|𝑀|𝑗⟩2𝑓𝑖(1 − 𝑓𝑖)𝛿(𝐸ј,𝑘 − 𝐸𝑖,𝑘 − 𝜔)𝑘𝑖,𝑗 𝑑3𝑘 (7.21)

It can be noticed from Figure 7.5 that peaks in Ԑ2 (ω) are in similar accordance with the

(DOS) of KXF3. Major peaks in Ԑ2(ω) occurs at about 23 eV for KXF3 compounds, that is

due to shift from unoccupied (X-d, K-d) states of conduction band to (X-d, F-p) states of

valence band. However, critical or threshold point occurs at about in the range of 0-5 eV for

respective KXF3 fluoroperovskites and then diversified peaks can be examined till 20 eV.

Next, the real part of dielectric function Ԑ1(ω) is given by the well-known Kramers-Kronig

relation via corresponding equation (Abelès 1972):

Ԑ1(𝜔) = 1 +2

𝜋𝑃 ∫

ὠԐ2(ὠ)𝑑ὠ

ὠ2−ὠ2

ⱷ

0 (7.22)

It can be analyzed from Figure 7.6 that values of Ԑ1 (ω) achieves maxima at about 8.12 eV

for KVF3, 7.89 eV for KFeF3, 7.81 eV for KCoF3 and 6.78 eV for KNiF3 respectively.

However static part of dielectric function, provides zero frequency limit Ԑ1(0), positioned at

2.12 eV for KVF3, 2.19 eV for KFeF3, 2.22 eV for KCoF3 and 2.32 eV for KNiF3

respectively. The corresponding curves receives trivial fluctuations till 23.87 eV, while peak

minimums are occurred in range at about 21-23 eV for KXF3. In this optical limit

propagation of photons are entirely attenuated.


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7.7.2 Calculation of energy loss function


function for the electromagnetic spectrum such as distinctive plasmon oscillations via

spectrum of electron energy loss spectrum L (ω), which can be determined by following

expressions (Murtaza and Ahmad 2012):

𝐿(𝜔) = 𝐼𝑚 (−1

Ԑ(𝜔)) (7.23)

Figure 7.7 illustrates that at 27 eV, a sharp plasmon peak can be observed. The trailing edge

of R (ω) can be associated with these peaks, as figured out clearly in Figure 7.10.

7.7.3 Calculation optical conductivity

The phenomenon of electronic conductivity due to electromagnetic radiation can be

described in terms of optical conductivity σ(ω) via following relation (Babu et al., 2014).


𝐸02 (7.24)

In equation 7.24 WCV is the transition probability between conduction and valence band. It

can be observed from Figure 7.8, that conduction starts at approximately 5 eV via small

rising peaks which eventually attains its maxima approximately at 23 eV.

7.7.4 Calculation of absorption coefficient

In this section, the plot of absorption coefficient is computed. The absorption coefficient α

(ω) can be calculated via following relations (Harmel et al., 2015):

𝛼(𝜔) =4𝜋ƙ(𝜔)

𝜆 (7.25)


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It can be analyzed from Figure 7.9 that compounds initiates phenomenon of absorption at

about 4.25 eV. The value of threshold point agrees well with the behavior of conductivity

plots and with the trend of bandgaps. The prominent peaks in absorption spectrum are

detected approximately at 23.5 eV, while the process of absorption attains maxima within 23-

26 eV for KXF3. Then the spectrum again going to decrease, while suffering trivial

variations. So, it can be concluded from absorption spectra that these compounds exhibit

wide capacity of absorption near Ultra-violet region, especially at 23.5 eV. Furthermore,

previously reported works are in similar accordance with above mentioned results (Mavin

2003).

7.7.5 Calculation of reflectivity

The key optical parameters such as reflectivity R can be determined by following expressions

(Wooten 1972):

𝑅 = |𝑛−1

𝑛+1|

2

(7.26)

The spectrum of reflectivity as shown in Figure 7.10, interprets about optical transition of

any material. These spectrum initiates high reflection, then achieves maximum value in the

range of 23-26 eV. So, in this particular energy range, material show transparency. In fact,

due to these reflective properties, KXF3 class of fluoroperovskite can be employed as

protective agents in ultra-violet region.

7.7.6 Calculation of refractive index

The key optical parameters such as refractive index n(ω), and reflectivity R can be

determined by following expressions (Wooten 1972):


Page | 250

𝑛(𝜔) = 1

√2[√Є1(𝜔)2 + Є2(𝜔)2 + Є1(𝜔) ]

1

2 (7.27)

The material’s transparency versus spectral radiation can be illustrates via Figure 6.11 in

terms of calculated refractive index n (ω). In many useful applications, knowledge of

refractive index plays a crucial role in several optoelectronic devices like solar cell, photonic

crystals, and detectors. For KXF3 the value of static refractive index can be found at about

1.5. However, maxima of refractive index achieve, within UV-spectrum energy range and

high value of refractive index is attained at low energy region.

7.7.7 Calculation of sum rule via neff




0ὠ 𝑑ὠ (7.28)

The extent of inter-band transition, via oscillator strength sum rule is shown in Figure 6.12.

The value of effective number of valence electron via sum rule is zero till 5 eV. Then there is

slow increase in trend-line, following advent of sharp peak, which saturates at approximately

25-27 eV.


Page | 251

Figure 7.1 (a): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u)3) for

KVF3.


Page | 252

Figure 7.1 (b): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u)3) for

KFeF3.


Page | 253

Figure 7.1 (c): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u)3) for

KCoF3.


Page | 254

Figure 7.1 (d): Variations of total energy (E, in Ry) with unit cell volume (V, in (a.u) 3) for

KNiF3.


Page | 255

Figure 7.2 (a): The LSDA (Spin up)-electronic band dispersion curves for KXF3

(X= V,Fe,Co,Ni).

En

ergy

(eV

)


Page | 256

Figure 7.2 (b): The GGA (Spin up)-electronic band dispersion curves for KXF3

(X= V,Fe,Co,Ni).

En

ergy

(eV

)


Page | 257

Figure 7.2 (c): The mBJ (Spin up)-electronic band dispersion curves for KXF3

(X= V,Fe,Co,Ni).

En

ergy

(eV

)


Page | 258

Figure 7.2 (d): The LSDA (Spin down)-electronic band dispersion curves for KXF3

(X= V,Fe,Co,Ni).

En

ergy

(eV

)


Page | 259

Figure 7.2 (e): The GGA (Spin down)-electronic band dispersion curves for KXF3

(X= V,Fe,Co,Ni).

En

ergy

(eV

)


Page | 260

Figure 7.2 (f): The mBj (Spin down)-electronic band dispersion curves for KXF3

(X= V,Fe,Co,Ni).

En

ergy

(eV

)


Page | 261

Energy (eV)

DO

S (

Sta

tes/

eV)

Figure 7.3: Spin-dependent total and partial density of states for (a) KVF3, (b) KFeF3,

(c) KCoF3 and (d) KNiF3.


Page | 262

Figure 7.4: Spin-dependent electron charge densities in (110) planes for KXF3 (X= V, Fe,

Co and Ni).


Page | 263

Figure 7.5: The calculated imaginary part Ԑ2 (ω) of the dielectric function for KXF3

(X= V,Fe,Co,Ni) compounds.


Page | 264

Figure 7.6: The Calculated real part Ԑ1(ω) of the dielectric function for KXF3



Page | 265

Figure 7.7: Calculated energy loss function L (ω) of the dielectric function for KXF3



Page | 266

Figure 7.8: Calculated conductivity σ (ω) of the dielectric function for KXF3



Page | 267

Figure 7.9: Calculated absorption coefficient α (ω) of the dielectric function for KXF3



Page | 268

Figure 7.10: Calculated reflectivity R (ω) of the dielectric function for KXF3



Page | 269

Figure 7.11: Refractive index n(ω) of the dielectric function for KXF3 (X= V,Fe,Co,Ni)

compounds.


Page | 270

Figure 7.12: Calculated sum rule (Neff) of the dielectric function for KXF3



Page | 271

Table 7.1: Comparison of experimental and calculated values of equilibrium lattice constants

(ao in Ǻ), ground state energies (Eo in Ry), bulk modulus (Bo in GPa) and its pressure

derivative (BP), and bond lengths of KXF3 (X = V, Fe, Co,Ni) compounds.

a)(Moreira and Dias 2007), f)(Travis et al., 2016), g)(Lee et al., 2003), i)(Verma and Kumar 2012), k)(Shafer 1967)

(Experimental Work); b)(Jiang 2007), c)(Majeed 2010), d)(Verma et al., 2008),e) (Pari 1994),h) (Dovesi et al., 1997) (Other

theoretical work).


——————— GGA

Present work

――—————

LDA

Present

Analytical work

——————— I.R method

Present

Analytical work

——————— V.J method

Experimental

work Other

theoretical

work

KVF3 ao (Å)

4.137 4.134 4.198 4.180 4.201a 4.134b,4.138c,

4.102c,3.847d Bo (GPa) 78.01 77.94 74.88e

BP(GPa) 4.95 Bond-lengths K-F(Å) 2.85 K-V(Å) 3.61 V-F(Å) 2.08 2.06f KFeF3 ao (Å)

4.064 4.059 4.089 4.042 4.130a,

4.129g

4.121c,4.124c,

4.120h,4.221i

Bo (GPa) 70.11 70.03 72.59e

BP(GPa) 3.68

Bond-lengths K-F(Å) 2.90 K-Fe(Å)

Fe-F(Å)

3.57

2.07 2.08f

KCoF3 ao (Å)

4.055 4.051 4.059 4.012 4.090i, 4.091a

4.092g

4.072b,4.076c

4.077d ,3.801d

Bo (GPa) 82.24 82.31 75.82e

BP(GPa) 4.89 5.06 Bond-lengths K-F(Å) 2.82 K-Co(Å) 3.47 Co-F(Å) 2.01 2.03k KNiF3 ao (Å)

4.018 4.013 4.117 4.110 4.020a,4.034g,

4.012f

4.015c,4.011c

Bo (GPa) 80.13 79.99 79.65e

BP(GPa) 5.05 Bond-lengths K-F(Å) 2.31 K-Ni(Å) 3.44 Ni-F(Å) 1.96 1.99f


Page | 272

Table 7.2: Calculated tolerance factor for KXF3 (X = V, Fe,Co,Ni) compounds.

a) (Travis et al., 2016), b) (Kocsis et al., 1999), c) (Moreira and Dias 2007)

Table 7.3: Calculated values of elastic constants (C11, C12 and C44 in GPa), for KXF3 (X =

V,Fe,Co,Ni) compounds.

a) (Aleksiejuk et al., 1975) (Experimental Work)

Tolerance

Factor Present work

——————————

Bond length formula

Present work

——————————

Goldschmidt’s formula

Experimental

work

Theoretical

work

KVF3 0.891 0.992 0.991a 0.997c

KFeF3 0.999 1.001 1.003b 1.002c

KCoF3 0.998 1.018 1.060b 1.019c

KNiF3 0.921 1.013 1.042a 1.047c

Sr.No. Parameters KVF3 KFeF3 KCoF3 KNiF3

1 C11 (GPa) 162.448 131.112 122.281

127.870a

121.231

2 C12 (GPa) 35.471 36.412 62.425

52.919a

58.989

3 C44 (GPa) 32.219 39.837 32.199

34.211a

46.639


Page | 273

Table 7.4: Calculated values of Bulk modulus (B0 in GPa), Young’s modulus (Y in GPa),

Voigt’s shear modulus (GV in GPa), Reuss’s shear modulus (GR in GPa), and Hill’s shear

modulus (GH in GPa) for KXF3 (X = V,Fe,Co,Ni) compounds.

a) (Verma and Kumar 2012) (Other theoretical work), b) (Dovesi et al., 1997) (Experimental Work)


1 Bo(GPa) 77.798

74.889a

68.211

69.110b

82.377

75.828a

80.217

79.651b

2 Gv(GPa) 43.212 42.012 31.291 40.886

3 GR(GPa) 41.299 42.532 31.256 39.067

4 GH(GPa) 42.213 42.277 31.273 39.971

5 Y (GPa) 107.231 105.661 183.274 102.824


Page | 274

Table 7.5: Calculated values of B/G ratio, Shear constant (C’), Cauchy pressure (C’’),

Lame’s coefficients (λ and μ), Kleinman parameter (ξ in GPa), Anisotropy constant (A in

GPa) and Poisson’s ratio (ѵ in GPa) for KXF3 (X = V,Fe,Co,Ni) compounds.


1 Bo/GH (GPa) 1.84 1.61 2.63 2.01

2 C′ 63.49 47.35 29.93 31.12

3 C′′ 3.26 -3.42 30.23 12.06

4 Ѵ (GPa) 0.27 0.24 0.31 0.29

5 A (GPa) 0.50 0.84 0.94 1.55

6 ξ(GPa) 0.37 0.43 0.63 0.64

7 λ 49.55 27.12 51.85 53.42

8 μ 42.22 42.23 31.78 39.97


Page | 275

Table 7.6: Comparison of experimental and calculated values of longitudinal (υl in Km/s),

transverse (υt in Km/s), average sound velocity (υm in Km/s), Debye temperature (θD in K)

and the melting temperature (TMelt in K) for KXF3 (X = V,Fe,Co,Ni) compounds.

Compound υl υt υm θD TMelt

KVF3

Present work

5.57 3.17 4.37 340 1500 ± 300

Experimental

work

1559a

KFeF3

Present work

5.37 3.13 4.25 325 1330 ± 300

Experimental

work

Other work

RbFeF3]1

4.95

2.59

340

1435b

1240 ± 300

KCoF3

Present work

5.49 2.75 4.12 312 1300 ± 300

Experimental

work

3.38c 247a 1305d

KNiF3

Present work

5.44 2.98 4.21 320 1225 ± 300

Experimental

work

Other work

RbNiF3]1

4.90

2.30

3.31c 262e

315

1400f

1173 ± 300

a) (Poirier 2000), b)(Hautier 2011), c) (Jong et al., 2015), d) (Holden 1971), e) Oleaga 2015, f)(Shafer et al., 1967)

(Experimental Work); 1) (Mubarak and Saleh 2015) (Other theoretical work)


Page | 276

Table 7.7: Comparison of calculated interstitial (minst), local and total magnetic moment

(MT) in μB of KXF3 (X= V,Fe,Co,Ni) compounds with available experimental and other

theoretical data.

Compound minst mK mX mF MT

KVF3 GGA 0.54199 0.00259 2.35991 0.02379 2.99981

Exp

Other

KFeF3 GGA 0.1629 -0.0017 3.55389 0.08897 4.0019

Exp

Other

3.43c

4.27a,4.49b

KCoF3 GGA 0.0497 -0.0016 2.6611 0.0903 3.0098

Exp

Other

2.3c

3.92a

KNiF3 GGA 0.0179 -0.0089 1.7104 0.0781 2.0001

Exp

Other

a) (Mackin et al., 1963), b) (Langley et al., 1984) (Experimental Work); c) (Dovesi et al., 1997) (Other theoretical work)


Page | 277

7.8 Conclusion

In this chapter detailed theoretical investigation of four transition metal based

fluoroperovskites KXF3 (X= V,Fe,Co,Ni) have been done successfully. The electronic

structure calculations demonstrate prominent decrease in lattice constant as transition metal

in KXF3 changes from V to Ni, in accordance with the decrease in atomic size of the

corresponding compounds. These elastically and mechanically stable compounds reveal

dominant ductile behavior with high directional bonding. The calculations of Debye

temperature θD helps to explore significant thermal parameters for KXF3. The spin based

magneto-electronic characteristics clarifies the phenomenon of exchange splitting, is due to

3d states of transition metals. However, the optimized magnetic phase calculations validate

the experimental findings at low temperature. Furthermore, the linear optical response

confers wide extent of reflection as well as absorption within region of high frequency.

Hopefully this investigation, as per best of concerned knowledge, is benchmarked

investigation on KXF3 and this contribution helps to stimulate an outlook on these

fluoroperovskites in specified areas of interest. In light of effective magneto-opto-electronic

implications, KXF3 class of fluoroperovskites can be applicable in spin based switching

devices.

Chapter 8 Results and discussion ΙV

Page | 278

Chapter 8: Results and discussion ΙV;

Effect of pressure variation on strontium and calcium

based fluoroperovskites

“Never say, I tried it once

and it did not work.”

Ernest Rutherford

8.1 Introduction

The purpose of this chapter is to investigate detailed information about effect of pressure

variation on electronic structure, mechanical stability, opto-electronic trend and

thermodynamic aspects of strontium and calcium based alkali earth fluoroperovskites. This

chapter comprises of six major sections. Following the introduction section, the second

section comprises on back ground of materials, including significance for the effect of

hydrostatic pressure on physical properties. In section 8.3, effect of pressure variation on

detailed physical properties of SrLiF3 have been investigated. Section 8.4, is dedicated to

explore influence of pressure variation on structural, elastic, mechanical, opto-electronic and

thermodynamic parameters. In section 8.5, the effect of pressure variation on stability,

structural parameters, elastic constants, mechanical, electronic and thermodynamic properties

of cubic SrKF3 fluoroperovskite have been investigated by using the Full-Potential

Linearized Augmented Plane Wave (FP-LAPW) method combined with Quasi-harmonic

Debye model in which the phonon effects are considered. In section 8.6, Density functional

theory (DFT) is employed to calculate the effect of pressure variation on electronic structure,

elastic parameters, mechanical durability, and thermodynamic aspects of SrRbF3. The next


Page | 279

section 8.7, is used to calculate the effect of pressure variation (0-50 GPa) on electronic

structure, elastic parameters, mechanical durability, and thermodynamic aspects of calcium

based CaLiF3. At the end, the last section is dedicated to draw conclusion from the present

investigation.

8.2 Background of investigation

Perovskite fluorides with general stoichiometry ABF3, where A and B cations stands for

alkali and alkaline earth metals and F is a fluoride anion, gains considerable attention in the

last few decades due to their technological benefits. Generally, this diverse group of

fluoroperovskites are gaining potential utilization in the field of electric ceramics, optical

parametric oscillators, astrophysics, geophysics, heterogenous catalysis, refractories and so

on (Wang and Kang 1998). The importance of these fluoroperovskites are also hidden is their

technical applications such as solar energy convertors, low birefringent lenses, optical wave

guides, light emitting and spintronics devices (Tilley 2016). Therefore, the search of new

wide bandgap fluoroperovskite compounds for thermodynamic and opto-electronic

applications becomes necessary. Among them wide bandgap strontium and calcium based

fluoroperovskites such as “SrLiF3, SrNaF3, SrKF3, SrRbF3 and CaLiF3” has gained

prominent attention because some of them are prospective candidate for vacuum ultraviolet-

transparent lens materials in optical lithography and anti-reflective coatings (Mousa et al.,

2013).

Early experimental studies on SrLiF3 have mainly focused on its crystal structure such as

Düvel and its fellows (Düvel et al., 2011) analyzed BaLiF3 and SrLiF3 by Magic Angle

Spinning (MAS) Nuclear Magnetic Resonance (NMR) spectroscopy. They investigated that

SrLiF3 compound is a highly metastable quaternary fluoride crystallizes in inverse perovskite


Page | 280

structure. A comparative experimental study between SrLiF3 and BaLiF3 is also done by

Nishimatsu with his coresearchers (Nishimatsu et al., 2002). The authors found that SrLiF3

may have a wider and direct bandgap than BaLiF3. In another pressure induced study, Korba

with his fellows (Korba et al., 2009), have calculated opto-electronic properties of BaLiF3

under the influence of pressure and found that the valence bandwidth increases

monotonically with the pressure. Theoretically, there are few computational studies based on

common density functional theory approximations such as Local Density Approximation

(LDA) and Generalized Gradient Approximation (GGA) within first principles technique

(Yalcin et al., 2016; Mubarak 2014 & Mousa et al., 2013). Mubarak and Mousa (Mubarak

and Mousa 2012) performed first principles calculations of BaXF3 (X= Li,Na,K,Rb) fluoride

perovskites and found wide and direct (Γ-Γ) band gap in these compounds. While from

different optical spectra these compounds can be utilized in high frequency opto-electronic

application including transparent optical coatings. In chapter 5, section 5.2 and 5.3 (Erum

and Iqbal 2016 & Erum and Iqbal, March 2017), on SrMF3 (M= Li,Na,K,Rb), we explore

structural, elastic and optoelectronic response of these compounds under constant (zero)

pressure by using different exchange and correlation schemes. Our elastic and mechanical

properties prove mechanical stability, brittle, ionic nature in these compounds which can

utilize them in lens material manufacturing discipline because they would not tolerate major

birefringence which can make design of lenses difficult. While our results of optoelectronic

properties suggest implementation of these materials in UV based devices.

From previous studies, it can be evaluated that there are few theoretical and some

experimental investigation is devoted to study “SrLiF3, SrNaF3, SrKF3, SrRbF3 and CaLiF3”

compounds. Especially experimental work on them is scarce due to their reactive and volatile


Page | 281

nature and as per best of our knowledge neither experimental nor theoretical effort have been

made to investigate pressure dependent physical properties. Though pressure imposes

significant variation on important physical properties such as lattice parameters, electronic

density of states (DOS), band structure curves, elastic moduli, cubic stability conditions,

complex dielectric coefficients, refractive index, reflectivity and so on. Generally,

fluoroperovskites are made up of the network of corner linked polyhedral, tilt or distortion in

polyhedral upon application of temperature or pressure plays a crucial role in their stability

because change in pressure transfer electron from s state to p state imparts an important role

in the stability of crystal structure in cubic phase (Flocken et al., 1986). Motivation of this

study is to enhance the rare experimental and existing theoretical literature by investigating

effect of pressure variation on structural, elastic, mechanical, electronic, optical and

thermodynamic properties of above mentioned compounds. We are in hope that such

pressure induced investigation allows future researchers to improve stable utilization of

“SrLiF3, SrNaF3, SrKF3, SrRbF3 and CaLiF3” compounds in manufacturing practical devices.

In addition to it, especially thermodynamic properties are reported for the first time.

8.3 Pressure variation on physical properties of SrLiF3

In this section, the structural, electronic, elastic, optical and thermodynamic properties of

cubic fluoroperovskite SrLiF3 at ambient and high-pressure are investigated by using first-

principles total energy calculations (Murnaghan 1944) within the framework of Generalized

Gradient Approximation (GGA) (Wu and Cohen 2006), combined with Quasi-harmonic

Debye model (Schwarz et al., 2010), in which the phonon effects are considered. The

pressure effects are determined in the range of 0-50 GPa.


Page | 282

8.3.1 Pressure variation on structural properties

The structural properties are determined via different volumes over a range ± 10% which are

selected to calculate minimum ground state energy (Eo) at zero pressure. Here Birch

Murnaghan’s equation of state (EOS) (Murnaghan 1944) is used to fit the minimum energy

(Eo) versus minimum volume (VO). It can be noticed from Table 8.1 that calculated ground

state lattice parameters (at zero pressure) such as lattice constant of the present work is 3.871

Å, which is in good agreement with previously reported work in chapter 5 (Erum and Iqbal,

March 2017 & Erum and Iqbal 2016). However, between experimental (Castro 2002 &

Mishra et al., 2011) and present lattice parameter (ao) there is some deviation because the

present work is done at the ground state while the experimental work was done at ambient

conditions.

In order to examine the crystal structure of SrLiF3 on different hydrostatic pressure (0 to 50

GPa), we attempt to study the effect of different pressure, with a step size of 10 GPa, on

lattice parameters. Figure 8.1 depicts calculated change in the value of lattice constant by

LDA and GGA approximations. It can be interpreted that lattice constant is going to decrease

both by LDA and GGA approximations. The variation of bond lengths Sr-F and Li-F with

pressure is also presented in Figure 8.2. However, with comparison of data at constant

pressure of the same series as shown in Table 8.2 (Erum and Iqbal 2016), it can be analyzed

that Li-F and Sr-F bond lengths compress to a reasonable extent within the limit that

constituent polyhedral of SrLiF3 do not become distorted with the change in pressure. The

reduction in values of bond lengths and lattice constant can be associated via difference of

bandgap and hybridization strength which will be discussed in association with electronic

properties in upcoming section.


Page | 283

8.3.2 Pressure variation on electronic properties

The electronic properties at constant pressure (Zero pressure) are explored in terms of band

dispersion curves as well as electronic density of available states to occupy. Figure 8.3 and

Figure 8.4 illustrates that at constant compression SrLiF3 has both conduction band minima

(CBM) and valence band maxima (VBM) lies at direct (Γ- Γ) symmetry points resulting

7.306 eV bandgap from GGA approximation which as compared to previously reported

work, are in reasonable agreement as shown in Table 8.2.

The electronic nature of the herein investigated compound at constant pressure is also

confirmed by Density of States (Partial as well as Total) as shown in Figure 8.5. The

difference between F-2p states within 0 to -3 eV and Sr-4p states within -13 to -15 eV range

explores the transition of states at zero pressure. It can be interpreted that as a whole Sr-3d

peaks are dominant in conduction band energy region.

Next, we have discussed the effect of pressure applications of electronic properties of SrLiF3

fluoroperovskite. The key issue in SrLiF3 compound is formation and widening of bandgap

with the increase in pressure. It can be analyzed from Figure 8.3-8.5 that as compression

increases from 0 to 50 GPa, calculated bandgap increases from 7.306 eV to 7.782 eV from

GGA. It is worthy to mention here that in this pressure range the cubic structure remains

intact. The bandgap does not change its nature under application of pressure, although the

increase in bandgap of SrLiF3 at high pressure shifts towards wider gap nature and the rate of

increase of bandgap with pressure shows a plateau like behavior up to 50 GPa. In general, the

opening of wide band gap in the fluoroperovskites is because of electro-negativity of fluoride

ion (Harmel et al., 2015; Babu et al., 2014 & Mishra et al., 2011).


Page | 284

In this section, we try to summarize broadening of bandgap and reduction in lattice

parameters from several points of view. From DOS point of view, it can be observed that

upon compression an increase in the ratio of splitting between Sr-5d, Sr-4d, Sr-3d, F-2s and

F-2p states because the bands broadened the energy of Sr-4d and Sr-5d states, thereby

resulting an ultimate rise in the bandgap and this increase in bandgap continues up to 50 GPa.

Since Mishra with his coresearchers (Mishra et al., 2011) and Korba with his coresearchers

(Korba et al., 2009) have experimentally and theoretically confirmed the high-pressure

structural stability of BaLiF3 class of fluoroperovskites which is in accordance with our

results. As pressure is increased from 0 to 50 GPa, from lattice parameter point of view, the

cubic lattice nature of SrLiF3 compound remains stable, with reduced lattice constant (ao),

reduced bonds length and increased band gap. The similar nature of compression is also

observed by Lee and their fellows (Lee et al., 2004) as lessening in bonds length increases

the bond energy, which increases bandgap consequently, reduces the strength of covalent

bond. In some valuable theoretical (Murtaza and Iftikhar 2012) and experimental (Kuo et al.,

2004), studies, the similar inverse relation between lattice constant and bandgap is reported.

On the other hand, we also shed light to a prominent fact that LDA and GGA

approximations, undervalues bandgaps in wide bandgap insulators and semiconductors. The

reason of this underestimation lies in falsified clarification of factual unoccupied states as

compared to Khon–Sham states of the system (Wu and Cohen 2006). Here we highlight

modified Becke-Johnson potential (mBJ), to attain expected accurate results near to

experimental value, which is usually formed by well-known GGA and LDA calculations.

Furthermore, through mBJ scheme a bandgap near to experimental values is obtained like

Yalcin and his fellows (Yalcin et al., 2016) for BaLiF3, which predicts 8.2 eV bandgap by


Page | 285

mBJ potential (Tran and Blaha 2009), in similar accordance with experimental bandgap of

8.41 eV with just divergence of 1.78%. Similar trend is also observed for SrLiF3. As a result,

the cubic SrLiF3 is by virtue, an insulator and with the increase in the pressure the strength of

hybridization increases with reduced bond lengths which give rise to antibonding

phenomenon among bonds. This antibonding creates high energy, which pushed up energy

level away from Ef, consequently widening of bandgap occurs which is previously reported

in Pseudo potential theory (Imada et al., 1998 & Harrison 1984) as well.

8.3.3 Pressure variation on elastic properties

To verify structural constancy of SrLiF3 with compression 0-50 GPa, we reconnoiter

significant elastic and mechanical parameters. The variation of elastic responses under

compression can provide reliable information about change in stiffness, stability and

hardness of any compound. The complete pressure dependent mechanical behavior of any

crystals can be easily interpreted by its pressure dependent elastic properties. The major

importance of elastic constants is hidden in its response towards an applied macroscopic

stress (Meziani and Belkhir 2012). In this subsection, our main aim is to calculate elastic and

mechanical properties such as elastic constants, elastic modulus, elastic stiffness coefficients,

Poisson’s ratio (ѵ), Cauchy pressure (C''), Anisotropy constant (A), Kleinman parameter (ξ)

and Melting temperature (Tm) according to some detailed mathematical relationships as

mentioned in the following reference (Sadd 2005).

The basic idea used to calculate elastic coefficient for cubic crystals, (C11, C12, C44) is the

application of homogenous deformations with a finite value using first-principles

investigation (Jamal et al., 2014). The computation of stress tensor (ϭ) is done by using

Charpin method (Charpin 2001). By applying cubic symmetry, 21 independent components


Page | 286

of elastic constants are condensed to C11, C12, and C44 elastic constants. In the present work,

our results regarding to elastic and mechanical parameters are tabulated in Table 8.3-8.6.

Unfortunately, for comparison there is unavailability of data at higher pressure ranges but our

investigated results are in reasonable accordance with previous work on same series

compound such as on BaLiF3 (Mishra et al., 2011). The condition for mechanical stability

for cubic crystals (Wang et al., 1993) are found to be satisfied. Table 8.4 shows that the

calculated elastic constants, for cubic crystals under finite strain, obey the modified stability

criteria (erum and Iqbal, November 2017) according to corresponding pressure of 50 GPa.

i.e.

𝑀1 =(𝐶11+2𝐶12)

3+

𝑃

3> 0 (8.1)

𝑀2 = 𝐶44 − 𝑃 > 0 (8.2)

𝑀3 =(𝐶11− 𝐶12)

2− 𝑃 > 0 (8.3)

However, it can be noticed that at 50 GPa pressure the stability condition from equation 8.2

is not fully satisfied (At this pressure M2 stability criteria is lower than zero i.e. -0.53) as

shown in Figure 8.6. Therefore, the cubic fluoroperovskite SrLiF3 is mechanically stable

against elastic deformation by the compression up to 40 GPa. A monotonic linear

dependence is found for all pressure ranges, as shown in Figure 8.7-8.9. The elasticity in

length, C11, expand with pressure which means that pressure enhance tensile strength of

SrLiF3 compound. Similar trend for elasticity in shape is perceived for C44 elastic constant.

So, C11 and C44 increases, as a result of bond length enhancement (Sr-F, Li-F) which predicts

bonds length reduction.


Page | 287

8.3.4 Pressure variation on mechanical properties

The important mechanical aspects, as shown in Table 8.5, such as different form of shear

modulus G, can be calculated by using following expressions (Shafiq et al., 2015):

𝑮𝑽 = 𝟏

𝟓(𝑪𝟏𝟏 − 𝑪𝟏𝟐 + 𝟑𝑪𝟒𝟒) (8.4)

𝑮𝑹 =𝟓𝑪𝟒𝟒(𝑪𝟏𝟏−𝑪𝟏𝟐)

𝟒𝑪𝟒𝟒+𝟑(𝑪𝟏𝟏−𝑪𝟏𝟐) (8.5)


𝟐 (8.6)

However, expression of bulk modulus B are mention from following equation (Kittel 2005):

𝑩 =𝟏

𝟑 (𝑪𝟏𝟏 + 𝟐𝑪𝟏𝟐) (8.7)

The response of a material towards linear strain can be well defined by Young’s modulus via

following relation (Jenkins & Khanna 2005):

𝒚 =𝟗𝑩𝑮

(𝟑𝑩+𝑮) (8.8)

From the trend of Young’s modulus (Y), Bulk modulus (B0), Reuss’s shear modulus (GR),

Voigt’s shear modulus (GV), and Hill’s shear modulus (GH), it can be observed that SrLiF3

has highest value of stiffness and rigidity at pressure of 50 GPa and lowest value of stiffness

and rigidity at 0 GPa pressure which means that material becomes stiffer and less

compressible when applied pressure is increased. While the value of shear constant reveals

that ionicity in SrLiF3 increases as pressure varies from 0 to 50 GPa because ionic materials

have low values of shear constant as shown in Table 8.6. These results are in reasonable

accordance with the previous work related to perovskite compounds under the influence of

varying pressure by Rai and his fellows (Rai et al., 2014).

Next, by using Cauchy’s law or Cauchy pressure, the angular characteristics of atomic

bonding is elucidated. It can be well defined as follows (Brik 2011):


Page | 288

𝑪′′ = 𝑪𝟏𝟐 − 𝑪𝟒𝟒 (8.9)

If the value of this pressure is negative, then the material tends towards, directional bonding

and if the value is positive then the material is expected to be metallic in nature. The

investigated perovskite SrLiF3 have negative value of Cauchy pressure at 0 GPa pressure

which is going to shift towards positive value upon increasing pressure, but insulating nature

of SrLiF3 as BaLiF3 (Nishimatsu et al., 2002) at high pressure shows, changed sign of

Cauchy pressure is unbiased pointer of reduced angular characteristics of the atomic bonding

as shown in Table 8.6.

The ratio of compression to relative expansion can be expressed in terms of Poisson’s ratio,

as follows (Pettifor 1992):

ѵ =(𝟑𝑩−𝟐𝑮)

𝟐(𝟑𝑩+𝑮) (8.10)

The poisson’s ratio also agrees well with behavior of Cauchy pressure, indicating that

application of pressure reduces brittleness and angular bonding nature of SrLiF3 compound.

In manufacturing disciplines, elastic anisotropy parameter A plays an imperative character.

The relation can be well defined as follows (Jamal et al., 2016):

𝑨 =𝟐𝑪𝟒𝟒

(𝑪𝟏𝟏−𝑪𝟏𝟐) (8.11)

From the calculated value of elastic anisotropy factor, the degree of deviation of anisotropic

behavior can be determined. Another significant parameter which was introduced by

Kleinman, used to quantify material’s behavior towards bond stretching or bond bending; if

minimum bond stretching then Kleinman parameter ξ=1 but If compound possess minutest

value for bond bending then ξ=0 as (Kleinman 1962):

𝝃 =𝑪𝟏𝟏+𝟖𝑪𝟏𝟐

𝟕𝑪𝟏𝟏−𝟐𝑪𝟏𝟐 (8.12)


Page | 289

As pressure increases the value of Kleinman parameter shift towards higher values which

implies that compression induces low resistance against bond bending or bond angle

distortion, from the value of 0.4563 (0 GPa) to 0.4567 (50 GPa) respectively. To calculate

melting tendency of SrLiF3 compounds, the next task is to explore melting temperature,

above which material changes from its solid phase to its liquid phase (Fine et al., 1984):

Tm = 607 + 9.3B + 555 (8.13)

From the calculation of melting temperature, it can be assessed, that an increase in pressure

induces less tendency of melting extent of SrLiF3 and eventually increases its melting

temperature as shown in Figure 8.9 and Table 8.6 respectively.

8.3.5 Thermodynamic properties

In this section, thermodynamic properties are investigated for SrLiF3, within 0 to 600 K

temperature and 0 – 50 GPa pressure ranges, by means of the Quasi-harmonic Debye model

as implemented in the Gibbs program (Blanco et al., 2004 & Francisco et al., 1998). In this

model the vibrations of the crystal are treated as a continuum isotopic, obtained from the

derivatives of the total electronic energy volume.

8.3.5.1 The Quasi-harmonic Debye model

To study the thermodynamic properties of the SrLiF3 compound quasi-harmonic Debye

model is applied, in which the non-equilibrium Gibbs function can be written as (Rached et

al., 2009):

𝐺∗(𝑉; 𝑃; 𝑇) = 𝐸(𝑉) + 𝑃𝑉 + 𝐴𝑣𝑖𝑏[𝛳𝐷(𝑉); 𝑇] (8.14)

Where E (V) is the total energy per unit cell, PV corresponds to the constant hydrostatic

pressure condition, θD (V) is the Debye temperature, T is the absolute temperature and AVib is


Page | 290

the contribution of vibratory term, which can be written using the Debye model by the

density of state of the phonons as follows (Greiner et al., 1995):

𝐴𝑣𝑖𝑏(𝛳𝐷; 𝑇) = 𝑛𝑘𝑇 [9𝛳𝐷

8𝑇+ 3𝑙𝑛 (1 − 𝑒−

𝛳𝐷𝑇 ) − 𝐷 (

𝛳𝐷

𝑇)] (8.15)

Where n is the number of atoms per chemical formula, D (ϴD / T) is the Debye integral. For

an isotropic solid, θD can be expressed as (Anderson 1963 & Gibbs 1873):

𝛳𝐷 =ħ

𝑘[6𝜋2𝑉

1

2𝑛]

1

3√

𝐵𝑆

𝑀𝑓(𝜎) (8.16)

Where M is the molecular weight per unit cell; BS is the adiabatic bulk modulus, which in

Debye model, is generally equal isothermal bulk modulus BT in the Debye model, leading to

the following equation (Francisco et al., 2001):

𝐵𝑆 ≅ 𝐵𝑇 = 𝑉𝑑2 𝐸(𝑉)

𝑑𝑉2 (8.17)

where E is the total energy of the crystal at 0 K.

f (σ) is given in references (Bouhemadou et al., 2009) as:

𝑓(𝜎) = {3 [2 (21+ 𝜎

31−2𝜎)

3

2+ (

11+ 𝜎

31− 𝜎)

3

2]

−1

}

1

3

(8.18)

The Poisson coefficient is taken to be equal to 0.25 (Francisco et al., 1998).

A minimization of Gibbs function G* makes it possible to obtain the thermal equation of

state, the volume V (P, T) and the corresponding chemical potential G (P, T) as (Blanco et

al., 2004):

[𝜕𝐺∗(𝑉;𝑃;𝑇)

𝜕𝑉]

𝑃,𝑇= 0 (8.19)


Page | 291

By solving the equation (8.19) with respect to V, a thermal equation of state can be achieved

which can be used to deduce the macroscopic properties: the heat capacity at constant

volume CV, the entropy, and the coefficient of thermal expansion are given as follows

(Rached et al., 2009):

𝐶𝑉 = 3𝑛𝑘 [4𝐷 (𝛳

𝑇) −

3𝛳

𝑇

𝑒𝛳𝑇−1

] (8.20)

𝑆 = 𝑛𝑘 [4𝐷 (𝛳

𝑇) − 3𝑙𝑛 (1 − 𝑒−

𝛳

𝑇)] (8.21)

α = 𝛾𝐶𝑉

𝐵𝑇𝑉 (8.22)

Anharmonic effect of the vibrating lattice is usually described in terms of Gruneisen

parameter, γ defined by (Greiner et al., 1995):

𝛾 = − 𝑑𝑙𝑛𝛳(𝑉)

𝑑𝑙𝑛𝑉 (8.23)

The heat capacity at constant pressure, CP can be expressed as (Roza 2011):

𝐶𝑃 = 𝐶𝑉(1 + 𝛼𝛾𝑇) (8.24)

8.3.5.2 Pressure and temperature variation on thermodynamic properties

The thermodynamic properties of SrLiF3 within 0 to 600 K temperature and 0 – 50 GPa

pressure ranges, has been determined using the Quasi-harmonic Debye model in which the

Debye temperature depends only on the volume of the crystal. This method has been

implemented in the Gibbs code and uses Only a set of points {V, E (V)} calculated in the

equilibrium state for T = 0 and P = 0 (Blanco et al., 2004).


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It can be noticed from Figure 8.10 (a-b) that at low temperatures heat capacities CV and CP

are proportional to T3, while at (T > 400K), CV tends to the Dulong-Petit limit (123.7 J mol-

1K-1), that is communal phenomenon in all solids at elevated temperatures, whereas CP

follows a linear increase (Ghebouli et al., 2012). The variation of the volume expansion

coefficient α(T) for the SrLiF3, is presented in Figure 8.10 (c). It can be observed that α

exhibits enhanced growth for low temperatures and then progressively inclines to rise

linearly at elevated temperatures. It should be distinguished that as the pressure increases, the

growth of α with the temperature becomes reduced. However, α decreases sharply with the

increase in pressure, for a given temperature.

The Debye cut-off frequency or Debye temperature (θD) is a significant form of temperature,

have used to quantify several thermodynamic properties in the solid. It is important due to

extraction of some useful physical quantities for example specific heat capacities and melting

point. (Anderson 1963). It can be perceived from Figure 8.10 (d) that θD is almost continual

since 0 to 100 K, then declines smoothly with the temperature for T > 200K. While the

Debye temperature increases linearly with pressure, for a constant temperature. At zero

pressure and ambient temperature, the calculated θD is 511.62 K.

8.3.6 Pressure variation on optical properties

One of the most significant property to discuss internal structure of any material is optical

property. These properties suggest material’s suitability and reliability in industrial

applications, specifically for opto-electronics (Erum and Iqbal, December 2017). The set of

complete optical properties such as complex dielectric function Ԑ(ω), absorption coefficient

α(w), refractive index n (ω) and reflectivity R (ω), optical conductivity σ(ω), energy loss

function L(ω), and effective number of electrons neff via sum rules are calculated within


Page | 293

pressure 0-50 GPa. All optical parameters which are calculated here, are based on some

proposed numerical relations. The real and imaginary part of dielectric function Ԑ(ω) can be

defined as follows (Wooten 1972):

Ԑ(𝜔) = Ԑ1(𝜔) + 𝑖Ԑ2(𝜔) (8.25)

In equation (8.25) Ԑ1(ω) and Ԑ2(ω) represents real and imaginary part of dielectric function

respectively. The imaginary part of dielectric function Ԑ2(ω) can be given as:

Ԑ2(𝜔) = (4𝜋2𝑒2

𝑚2𝜔2) ∑ ∫ ⟨𝑖|𝑀|𝑗⟩2𝑓𝑖(1 − 𝑓𝑖)𝛿(𝐸ј,𝑘 − 𝐸𝑖,𝑘 − 𝜔)

𝑘𝑖,𝑗 𝑑3𝑘 (8.26)

As shown in Figure 8.11 (a), the interpretation of imaginary part of dielectric function plot,

describes complete response of material due to applied electromagnetic field which is

influenced by intraband as well as interband transitions (Monkhorst and Pack 1976). It can

be noticed that at zero pressure the threshold point is at about 7.306 eV which is critical point

in band gap edge of the SrLiF3 compound also known as fundamental absorption edge.

However, by increasing pressure from 0 to 50 GPa, the fundamental absorption edge shift

towards higher energy. This shift of threshold point is due to increase in bandgap (as detailed

explanation is mentioned in electronic property section). The main cause of this shift is

transition of electrons from valence band maxima to conduction band minima in SrLiF3

compound. These peaks positioned just below the zero energy Fermi level (EF), are ascribed

to transitions of F-2p state along with minor contribution of Sr-3d and Li-states. However,

within pressure range 0-50 GPa, this compound work well in ultraviolet region of

electromagnetic spectrum because of larger value of band gap than 3.1 eV (Murtaza and

Iftikhar 2012). Next, the real part of dielectric function Ԑ1(ω) is given by the well-known

Kramers-Kronig relation via corresponding equation (Abelès 1972):


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Ԑ1(𝜔) = 1 +2

𝜋𝑃 ∫

ὠԐ2(ὠ)𝑑ὠ

ὠ2−ὠ2

ⱷ

0 (8.27)

It describes Ԑ1(ω) defines electric polarizability and absorptive behavior of the material. The

calculated plot of real dielectric function Ԑ1(ω) is displayed in Figure 8.11 (b). Ԑ1(0) has value

of about 2.12 for 0.00 GPa pressure and is found to be increased with increasing pressure

while at 50 GPa, the value of Ԑ1(0) reaches up to 2.38. This trend in pressure variation study

reveal the inverse behavior of Ԑ1(0) with electronic band gap, which has also been observed

for other fluoroperovskites (Harmel et al., 2015 & Babu et al., 2014).


function for the electromagnetic spectrum such as refractive index n(ω), reflectivity R,

optical conductivity σ(ω), and electron energy loss spectrum L(ω) which can be determined

by following expressions (Fox 2001):

𝑛(𝜔) = 1

√2[√Є1(𝜔)2 + Є2(𝜔)2 + Є1(𝜔) ]

1

2 (8.28)

𝑅 = |𝑛−1

𝑛+1|

2

(8.29)


𝐸02 (8.30)

𝐿(𝜔) = 𝐼𝑚 (−1

Ԑ(𝜔)) (8.31)

The calculated n(ω) plot from Figure 8.11(c) have similar shape as of Ԑ1(ω) curve. While

peaks shift towards higher energies which suggests higher values of refractive index which is

beneficial for successful utilization of this material in photonic applications. The computed

value of zero frequency reflectivity is 0.035, which can be clearly observed from Figure

8.11(d) that pressure encounter broadening of the maximum reflectivity range, in accordance


Page | 295

with the trend of Ԑ1(0). Our calculated value of optical conductivity σ(ω) for SrLiF3

compound is shown in Figure 8.11(e) The figure reveals that optical conduction at zero

pressure starts at approximately, 5eV originating after minor rising crests then finally shifts

towards higher energy ranges as pressure is increased and gains maximum peaks within

energy range of 12-14 eV. It is obvious as of this investigation that optical conductivity

spectrum swings near to increased energy ranges from 0 to 50 GPa due to increased bandgap

with pressure and this abrupt increase in conduction is due to trailing edge of Ԑ1(ω) curve. A

similar trend of behavior like optical conductivity is detected for coefficient of absorption

α(w), as portrayed in Figure 8.11(f). The energy loss function per moving electron is shown

in Figure 8.11(g) via energy loss function L(w). These peaks give us brief characteristics

related to phenomenon of plasma resonance. It can be observed that no energy loss occurs for

photon energy less than at about 7 eV for 0-50 GPa pressure but as soon as the photons

exceeds from this limit, the energy loss will start increasing and get maximum peaks.

However, it can be clearly seen from Figure 8.11(g) that pressure shifts energy loss function

towards higher energy region.




0ὠ 𝑑ὠ (8.32)

As shown in Figure 8.11(h), for SrLiF3 compound explored that electrons at zero pressure

initiates interband transitions about 5.5 eV. These peaks intensify gradually but at about 14-

16 eV range, rapid increase in saturation ratio of electrons can be observed. So, it can be

concluded that with the increase in pressure, the peaks move towards higher energies so the

number of effective electrons taking part in intraband as well as interband transitions


Page | 296

decrease. To best of our information there is lack of investigated information on pressure

dependent optical behavior of SrLiF3 in cubic phases we therefore hope that our work

provides better beneficial understanding about pressure dependent behavior of this material.

So, it can be concluded that with the increase in pressure, the peaks move towards higher

energies so the number of effective electrons taking part in intraband as well as interband

transitions decrease.

To best of our information there is lack of investigated information on pressure dependent

physical behavior of SrLiF3 in cubic phases so hopefully this work will motivate research

scholars to done theoretical as well as experimental studies in this direction, so they can

compare their results with our work to get better beneficial understanding about pressure

dependent behavior of this material.


Page | 297

Figure 8.1: The Pressure variation of Lattice Constant (a) GGA (b) LDA

Figure 8.2: The Pressure variation of Bonds length (a) Sr-F (b) Li-F


Page | 298

Figure 8.3: The Pressure dependence of Band Gap (a) GGA (b) mBj


Page | 299

Figure 8.4: The electronic band structures of SrLiF3 under the application of pressure

(0, 10, 20, 30, 40 and 50 GPa) calculated using GGA Approximation.

En

ergy

(eV

)


Page | 300

Figure 8.5: The Total and Partial Density of states (TDOS & PDOS) of SrLiF3 at 0 GPa

using GGA Approximation.

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 301

Figure 8.6: Stability criteria for cubic SrLiF3 compound as a function of

pressure.

Figure 8.7: Calculated pressure dependence of elastic constant/moduli

(a) C11 (b) C12 for SrLiF3 compound.


Page | 302

Figure 8.8: Calculated pressure dependence of (a) Elastic constant/moduli (C44)

(b) Bulk modulus (B) for SrLiF3 compound.

Figure 8.9: Calculated pressure dependence of Kleinman parameter (ξ), and

Melting temperature (Tm) for SrLiF3 compound.


Page | 303

Figure 8.10 (a): Variation of the specific heat capacities (Cp) versus temperature at

different pressures for SrLiF3 compound.


Page | 304

Figure 8.10 (b): Variation of the heat capacities (CV) versus temperature at different

pressures for SrLiF3 compound.


Page | 305

Figure 8.10 (c): Temperature dependence of the volume expansion coefficient α(T) at



Page | 306

Figure 8.10 (d): Variation of the Debye temperature (θD) as a function of temperature at



Page | 307

Figure 8.11 (a): Calculated Imaginary part Ԑ2 (ω) of the dielectric function as a function of

pressure for SrLiF3 compound.


Page | 308


pressure for SrLiF3 compound.


Page | 309

Figure 8.11 (c): Calculated Refractive index n (ω) as a function of pressure for SrLiF3

compound.


Page | 310

Figure 8.11 (d): Calculated Reflectivity R(ω) as a function of pressure for SrLiF3 compound.


Page | 311

Figure 8.11 (e): Calculated Conductivity σ (ω) as a function of pressure for SrLiF3

compound.


Page | 312

Figure 8.11 (f): Calculated Absorption coefficient α (w) as a function of pressure for SrLiF3

compound.


Page | 313

Figure 8.11 (g): Calculated Energy loss function L (ω) as a function of pressure for SrLiF3

compound.


Page | 314

.

Figure 8.11 (h): Calculated Sum rule as a function of pressure for SrLiF3 compound.


Page | 315


theoretical values for the lattice constants (ao), ground state energies (Eo), bulk modulus (Bo)

and its pressure derivative (B′) at ambient pressure of SrLiF3 compound.

Compound: SrLiF3 Present work

Experimental work/ Other work

ao (Å) 3.871 4.449a/ 3.879c, 3.871d, 3.762e, 3.754 f

Bo (GPa) 72.055 72.071 b/72.861c, 71.350 d

B′ (GPa) 4.353 4.356e

a) (Castro 2002), b) (Mishra et al., 2011) (Experimental Work) c) (Mousa et al., 2013), d) (Mubarak and Mousa

2012),e) (Erum and Iqbal 2016 & Erum and Iqbal, March 2017) (Other theoretical work)

Table 8.2: Comparison of previous and calculated values of Pressure (P in GPa), Energies (E

in Ry), Volume of unit cell (V in (a.u.)3), Energy Gap (eV), and Bond length (dSr-F, dLi-F).

a) (Mubarak and Mousa 2012) b) (Sarukura et al., 2007), c) (Nishimatsu et al., 2002), d) (Erum and Iqbal 2016),

e) (Erum and Iqbal, March 2017)

Pressure

(GPa)

Energies

(Ry)

Volume

(a.u.)3

Energy Gap (eV) dSr-F

(Ǻ)

dLi-F

(Ǻ)

dX-F (Ǻ) (X=

Li,Na,K,Rb)d Present (GGA)

Present (mBJ)

Previous (0.00 GPa)

0 -6974.891 394.1935664 7.306 9.201 7.21a(GGA)

2.521 1.849 dSr-F = 2.52

(0.00 GPa)

10 -6974.863 353.0442656 7.455 9.337 7.30b(GGA) 2.517 1.843 dLi-F = 1.85

(0.00 GPa)

20 -6974.855 328.1655202 7.591 9.411 7.19c(GGA) 2.511 1.838 dNa-F = 2.23

(0.00 GPa)

30 -6974.839 367.3939371 7.646 9.463 7.28d(GGA) 2.508 1.833 dK-F = 2.60

(0.00 GPa)

40 -6974.812 409.6302366 7.768 9.524 9.20e(mBj) 2.501 1.828 dRb-F = 2.74

(0.00 GPa)

50 -6974.798 454.9855278 7.782 9.501 2.503 1.830


Page | 316

Table 8.3: Calculated values of elastic constants (C11, C12, C44) of SrLiF3 at pressure from 0-

50 GPa.

a) (Erum and Iqbal 2016), b) (Mishra et al., 2011)

Table 8.4: Derived elastic constants characterizing mechanical stability (Equations 8.1-8.3)

of SrLiF3 at pressure from 0-50 GPa.

Pressure (GPa) M1 M2 M3

0

BaLiF3] a

75.48

73.90

49.45

47.13

57.19

46.79

10 78.82 39.45 47.19

20 82.16 29.45 37.19

30 85.49 19.46 27.19

40 88.83 9.46 17.20

50 92.16 -0.53 7.20

a) (Mishra et al., 2011)

Pressure

(GPa)

0 10 20 30 40 50 Previous

Work

(0.00 GPa)

C11 (GPa) 151.741 151.750 151.758 151.765 151.771 151.780 151.7a, 154.2b

C12 (GPa) 37.353 37.365 37.372 37.386 37.395 37.404 37.3a, 38.5b

C44 (GPa) 49.447 49.455 49.466 49.478 49.489 49.496 49.4a, 48.1b


Page | 317

Table 8.5: Calculated values of elastic moduli Bulk modulus (B0), Voigt’s shear modulus

(GV), Reuss’s shear modulus (GR) and Hill’s shear modulus (GH), and Young’s modulus (Y)

of SrLiF3 at pressure from 0-50 GPa.

a) (Erum and Iqbal 2016), b) (Mubarak and Mousa 2012), c) (Mishra et al., 2011)

Table 8.6: Calculated values of Shear constant (C’), Cauchy pressure (C’’), Poisson’s ratio

(ѵ) Anisotropy constant (A), Kleinman parameter (ξ), and melting temperature (Tm) of

SrLiF3 at pressure from 0-50 GPa.

a) (Erum and Iqbal 2016), b) (Mishra et al., 2011)

Pressure

(GPa)

0 10 20 30 40 50 Previous

Work (0.00

GPa)

Bo(GPa) 74.481 74.489 74.495 74.504 74.513 74.524 74.4a 72.2b

Gv(GPa) 52.546 52.550 52.557 52.563 52.569 52.573 52.5a,

GR(GPa) 52.280 52.284 52.292 52.299 52.306 52.310 52.281

GH(GPa) 52.413 52.417 52.424 52.431 52.437 52.442 52.4a, 52.2c

Y(GPa) 127.363 127.374 127.390 127.406 127.421 127.434 127.362 a

Pressure (GPa) 0 10 20 30 40 50 Previous

Work (0.00

GPa)

C' 57.194 57.193 57.193 57.190 57.188 57.188 57.194a

C'' -12.094 -12.090 -12.088 -12.085 -12.081 -12.079 -12.094a

Ѵ (GPa) 0.2151 0.2163 0.2174 0.2172 0.2184 0.2187 0.2150a

A (GPa) 0.8645 0.8647 0.8649 0.8652 0.8654 0.8655 0.86a, 0.83b

ξ(GPa) 0.4563 0.4564 0.4564 0.4565 0.4566 0.4567 0.4563a

Tm(K) 1854.67 1854.75 1854.80 1854.89 1854.97 1855.07 1854.65a


Page | 318

8.4 Effect of pressure variation on physical properties of SrNaF3

In this section, the effect of pressure variation on structural, electronic, elastic, mechanical,

optical and thermodynamic characteristics of cubic SrNaF3 fluoroperovskite have been

investigated by employing first-principles method. For the total energy calculations, the Full-

Potential Linearized Augmented Plane Wave (FP-LAPW) method (Schwarz et al., 2010) is

employed. Thermodynamic properties are computed in terms of Quasi-harmonic Debye

model (Blanco et al., 2004), within 0-25 GPa pressure and 0-600 K temperature.


The herein investigated compound SrNaF3 crystallize itself in ideal cubic perovskite structure

with space group Pm-3m (no. 221). The details of various lattice parameters, tolerance factor,

bond lengths, bulk modulus, and its pressure derivative at zero pressure are previously

reported in chapter 5 (Erum and Iqbal, March 2017 & Erum and Iqbal 2016). However, for

the continuation of thesis, brief form of previously calculated lattice parameters is mentioned

in Table 8.7, at constant (zero) pressure. The main aim of this section is to explore the

influence of external pressure on the electronic structure of SrNaF3 fluoroperovskite. Here

we examine effect of pressure variation in the range of 0-25 GPa with a step size of 5 GPa. It

can be observed from Figure 8.12 that calculated lattice constants by LDA and GGA

approximations are going to decrease under the influence of increase in pressure while the

calculated change in bond lengths Sr-F and Na-F under the effect of increasing pressure is

shown in Figure 8.13. It can be noticed from Table 8.8, that bond lengths are also going to

decrease as pressure is increased. As compared to our previous work / other work (Erum and

Iqbal, March 2017 & Erum and Iqbal 2016; Düvel et al., 2011 & Korba et al., 2009) (at


Page | 319

fixed pressure), it can be evaluated, that lattice constants, bond lengths and volume of unit

cell compress to a reasonable extent. As a result, the constituent polyhedral of SrNaF3 do not

become distorted by the change in pressure from 0-25 GPa. However, in the next section we

discuss the reason of this decrease via lattice associated parameters in conjunction with

electronic band structure and density of states (DOS).


The concept of electronic nature of SrNaF3 are discussed in terms of Density of States, Total

as well as Partial (TDOS & PDOS) and electronic band structure calculations are figured out

in Figure 8.14-8.16. At zero pressure, SrNaF3 has a direct energy bandgap with the valence

band maxima (VBM) and conduction band minima (CBM) both located at the Γ symmetry

point resulting wide and direct (Γ- Γ) bandgap of 5.551 eV from GGA approximation while

from TB-mBj functional larger bandgaps of 8.298 eV is observed as shown in Table 8.8 and

in Figure 8.14-8.15. This change in the value of bandgap is in continuation with previous

study (Erum and Iqbal, March 2017 & Yalcin et al., 2016), that GGA and LDA schemes

undervalues energy bandgap of wide bandgap semiconductors and insulators. The cause of

this underestimation is discussed in detail by Tran and Blaha (Tran and Blaha 2009 & Grabo

et al., 1997). The Total and Partial Density of States (TDOS & PDOS) occupies energy

interval from (EF - 25 eV) up to (EF + 15 eV) at 0 GPa and 25 GPa pressure ranges as shown

in Figure 8.16. It can be observed that above fermi level slightly wide peak is dominated by

Sr-3d state at about (EF + 8 eV). Next energy interval (EF - 3 eV) is governed by F-2p state.

The upper valence band situated in a range from (EF - 12 eV) to (EF - 25 eV) is due to

hybridized state between Sr-4p, Na-2p and F-2s states respectively. Next, we have discussed

influence of increase in pressure from (0-25 GPa) range, (below which compound remains


Page | 320

stable in cubic phase) in terms of electronic band structure. The motivating fact of SrNaF3

compound is formation and widening of bandgap by the increase in pressure, illustrating

potential application of this compound in modern opto-electronic devices. The variation of

energy gap as a function of pressure are shown in Figure 8.14 and Figure 8.15. It can be

clearly observed that as the value of compression increases from 0 to 25 GPa, calculated

bandgap energy increase from 5.551 eV to 7.063 eV. It is worthy to mention here that

bandgap does not change its nature under pressure, although conduction band at Γ symmetry

point of Brillouin Zone shift towards higher energy ranges. However, the rate of widening of

bandgap with the increase in pressure shows a plateau like behavior up to 25 GPa. In general,

the widening of bandgap in fluorine based perovskites are due to the dominant

electronegative influence of fluoride ion (Harmel et al., 2015; Babu et al., 2014 & Mishra et

al., 2011).

The important issue to discuss here is that, what is the reason behind broadening of bandgap

and reduction in lattice parameters. In view of lattice parameter with the increase in pressure

from 0 to 25 GPa, there is attributed decrease in nearest neighbor distance, and in bond

lengths as well. The inverse relation between above mentioned parameters and bandgap is

already reported experimentally (Mishra et al., 2011) and theoretically (Korba et al., 2009).

Another similar behavior of compression is also observed by Lee and their fellows (Lee et

al., 2004) as decrease in bond lengths increases the bond energy, which reduces the strength

of covalent bond, consequently a wider bandgap. In fact, broadening of band energy upon

compression is due to broadened energy of Sr (4d & 5d) states which increases the ratio of

splitting between Sr-3d, Sr-4d, Sr-5d, F-2s and F-2p states respectively. This shift ultimately

moves bandgap towards higher energies and this increase in bandgap continues up to 25 GPa.


Page | 321

Since Mishra with his coresearchers (Mishra et al., 2011) and Korba with his coresearchers

(Korba et al., 2009) have experimentally and theoretically confirmed the high-pressure

structural stability which is in suitable accordance with band structures and DOS results. As a

result, with the increase in pressure of SrNaF3 the strength of hybridization increases with

reduced bond length which give rise to antibonding phenomenon among bonds. This

antibonding creates high energy, which pushed up energy level away from EF, consequently

widening of bandgap occurs which is previously reported in Pseudo potential theory (Imada

et al., 1998 & Harrison 1984) as well.

8.4.3 Pressure variation on elastic and mechanical properties

In order to verify structural, mechanical and cubic stability of SrNaF3 fluoroperovskite, we

shed light on the pressure dependence within range 0-25 GPa of the significant elastic and

mechanical parameters. To study the change in behavior of elastic properties, we performed

calculations at several values of reduced volumes, each of which corresponds to the system at

fixed hydrostatic pressure up to 25 GPa with the step size of 5 GPa. This is done by

performing complete optimization against each pressure value. Furthermore, the

computation of stress tensor (ϭ) is done with the help of Charpin method (Charpin 2001), by

applying cubic symmetry to 21 independent components of elastic constants which are

condensed to C11, C12, and C44 elastic constants respectively. The variation of elastic

constants under pressure gives reliable information regarding to change in stability, stiffness

and hardness of SrNaF3 compound. Here we calculate elastic and mechanical properties such

as elastic constants, elastic modulus, elastic stiffness coefficients, according to some

proposed mathematical relationships as mentioned in section 8.3.3-8.3.4. The complete list of

calculated elastic and mechanical parameters under compression are tabulated in Table 8.9-


Page | 322

8.11. Unluckily for SrNaF3, there is lack of availability for data to compare at high pressure

ranges so we only compare data of relevant compounds at zero pressure, which are found to

be in good agreement (Erum and Iqbal, March 2017 & Erum and Iqbal 2016; Düvel et al.,

2011 & Korba et al., 2009). Hence these results can serve as a reference for future

investigations.

A monotonic linear dependence of all curves of elastic constant/moduli can be observed from

Figure 8.17 and Table 8.9-8.10. The elastic constant, C11 which is related to longitudinal

distortion, linearly increases as with the change in pressure from 145.490 GPa to 429.043

GPa from 0 to 25 GPa. This is due to the fact that C11 increases as a result of bond strength

enhancement between Sr-F and Na-F which in actual decrease bond lengths and increase

charge density of the bonds. On the other hand, as compared to C11, the value of C12 and C44

elastic constants are less sensitive by the variation of pressure. In fact, the elastic constant,

C44 which is in actual related to transverse distortion are observed to be almost flat as shown

in Figure 8.17 which reveals that the enhancement of the bond strength has a very little

influence on C44. Our results indicate that at higher pressure ranges SrNaF3 compound have

more resistance to compression rather than shear deformation. Similar behavior is observed

for C12 elastic constant. Furthermore, the value of bulk modulus (B) which describes

hardness of a material, is going to increase from 55.482 GPa (at 0 GPa pressure) to 151.839

GPa (at 25 GPa pressure), suggests that SrNaF3 fluoroperovskite becomes harder and less

compressible upon application f hydrostatic pressure.

To verify the stability of cubic SrNaF3, we calculate mechanical stability conditions for

strontium based fluoroperovskite as shown in Table 8.12 within pressure range from 0-25


Page | 323

GPa. It is found that SrNaF3 compound obey the modified stability criteria (Wang et al.,

1993) for cubic crystals under finite strain till 20 GPa i.e.

𝑀1 =(𝐶11+2𝐶12)

3+

𝑃

3> 0 (8.33)

𝑀2 = 𝐶44 − 𝑃 > 0 (8.34)

𝑀3 =(𝐶11− 𝐶12)

2− 𝑃 > 0 (8.35)

It can be noticed from Figure 8.18 that calculated elastic constants do not satisfy stability

condition from equation 8.34 because at 25 GPa pressure stability criteria M2 value is lower

than zero (-2.57), which implies that calculated elastic constants cannot satisfy all mechanical

stability conditions, hence the compound is not mechanically stable above 20 GPa which is

in agreement with our electronic structure calculations that the compound is mechanically

stable against pressure less than 25 GPa.

From the trend of Voigt’s shear modulus (GV), Reuss’s shear modulus (GR), Hill’s shear

modulus (GH), and Young’s modulus (Y), as shown in Table 8.10, it can be observed that

SrNaF3 compound act as a stiffest material at 25 GPa, which implies more sharing of charge

transfer among cation and anion and material becomes less compressible at elevated

pressure. In order to examine ductile/brittle behavior of SrNaF3 at different pressure, we

calculate B/G ratio. It can be evaluated from Table 8.10, that the ratio is going towards

higher values from 1.571 to 2.359 as pressure is going to shift from 0 to 25 GPa. As reported

earlier in our recent work (Erum and Iqbal, Feburary 2017), the B/G >1.75 classify material

as ductile, while B/G < 1.75 classify material as brittle. So, with the increase in pressure the

behavior of this compound tends towards ductile in nature. Similar trend is also observed for

pressure induced behavior of Poisson’s ratio (ѵ), as shown in Table 8.11. The angular


Page | 324

characteristics of atomic bonding is explained by the Cauchy’s law or Cauchy pressure

(Jenkins & Khanna 2005). If the value of this pressure is negative, then the material tend

towards, directional bonding and if the value is positive then the material is expected to be

metallic in nature (Brik 2011). The investigated perovskite SrNaF3 have negative value of

Cauchy pressure at 0 GPa pressure which is going to shift towards positive value upon

increasing pressure, as shown in Table 8.11 but wide bandgap nature of SrNaF3 as SrLiF3,

and BaLiF3 (Erum and Iqbal, March 2017 & Mishra et al., 2011) at high pressure shows that

this change in sigh of Cauchy pressure is just an indicator of reduction in its angular

characteristics of the atomic bonding. Furthermore, from the calculated value of elastic

anisotropy factor (A), it can be noticed from Table 8.11, that the degree of deviation of

anisotropic behavior is increased with the increase in pressure. Next, we explore Kleinman

parameters which can be used to fix the relative positions of cation and anion under volume-

conserving distortions in the strain. As pressure increases the value of Kleinman parameter

shift towards higher values which implies that compression induces low resistance against

bond bending or bond angle distortion, (Kleinman 1962) from the value of 0.244 (0 GPa) to

0.319 (25 GPa) respectively. At the end, we acknowledge the material’s property above

which a compound or a substance changes from its solid phase to its liquid phase. From the

calculation of melting temperature, it can be assessed that an increase in pressure induces less

tendency of melting extent of SrNaF3 and eventually increases its melting temperature as

shown in Table 8.11.

8.4.4 Pressure and temperature variation on thermodynamic properties

In order to investigate significant thermodynamic properties of cubic SrNaF3 fluoro-

perovskite, we employ Quasi-harmonic Debye model as implemented in the Gibbs program


Page | 325

(Rached et al., 2009; Blanco et al., 2004 & Francisco et al., 1998). Details of calculations

can be found in section 8.3.5. The thermal properties are studied in the temperature range

from 0 to 600 K, and the pressure effects are determined in the range of 0-25 GPa, in which

cubic stability of SrNaF3 fluoroperovskite remains valid.

Initially, we study the evolution of lattice parameters such as lattice constant, bulk modulus

and volume expansion under the effect of temperature for different values of pressure as

shown in Figure 8.19 (a-c) respectively. It can be observed that the effects of pressure and

temperature are inversely proportional to each other. The lattice parameters such as lattice

constants and volume increase with increasing temperature at a given pressure. On the other

side, at a given temperature, pressure has a tendency to decrease lattice constant. In fact, the

thermal effect on lattice parameters becomes weaker at high pressure. Figure 8.19 (c) depicts,

the variation of bulk modulus versus temperature at different levels of pressure. The increase

in bulk modulus follows the increase in pressure from (0 to 25 GPa) at a given temperature.

This is attributed to the fact that an increase in temperature of material causes a significant

reduction in its hardness.

In Figure 8.19 (d) we present the evolution of the Debye temperature θD, at several values of

pressures. It can be noticed from the figure that, for a fixed pressure, θD decreases with

increase in temperature and for a fixed temperature, θD increases with the increasing

pressure. As a result, both the increase in pressure and decrease in temperature lead to an

increase in θD which is in exact accordance with effect of bulk modulus on various levels of

temperature and pressure. Hence, it is authenticated that hard materials exhibits high θD

because as bulk modulus increases with pressure, a phonon softening takes place and the

Debye temperature will increase.


Page | 326

Next, we fix the behavior of heat capacity, which depicts the measure of how well the

material stores heat. The variation of heat capacities CV and CP with temperature at 0, 5, 10,

15, 20, and 25 GPa pressures are shown in Figure 8.19 (e) and Figure 8.19 (f) respectively. It

can be observed that as temperature increases the variation of heat capacity at constant

volume CV and constant pressure CP are similar to each other. The trend of CV increases

slowly and tends to shift towards Dulong-Petit limit (123.7 J mol-1K-1), which is common

phenomenon to all solids at high temperature (Ghebouli et al., 2012). Moreover, the

anharmonic effects are suppressed at high temperate. However, CP decreases as increase in

pressure from (0-25GPa) and deviation is observed from a constant value. So, at ambient

pressure, the values of CP increase rapidly at higher temperature.

8.4.5 Effect of pressure variation on optical properties

To quantify the internal behavior of any material optical properties are employed. In this

regard, the computation of complex dielectric function Ԑ (ω) can be best described by

Ԑ(𝜔) = Ԑ1(𝜔) + 𝑖Ԑ2(𝜔), provides useful information concerning the optical response of a

material. These responses suggest material’s suitability and durability, and reliability in

industrial opto-electronics applications (Wooten 1972). We already report SrMF3

(M=Li,Na,K,Rb) optoelectronic response at zero pressure in chapter 5 but in this section of

thesis, keeping in mind the significant possible applications of SrNaF3 compound, our main

aim is to investigate the set of pressure induced (0-25 GPa) complete optical properties such

as complex dielectric function Ԑ(ω), absorption coefficient α(w), refractive index n (ω) and

reflectivity R (ω), Optical conductivity σ(ω), energy loss function L(ω), and effective number

of electrons neff via sum rules. All optical parameters which are calculated here, are based on

some proposed numerical relations as mentioned section 8.3.6.


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The calculated values of imaginary and real parts of complex dielectric tensor are plotted in

Figure 8.20 (a) and Figure 8.20 (b), for various value of applied hydrostatic pressure with 0-

25 GPa with a step size of 5 GPa. The Ԑ2(ω) spectra reveals that threshold energy point

occurs at 5.563 eV at 0 GPa and it move towards higher energy ranges and at 25 GPa it

becomes 7.064 which is exactly in accordance with fundamental bandgap of the material at

constant and elevated pressure ranges. The occurrence of main peak is also in the same

manner. The main cause of this shift is transition of electrons from valence band maxima to

conduction band minima in SrNaF3 compound. These peaks are ascribed to transitions of F-

2p state along with minor contribution of Sr-3d and Na-states positioned just below zero

energy Fermi level (EF). The calculated static dielectric constant Ԑ1(0) without any

contribution to lattice vibration corresponds to low energy limit in Ԑ1(ω) as shown from

Figure 8.20 (b). The main peak of Ԑ1(ω) are shifted to higher energy values with increasing

compression, therefore the transparency of SrNaF3 fluoroperovskite for incident

electromagnetic radiation of lower energies increases as with the application of compression.

Furthermore, an increase in incident energy of photon decrease Ԑ1 (ω) peaks which finds a

minimum value beyond 10 eV. The negative values of Ԑ1 (ω) finds between 10-13.5 eV that

attenuates incident electromagnetic waves. However, the onset of this attenuation is

remarkably shifted towards higher energy values upon increasing pressure from 0 to 25 GPa

which retains again its positive behavior between 14-18 eV. This systematic shift of the Ԑ1(ω)

values against incident photon’s energy suggests that pressure variation phenomenon can be

utilized for tuning the optical parametric responses of SrNaF3 fluoroperovskite compound.

It can be noticed from Figure 8.20 (c) that calculated plot of refractive index n(ω) curve

(which dictates the capability to allow electromagnetic radiation to pass through it) have


Page | 328

similar shape as of Ԑ1(ω) curve, as shown in Figure 8.20 (b) and with the application of

pressure n(ω) peaks shift towards higher energies which suggests higher values of refractive

index which is beneficial for successful utilization of this material in photonic applications

(Murtaza and Iftikhar 2012 & Monkhorst and Pack 1976). It is clear from calculated

variation of reflectivity, as shown in Figure 8.20 (d) that as pressure changes reflectivity

possesses maximum value where Ԑ1(ω) adopts minimum or negative value. The shift of

reflectivity towards higher energy values suggests that SrNaF3 reflectivity can be tuned with

the application of hydrostatic pressure.

The pressure induced variation of conductivity σ (ω) and absorption coefficient α (w) as

shown in Figure 8.20 (e) and Figure 8.20 (f) respectively. As optical conductivity appears as

a result of optical absorption, so these two phenomena can be closely related with the

application of pressure. On increasing external pressure, the absorption edge shift towards

ultraviolet part of electromagnetic spectrum. Interestingly from 0-25 GPa an approximate

linear absorption/ conduction is observed with incident energy confirming the direct bandgap

nature of SrNaF3 compound. In actual, the peak in Figure 8.20 (e) and Figure 8.20 (f)

represent pressure induced optical transition between different states of occupied valence

band and unoccupied conduction band. The characteristics of energy loss function L(w) is

related to plasma resonance phenomenon as shown in Figure 8.20 (g). It can be observed that

effect of increase in pressure shifts energy loss function towards higher energy region. The

oscillator strength sum rule is shown in Figure 8.20 (h) reveals that electron starts taking part

in interband transition at about 5.2 eV and with the increase in pressure, the peaks move

towards higher energies so the number of effective electrons taking part in intraband as well

as interband transitions decrease.


Page | 329


physical behavior of SrNaF3 in cubic phase so hopefully this work will motivate research





Page | 330

Figure 8.13: The Pressure variation of Bond lengths (a) Sr-F (b) Na-F



Page | 331



Page | 332

Figure 8.15: The electronic band structures of SrNaF3 under the application of pressure

(0, 5, 10, 15, 20 and 25 GPa) calculated using GGA Approximation.

En

ergy

(eV

)


Page | 333

Figure 8.16: The Total and Partial Density of states (TDOS & PDOS) of SrNaF3 at 0 and

25 GPa using GGA Approximation.

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 334

Figure 8.17: Calculated pressure dependence of elastic constants/moduli (a) C11

(b) C12 (c) C44 (d) Bulk modulus, B for SrNaF3 compound.

Figure 8.18: Stability criteria for cubic SrNaF3 compound as a function of

pressure.


Page | 335

Figure 8.19 (a): Variation of the Lattice constant as a function of temperature at different

pressures for SrNaF3 compound.


Page | 336


different pressures for SrNaF3 compound.


Page | 337

Figure 8.19 (c): Variation of the Bulk modulus as a function of temperature at

different pressures for SrNaF3 compound.


Page | 338


at different pressures for SrNaF3 compound.


Page | 339


temperature at different pressures for SrNaF3 compound.


Page | 340


temperature at different pressures for SrNaF3 compound.


Page | 341

Figure 8.20 (a): Calculated Imaginary part Ԑ2(ω) of the dielectric function as a

function of pressure for SrNaF3 compound.


Page | 342

Figure 8.20 (b): Calculated Real part Ԑ1(ω) of the dielectric function as a function of

pressure for SrNaF3 compound.


Page | 343

Figure 8.20 (c): Calculated Refractive index n (ω) as a function of pressure for

SrNaF3 compound.


Page | 344

Figure 8.20 (d): Calculated Reflectivity R(ω) as a function of pressure for SrNaF3

compound.


Page | 345

Figure 8.20 (e): Calculated Conductivity σ(ω) as a function of pressure for SrNaF3

compound.


Page | 346

Figure 8.20 (f): Calculated Absorption coefficient α(w) as a function of pressure for

SrNaF3 compound.


Page | 347

Figure 8.20 (g): Calculated Energy loss function L(ω) as a function of pressure for

SrNaF3 compound.


Page | 348

Figure 8.20 (h): Calculated Sum rule as a function of pressure for SrNaF3

compound.


Page | 349



and its pressure derivative (B′) at ambient pressure of SrNaF3 compound.

Compound: SrNaF3 Present work Experimental work/

Theoretical work

ao (Å) 4.178 4.440a/ 4.181b, 4.179c

Bo (GPa) 55.482 55.811 b/55.481c

B′ (GPa) 4.661 4.642c

a) (Castro 2002) (Experimental Work) b) (Duvel et al., 2011), c) (Erum and Iqbal 2016 & Erum and Iqbal, March

2017) (Other theoretical work)

Table 8.8: Comparison of previous and calculated values of Pressure (P in GPa), Energies (E

in Ry), Volume of unit cell (V in (a.u.)3), Energy Gap (eV), and Bond length (dSr-F, dNa-F).

a) (Duvel et al., 2011), b) (Korba et al., 2009), c) (Erum and Iqbal, March 2017), d) (Erum and Iqbal 2016)

Pressure

(GPa)

Energies

(Ry)

Volume

(a.u.)3

Energy Gap (eV)

Present Present Previous (GGA) (mBJ) (0.00 GPa)

dSr-F

(Ǻ)

dNa-F

(Ǻ)

dX-F (Ǻ) (X= Li,Na,K,Rb)d

0 -7284.547 492.8822751 5.551 8.298 5.58a(GGA)

2.552 2.233 dSr-F = 2.52

(0.00 GPa)

5 -7284.544 458.3474763 5.986 8.732 5.94b(LDA) 2.550 2.230 dLi-F = 1.85

(0.00 GPa)

10 -7284.542 425.4647373 6.476 9.035 5.61c(GGA) 2.547 2.228 dNa-F = 2.23

(0.00 GPa)

15 -7284.538 394.1935664 6.912 9.551 8.30c(mBj) 2.544 2.225 dK-F = 2.60

(0.00 GPa)

20 -7284.535 364.4934721 7.007 10.031 2.541 2.222 dRb-F = 2.74

(0.00 GPa)

25 -7284.533 336.3239625 7.063 10.095 2.537 2.219


Page | 350

Table 8.9: Calculated values of elastic constants (C11, C12, C44), of SrNaF3 at pressure from

0-25 GPa.

a) (Erum and Iqbal 2016)

Table 8.10: Calculated values of Bulk modulus (B0), Voigt’s shear modulus (GV), Reuss’s

shear modulus (GR) and Hill’s shear modulus (GH), Young’s modulus (Y), and B/G ratio, of

SrNaF3 at pressure from 0-25 GPa.

a) (Erum and Iqbal 2016), b) (Duvel et al., 2011), c) (Erum and Iqbal, March 2017)


Work (0.00 GPa)

C11 (GPa) 145.490 188.086 242.725 303.584 361.798 429.043 145.490a

C12 (GPa) 12.140 21.987 30.398 42.125 54.302 61.098 12.139a

C44 (GPa) 22.411 23.413 23.817 24.428 25.789 25.98 22.410a

Pressure

(GPa)

0 5 10 15 20 25 Previous

Work

(0.00 GPa)

Bo(GPa) 55.482 73.509 89.875 110.208 129.777 151.839 55.81a, 55.98b,

56.59c

Gv(GPa) 40.116 47.268 56.756 66.949 76.973 89.177 40.114a,

GR(GPa) 30.513 32.848 34.531 36.203 38.659 39.574 30.514a

GH(GPa) 35.314 40.058 45.643 51.576 57.816 64.376 35.314a

Y(GPa) 87.400 101.700 117.105 133.848 151.020 169.213 87.397a

B/G (GPa) 1.571 1.835 1.969 2.137 2.245 2.359 1.575a


Page | 351


constant (A), Kleinman parameter (ξ), and melting temperature (Tm) of SrNaF3 at pressure

from 0-25 GPa.


Table 8.12: Derived elastic constants characterizing mechanical stability (Equations 8.33-

8.35) of SrNaF3 at pressure from 0-25 GPa.


0 56.59 22.41 66.68

5 58.26 17.41 61.68

10 59.93 12.42 56.68

15 61.60 7.42 51.68

20 63.27 2.42 46.68

25 64.94 -2.57 41.68


Work

(0.00 GPa)

Ѵ (GPa) 0.237 0.269 0.283 0.298 0.306 0.314 0.2373a

C'' -10.270 -1.426 6.581 17.697 28.513 35.118 -10.267a

A (GPa) 0.336 0.282 0.224 0.187 0.168 0.141 0.3361a

ξ(GPa) 0.244 0.286 0.297 0.314 0.328 0.319 0.2443a

Tm(K) 1677.98 1845.63 1997.84 2186.93 2368.93 2574.10 1677.96a


Page | 352

8.5 Pressure variation on physical properties of SrKF3

In this section, the effect of pressure variation (0-25 GPa) on electronic structure, elastic

constants, mechanical stability, and thermodynamic aspects of cubic SrKF3 fluoroperovskite

have been investigated, by using the Full-Potential Linearized Augmented Plane Wave (FP-

LAPW) method (Schwarz et al., 2010) combined with Quasi-harmonic Debye model (Blanco

et al., 2004), in which the phonon effects are considered. We have successfully computed

variation of lattice constant, volume expansion, bulk modulus, Debye temperature and

specific heat capacities at pressure and temperature in the range of 0-25 GPa and 0-600 K.


The ideal cubic structure of SrKF3 fluoroperovskite retains with space group Pm-3m (no.

221). In chapter 5, calculated details of optimized structural parameters are given in detail at

fixed pressure. Yet for convenience, precise form of specific structural parameters are

presented in Table 8.13. For the lattice constant (ao) about 4% deviation is observed between

experimental and present calculation that is obvious, because the experimental work was

done at ambient conditions while the present work is done at zero kelvin and Castro (Castro

2002), uses large volume of unit cell as compared to the present first principle investigation.

In fact, the actual target, of the current section is to explore consequences of hydrostatic

pressure, with a step size of 5 GPa on electronic structure of SrKF3, within 0-25 GPa range.

The hydrostatic pressure of 0.00 GPa is considered as a base for ground state stable structure

and increase in pressure further decreases the volume of the crystal structure as shown in

Table 8.14. The calculated lattice constants by LDA and GGA approximation schemes and

bond lengths are plotted in Figure 8.21 and Figure 8.22 respectively. It can be analyzed from

the figures that both lattice constants and bond lengths are going to decrease as pressure


Page | 353

increases in accordance with our previous work / Other work (Erum and Iqbal, March 2017

& Erum and Iqbal 2016; Mubarak and Mousa 2012 & Nishimatsu et al., 2002) (at ambient

pressure). The calculated lattice constants, bond lengths and volume of unit cell compress to

a reasonable extent that the constituent polyhedral of SrKF3 do not become distorted by the

change in pressure from 0-25 GPa. In general, the opening of wide bandgap in

fluoroperovskites are due to electro-negativity of fluorine ion. Similar to oxide perovskites,

fluorine based perovskites also form the weak covalent bond (Harmel et al., 2015; Babu et

al., 2014 & Mishra et al., 2011). In this regard, the pressure induced behavior of SrKF3 is in

good agreement with previously reported work by Lee and their fellows (Lee et al., 2004),

that decrease in bond length increases the bonding energy as strength of covalent bond

decreases, consequently wider bandgap. In this work, we have examined that as compression

increases there is decrease in bond lengths both for Sr-F and K-F as shown in Figure 8.22

respectively. While according to Table 8.14, the calculated bond lengths are in reasonable

agreement with the previously published work at 0 GPa.


The concept of electronic nature of SrKF3 are discussed in terms of Density of States, Total

as well as Partial (TDOS & PDOS) and electronic band structure calculations. The key issue

to discuss the electronic structure of SrKF3 is the formation and widening of bandgap with

the increase in pressure. As it can be observed from Table 8.14 that compression on the

system (0 GPa to 25 GPa) increases the bandgap from 3.307 eV to 3.823 eV from GGA

scheme and 6.799 eV to 7.302 eV from mBj potential as shown in Figure 8.23. This change

in the value of bandgap is in continuation with our previous study as mentioned in section 8.3

and 8.4 (Erum and Iqbal, November 2017 & Erum and Iqbal, December 2017), that GGA


Page | 354

and LDA schemes undervalues energy bandgap of wide bandgap semiconductors and

insulators. The cause of this underestimation is discussed in detail by Tran and Blaha (Tran

and Blaha 2009 & Grabo et al., 1997). The density of states (DOS) and energy band

structures at different pressure ranges from 0- 25 GPa are shown in Figure 8.23-8.25. It can

be observed that throughout the calculations the position of maxima of valence band F-2p

states remains almost same. The Total and Partial Density of States (TDOS & PDOS)

occupies energy interval from (EF - 30 eV) up to (EF + 15 eV) at ambient pressure as shown

in Figure 8.25. It can be observed that above fermi level slightly wide peak is dominated by

Sr-3d state at about (EF + 8 eV). Next energy interval (EF - 3 eV) is governed by F-2p state.

The upper valence band situated in a range from (EF - 12 eV) to (EF - 20 eV) is due to

hybridized state of K-2p, F-2s and some 4p states of Sr respectively. So, in cubic SrKF3, with

the increase in the pressure the strength of hybridization increases with reduced bond lengths

which give rise to antibonding phenomenon among bonds. This antibonding creates high

energy, which pushed up energy level away from Ef, consequently widening of bandgap

occurs which is previously reported in Pseudo potential theory (Imada et al., 1998 &

Harrison 1984) as well.


The mechanical stability of cubic crystals under pressure can be determined by elastic

constants (Sadd 2005). In order to verify the change in behavior of mechanical parameters

under pressure of (0-25GPa), we used to perform calculations at different values of reduced

volumes, each of them corresponds to specific value of hydrostatic pressure. The complete

optimization against each value of pressure is done with the step size of 5 GPa pressure.

Furthermore, for computing stress tensor, we employ Charpin method (Charpin 2001). For


Page | 355

cubic system, there are three independent elastic constants namely C11, C12, and C44

respectively. Here we calculate elastic and mechanical properties such as elastic constants,

elastic modulus, elastic stiffness coefficients, according to some proposed mathematical

relationships as mentioned in section 8.3.3-8.3.4 as tabulated in Table 8.15-8.18. Figure 8.26

presents the variation of elastic constants/moduli such as C11, C12, and C44 and B with regard

to different values of pressure. From Figure 8.26 and Table 8.15 it can be observed that

elastic constants increase with the rise in compression. However, a prominent increase is

observed in C11 elastic constant, which is concerned with elasticity in length, in accordance

with bulk modulus, which is measure of hardness for the solid. The change in the value of

C11 is from 98.101 GPa to 238.746 GPa and the change in bulk modulus value is from 38.71

GPa to 119.74 GPa respectively. The increase in C11 can be best explained in relation with

bond strength enhancement (Sr-F, and K-F) because at elevated pressures there is decrease in

the bond length and increase of charge density as mentioned in the previous sections. The

elastic moduli of shape deformation explore less effect against variation in pressure, which

indicates that enhancement of bond strength is less effected by shear deformation. To verify

the stability of cubic SrKF3, we calculate modified stability criteria of elastic constants, as

shown in Table 8.18, within pressure range from 0-25 GPa. It is found that SrKF3 compound

obey the modified stability criteria (Wang et al., 1993) for cubic crystals under finite strain

till 20 GPa i.e.

𝑀1 =(𝐶11+2𝐶12)

3+

𝑃

3> 0 (8.36)

𝑀2 = 𝐶44 − 𝑃 > 0 (8.37)

𝑀3 =(𝐶11− 𝐶12)

2− 𝑃 > 0 (8.38)


Page | 356

It can be noticed from Figure 8.27 and Table 8.18 that calculated elastic constants do not

satisfy stability condition from equation 8.37 because at 25 GPa pressure stability criteria M2

(from equation 8.37) value is lower than zero (-0.49), which implies that calculated elastic

constants cannot satisfy all mechanical stability conditions, hence the compound is not

mechanically stable above 20 GPa which is in agreement with our electronic structure

calculations that the compound is mechanically stable against pressure less than 25 GPa.


In this section, our main aim is to analyze effect of pressure variation on tensile strength,

shear strength, rigidity, brittle/ductile behavior, and trends of directional/ non-directional

bonding, average sound velocities, and Debye temperature of cubic SrKF3 fluoroperovskite.

The details of all calculated mechanical parameters can be found in section 8.3.3-8.3.4 as

tabulated in Table 8.16-8.17. The Shear modulus (GH), and Young’s modulus (Y) also

increase with the elevated pressure in similar trend with the value of Bulk modulus (B). For

example, the value of GV, GR, GH, and Y at 25 GPa is 51.579 GPa, 34.702 GPa, 43.140

GPa, and 115.545 GPa are several times greater than at 0 GPa respectively, indicating that

SrKF3 compound becomes less compressible at higher pressures ranges as shown in Table

8.16. In addition, we observed that with the change in pressure from 0 to 25 GPa the

transition is observed from brittle to ductile behavior because B/G value shift from 1.39 to

2.78 and it is previously reported in our own work (Erum and Iqbal, Feburary 2017) that, the

B/G >1.75 classify material as ductile, while B/G <1.75 classify material as brittle. Similar

effect of pressure variation is observed for poisson’s ratio (ѵ). The angular characteristics of

atomic bonding via Cauchy’s law or Cauchy pressure (Jenkins & Khanna 2005)

demonstrates comparable transition from negative value of -11.68 to the positive value of


Page | 357

29.87 respectively as with the increase in pressure. But according to widening trend of

bandgap this change in sign is just an indicator in reduction of its angular characteristics

(Brik 2011), as shown in Table 8.17. The similar kind of pressure induced behavior is

observed for other fluoroperovskite of the same series such as SrLiF3 and BaLiF3 (Erum and

Iqbal, November 2017 & Mishra et al., 2011). Next, we interpret value of elastic anisotropy

factor (A), it can be noticed that the degree of deviation of anisotropic behavior is increased

with the increase in pressure. The effect of varying pressure on bond bending trend, as given

by Kleinman parameter (Kleinman 1962), agrees well with the above-mentioned responses.

The trend of melting temperature depicts that an increase in pressure induces less tendency of

melting extent of SrKF3 compound, which eventually increases its melting temperature, as

shown in Table 8.17.

8.5.5 Pressure variation on Debye temperature (θD)

The Debye temperature (θD) or Debye cut-off frequency is a significant form of temperature,

which used to quantify several thermodynamic properties in the solid. It is basically a

measure of the vibrational response of the crystal. In actual, it is the temperature above which

the crystal behaves classically. There are two main methods to calculate Debye temperature

(θD) including elastic constant method and specific heat measurement method (Rached et al.,

2009). At low temperature, the vibrational excitations result only from acoustic vibrations.

Thus, the Debye temperature calculated from the elastic constants is the same as that

determined from specific measurements. The standard method for calculating Debye

temperature (θD) and associated parameters from the elastic constants is derived by Anderson

(Anderson 1963), which expresses the link between θD and the mean elastic wave velocity

(Wachter et al., 2001) as:


Page | 358

𝛳𝐷 = ℎ

𝑘𝐵[

3𝑛

4𝜋𝑉𝑎]

1

3ѵ𝑚 (8.39)

where h is Planck’s constant, kB is Boltzmann’s constant, Va is the atomic volume, and n is

the number of atoms per unit volume while the average propagation velocity of the acoustic

wave is given by (Anderson 1963):

ѵ𝑚 = [1

3(

2

ѵ𝑡3 +

1

ѵ𝑙3)]

−1

3 (8.40)

Furthermore, the propagation velocities of the transverse and longitudinal acoustic waves of

a polycrystalline material can be obtained by the following relations (Schreiber and Anderson

1973):

ѵ𝑙 = (3𝐵+4𝐺

3𝜌)

1

2 (8.41)

ѵ𝑡 = (𝐺

𝜌)

1

2 (8.42)

Where B is the bulk modulus, G is the shear modulus and ρ is the density of the material. The

calculated sound velocities and Debye temperature for SrKF3 are given in Table 8.17. It can

be observed that wave velocities show a quadratic variation which increases monotonically

with rising pressure. However, they can be divided into two groups because longitudinal

wave velocity increases rapidly with pressure and is greater than the shear wave velocity. As

Debye temperature is directly derived from the elastic wave velocity (Wachter et al., 2001),

so the similar trend of rise is also observed for Debye temperature (θD). This monotonic

increase can be attributed to the increase in the bulk modulus under elevated pressure ranges

because it is well known that the Debye temperature is proportional to the bulk modulus and


Page | 359

increase in bulk modulus induces a phenomenon of phonon softening under pressure (Rached

et al., 2009). As per best of our knowledge, there is lack of available experimental data to

compare elastic wave velocity and Debye temperature under pressure, and we hope our study

can provide useful guidance for future investigations.

8.5.6 Pressure and temperature variations on thermodynamic properties

In this section, thermodynamic properties are investigated for SrKF3, within 0 to 600 K

temperature and 0 – 25 GPa pressure ranges, by means of the Quasi-harmonic Debye model

as implemented in the Gibbs program (Blanco et al., 2004 & Francisco et al., 1998). In this

model the vibrations of the crystal are treated as a continuum isotopic, obtained from the

derivatives of the total electronic energy volume. Details of calculations can be found in

section 8.3.5 and in the previously published work (Erum and Iqbal, September 2017 &

Francisco et al., 2001). Figure 8.28 (a) and Figure 8.28 (b) presents the relation between

lattice parameters (lattice constant and unit cell volume) and Figure 8.28 (c) presents

isothermal bulk modulus with temperature T up to 600 K at P = 0 GPa, 5 GPa, 10 GPa, 15

GPa, 20 GPa, and 25 GPa respectively. It can be noticed that both temperature and pressure

have inverse relation with lattice parameters and bulk modulus because at a given pressure

lattice parameters such as lattice constants and volume expansion increase with the

increasing temperature and bulk modulus decreases with increasing temperature at a given

pressure and vice-versa. In fact, at higher pressure thermal effects becomes weaker and

stronger for lattice parameters and bulk modulus respectively, which reveals that effect of

pressure and temperature are inversely proportional to each other. So, increase in temperature

causes a significant reduction in the hardness of SrKF3.


Page | 360

Next, we explore as important fundamental parameter of Debye temperature θD that depicts

many physical properties such as specific heat, elastic constants and melting extent of a

material (Roza 2011). From the plot of Debye temperature θD as shown in Figure 8.28 (d). It

should be noted that the static values of the Debye temperature (at T = 0 and P = 0)

calculated from the quasi-harmonic Debye model is 396.989 K which is very close to the

value computed from elastic properties (396.512) as listed in Table 8.17. However, to the

best of our knowledge there is no experimental data for comparison with our calculated

values. One can notice that (a) for a fixed pressure, θD decreases with increase in temperature

and for a fixed temperature, θD increases with the increasing pressure. (b) The process of

increase in pressure and decrease in extent of melting temperature will lead to increase

Debye temperature (θD) and bulk modulus because at higher pressure ranges bulk modulus

increases, as a result a phenomenon of phonon softening takes place and θD will increase in

accordance with previous temperature ranges (Bouhemadou et al., 2009).

At the end, we analyze material’s extent to store heat in terms of specific heat capacities at

constant volume and constant pressure CV and CP respectively. It can be observed from

Figure 8.28 (e) and Figure 8.28 (f) that variation of CV and CP are similar to each other as

temperature increases. However, the trend of CV increases slowly following a common

phenomenon in all solids at higher temperature ranges by approaching towards Dulong-Petit

limit (123.7 J mol-1K-1) (Ghebouli et al., 2012) where anharmonic effects prominently

suppressed. However, CP behavior deviates from T> 300 K and does not converge to a

constant value. In particular, when pressure vanishes, the values of CP increases rapidly at

about higher temperature.


Page | 361


physical behavior of SrKF3 in cubic phases so hopefully this work will motivate research





Page | 362


Figure 8.22: The Pressure variation of Bond lengths (a) Sr-F (b) K-F


Page | 363



Page | 364

Figure 8.24: The electronic band structures of SrKF3 under the application of

pressure (0, 5, 10, 15, 20 and 25 GPa) calculated using GGA Approximation.

En

ergy

(eV

)


Page | 365

Figure 8.25: The Total and Partial Density of states (TDOS & PDOS) of SrKF3 at

0 and 25 GPa using GGA Approximation.

Energy (eV)

DO

S (

Sta

tes/e

V)


Page | 366

Figure 8.26: Calculated pressure dependence of elastic constants/moduli

(a) C11 (b) C12 (c) C44 (d) Bulk modulus, B for SrKF3 compound.

Figure 8.27: Stability criteria for cubic SrKF3 compound as a

function of pressure.


Page | 367


different pressures for SrKF3 compound.


Page | 368




Page | 369

Figure 8.28 (c): Variation of the Bulk modulus as a function of temperature at



Page | 370


at different pressures for SrKF3 compound.


Page | 371

Figure 8.28 (e): Variation of the specific heat capacities of Cv as a function of temperature



Page | 372

Figure 8.28 (f): Variation of the specific heat capacities of Cp as a function of temperature



Page | 373



and its pressure derivative (B′) at ambient pressure of SrKF3 compound.

Compound: SrKF3 Present work Previous work

ao (Å) 4.678 4.680a, 4.656b, 4.629c, 4.49d

Bo (GPa) 38.698 38.692a, 38.121b, 37.125c

B′ (GPa) 2.961 2.991a, b

a) (Mubarak and Mousa 2012), b) (Erum and Iqbal 2016), c) (Erum and Iqbal, March 2017) (Other theoretical

work), d) (Castro 2002) (Experimental Work)

Table 8.14: Comparison of previous and calculated values of Pressure (P in GPa), Energies

(E in Ry), Volume of unit cell (V in (a.u.)3), Energy Gap (eV), and Bond length (dSr-F, dK-F).

a) (Mubarak and Mousa 2012), b) (Erum and Iqbal, March 2017), c) (Erum and Iqbal 2016)

Pressure

(GPa)

Energies

(Ry)

Volume

(a.u.)3

Energy Gap (eV)

Present Present Previous (GGA) (mBJ) (0.00 GPa)

dSr-F

(Ǻ)

dK-F (Ǻ) dX-F (Ǻ) (X= Li,Na,K,Rb)c

0 -8163.848 691.75437 3.307 6.799 3.31a(GGA)

2.579 2.601 dSr-F = 2.57

(0.00 GPa)

5 -8163.820 644.11432 3.537 6.982 3.23b(LDA) 2.572 2.598 dLi-F = 1.85

(0.00 GPa)

10 -8163.804 598.71334 3.633 7.120 3.30b(GGA) 2.564 2.592 dNa-F = 2.23

(0.00 GPa)

15 -8163.795 555.49752 3.727 7.198 6.79b(mBj) 2.558 2.586 dK-F = 2.60

(0.00 GPa)

20 -8163.782 514.41299 3.755 7.235 2.546 2.581 dRb-F = 2.74

(0.00 GPa)

25 -8163.771 475.40584 3.823 7.302 2.539 2.577


Page | 374

Table 8.15: Calculated values of elastic constants (C11, C12, C44) of SrKF3 at pressure from

0-25 GPa.



shear modulus (GR) and Hill’s shear modulus (GH), Young’s modulus (Y), and B/G ratio of

SrKF3 at pressure from 0-25 GPa.

a) (Erum and Iqbal 2016), b) (Mubarak and Mousa 2012), c) (Erum and Iqbal, March 2017)


Work

(0.00 GPa)

C11 (GPa) 98.101 123.268 152.542 179.842 207.197 238.746 98.230a

C12 (GPa) 9.021 17.289 26.598 37.128 44.895 54.385 9.019a

C44 (GPa) 20.189 21.221 22.985 23.682 23.998 24.512 20.170a

Pressure

(GPa)

0 5 10 15 20 25 Previous

Work

(0.00 GPa)

Bo(GPa) 38.71 53.71 71.12 85.26 104.07 119.74 38.423a, 37.691b,

37.925c

Gv(GPa) 29.929 33.928 38.980 42.752 46.859 51.579 29.854a,

GR(GPa) 25.840 27.915 30.811 32.319 33.410 34.702 25.794a

GH(GPa) 27.885 30.922 34.895 37.536 40.135 43.140 27.884a

Y(GPa) 67.457 77.830 89.971 98.197 106.689 115.545 67.397a

B/G (GPa) 1.388 1.737 2.038 2.271 2.593 2.776 1.375a


Page | 375


constant (A), Kleinman parameter (ξ), melting temperature (Tm), longitudinal (υl in m/s),

transverse (υt in m/s), average sound velocity (υm in m/s), and Debye temperature (θD in K)

of SrKF3 at pressure from 0-25 GPa.


Pressure

(GPa)

0 5 10 15 20 25 Previous

Work

(0.00 GPa)

Ѵ (GPa) 0.210 0.258 0.289 0.308 0.329 0.339 0.210a

C'' -11.168 -3.932 3.613 13.446 20.897 29.873 -11.167a

A (GPa) 0.453 0.400 0.365 0.332 0.296 0.266 0.451a

ξ(GPa) 0.255 0.316 0.360 0.403 0.416 0.431 0.256a

Tm(K) 1522.00 1661.50 1823.42 1954.92 2129.85 2275.58 1522.26a

υl (m/s) 2597.780 2733.618 2902.544 3007.808 3106.828 3217.591 ----

υt (m/s) 4285.592 4789.912 5329.490 5710.698 6156.192 6522.205 ----

υm (m/s) 3288.003 3520.575 3783.154 3951.142 4118.743 4284.748 ----

θD(K) 396.592 442.814 481.950 518.056 549.218 578.968 ----


Page | 376

Table 8.18: Derived elastic constants characterizing mechanical stability (equation 8.36-

8.38) of SrKF3 at pressure from 0-25 GPa.


0 38.71 20.19 44.54

5 54.28 16.22 47.99

10 71.91 12.99 52.97

15 89.70 8.68 56.36

20 105.66 4.00 61.15

25 124.17 -0.49 67.18


Page | 377

8.6 Pressure variation on physical properties of SrRbF3

In this section, Density functional theory (DFT) is employed to calculate the effect of

pressure variation on electronic structure, elastic parameters, mechanical durability, and

thermodynamic aspects of SrRbF3. For the total energy calculations, the Full-Potential

Linearized Augmented Plane Wave (FP-LAPW) method (Schwarz et al., 2010) is employed.

Thermodynamic properties are computed in terms of Quasi-harmonic Debye model (Blanco

et al., 2004), within 0-25 GPa pressure and 0-600 K temperature.


The fluoroperovskite SrRbF3 compound has an ideal cubic structure with space group Pm-3m

(no. 221) at ambient conditions. To determine the structural properties the total energy is

calculated at different volumes (Murnaghan 1944). The details of various lattice parameters,

tolerance factor, bond lengths, bulk modulus, and its pressure derivative at zero pressure are

previously reported in chapter 5 (Erum and Iqbal, March 2017 & Erum and Iqbal 2016).

However, for the continuation of thesis, brief form of previously calculated lattice parameters

is mentioned in Table 8.19, at constant (zero) pressure. The main aim of this section is to

investigate pressure variation consequences on structural properties and mechanic aspects of

SrRbF3 within 0-25 GPa, with step size of 5 GPa as revealed in Table 8.20. Plot of lattice

constant by GGA and LDA schemes are shown in Figure 8.29. Here, compression results

decrease in lattice constants and bond lengths both for Sr-F and Rb-F as displayed in Figure

8.30 respectively, in similar accordance with our/others published work (Erum and Iqbal,

December 2017 , Erum and Iqbal, November 2017, Erum and Iqbal, September 2017, Erum

and Iqbal, March 2017, Erum and Iqbal 2016; Düvel et al., 2011 & Korba et al., 2009).


Page | 378


The elastic properties have been investigated by using Charpin method. Furthermore, within

the Charpin method the elastic properties of cubic crystals are elucidated by the elastic

constants namely C11, C12, and C44 respectively (Charpin 2001). The computed values of the

C11, C12, and C44 are presented in Table 8.21, which shows reasonable agreement with

previous work (Erum and Iqbal 2016) at fixed (zero) pressure. The focus of this section is to

examine the effect of pressure variation on shear strength, tensile strength, rigidity of the

investigated compound whom calculational details can be found in previous section 8.3.3-

8.3.4 and in the following references (Shafiq et al., 2015; Brik 2011 & Kleinman 1962).

The deformation of material under any small stresses can characterized by elastic constants

(Pettifor 1992). Figure 8.31 presents effect of pressure variation on elastic moduli such as

C11, C12, and C44 respectively. From Figure 8.31 and Table 8.21, it can be observed that with

the rise in compression the value of all elastic constants increases. The C11, which is related

to elasticity in length improve with pressure which means that pressure enhance tensile

strength of SrRbF3 compound. The increase in C11, can be described in terms of enhancement

in bond strength (Sr-F, and Rb-F) because there is decrease in the bonds length and increase

in charge density at elevated pressure ranges. The C44 elastic constant, which is related to

shape deformation, becomes less effected against elevated pressure that indicates less bond

strength enhancement. To check cubic phase stability of SrRbF3, the derived elastic stability

conditions are shown in Table 8.22 within pressure range from 0-25 GPa. It is evident that

compound obey the cubic stability conditions (Wang et al., 1993) and modified stability

criteria (Sadd 2005) for cubic crystals under finite strain till 20 GPa. It can be perceived from

Figure 8.32 that considered elastic constants do not obey stability condition, M2 as mentioned


Page | 379

in section 8.3.3, 8.4.3, and 8.5.3 because at 25 GPa pressure stability criteria of M2 value is

lower than zero (-2.84), which implies that calculated elastic constants cannot satisfy all

mechanical stability conditions, hence the compound is not mechanically stable above 20

GPa. As a result, it can be ensured that SrRbF3 undergo no structural phase transition for

pressure up to 20 GPa.


This section is dedicated to calculate effect of hydrostatic pressure on mechanical properties

of SrRbF3 compound according to some proposed mathematical relationships as cited in the

following reference (Shafiq et al., 2015; Brik 2011 & Kleinman 1962) and whom

calculational details can be found in previous section 8.3.3-8.3.4 as well. From the trend of

elastic moduli, as shown in Figure 8.33 and Table 8.23-8.24, such as Young’s modulus (Y),

Bulk modulus (B0), Reuss’s shear modulus (GR), Voigt’s shear modulus (GV), and Hill’s

shear modulus (GH), it can be observed that SrRbF3 has highest value of stiffness and rigidity

at pressure of 25 GPa and lowest value of stiffness and rigidity at 0 GPa pressure. It depicts

material becomes stiffer and less compressible when applied pressure is increased. These

results are in reasonable accordance with the previous work related to perovskite compounds

under the influence of varying pressure by Rai and his fellows (Rai et al., 2014).

Furthermore, SrRbF3 changes its behavior from brittle to ductile character, one as pressure

changes from 0 to 25 GPa. In accordance with our previous work (Jenkins & Khanna 2005)

B/G less than 1.75 confirms, compound is brittle in character, on the other hand, its value

greater than 1.75 classifies compound as ductile. Poisson’s ratio (ѵ) reveals similar trend of

behavior with the change in pressure. The anisotropy elastic factor (A), interprets an increase

in degree of deviation with the increased pressure. The Kleinman parameter can be used to


Page | 380

analyze bond bending (Kleinman 1962), that is basically consistent with above mentioned

results. The trend of melting temperature depicts that an increase in pressure induces less

tendency of melting extent for SrRbF3 compound, which eventually increases its melting

temperature.


The Debye temperature (θD) explores the crystal’s response towards vibration. Above this

temperature crystal behaves classically. In this investigation we use elastic constant method

to perform these calculations as show in Table 8.24. The details of this method and about

their related formulas can be found in section 8.5.5 and in following references (Rached et

al., 2009; Schreiber and Anderson 1973 & Anderson 1963) as well. Figure 8.34 (a) depicts

effect of pressure variation within (0-25 GPa) on average, longitudinal, and transverse wave

velocities as a function of pressure (0- 25 GPa). A prominent quadratic variation in wave

velocities can be observed which monotonically increases with the increase in pressure. In

fact, longitudinal wave velocity increases rapidly as compared to shear wave velocity. The

increasing trend of θD, as shown in Figure 8.34 (b) and Table 8.24, is in similar accordance

with elastic wave velocity. Furthermore, increase in θD is in direct relation with bulk modulus

because of the fact that its high value persuades phonon softening within the applied

pressure.


The thermodynamic properties of SrRbF3 are determined using Quasi-harmonic Debye

model as implemented in the Gibbs program (Blanco et al., 2004). Through this model, we

can get all thermodynamic quantities from the calculated energy-volume data. The calculated

details can be seen in the previously published work (Francisco et al., 2001; Francisco et al.,


Page | 381

1998 & Greiner et al., 1995) and in section 8.3.5 as well. The pressure effects are determined

in the range of 0-25 GPa, in which cubic stability of SrRbF3 fluoroperovskite remains valid

and thermal properties are studied in temperature range from 0 to 600 K.

Figure 8.35 (a) and Figure 8.35 (b) presents the relation between lattice parameters (lattice

constant and unit cell volume) and Figure 8.35 (c) presents isothermal bulk modulus with

temperature T up to 600 K at P = 0 GPa, 5 GPa, 10 GPa, 15 GPa, 20 GPa, and 25 GPa

respectively. It can be noticed that both temperature and pressure have inverse relation with

lattice parameters and bulk modulus because at a given pressure lattice parameters such as

lattice constants and volume expansion increase with the increasing temperature and bulk

modulus decreases with increasing temperature at a given pressure and vice-versa. In fact, at

higher pressure thermal effects becomes weaker as well as stronger for lattice parameters and

bulk modulus respectively (Ghebouli et al., 2012), which reveals that effect of pressure and

temperature are inversely proportional to each other. So, increase in temperature causes a

significant reduction in the hardness of SrRbF3.

The Debye cut-off frequency or Debye temperature (θD) is important due to extraction of

some useful physical quantities for example specific heat capacities and melting point

(Bouhemadou et al., 2009). Figure 8.35 (d) reveals that value of θD remains smooth till 100 K

and for temperature greater than 200 K it decreases monotonically. While for a constant

temperature, θD rises linearly with pressure.

However, at zero pressure and ambient temperature, our calculated θD from elastic constant

method is 362.019 K, as shown in Table 8.24, which are closer from calculated values to the

quasi-harmonic model as shown in Figure 8.35 (d). The specific constant volume heat


Page | 382

capacity CV and constant pressure heat capacity CP are shown in Figure 8.35 (e) and Figure

8.35 (f). It can be observed from figures that variation of CV and CP with the increase in

temperature are similar to each other. Further, the trend of CV rises gradually and inclines to

move in the direction of Dulong-Petit limit which is 123.7 J mol-1K-1, a common

phenomenon in all solids at high temperature ranges (Rached et al., 2009). Moreover, the

anharmonic effects are also suppressed at high temperature. However, CP decreases as

increase in pressure from (0- 25 GPa) and deviation is observed with a constant value. So, at

ambient pressure, the values of CP increase quickly at elevated temperature ranges.


physical behavior of SrRbF3 in cubic phases so hopefully this work will motivate research





Page | 383


Figure 8.30: The Pressure variation of Bond lengths (a) Sr-F (b) Rb-F


Page | 384

Figure 8.32: Stability criteria for cubic SrRbF3 compound

as a function of pressure.

Figure 8.31: Calculated pressure dependence of elastic constants

(a) C11 (b) C12 (c) C44 for SrRbF3 compound.


Page | 385

Figure 8.33: Calculated pressure dependence of elastic parameters (a) Bulk

modulus (B) (b) Shear modulus (G) (c) Young’s modulus (Y) (d) B/G Ratio

for SrRbF3 compound.


Page | 386

Figure 8.34 (a): Calculated pressure dependence of elastic wave

velocities (a) υl (b) υt (c) υm for SrRbF3 compound.

Figure 8.34 (b): Calculated pressure dependence of Debye

temperature (θD) for SrRbF3 compound.


Page | 387

Figure 8.35 (a): Variation of the Lattice constant as a function of temperature

at different pressures for SrRbF3 compound.


Page | 388

Figure 8.35 (b): Variation of the unit cell volume as a function of temperature



Page | 389

Figure 8.35 (c): Variation of the Bulk modulus as a function of temperature



Page | 390




Page | 391

Figure 8.35 (e): Variation of the specific heat capacities of Cv as a function of temperature



Page | 392

Figure 8.35 (f): Variation of the specific heat capacities of Cp as a function of temperature



Page | 393



and its pressure derivative (B′) at ambient pressure for SrRbF3 compound.

a) (Mubarak and Mousa 2012), b) (Erum and Iqbal 2016), c) (Erum and Iqbal, March 2017) (Other theoretical

work), d) (Castro 2002) (Experimental Work)


(E in Ry), Volume of unit cell (V in (a.u.)3) and Bond length (dSr-F, dRb-F) of SrRbF3

compound.

a) (Erum and Iqbal, September 2017), b) (Erum and Iqbal 2016)

Compound: SrRbF3 Present work Previous work

ao (Å) 4.947 4.952a, 4.938b, 4.940c, 4.479d

Bo (GPa) 33.098 32.902a, 33.121b, 33.105c

B′ (GPa) 4.009 4.010a, b

Pressure

(GPa)

Energies

(Ry)

Volume

(a.u.)3

dSr-F (Ǻ) dSr-F (Ǻ)

(SrKF3) a

dRb-F

(Ǻ)

dK-F (Ǻ)

(SrKF3) a dX-F (Ǻ)

(0.00 GPa) (X= Li,Na,K,Rb)b

0 -12922.44 818.52124 2.601 2.579 2.742 2.601 dSr-F = 2.57

5 -12922.41 769.90931 2.598 2.572 2.738 2.598 dLi-F = 1.85

10 -12922.37 727.83882 2.586 2.564 2.732 2.592 dNa-F = 2.23

15 -12922.34 678.53650 2.581 2.558 2.729 2.586 dK-F = 2.60

20 -12922.29 644.11432 2.575 2.546 2.724 2.581 dRb-F = 2.74

25 -12922.25 606.80395 2.569 2.539 2.718 2.577


Page | 394

Table 8.21: Calculated values of elastic constants (C11, C12, C44) of SrRbF3 at pressure from

0-25 GPa.

a) (Erum and Iqbal 2016), b) (Erum and Iqbal, September 2017), c) (Erum and Iqbal, December 2017)

Table 8.22: Calculated values of derived elastic constants characterizing mechanical stability

of SrRbF3 at pressure from 0-25 GPa.

a) (Erum and Iqbal, September 2017), b) (Erum and Iqbal, December 2017)

Pressure

(GPa)

0 5 10 15 20 25 Previous

Work (0.00 GPa)

(SrRbF3) a

Previous

Work (25.00 GPa)

(SrKF3) b

Previous

Work (25.00 GPa)

(SrNaF3) c

C11

(GPa)

85.251 112.525 150.256 175.021 205.895 225.284 85.230 238.746 429.043

C12

(GPa)

6.881 12.102 16.258 21.121 26.598 32.427 6.819 54.385 61.098

C44

(GPa)

14.091 15.289 17.021 18.992 20.021 22.158 14.170 24.512 25.98


Work (25.00 GPa)

(SrKF3) a

Previous

Work (25.00 GPa)

(SrNaF3) b

M1 33.05 47.24 64.26 77.42 93.03 105.05 124.17 64.94

M2 14.09 10.29 7.02 3.99 0.02 -2.84 -0.49 -2.57

M3 39.26 45.21 57.00 61.95 69.65 71.43 67.18 41.68


Page | 395


shear modulus (GR) Hill’s shear modulus (GH), Young’s modulus (Y), and B/G ratio of

rRbF3 at pressure from 0-25 GPa.

a)(Erum and Iqbal 2016), b)(Mubarak 2014), c)(Erum and Iqbal, March 2017), d)(Erum and Iqbal, September

2017)


constant (A), Kleinman parameter (ξ), melting temperature (Tm), longitudinal (υl in m/s),

transverse (υt in m/s), average sound velocity (υm in m/s), and Debye temperature (θD in K)

of SrRbF3 at pressure from 0-25 GPa.

a) (Erum and Iqbal 2016), b)(Mubarak 2014), c)(Erum and Iqbal, March 2017), d)(Erum and Iqbal, September

2017)

Pressure

(GPa)

0 5 10 15 20 25 Previous

Work

(0.00 GPa)

Previous

Work (25.00 GPa)

(SrKF3) d

Bo(GPa) 32.129 48.125 68.254 82.84 101.245 114.256 32.123a,

33.691b,

32.125c

119.74

Gv(GPa) 24.158 29.258 37.012 42.175 47.872 51.866 24.154a, 51.579

GR(GPa) 18.750 21.182 24.260 27.181 29.044 32.024 18.794a 34.702

GH(GPa) 21.554 25.220 30.636 34.678 38.458 41.945 21.884a 43.140

Y(GPa) 52.845 64.409 79.946 91.295 102.408 112.116 52.897a 115.545

B/G (GPa) 1.491 1.908 2.228 2.389 2.633 2.724 1.495a 2.776

Pressure

(GPa)

0 5 10 15 20 25 Previous

Work (0.00 GPa)

(SrRbF3) a

Previous

Work (25.00 GPa)

(SrKF3) d

Ѵ (GPa) 0.226 0.277 0.305 0.316 0.331 0.336 0.225 0.339

C'' -7.210 -3.187 -0.763 2.129 6.577 10.269 -7.167 29.873

A (GPa) 0.359 0.304 0.254 0.247 0.223 0.230 0.358 0.266

ξ(GPa) 0.240 0.274 0.275 0.291 0.302 0.321 0.242 0.431

Tm(K) 1460.80 1609.56 1796.76 1932.41 2103.58 2224.58 1461.11 2275.58

υl (m/s) 2326.274 2513.802 2767.831 2907.567 3024.201 3156.458 ---- 3217.591

υt (m/s) 3909.211 4525.923 5223.247 5609.545 6022.603 6357.950 ---- 6522.205

υm (m/s) 2961.153 3260.925 3631.182 3833.188 4013.397 4198.241 ---- 4284.748

θD(K) 362 366 371 374 380 382 ---- 578.968


Page | 396

8.7 Pressure variation on physical properties of CaLiF3

In this section, the effect of pressure variation on electronic structure, elastic constants,

mechanical stability, and thermodynamic characteristics of cubic CaLiF3 fluoroperovskite

have been investigated by employing first-principles method. For the total energy

calculations, the Full-Potential Linearized Augmented Plane Wave (FP-LAPW) method

(Schwarz et al., 2010) is employed. Thermodynamic properties are computed in terms of

Quasi-harmonic Debye model (Blanco et al., 2004), within 0-50 GPa pressure and 0-600 K

temperature.


The structural properties are determined via different volumes over a range ± 10% which are

selected to calculate minimum ground state energy (EO) at zero pressure. Here Birch

Murnaghan’s equation of state (EOS) (Murnaghan 1944) is used to fit the minimum energy

(EO) versus minimum volume (VO). The calculated ground state lattice parameters (at zero

pressure) such as lattice constant of the present work is 3.687 Å, as shown in Table 8.25, is in

good agreement with previously reported theoretical/ experimental work (Mousa et al., 2013;

Mishra et al., 2011; Ouenzerfi 2004 & Castro 2002).

In order to examine the crystal structure of CaLiF3 on different hydrostatic pressure (0 to 50

GPa), we attempt to study the effect of different pressure, with a step size of 10 GPa, on

lattice parameters. Figure 8.36 depicts calculated change in the value of lattice constant by

LDA and GGA approximations. It can be interpreted that lattice constant is going to decrease

both by LDA and GGA approximations. The variation of bond lengths Ca-F and Li-F with

pressure is also presented in Figure 8.37. However, with comparison of data at constant

pressure of the same series as shown in Table 8.26 (Erum and Iqbal 2016), it can be analyzed


Page | 397

that Li-F and Ca-F bond lengths compress to a reasonable extent within the limit that

constituent polyhedral of CaLiF3 do not become distorted with the change in the pressure.

The reduction in values of bond lengths and lattice constant can be associated via difference

of bandgap and hybridization strength which is previously discussed in section 8.3.


The elastic properties have been investigated by using Charpin method. Furthermore, within

the Charpin method the elastic properties of cubic crystals are elucidated by the elastic

constants namely C11, C12, and C44 respectively (Charpin 2001). The computed values of the

C11, C12, and C44 are presented in Table 8.27, which shows reasonable agreement with

previous work (Erum and Iqbal 2016) at fixed (zero) pressure. The focus of this section is to

examine the effect of pressure variation on shear strength, tensile strength, rigidity of the

investigated compound whom calculational details can be found in previous section 8.3.3-

8.3.4 and in the following references (Shafiq et al., 2015; Brik 2011 & Kleinman 1962).

The deformation of material under any small stresses can characterized by elastic constants

(Pettifor 1992). Figure 8.38 presents effect of pressure variation on elastic moduli such as

C11, C12, and C44 respectively. From Figure 8.38 and Table 8.27 it can be observed that with

the rise in compression all elastic constants increase. Elasticity in length C11, is observed with

increase in pressure. The growth in C11 can be described in terms with enhancement in bond

strength (Ca-F, and Li-F) because there is decrease in the bond lengths and increase of charge

density at elevated pressure ranges. The shape deformation elastic moduli become less

effected against elevated pressure that indicates less bond strength enhancement. It is evident

that compound obey the cubic stability conditions (Wang et al., 1993) and modified stability

criteria (Sadd 2005) for cubic crystals under finite strain till 40 GPa. It can be perceived from


Page | 398

Figure 8.39 that considered elastic constants do not obey stability condition, M2 as mentioned

in section 8.3.3, 8.4.3, and 8.5.3 because at 50 GPa pressure stability criteria of M2 value is

lower than zero (-4.55), which implies that calculated elastic constants cannot satisfy all

mechanical stability conditions, hence the compound is not mechanically stable above 40

GPa. As a result, it can be ensured that CaLiF3 undergo no structural phase transition for

pressure up to 40 GPa.


The elastic parameters (calculated in previous section) relates thermal and mechanical

behavior of compound (Shafiq et al., 2015). This section is dedicated to calculate effect of

hydrostatic pressure on mechanical properties of CaLiF3 compound according to some

proposed mathematical relationships as cited in the following reference (Brik 2011 &

Kleinman 1962) and whom calculational details can be found in previous section 8.3.3-8.3.4

as well. From the trend of Young’s modulus (Y), Bulk modulus (B0), Reuss’s shear modulus

(GR), Voigt’s shear modulus (GV), and Hill’s shear modulus (GH), it can be observed that

CaLiF3 has highest value of stiffness and rigidity at pressure of 50 GPa and lowest value of

stiffness and rigidity at 0 GPa pressure which means that material becomes stiffer and less

compressible when applied pressure is increased, as shown in Figure 8.40. These results are

in reasonable accordance with the previous work related to perovskite compounds under the

influence of varying pressure by Rai and his fellows (Rai et al., 2014).

Furthermore, CaLiF3 changes its behavior from brittle character into ductile one as pressure

changes from 0 to 50 GPa in accordance with work done by Jenkins and Khanna (Jenkins &

Khanna 2005) that, the B/G less than 1.75 conforms material’s brittle character, on the other

hand, its value greater than 1.75 makes compound as ductile. Poisson’s ratio (ѵ) reveals the


Page | 399

alike trend with pressure change. The anisotropy elastic factor (A), interprets an increase in

the degree of deviation with increased pressure. The Kleinman parameter can be used to

analyze bond bending (Kleinman 1962), that is basically consistent with above mentioned

results. Next, we found that an application of hydrostatic compression induces less tendency

of melting extent for calcium based fluoroperovskite.


The Debye temperature (θD) explores the crystal’s response towards vibration. In this

investigation we use elastic constant method to perform these calculations. Above θD crystal

behaves classically. Here, method based on elastic constant is employed to calculate θD. The

details of this method and about their related formulas can be found in section 8.5.5 and in

following references (Rached et al., 2009; Schreiber and Anderson 1973 & Anderson 1963)

as well. Figure 8.41 (a) depicts effect of pressure variation within (0-50 GPa) on average,

longitudinal, and transverse wave velocities as a function of pressure (0- 50 GPa). A

prominent quadratic variation in wave velocities can be observed which monotonically

increases with the every 10 GPa increase in pressure. In fact, longitudinal wave velocity

increases rapidly as compared to shear wave velocity. The increasing trend of θD is also in

similar accordance with elastic wave velocity as shown in Figure 8.41 (b). Furthermore,

increase in θD is in direct relation with bulk modulus because of the fact that its high value

persuades phonon softening within the applied pressure.


These parameters of a material provide a way that gives contributing factors in development

of thermodynamic chemistry and solid-state physics. These aspects of CaLiF3 are determined


Page | 400

using Quasi-harmonic Debye model as implemented in the Gibbs program (Blanco et al.,

2004 & Francisco et al., 1998). In this model, from the calculated energy-volume data all

thermodynamic quantities can be evaluated. The details of this method and about their related

formulas can be found in section 8.5.5 and in following references (Rached et al., 2009;

Schreiber and Anderson 1973 & Anderson 1963) as well. In this study thermal properties are

analyzed within 0 to 600 K range with pressure variation within 0-50 GPa range. The specific

constant volume heat capacity CV and constant pressure heat capacity CP are shown in Figure

8.42 (a) and Figure 8.42 (b). It can be observed from figures that variation of CV and CP with

the increase in temperature are similar to each other. Further, the trend of CV rises gradually

and inclines to move in the direction of Dulong-Petit limit which is 123.7 J mol-1K-1, as is

common phenomenon in all solids at high temperature (Bouhemadou et al., 2009). Moreover,

the anharmonic effects are suppressed at high temperature. However, CP decreases as

increase in pressure from (0- 50 GPa) and deviation is observed from a constant value. So, at

ambient pressure, the values of CP increase quickly at elevated temperature. The discrepancy

of the volume (expansion) coefficient α(T) for CaLiF3, is given by the Figure 8.42 (c). It can

be clearly seen that α exhibits enhanced growth for low temperature values and then

progressively tends to rise linearly at elevated temperature ranges. It should be noted that the

progress of α with the temperature becomes smaller as the pressure rises. However, for a

given temperature, α decreases sharply with increasing pressure. The Debye cut-off

frequency or Debye temperature (θD) is important due to extraction of some useful physical

quantities for example specific heat capacities and melting point (Anderson 1963). Figure

8.42 (d) reveals that value of θD remains smooth till 100 K and for temperature greater than

200 K it decreases monotonically. While for a constant temperature, θD rises linearly with


Page | 401

pressure. However, at zero pressure and ambient temperature, our calculated θD from elastic

constant method is 649.019 K which are closer to calculated values from the quasi-harmonic

model.


physical behavior of CaLiF3 in cubic phases so hopefully this work will motivate research





Page | 402

Figure 8.36: The Pressure variation of Lattice Constant (a) LDA (b) GGA

Figure 8.37: The Pressure variation of Bond lengths (a) Ca-F (b) Li-F


Page | 403

Figure 8.39: Stability criteria for cubic CaLiF3 compound as a function of pressure.


for CaLiF3 compound.


Page | 404

Figure 8.40: Calculated pressure dependence of isotropic elastic parameters (a) Bulk

modulus (B) (b) Shear modulus (G) (c) Young’s modulus (Y) (d) B/G ratio for CaLiF3

compound.


Page | 405

Figure 8.41 (b): Calculated pressure dependence of Debye temperature (θD)

for CaLiF3 compound.

Figure 8.41 (a): Calculated pressure dependence of elastic wave velocities

(a) υl (b) υt (c) υm for CaLiF3 compound.


Page | 406

Figure 8.42 (a): Variation of the specific heat capacities of Cv as a function of temperature

at different pressures for CaLiF3 compound.


Page | 407

Figure 8.42 (b): Variation of the specific heat capacities of Cp as a function of temperature



Page | 408

Figure 8.42 (c): Temperature dependence of the volume expansion coefficient α(T)



Page | 409




Page | 410



and its pressure derivative (B′) at ambient pressure of CaLiF3 compound.

Compound: CaLiF3 Present

work (LDA)

Present

work (GGA)

Previous work

ao (Å) 3.687 3.772 3.672a, 3.606b, 3.760c

Bo (GPa) 98.87 81.28 98.78b

B′ (GPa) 4.51 4.25 4.43c

a)( Ouenzerfi 2004), b)( Castro 2002) (Experimental Work) c)( Mousa et al., 2013) (Other theoretical work)


(E in Ry), Volume of unit cell (V in (a.u.)3) and Bond length (dCa-F, dLi-F) of CaLiF3

compound.


Pressure (GPa) Energies (Ry) Volume (a.u.)3 dCa-F (Ǻ) dLi-F (Ǻ) dX-F (Ǻ)

(X=Li,Na,K,Rb)a

0 -6774.891 362.18412 2.531 1.851 dSr-F = 2.57

(0.00 GPa)

10 -6774.863 359.02472 2.527 1.845 dLi-F = 1.85

(0.00 GPa)

20 -6774.855 357.02379 2.521 1.841 dNa-F = 2.23

(0.00 GPa)

30 -6774.839 354.17824 2.518 1.838 dK-F = 2.60

(0.00 GPa)

40 -6774.812 350.50168 2.511 1.832 dRb-F = 2.74

(0.00 GPa)

50 -6774.798 347.41069 2.509 1.828


Page | 411

Table 8.27: Calculated values of elastic constants (C11, C12, C44) of CaLiF3 at pressure from

0-50 GPa.

a) (Erum and Iqbal, November 2017)

Table 8.28: Calculated values of derived elastic constants characterizing mechanical stability

of CaLiF3 at pressure from 0-50 GPa.

a) (Mishra et al., 2011)


Work (0.00 GPa)

(SrLiF3) a

C11 (GPa) 156.11 157.254 158.251 160.001 161.017 162.548 157.741

C12 (GPa) 34.351 34.38 34.998 35.105 35.35 36.12 37.353

C44 (GPa) 43.31 43.851 44.014 44.12 45.125 45.452 49.447


Work (0.00 GPa)

(BaLiF3) a

M1 74.94 78.67 82.75 86.74 90.57 94.93 73.90

M2 43.31 33.85 24.01 14.12 5.13 -4.55 44.13

M3 60.88 51.44 41.63 32.45 22.83 13.21


Page | 412


shear modulus (GR), Hill’s shear modulus (GH), Young’s modulus (Y), and B/G ratio, of

CaLiF3 at pressure from 0-50 GPa.

a) (Mousa et al., 2013), b) (Erum and Iqbal, 2016), c) (Mishra et al., 2011)

Table 8.30: Calculated values of Poisson’s ratio (ѵ), Anisotropy constant (A), Kleinman

parameter (ξ), melting temperature (Tm) longitudinal (υl in m/s), transverse (υt in m/s),

average sound velocity (υm in m/s), and Debye temperature (θD in K) of CaLiF3 at pressure

from 0-50 GPa.

a) (Erum and Iqbal, 2016), b) (Mishra et al., 2011)


Work

(0.00 GPa)

Bo(GPa) 76.252 77.512 79.285 82.159 86.214 89.128 77.510a

Gv(GPa) 50.338 50.885 51.059 51.451 52.208 52.557 52.515 (SrLiF3) b

GR(GPa) 48.962 49.521 49.695 49.988 50.858 51.207 52.281(SrLiF3) b

GH(GPa) 49.650 50.203 50.377 50.720 51.533 51.882 52.401 (SrLiF3) b,

52.202c

Y(GPa) 122.387 123.867 124.716 126.192 128.915 130.353 127.362 (SrLiF3) b

B/G (GPa) 1.536 1.544 1.574 1.620 1.673 1.718 1.49 (SrLiF3) b

Pressure

(GPa)

0 10 20 30 40 50 Previous

Work (0.00 GPa)

(SrLiF3) a,

b

Ѵ (GPa) 0.232 0.234 0.238 0.244 0.251 0.256 0.215a

A (GPa) 0.711 0.714 0.714 0.707 0.718 0.719 0.81a

0.83b

ξ(GPa) 0.421 0.419 0.422 0.422 0.423 0.424 0.456a

Tm(K) 1871.14 1882.86 1899.35 1926.08 1963.79 1990.89 1854.65a

υl (m/s) 3935.274 4011.802 4142.831 4344.567 4544.201 4656.458 ----

υt (m/s) 6639.011 6798.923 6923.247 7029.545 7122.603 7357.950 ----

υm (m/s) 5054.102 5211.925 5351.182 5533.188 5813.397 5918.241 ----

θD(K) 649 652 657 661 664 667 ----


Page | 413

8.8 Conclusion

In this chapter, effect of hydrostatic pressure on physical properties of five fluoroperovskites

(SrLiF3, SrNaF3, SrKF3, SrRbF3 and CaLiF3) have been carried out successfully.

In Section 8.3, detailed physical properties of SrLiF3 are studied under ambient and high-

pressure ranges. The calculated equilibrium lattice parameters at 0 GPa are in good

agreement with earlier reports. The calculation of elastic properties under various pressure

ranges confirms that the compound is mechanically stable in cubic structure up to 40 GPa.

All elastic and mechanical parameters are linearly dependent on applied pressure. Moreover,

an increase in pressure improves tensile strength and stiffness, on the other hand, reduces

brittleness and compressibility of cubic fluoroperovskite SrLiF3. It is observed that an

increase in pressure considerably improves the wide and direct (Γ-Γ) electronic bandgap

because upon compression bands broadened the energy of Sr-4d and Sr-5d states thereby

resulting an increase in the ratio of splitting between Sr-5d, Sr-4d, Sr-3d, F-2s and F-2p states

which ultimately increase in the bandgap and this increase in bandgap continues upto 50

GPa. All optical responses shift towards higher energy ranges which reveals that SrLiF3 is

more suitable for optoelectronic devices at higher pressure ranges. Finally, thermodynamic

effects on macroscopic properties are predicted to verify application of this compound in

thermodynamic devices in the range 0-50 GPa and 0-600 K. Since SrLiF3 fluoroperovskite

do not undergo any structural phase transition at high pressure so it can be used as an

alternative pressure marker for other materials.

In Section 8.4, we have reported for the first time, detailed theoretical results on effect of

pressure dependence on structural, electronic elastic, mechanical, optical and thermodynamic


Page | 414

properties of cubic SrNaF3 compound based on Density functional theory (DFT). The

calculated lattice parameters at fixed (zero) pressure and temperature are in good agreement

with previous experimental work. The pressure dependence of elastic constants and

significant mechanical parameters confirm compound’s mechanically stability in cubic

structure till 20 GPa. Moreover, an increase in pressure improves tensile strength and

stiffness, on the other hand, reduces brittleness and compressibility of SrNaF3 compound. It

is observed that an increase in pressure considerably improves the wide and direct (Γ-Γ)

electronic bandgap because upon compression bands broadened the energy of Sr-4d and Sr-

5d states thereby resulting in an increase in the ratio of splitting between Sr-5d, Sr-4d, Sr-3d,

F-2s and F-2p states which ultimately increase in the bandgap. All the calculated optical

properties such as the complex dielectric function ε(ω), optical conductivity σ(ω), energy

loss function L(ω), absorption coefficient α(w), refractive index n(ω), reflectivity R(ω), and

effective number of electrons neff, via sum rules shift towards the higher energies under the

application of pressure. Finally, thermodynamic effects on macroscopic properties are

predicted to verify application of this compound in thermodynamic devices using Quasi-

harmonic Debye model in the range 0-25 GPa and 0-600 K. Since increase in pressure

improves elastic and mechanical behavior of SrNaF3 compound, so it can effectively be used

in lens materials. Consequently, we believe that our theoretical results have benchmarked

various quantum mechanical effects at different pressures, which must be taken into account

to understand and utilize in fabricating practical devices.

In Section 8.5, effect of pressure variation on detailed physical properties have investigated

for the first time by ab-inito Density Functional Theory (DFT) method for cubic phase of

SrKF3 fluoroperovskite compound. The calculated equilibrium lattice parameters are in good


Page | 415

agreement with previous theoretical and experimental reports at 0 GPa. It is observed that an

increase in pressure considerably improves the wide and direct (Γ-Γ) electronic nature of

bandgap because at elevated pressure ranges bands broadened the energy of Sr-3d and Sr-4d

states thereby resulting in an increase in the ratio of splitting between Sr-4d, Sr-3d, K-2p, F-

2s and F-2p states which ultimately results an increase in the bandgap of the material. The

pressure dependence of elastic constants and significant mechanical parameters confirm

compound’s mechanically stability in cubic structure till 20 GPa. Moreover, an increase in

pressure improves tensile strength and stiffness, on the other hand, reduces brittleness and

compressibility of the SrKF3 compound. The effect of thermodynamic parameters on

macroscopic properties are predicted to utilize this material in temperature dependent

applications implementing Quasi-harmonic Debye model within the range 0-25 GPa and 0-

600 K with the step size of 5 GPa and 100 K respectively. Consequently, we believe that our

work will motivate research scholars to done theoretical as well as experimental studies in

this direction, which must be taken into account to understand and utilize this material in

fabricating practical devices.

In Section 8.6, physical properties of SrRbF3 under varying pressure are investigated by

using ab-initio study. The calculation of elastic properties under pressure confirms that the

compound is mechanically stable in cubic structure and compound undergo no structural

phase transition, till 20 GPa. All elastic and mechanical parameters are linearly dependent on

applied pressure. Moreover, an increase in pressure reduces brittleness and compressibility

while improves tensile strength and stiffness. Furthermore, macroscopic thermodynamic

properties are successfully evaluated to apply this material in pressure and temperature

dependent applications, within 0-25 GPa and 0-600 K ranges. The static value of Debye


Page | 416

temperature using Quasi-harmonic Debye model are in good agreement with that one

calculated from the elastic constant method. So, we hope that our work will motivate

research scholars to produce valuable studies in this direction for exploring various device

applications.

In Section 8.7, the effect of pressure variation on physical properties of CaLiF3 are

investigated. The calculation of elastic properties under pressure confirms that the compound

is mechanically stable in cubic structure till 40 GPa. All elastic and mechanical parameters

are linearly dependent on applied pressure. Moreover, an increase in pressure reduces

brittleness, compressibility and improves tensile strength as well as stiffness. Furthermore,

macroscopic thermodynamic properties are successfully evaluated to apply this material in

temperature dependent applications within the range 0-50 GPa and 0-600 K with the step size

of 10 GPa and 100 K respectively. The static value of Debye temperature using Quasi-

harmonic Debye model are in reasonable accordance with that one calculated from the elastic

constant method. So, we hope that our work will motivate research scholars to produce

theoretical as well as experimental studies in this direction, which must be taken into account

to understand and utilize this material in low birefringence lens fabrication technology.

Chapter 9 Conclusions and future work

Page | 417

Chapter 9

Conclusions and future work

“Now this is not the end

It is not even the beginning of the end

but it is, perhaps, the end of the

Beginning”

Sir Winston Churchill (1942)

9.1 Conclusions

Materials with varied types of bonding interactions and therefore some computational

approximations are better suited for theoretical simulations, before expensive

experimentations. The work in this thesis has focused on understanding pressure and

temperature dependent computational and quantum mechanical interrelations between

composition, structure and physical properties in new fluoride and oxide related perovskite

materials, such as SrLiF3, CaLiF3, SrNaF3, SrKF3, SrRbF3, KVF3, KFeF3, KCoF3, KNiF3,

KPaO3, RbPaO3, BaPaO3 and BaUO3 compounds. All these calculations are carried out using

first principles density-functional theory, using WIEN2K code. The main emphasize of each

property is to evaluate its application because for technological needs, the importance of the

ability to control the properties is crucial in order to maintain high efficiency while avoiding

material degradation at various temperatures and pressures.

During this research project 11-12 research articles (as first author and corresponding author)

have already been published in well reputed international journals such as Solid State

Communications, Physica B, Materials Research Express, Chinese Physics B,

Communications in Theoretical Physics, and Chinese Journal of Physics are worthy to


Page | 418

mention. The purpose of this chapter is to summarize significant conclusion and our

contributions to new knowledge about “Perovskite family” via theoretical simulations. In the

end, future recommendations are briefly stated.

9.2 Mechanical and opto-electronic properties of Perovskites

This section is devoted to express our contributions to novel knowledge on “Perovskite

family” specially on fluoride and oxide perovskite family of compounds.

Comprehensive results of structural, elastic, mechanical and opto-electronic properties of

SrMF3 (M= Li, Na, K, Rb), reveals increase in value of lattice constants, in accordance with

the experimental studies, as cation shift from Lithium to Rubidium, while value of bulk

modulus decreases, that can be attributed to higher extent of atomic radii of Rubidium. These

elastically and mechanically stable compounds have dominant brittle and ionic behavior.

Furthermore opto-electronic trends, via various exchange and correlation schemes, provides

accurate description of band profiles, which permits to investigate reliable predictions of

electronic charge density and density of states. These calculations argue against the existence

of low bandgap values that have been studied previously with less reliable LDA and GGA

schemes. but there is lack of experimental data so in description we compare (TB-mBJ) band

gap results that are generally similar to experimental band profile of BaLiF3 compounds. As a

whole, strontium based alkali earth fluoroperovskites need an extensive experimental

research for their possible utilization in Ultra-Violet (UV) transparent lens material and in

advanced lithographic technology (Publication 1 and 2).

Structural, and opto-electronic studies of RbHgF3 verifies indirect narrow energy bandgap

(M–Γ) semi-conductive nature, following mixed covalent and ionic behavior. The valuable


Page | 419

optical responses in high frequency region authenticates that compound can be efficiently

utilized in manufacturing high class lens material with low birefringence (Publication 3).

In this work, all electron self-consistent full potential-linearized augmented plane wave (FP-

LAPW) method is used to explore structural, electronic, mechanical and optical properties of

XPaO3 (X= K, Rb) within Generalized Gradient Approximation (GGA), Local Density

Approximation (LDA) and Tran-Blaha modified Becke–Johnson (TB-mBJ) potential. The

estimated structural parameters are found to be comparable with available data. Energy band

profile confirms that the investigated materials are (Γ– Γ) direct bandgap semiconductors.

The curves of total and partial density of states are used to determine the contribution of

different bands. The detailed studies of elastic and mechanical parameters prove flexible,

anisotropic and covalent nature of the herein compounds. These results are in favorable

agreement with previous theoretical and existing experimental data. The optical properties

are discussed in terms of complex dielectric function Ԑ(ω) and the analysis is carried out by

interband contribution that shows the XPaO3 (X= K, Rb) compounds possess wide ranges of

absorption and reflection in high frequency regions and these characteristics make them

useful for flexible opto-electronic applications. Hence, these perovskites are efficiently

employed in scientific investigation and need an extensive experimental research for their

possible technological benefit (Publication 4).

The structural parameters of BaXO3 (X= Pa, U) oxide perovskites has shown to be in good

agreement with previous experimental reports. Type of chemical bonding is analyzed with

the help of variations in electron density difference distribution that is induced due to

changes of second cation. Detailed analysis of opto-electronic responses by LDA, GGA

approximations and TB-mBJ potential reveals (Γ-Γ) direct bandgap semi-conductive nature


Page | 420

in both compounds. Finally, prominent variation of optical responses suggests that BaPaO3

and BaUO3 are applicant materials for micro as well as nano-electronic devices.

In summary, these actinoid based oxide perovskites, have valuable features in one aspect or

another, so by extensive experimental research and via properly handling their radioactive

nature, versatile outcomes can be achieved for their possible technological benefits.

Furthermore, this investigation can be upgraded if the two materials can be doped with

another magnetic semiconductor element to make BaPaO3 and BaUO3 semiconductor

compounds. (Publication 5).

9.3 Magneto-opto-electronic properites of fluoroperovskites

In this thesis the unique theoretical strategy is used to investigate KVF3, KFeF3, KCoF3, and

KNiF3 fluoroperovskites by LSDA, GGA approximation and TB-mBJ potential based on

DFT. Structural properties are calculated by DFT as well as by analytical methods and are

found in close agreement with each other. A complete description of elastic, mechanical, and

some of thermal parameters confirms anisotropic and mixed covalent-ionic nature. From the

findings of elastic and mechanical properties, it can be inferred that these compounds are

elastically stable and anisotropic while KCoF3 is harder than rest of the compounds. Our

calculations show that the bandgap results of TB-mBJ potential are much consistent with the

available experimental data. The stable magnetic phase optimizations verify the experimental

observations at low temperature. Furthermore, the calculated spin dependent magneto-

electronic properties in these compounds reveal that exchange splitting is dominated by N-3d

orbital. Optical properties, show that these compounds have wide range of absorption and

reflection in high frequency regions. Consequently, the present methodology represents


Page | 421

detailed set of structural and magneto-opto-electronic parameters, which will explore an

opportunity to understand numerous physical phenomena and permit material scientists for

implementing these materials in spintronic applications, for their possible technological

benefits (Publication 6).

9.4 Pressure and temperature dependent physical aspects of

fluoroperovskites

The structural, electronic, elastic, optical and thermodynamic properties of cubic

fluoroperovskite SrLiF3 and SrNaF3 at ambient and high-pressure are investigated by using

first-principles total energy calculations within the framework of Generalized Gradient

Approximation (GGA), combined with Quasi-harmonic Debye model in which the phonon

effects are considered. The pressure effects are determined in the range of 0-50 GPa, and 0-

25 GPa, in which cubic stability of SrLiF3 and SrNaF3 fluoroperovskite remains respectively

valid. The computed lattice parameters agree well with experimental and previous theoretical

results. Decrease in lattice constant and bonds length is observed with the increase in

pressure. The effect of increase in pressure on electronic band structure calculations with

GGA and GGA plus Tran-Blaha modified Becke–Johnson (TB-mBJ) potential reveals a

predominant characteristic associated with widening of bandgap. It is observed that an

increase in pressure considerably improves the wide and direct (Γ-Γ) electronic bandgap

because upon compression bands broadened the energy of Sr-4d and Sr-5d states thereby

resulting in an increase in the ratio of splitting between Sr-5d, Sr-4d, Sr-3d, F-2s and F-2p

states which ultimately increase in the bandgap. Moreover, an increase in pressure improves

tensile strength and stiffness, on the other hand, reduces brittleness and compressibility of

both fluoroperovskites. All optical responses shift towards higher energy ranges which


Page | 422

reveals that SrLiF3 and SrNaF3 are more suitable for optoelectronic devices at higher

pressure ranges. Finally, thermodynamic effects on macroscopic properties are predicted to

verify application of these compounds in thermodynamic devices using Quasi-harmonic

Debye model within wide temperature and pressure ranges. Since these fluoroperovskites do

not undergo any structural phase transition at high pressure so it can be used as an alternative

pressure marker for other materials (Publication 7 and 8).

The effect of pressure variation on stability, structural parameters, elastic constants,

mechanical, electronic and thermodynamic properties of cubic SrKF3, and SrRbF3

fluoroperovskites are investigated by ab-initio Density functional theory (DFT) method. The

calculated equilibrium lattice parameters are in good agreement with previous theoretical and

experimental reports at 0 GPa. It is observed that an increase in pressure considerably

improves the wide and direct (Γ-Γ) electronic nature of bandgap because at elevated pressure

ranges bands broadened the energy of Sr-3d and Sr-4d states thereby resulting in an increase

in the ratio of splitting between Sr-4d, Sr-3d, K/Rb-2p, F-2s and F-2p states which ultimately

results an increase in the bandgap of the material. The pressure dependence of elastic

constants and significant mechanical parameters confirm compound’s mechanically stability

in cubic structure till 20 GPa. Moreover, an increase in pressure improves tensile strength

and stiffness, on the other hand, reduces brittleness and compressibility of both compounds.

Significant influence of compression on wide range of elastic parameters and related

mechanical properties have been discussed, to utilize this material in low birefringence lens

fabrication technology. The effect of thermodynamic parameters on macroscopic properties

are predicted to utilize this material in temperature dependent applications implementing

Quasi-harmonic Debye model within the range 0-25 GPa and 0-600 K with the step size of 5


Page | 423

GPa and 100 K respectively. The static value of Debye temperature using Quasi-harmonic

Debye model are in good agreement with that one calculated from the elastic constant

method. Consequently, we believe that our work will motivate research scholars to produce

theoretical as well as experimental studies in this direction which must be considered to

understand and utilize this material in fabricating practical devices (Publication 9 and 10).

The effect of pressure variation (0-50 GPa) on electronic structure, elastic parameters,

mechanical durability, and thermodynamic aspects of calcium based CaLiF3 in combination

with Quasi-harmonic Debye model are studied. A prominent decrease in the value of lattice

constant and bonds length is observed with the increase in pressure. The presently calculated

lattice parameters are in good agreement with the previous experimental reports. The

calculation of elastic properties under pressure confirms that the compound is mechanically

stable in cubic structure till 40 GPa. All elastic and mechanical parameters are linearly

dependent on applied pressure. The transition from brittle to ductile behavior is also observed

with the intent of increase in pressure. Furthermore, macroscopic thermodynamic properties

are successfully evaluated to apply this material in temperature dependent applications within

the range 0-50 GPa and 0-600 K with the step size of 10 GPa and 100 K respectively.

Consequently, we believe that our theoretical results have benchmarked various quantum

mechanical effects at different pressures, which must be considered to understand and utilize

in fabricating practical devices (Publication 11).

To conclude in brief, the work in this thesis hopefully emphasizes the need for more

systematic studies of fundamental properties in complex fluoride and oxide perovskites.

These types of investigations are essential in developing today’s technology based on

material science. More detailed investigations, in particular of local structures and their


Page | 424

relations to the average structures and properties, are therefore necessary for quantum

mechanical investigations in order to understand how computational simulations, structure

and properties are interrelated and how they can be effectively controlled for specific

applications.

Our detailed theoretical studies on several series of perovskites, with various computational

tools showed that several theoretical methods are needed in order to characterize the physical

properties correctly, otherwise one will end up with wrong conclusions. Drawbacks involved

in our calculational procedure includes that, in electronic and magnetic properties we give

more emphasize on LDA, GGA and mBj exchange correlation schemes on transition metal

based fluoroperovskites. However, the work can be crosschecked from LDA/GGA+U

method for proper exploration of strongly correlated systems (Anisimov et al., 1997).

Furthermore, the possibility of band gap engineering by the addition of suitable dopant

element can enhance material’s properties. Very few experimental results are available to

compare with our calculations because most of the compounds are explored for the first time.

So, hopefully this work will motivate research scholars to done theoretical as well as

experimental studies in this direction, so they can compare their results with our work to get

better beneficial understanding about specific application on these materials.

9.5 Future work plan

In future, our work plan is to elaborate current investigation from cubic oxide and

fluoride perovskites to explore physical aspects of non-cubic oxide and halide

perovskites, anti-perovskites, inverse perovskites and double perovskites.


Page | 425

To explore thermoelectric properties, vibrational properties and various properties

based on spectroscopic analysis.

Effect of doping on perovskites.

Surface and bulk studies of nanoparticle perovskites.

Study of phase transitions occurred in perovskites.

To explore magnetic properties of strongly correlated systems by adapting different

(DFT+U) schemes.

Multiferric properties and electric polarization may also be studies in future.

Particle size variation study of perovskite family for technological benefits.

From application point of view, our future effort is to search for new, cheaper and

multipurpose materials with higher saturation magnetization and controllable via

carrier induced ordering process for set of complete halide series.

References

Page | 426

REFERENCES

Abbes, L. and Noura, H., “Perovskite oxides MRuO3 (M = Sr, Ca and Ba): Structural

distortion, electronic and magnetic properties with GGA and GGA-modified Becke

Johnson approaches”. Results in Physics, 5 (2015) 38-52.

Abelès, F., “Optical Properties of Solids”. North Holland Publishing Company, (1972).

Ahmad, A. M., “First-principles study of structural, electronic and optical properties of the

KCaX3 (X = F and Cl) compounds”. International Journal of Modern Physics B,

28(21) (2014) 1450139.

Ahmad, M., Naeemullah, Murtaza, G., Khenata, R., Omran, S. B. and Bouhemadou, A.,

“Structural, elastic, electronic, magnetic and optical properties of RbSrX (C, SI, Ge)

half-Heusler compounds”. Journal of Magnetism and Magnetic Materials, 377 (2015)

204-210.

Alder, B. J. and Wainwright, T. E., "Studies in Molecular Dynamics. I. General Method".

Journal of Chemical Physics, 31(2) (1959) 459.

Aleksiejuk, M. and Kraska, D., “Elastic properties of potassium cobalt (II) trifluoride

(KCoF3)”. Physica status solidi (a), 31(1) (1975) K65–K68.

Ali, Z., Sattar, A., Asadabadi, S. J. and Ahmad, I., “Theoretical studies of the osmium based

perovskites AOsO3 (A=Ca, Sr and Ba)”. Journal of Physics and Chemistry of Solids,

86 (2015) 114–121.

Alonso, J. A., Benitez, J. S., Andres, A. D., Lope, M. J. M., Casais, M. T. and Martinez, J. L.,

“Enhanced magnetoresistance in the complex perovskite LaCu3Mn4O12.” Applied

Physics Letters, 83(13) (2003) 2623.

Alouani M., Albers C. and Methfessel M., “Calculated elastic constants and structural

properties of Mo and MoSiZ”. Physical Reviews B, 43 (1991) 6500- 6509.

References

Page | 427

Alvarez, G., Conde-Gallardo, A., Montiel, H. and Zamorano, R., “About room temperature

ferromagnetic behavior in BaTiO3 perovskite”. Journal of Magnetism and Magnetic

Materials, 401 (2016)196–199.

Amin, B., Ahmad, I., Maqbool, M., Goumri-Said, S. and Ahmad, R., “Ab-initio study of the

bandgap engineering of AlxGa1-xN for optoelectronic applications". Journal of

Applied Physics, 109 (2011) 024509.

Anderson, O. L., “A simplified method for calculating the debye temperature from elastic

constants”. J. Phys. Chem. Solids, 24(7) (1963) 909-917.

Androulakis, J., Migiakis, P. and Giapintzakis, J., “La0.95Sr0.05CoO3: An efficient room-

temperature thermoelectric oxide”. Applied physics letters, 84(7) (2004) 1099-1101.

Anisimov, V. I., Aryasetiawan, F. and Lichtenstein, A. I., “First- principles calculations of

the electronic structure and spectra of strongly correlated systems: the LDA+U

method”. Journal of Physics: Condensed Matter, 9 (1997) 767-808.

Anisimov, V. I., Aryasetiawan, F. and Lichtenstein, A. I., “First-principles calculations of the

electronic structure and spectra of strongly correlated systems: the LDA + U

method”. Journal of Physics: Condensed matter, 9 (1997) 767-808.

Anthony, R. W., “Solid State Chemistry and its Applications”. John Wiley & Sons 2 (2003).

Anzai, M., Kawakami, H., Saito, M., and Yamamura, H., “Thermoelectric properties of p-

type perovskite compounds LaCoO3 systems containing the A-site vacancy”.

Materials Science and Engineering, 18 (2011) 142005.

Arlt, G. and Hennings, D., “Dielectric properties of fine‐grained barium titanate ceramics”.

Journal of applied physics, 58(4) (1985) 1619-1625.

Ascher, E., Rieder, H., Schmid, H. et al., “Some properties of ferromagnetoelectric Nickle-

Iodine Boracite Ni3B7O13I”. Journal of Applied Physics, 37(3) (1966) 1404-1405.

Babu, K. E., Murali, N., Babu, K. V., Shibeshi, P. T. and Veeraiah, V., “Structural, Elastic,

Electronic, and Optical Properties of Cubic Perovskite CsCaCl3 Compound: An ab

initio Study”. Acta Physica Polonica A, 125(5) (2014) 1179-1185.

References

Page | 428

Babu, K. E., Veeraiah, A., Swamy, D. T. and Veeraiah, V., “First-principles study of

electronic and optical properties of cubic perovskite CsSrF3”. Materials science

Poland, 30(4) 2012 359-367.

Bayindir, M., Sorin, F., Abouraddy, A. F., Viens, J., Hart, S.D., Joannopoulos, J.D. and Fink,

Y., “Metal-Insulator-Semiconductor optoelectronic fibres”. Nature, 431(7010)

(2004) 826-829.

Becke, A. D. and Johnson, E. R. A., “Simple effective potential for exchange”. The Journal

of Chemical Physics, 124(22) (2006) 221101.

Becke, A. D., “Density-functional exchange-energy approximation with correct asymptotic

behavior”. Physical Review A., 38(6) (1988) 3098-3100.

Bednorz, J. G. and Muller, K. A., “Possible high Tc Superconductivity in the

Ba-La-Cu-O system,” Physik B Condensed Matter, 64 (1986) 189-193.

Bellis, M., “Teflon ® - Roy Plunkett”. About.com (2013).at

<http://inventors.about.com/library/inventors/blteflon.htm>

Bellis, M., “Vulcanized Rubber Charles Goodyear received two patents for methods of

making rubber better”. About.com (2013).at

<http://inventors.about.com/od/gstartinventors/a/CharlesGoodyear.htm>

Benedek, N.A. and Fennie, C. J., “Why are there so few perovskite ferroelectrics?” Journal

of Physical Chemistry C, 117(26) (2013) 13339–13349.

Besnalah, A., Shimamura, K., Fujita, T., Sato, H., Nikl, M. and Fukuda, T., “Growth and

characterization of BaLiF3 single crystal as a new optical material in the VUV

region”. Journal of Alloys and compounds, 348(1-2) (2003) 258.

Bhalla, A. S., Guo R. and Roy, R., “The perovskite structure – a review of its role in ceramic

science and technology”. Materials Research Innovations, 4 (2000) 3-26.

Bhalla, A. S., Guo R. Y. and Roy, R., “The perovskite structure, a review of its role in

ceramic science and technology,” Materials Research Innovations, 4(1) (2000) 3-26.

References

Page | 429

Blaha, P., Schwarz, K. and Madsen, G. K. H., “Electronic structure calculations of solids

using the WIEN2k package for material sciences”. Computer Physics

Communications, 147(1-2) (2002) 71–76.

Blaha, P., Schwarz, K., Sorantin, P., and Trickey, S. B. “Full-potential, linearized augmented

plane wave programs for crystalline systems”. Computer Physics Communications,

59(2) (1990) 399.

Blaha., P, Schwarz, K., Madsen, G. K. H., Kvasnicka, D., Luitz, J. and Schwarz, K., “An

Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal

Properties”. WIEN2K User’s Guide, (2008) Techn. Universitat Wien, Wien.

Blanco, M. A., Francisco, E. and Luaña, V., “GIBBS: isothermal-isobaric thermodynamics of

solids from energy curves using a quasi-harmonic Debye model”. Computer Physics

Communication, 158(1) (2004) 57.

Blundell, S., “Magnetism in Condensed Matter”. New York: Oxford University Press,

(2001).

Bordet, P., Darie, C., Goujon, C., Bacia, M., Klein H. and Suard, E., “Magnetic and crystal

structure of the BiCrO3 multiferroic compound”. Acta Crystallographica, A63 (2007)

s45.

Born, M. and Oppenheimer, R., “Zur Quantentheorie derMolekeln”. Annalen der Physik,

(Leipzig) 389(20), 457-484 (1927).

Born, M., "Physical aspects of quantum mechanics". Nature, 119 (1927) 354–357.

Bouhemadou, A., Khenata, R. and Amrani, B., “Structural and thermodynamic properties of

the cubic perovskite BiAlO3”. Physica B, 404(20) (2009) 3534-3538.

Brik, M. G., “Comparative first-principles calculations of electronic, optical and elastic

anisotropy properties of CsXBr3 (X=Ca,Ge,Sn ) crystals”. Solid State

Communications, 151(23) (2011) 1733-1738.

References

Page | 430

Bringley, J. F., Scott, B. A., Placa, S. J. L., McGuire, T. R., Mehran, F., McElfresh, M. W.

and Cox D. E., “Structure and properties of the LaCuO3−δ perovskites”. Physical

review, 47 (1993) 15269.

Brittman, S., Adhyaksa, G. W. P. and Garnett, E. C., “The expanding world of hybrid

perovskites: materials properties and emerging applications”. MRS Communications

5 (2015) 7–26.

Campbell, P. K., Jones, K. E., Huber, R. J., Horch, K. W. and Normann, R. A., “A silicon-

based, three-dimensional neural interface: manufacturing processes for an

intracortical electrode array”. IEEE Transactions in Biomedical Engineering, 38(8)

(1991) 758-768.

Carignano, M. A., Kachmar, A. and Hutter, J., “Thermal effects on CH3NH3PbI3 perovskite

from ab-initio molecular dynamics simulations”. Journal of chemical physics (2014)

arXiv. org., arXiv: 1409.6842v2.

Castro, M.A.C., “Estabilidad Estructural y Quimica de Halurocompuestos Binaries y

Ternaries”. PhD. Thesis, University of Oviedo, Asturias (2002).

Catherine, E. H. and Sharpe, A. G., “Inorganic Chemistry” Pearson Education Limited, 3

(2008).

Ceperley, D. M. and Alder, B. J., “Ground State of the Electron Gas by a Stochastic

Method". Physical Review Letters, 45(7) (1980) 566.

Chakhmouradian, A. R. and Mitchell, R. H., "Compositional variation of perovskite-group

minerals from the Khibina Complex, Kola Peninsula, Russia". The Canadian

Mineralogist, 36 (1998) 953–969.

Chan, W. H., Xu, Z., Zhai, J. and Chen, H., “Uncooled tunable pyroelectric response of

antiferroelectric Pb0.97La0.02 (Zr0.65Sn0.22Ti0.13)O3 perovskite”. Applied Physics Letters, 87

(2005).

References

Page | 431

Charmola, A., Singh, H. and Naithani, U. C., “Study of Pb(Zr0.65Ti0.35) O3(PZT (65/35) doping

on structural, dielectric and conductivity properties of BaTiO3 (BT) ceramics,”

Advanced Materials Letters, 2(2) (2011) 148-152.

Charpin, T., “A Package for Calculating Elastic Tensors of Cubic Phases Using WIEN”.

(Paris: Laboratry of Matrix) (2001)

Chen, B. A., Liu, G., Wang, R. H., Zhang, J. Y., Jiang, L., Song. J. J. and Sun, J., “Effect of

interfacial solute segregation on ductile fracture of Al–Cu–Sc alloys”.Acta Materialia,

61(5) (2013) 1676-1690.

Chen, F., Ewing, R. C. and Clark, S. B. “The Gibbs free energies and enthalpies of formation

of U6+ phases: an empirical method of prediction”. Amer Miner, 84 (1999) 650-684.

Chen, Q., De Marco, N., Yang, Y., Song, T. B., Chen, C. C., Zhao, H., Hong, Z., Zhou, H.

and Yang, Y., “Under the spotlight: the organic–inorganic hybrid halide perovskite

for optoelectronic applications”. Nano Today 10(3) (2015) 355–396.

Choi, K. J., Biegalski, M., Li, Y., Sharan, A. et al., “Enhancement of ferroelectricity in

strained BaTiO3 thin films”. Science, 306 (2004) 1005-1009.

Chou, C. C. and Chen, C. S., “Banded Structure and Domain Arrangements in PbTiO3 Single

Crystals”. Japanese Journal of Applied Physics, 37 (1998) 5394-5396.

Chu, C. W., Gao, L., Chen, F., Huang, Z. J., Meng, R. L. and Xue, Y. Y.,

“Superconductivity above 150 K in HgBa2Ca2Cu3O8+δ at high pressures” Nature,

365 (1993) 323–325.

Clementi, E., Raimondi, D. L. and Reinhardt, W. P., “Atomic Screening Constants from SCF

Functions”. The Journal of Chemical Physics, 38(11) (1963) 2686.

Cora, F. and Catlow, C. R. A., “QM investigations on perovskite-structured transition metal

oxides: bulk, surfaces and interfaces”. Faraday Discussions, 114 (1999) 421-442.

Coulaud, O., Bordat, P., Fayon, P., LeBris, V., Baraille, I. and Brown, R., “Extensions of the

siesta DFT code for simulation of molecules”. Computational physics, (2013) arXiv.

org., arXiv: 1302.4617v1.

References

Page | 432

Cox, P. A., “Transition metal oxides; An introduction to their electronic structure and

properties”. International Series of Monographs on Chemistry, Oxford University

Press, Incorporated, 27 (1992).

Csavinszky, P., “Approximate Variational Solution of the Thomas-Fermi Equation for

Atom”. Physical Review, 166 (1968) 53.

Cullity, B. D., “Elements of X-ray diffraction”. Addison-Wesley, 2 (1977).

Curie, P., “Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique

d'un champ magnétique”. Journal de Physique, 3(1) (1894) 393.

Das, N., “Application of perovskites towards remediation of environmental pollutants: An

overview”. International Journal of Environmental Science and Technology, 14(7)

(2017) 1559–1572.

Deacon-Smith, D., “Structure prediction of perovskite surfaces and nanoclusters”. Ph.D.

dissertation University College, London (2015).

Demetriou, D. Z., Catlow, C. R. A., Chadwick, A. V., Mclntyre G. J. and I. Abrahams, “The

Anion Disorder in the Perovskite Fluoride KCaF3”. Solid State Ionics, 176(17)

(2005)1571-1575.

Dimos, D. and Mueller, C., “Perovskite Thin Films for High-Frequency Capacitor

Applications 1”. Annual Review of Materials Science, 28 (1998) 397-419.

Dogan, F., Lin, H., Guilloux-Viry M. and Pena, O., 2015 “Focus on properties and

applications of perovskites”. Science and Technology of Advanced Materials, 16(2)

(2015) 020301.

Dotzler C., Williams G. V. M., Edgar A., “Radiation-induced optically and thermally

stimulated luminescence in RbCdF3 and RbMgF3”. Current Applied Physics 8(3-4)

(2008) 447-450.

Dovesi, R., Fava, F. F., Roetti C. and Saunders, V. R., “Structural, Electronic and Magnetic

Properties of KMF3 (M = Mn, Fe, Co, Ni)”. Faraday Discussions - Royal Society of

Chemistry, 106 (1997) 173.

References

Page | 433

Dreizler, Reiner R. G. and Eberhard, K. U., “Density Functional Theory”. Springer-Verlag

Berlin, (1990).

Dressel, M. and Gruner, G., “Electrodynamics of Solids: Optical Properties of Electrons in

Matter”. Cambridge University Press, UK, (2002).

Duan, R. R., Speyer, R. F., Alberta, E. and Shrout, T. R., “High curie temperature perovskite

BiInO3 PbTiO3 ceramics”. Journal of Materials Research, 19 (2004) 2185-93.

Düvel, A., Efimov, K., Feldhoff, A., Šepelák, V., Heitjans, P. and Wilkening, M., “Access to

metastable ion conductors via mechanosynthesis: Preparation, structure and ionic

conductivity of mixed (Ba,Sr)LiF3 with inverse perovskite structure”. Journal of

Materials Chemistry, 21(17) (2011) 6238-6250.

Duvel, A., Wegner, S., Efimov, K., Feldhoff, A., Heitjans, P. and Wilkening, M., “Access to

metastable complex ion conductors viamechanosynthesis: preparation,

microstructure and conductivity of (Ba,Sr)LiF3 with inverse perovskite structure”.

Journal of Materials Chemistry, 21(17) (2011) 6238.

Düvel, M., Wegner, W. S., Feldhoff, A., Šepelák, V. and Heitjans, P., “Ion conduction and

dynamics in mechanosynthesized nanocrystalline BaLiF3”. Solid State Ionics, 184

(2011) 65.

Ebbing, D. D. and Wrighton, M. S., “General Chemistry”. Houghton Mifflin, Boston (1984).

Ederer, C. and Spaldin, N. A., “Influence of strain and oxygen vacancies on the

magnetoelectric properties of multiferroic bismuth ferrite”. Physical Review B,

71(22) (2005) 224103.

Eitel, R.E., Randall, C.A., Shrout, T.R., Rehrig, P. W., Hackenberger, W. and Park, S. E.,

“New high temperature morphotropic phase boundary piezoelectrics based on Bi

(Me) O3-PbTiO3 ceramics”. Japanese Journal of Applied Physics, 40 (2001) 5999-

6002.

References

Page | 434

Erum, N. and Iqbal, M. A., “A novel pressure variation study on electronic structure,

mechanical stability and thermodynamic properties of potassium based

fluoroperovskite”. Materials Research Express, 4 (2017) 096302.

Erum, N. and Iqbal, M. A., “Ab initio study of high dielectric constant oxide-perovskites:

Perspective for miniaturization technology”. Materials Research Express, 4(2) (2017)

025904.

Erum, N. and Iqbal, M. A., “Effect of hydrostatic pressure on physical properties of

strontium based fluoroperovskites for novel applications”. Materials Research

Express, 5 (2018) 025904.

Erum, N. and Iqbal, M. A., “Effect of pressure variation on structural, elastic, mechanical,

optoelectronic and thermodynamic properties of SrNaF3 fluoroperovskite”. Materials

Research Express, 4 (2017) 126311.

Erum, N. and Iqbal, M. A., “Effect of pressure variation on structural, elastic, mechanical,



Erum, N. and Iqbal, M. A., “First Principles Investigation of Fluorine Based Strontium Series

of Perovskites”. Communications in Theoretical Physics, 66(5) (2016) 571.

Erum, N. and Iqbal, M. A., “First principles investigation of protactinium-based oxide-

perovskites for flexible opto electronic devices”. Chinese Physics B, 26(4) (2017)

047102.

Erum, N. and Iqbal, M. A., “Mechanical and magneto-opto-electronic investigation of

transition metal based fluoro-perovskites: An ab-initio DFT study”. Solid State

Communications, 264 (2017) 39–48.

Erum, N. and Iqbal, M. A., “Opto-electronic investigation of Rubidium based Fluoro-

Perovskite for low birefringent lens materials”. Scientific Inquiry Review, 1(1) (2016)

1- 4.

References

Page | 435

Erum, N. and Iqbal, M. A., “Physical properties of fluorine based perovskites for vacuum-

ultraviolet-transparent lens materials”. Chinese Journal of Physics, 55(3) (2017) 893–

903.

Erum, N. and Iqbal, M. A., “Physical properties of fluorine based perovskites for vacuum-

ultraviolet-transparent lens materials”. Chinese Journal of Physics, 56(4) (2018)

1353-1361.

Erum, N. and Iqbal, M. A., “Study of pressure variation effect on structural, opto-electronic,

elastic, mechanical, and thermodynamic properties of SrLiF3”. Physica B, 525 (2017)

60–69.

Fergus, J. W., “Perovskite oxides for semiconductor-based gas sensors”. Sensors and

Actuators B: Chemical, 123(2) (2007) 1169–1179.

Filippi, C., Umrigar, C. J. and Taut, M., “Comparison of exact and approximate density

functionals for an exactly soluble model”. Journal of Chemical Physics, 100 (1994)

1295-7.

Fine, M. E., Brown, L. D. and Marcus, H. L., “Elastic constants versus melting temperature

in metals”. Scripta Metallurgica, 18(9) (1984) 951-956.

Flocken, J. W., Guenther, R. A., Hardy, J. R. and Boyer, L. L., “A Priori Predictions of Phase

Transitions in KCaF3 and RbCaF3: Existence of a New Ground State”. Physical

Review Letters, 56 (1986) 1738-1741.

Fock, V., “Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems”.

Zeitschrift für Physik, 61(1-2) (1930) 126-128.

Fox, M Optical Properties of Solids, Oxford University Press, New York, 2001 (Second

edition).

Francisco, E., Recio, J. M., Blonco, M. A. and Pendas, A. M., “Quantum-Mechanical Study

of Thermodynamic and Bonding Properties of MgF2”. Journal of Physical

Chemistry, 102(9) (1998) 1595-1601.

References

Page | 436

Francisco, M. A. and Blonco, G. S., “Atomistic simulation of SrF2 polymorphs”. Physical

review B, 63 (2001) 094107.

Frantsevich, I. N., Voronov, F. F. and Bokuta, S. A., “Elastic constants and elastic moduli of

metals and insulators handbook”.Naukova Dumka, Kiev, (1983) 60–180.

Gelatt, Jr. C. D., Williams, A. R. and Moruzzi V. L., “Theory of bonding of transition metals

to nontransition metals”. Physical Reviews B, 27 (1983) 2005-2013.

Geusic, J., Marcos, H. and Uitert, L.V., “Laser oscillations in Nd‐doped yttrium

aluminum, yttrium gallium and gadolinium garnets”. Applied Physics Letters, 4

(1964)182-184.

Ghebouli B., Ghebouli M. A., Bouhemadou A., Fatmi M., Khenata R., Rached D., Ouahrani

T. and Bin-Omran S., “First principles study of the structural, elastic, electronic,

optical and thermodynamic properties of the cubic perovskite CsCdCl3 under high

pressure”. Solid State Science, 14 (2012) 903.

Giannozzi, P et al., “Quantum ESPRESSO: A modular and open-source software project for

quantum simulations of materials”. Journal of physics: Condensed Matter, 21 (2009)

395502.

Gibbs, J. W., “A Method of Geometrical Representation of the Thermodynamic Properties of

Substances by Means of Surfaces”. Transactions of the Connecticut Academy of Arts

and Sciences, 2 (1873) 382-404.

Gill, P. M. W., Johnson, B. G., Pole, J. A. and Frisch, M. J., “An investigation of the

performance of a hybrid of Hartree-Fock and density functional theory”.

International Journal of Quantum Chemistry, 44(S26) (1992) 319-331.

Goldschmidt, V. M., “Die gesetze der krystallochemie”. Die Naturwissenschaften, 14(21)

(1926) 477–485.

Goodenough, J. B., “Magnetism and the chemical bond”. Interscience monograph on

chemistry, 1 (1963) 393.

References

Page | 437

Goodenough, J. B., “Metallic oxides”. Progress in Solid State Chemistry, 5(C) (1971)145–

399.

Goodenough, J., “Narrow-band electrons in transition-metal oxides”. Czechoslovak

Journal of Physics, 17 (1967) 304–336.

Grabo, T., Kreibich, T. and Gross, E. K. U., “Optimized Effective Potential for Atoms and

Molecules”. Molecular Engineering, 7(1-2) (1997) 27–50.

Green, M. A., Ho-Baille, A. and Snaith, H. J., “The emergence of perovskite solar cells”.

Nature Photon, 8(7) (2014) 506–514.

Greiner, W., Ludwig N. and Horst, S., “Thermodynamics and statistical mechanics”.

Springer-Verlag, (1995) 101.

Griffiths, D. J., “Introduction to Quantum Mechanics”. Prentice Hall, (2nd ed.) (2004).

Gu, F., Chen, Y. Y., Zhang, X. L., Zhang, J. and Liu, Q., “First-principles calculations for the

structural, elastic, and thermodynamic properties of cubic perovskite BaHfO3 under

pressure”. Physica Scripta, 89 (2014)105703-12.

Haines, J., Lger, J. M. and Bocquillon, G., “Synthesis and Design of Superhard Materials”.

Annual Review of Materials Research, 31(1) (2001) 1-23.

Handley, R.C.O., “Modern magnetic materials: principles and applications”. John Wiley &

Sons. (2000) 83.

Harmel, M., Khachai, H., Ameri, M., Khenata, R., Baki, N., et al., “Dft-based ab initio study

of the electronic and optical properties of cesium based fluoro-perovskite CsMF3 (M

= Ca and Sr)”. International Journal of Modern Physics B, 26(32) (2012) 1250199

(13 pages).

Harmel, M., Khachai, H., Haddou, A., Khenata,R., Murtaza, G., Abbar, B., Omran, S. B. and

Khalfa, M., “Ab Initio Study of the Mechanical, Thermal and Optoelectronic

Properties of the Cubic CsBaF3”. Acta Physica Polonica A, 128(1) (2015).

https://en.wikipedia.org/wiki/John_Wiley_%26_Sons

https://en.wikipedia.org/wiki/John_Wiley_%26_Sons

References

Page | 438

Harrison, W. A., “Electronic structure and the properties of interfaces”. Ultramicroscopy, 14

(1-2) (1984) 85-87.

Hartree, D., “The Wave Mechanics of an Atom with a Non-Coulomb Central Field”. Part I.

Theory and Methods. Mathematical Proceedings of the Cambridge Philosophical

Society, 24(1), (1928) 89-110.

Hastings, W. K., "Monte Carlo sampling methods using Markov chains and their

applications". Biometrika, 57(1) (1970) 97–109.

Hautier, G., Fischer, C., Ehrlacher, V., Jain, A. and Ceder, G., “Data Mined Ionic

Substitutions for the Discovery of New Compounds”. Inorganic chemistry;

American Chemical Society, 50(2) (2011) 656–663.

Hayatullah, Murtaza, G., Khenata, R., Mohammad S., Naeem, S., Khalid, M. N. and Manzar,

A., “Structural, elastic, electronic and optical properties of CsMCl3 (M= Zn, Cd)”.

Physica B, 420 (2013) 15–23.

Hayatullah, Murtaza, G., Muhammad, S., Naeem, S., Khalid, M. N. and Manzar, M.,

“Physical properties of CsSnM3 (M = Cl, Br, I): A First Principle Study”. Acta

Physica Polonica A, 124 (2013) 1113-1118.

He, J. and Franchini, C., “Screened hybrid functional applied to 3d0→3d8 transition-metal

perovskites LaMO3 (M = Sc–Cu): Influence of the exchange mixing parameter on the

structural, electronic, and magnetic properties”. Physical review, B 86 (2012)

235117.

He, Y. and Galli, G., “Perovskites for solar thermoelectric applications: A First Principle

study of CH3NH3AI3 (A=Pb and Sn)”. Chemistry of Materials, 26(18) (2014)

5394−5400.

Hill, R., “The Elastic Behavior of a Crystalline Aggregate”. Proceedings of the Physical

Society, Section A 65(5) 1952 349.

Hinatsu, Y., “The Magnetic Susceptibility and Structure of BaUO3”. Journal of Solid State

Chemistry, 102(2) (1993) 566.

References

Page | 439

Hoffmann, R., “How chemistry and physics meet in the solid state”. Angewandte Chemie

International Edition, 26(9) (1987) 846–878.

Hohenberg, P. C. and Kohn, W., “Inhomogenous electron gas”. Physical Review, 136(3B)

(1964) 864–871.

Holden, T. M., Buyers, W. J. L., Svensson, E. C., Cowley, R. A., Hutchings, M. T., Hukin,

D. and Stevenson, R. W. H., “Excitations in KCoF3-I. Experimental”. Journal of

Physics C: Solid State Physics, 4(14) (1971) 2127.

Holgate, S., “Understanding Solid State Physics”. CRC Press, (2009) 177-178.

Homes, C. C., Vogt, T., Shapiro, S. M., Wakemoto S. and Ramirez A. P., “Optical Response

of High-Dielectric-Constant Perovskite-Related Oxide,” Science 293(5530) (2001)

673-676.

Imada, M., Fujimori, A. and Tokura, Y., “Metal-insulator transitions”. Reviews of Modern

Physics, 70(4) (1998) 1039.

Irwin, T. H., “Aristotle's First Principles”. Oxford University Press, Oxford, (1988).

Ishihara, T., “Perovskite Oxide for Solid Oxide Fuel Cells”. Springer, (2009).

Ito, A. and Morimoto, S., “Mössbauer Study of Strain Sensitive Properties of KCoF3 and

KFeF3”. Journal of the physical society of Japan, 16(1) 1977.

Iyer, P. N. and Smith, A. J., “Double oxides containing niobium, tantalum or protactinium.

IV. Further systems involving alkali metals”. Acta Crystallographica, B27 (1971)

731-734.

Jaffe, B., Cook, W. R. and Jaffe, H., “Piezoelectric Ceramics”. Academic Press

Incorporation, London, 1 (1971).

Jaffe, B.; Cook, W. R. & Jaffe, H. (1971). Piezoelectric Ceramics, Academic Press Inc

Jain, A., Ong, S. P., Hautier, G., Chen, W., Richards, W. D., Dacek, S., Cholia, S., Gunter,

D., Skinner, D., Ceder, G., Persson, K. A., "The Materials Project: A materials

genome approach to accelerating materials innovation”. APL Materials, (2013) 1(1).

https://books.google.com/books?id=eefKBQAAQBAJ&pg=PA178

References

Page | 440

Jamal, M., Abu-Jafar, M. S. and Reshak, A. H., “Revealing the structural, elastic and

thermodynamic properties of CdSexTe1−x (x = 0, 0.25, 0.5, 0.75, 1)”. Journal of

Alloys and Compounds, 667 (2016) 151-157.

Jamal, M., Asadabadi, S. J., Ahmad, I. and Rahnamaye, H. A. A., “Elastic constants of cubic

crystals”. Computational Materials Science, 95 (2004) 592–599.

Jana, L., Petra, S. and Trojan, M., "Study of Perovskite”. Journal of Thermal Analysis and

Calorimetry, 93 (3) (2008) 823–827.

Jenkins, C. and Khanna, S., “A Modern Integration of Mechanics and Materials in Structural

Design”. Elsevier Academic Press, USA (2005).

Jeon, N. J., Noh, J. H., Yang, W. S., Kim, Y. C., Ryu, S., Seo, J. and Seok, S.,

“Compositional engineering of perovskite materials for high-performance solar

cells”. Nature,517(7535) (2015) 476–480.

Jiang, L. Q., Guo, J. K., Liu, H. B., Zhu M., Zhou, X., Wu, P. and Li, C. H., “Prediction of

lattice constant in cubic perovskites,” Journal of Physics and Chemistry of Solids, 67

(7) (2006) 1531-1536.

Jong, M. D., Chen, W., Angsten, T., Jain, A., Notestine, R., Gamst, A., Sluiter, M., Ande, C.

K., Zwaag, S. V. D., Plata, J. J. et al., “Charting the complete elastic properties of

inorganic crystalline compounds”. Scientific Data, 2(2015)150009.

Kaczkowski, J. and Jezierski, A., “Electronic Structure of the Cubic Perovskites BiMO3 (M =

Al, Ga, In, Sc)”. Acta Physica Polonica A, 124 (5) (2013) 852-854.

Kanamori, J., “Superexchange interaction and symmetry properties of electron orbitals”.

Journal of Physics and Chemistry of Solids, 10(2) (1959) 87–98.

Katzir, S., “The discovery of the piezoelectric effect”. Archive for history of exact sciences,

57 (2003) 61-91.

Kawada, T. and Yokokawa, H., “Materials and characterization of solid oxide fuel cell”. Key

Engineering Materials, 125-126 (1997) 187–248.

References

Page | 441

Kay, H. and Bailey, P., “Structure and properties of CaTiO3”. Acta Crystallographica, 10 (3)

(1957) 219-226.

Keller, C., “Die reaktion der dioxide der elemente thorium bis americium mit niob- und

tantalpentoxid”. Journal of Inorganic and Nuclear Chemistry, 27(6) (1965) 1233-

1246.

Khan, B., Rahnamaye Aliabad H. A., Razghandi, N., Maqbool, M., Asadabadi, S. J. and

Iftikhar, Ahmad., “Structural and thermoelectric properties of pure and La, Y doped

HoMnO3 for their use as alternative energy materials”. Computer Physics

Communications, 187 (2015) 1-7.

Khan, B., Rahnamaye, H. A., Razghandi, N., Maqbool, M., Jalali, S. A. and Ahmad, I.,

“Structural and thermoelectric properties of pure and La, Y doped HoMnO3 for their

use as alternative energy materials”. Computer Physics Communications, 187 (2015)

1–7.

Kim, D. H., Wiler, J. A., Anderson, D. J., Kipke, D. R. and Martin, D. C., “Conducting

polymers on hydrogel-coated neural electrode provide sensitive neural recordings in

auditory cortex”. Acta Biomaterialia, 6(1) (2010) 57-62.

Kim, Y. I. and Woodward, P. M., “Crystal structures and dielectric properties of ordered

double perovskites containing Mg2+ and Ta5+”. Journal of Solid State Chemistry, 180 (2007)

2798–2807.

Kingon, A., Streiffer, S., Basceri, C. and Summerfelt, S., “High-permittivity perovskite

thin films for dynamic random-access memories”. Mrs Bulletin, 21(7) (1996) 46-52.

Kittel, C., “Introduction to Solid State Physics.”, eighth ed.,John Wiley & Sons Inc., USA

(2005).

Kleinman, L., “Deformation Potentials in Silicon. I. Uniaxial Strain.” Physical Reviews B,

128 (1962) 2614.

Knížek, K., Jirák, Z., Hejtmánek, J. and Novák, P., “Character of the excited state of the Co3+

ion in LaCoO3”. Journal of Physics: Condensed Matter, 18 (2006) 3285–3297.

References

Page | 442

Knizhnik, A., "Interrelation of preparation conditions, morphology, chemical reactivity and

homogeneity of ceramic YBCO". Physica C: Superconductivity. 400(1-2) (2003) 25-

35.

Kobayashi, K., Yamaguchi, S., Mukaida, M. and Tsunoda, T., “Thermoelectric power of

titanate and ferrite with the cubic perovskite structure”. Solid State Ionics,

144(3-4) (2001) 315– 320.

Kobayashi, T., Takizawa, H., Endo, T., Sato, T., Shimada, M., Taguchi, H. and Nagao, M. J.,

“Metal-insulator transition and thermoelectric properties in the system ( R1-

xCax)MnO3- δ ( R: Tb, Ho, Y)”. Journal of Solid State Chemistry, 92 (1991) 116-129.

Kocsis, M., “Ieee and Ieee, in 1999 Ieee Nuclear Science Symposium “. Conference Record,

1-3 Ieee, New York (1999) 741-743.

Koferstein, R., “Synthesis, phase evolution and properties of phase-pure nanocrystalline

BiFeO3 prepared by a starch-based combustion method”. Journal of Alloys and

Compounds, 590 (2014) 324-330.

Kohanoff, J. and Gidopoulos, N. I., “Density Functional Theory: Basics, New Trends and

Applications”. John Wiley & Sons, Ltd, Chichester, 2(5) (2003) 532–568.

Kohn, W. and Sham, L. S., “Self-consistent equations including exchange and correlation

effects”. Physical Review, 140 (1965) A1133.

Koller, D., Tran, F. and Blaha, P., “Merits and limits of the modified Becke-Johnson

exchange potential’. Physical Review B, 83 195 (2011) 134-10.

Korba S. A., Meradji, H., Ghemid, S. and Bouhafs, B., “First principles calculations of

structural, electronic and optical properties of BaLiF3” Computational Materials

Science, 44(4) (2009) 1265–1271.

Kumar, A., Verma A. S. and Bhardwaj, S. R., “Prediction of Formability in Perovskite-Type

Oxides” The Open Applied Physics Journal 1(1) (2008) 11-19.

References

Page | 443

Kuo, Y. K., Liou, B. T., Yen, S. H. and Chu, H. Y., “Vegard's law deviation in lattice

constant and band gap bowing parameter of zincblende InxGa1− xN”. Optics

Communications, 237(4-6) (2004) 363.

Kurosaki, K., Matsuda, T., Uno, M., Kobayashi, S. and Yamanaka, S., “Thermoelectric

properties of BaUO3”. Journal of Alloys and Compounds, 319(1-2) (2001) 271.

Kutzelnigg, W., “Density Functional Theory in Terms of a Legendre Transformation for

Beginners”. Journal of Molecular Structure: THEOCHEM, 768(1-3) (2006)163-173.

Lang, L., Yang, J., Liu, H., Xiang, H. J. and Gong, X. G. “First-principles study on the

electronic and optical properties of cubic ABX3 halide perovskites”. Physics Letters

A, 378(3) (2014) 290–293.

Langley, R. H., Schmitz, C. K., Langley, M. B., “The synthesis and characterization of some

fluoride perovskites, an undergraduate experiment in solid state chemistry”. Journal

of Chemical Education, 61(7) (1984) 643–645.

Lany, S. and Zunger, A., “Accurate prediction of defect properties in density functional

supercell calculations”. Modelling and Simulation in Materials Science and

Engineering, 17(8) (2009) 084002.

Laskowski, R., Madsen, G. K. H., Blaha, P. and Schwarz, K., “Magnetic structure of

electric-field gradients of uranium dioxide: An ab initio study”. Physical Review B,

69 (2004) 140408.

Lee, J. S., Lee, Y. S. and Noh, T. W., “Bond-length dependence of charge-transfer

excitations and stretch phonon modes in perovskite ruthenates: Evidence of strong of

p-d hybridization effects”. Physical Review B, 70(8) (2004) 085103.

Lee, J., Shin, H., Chung, H., Zhang, Q., Saito, F., “Mechanochemical Syntheses of

Perovskite KMIIF3 with Cubic Structure (MII = Mg, Ca, Mn, Fe, Co, Ni, and Zn)”.

Materials Transactions. 44(7) (2003) 1457-1460.

Lejaeghere, K., Speaybroeck, V. V., Oost, G, V. and Cottentier, S., “Error estimates for

solid-state density-functional theory predictions: An overview of the ground-state

References

Page | 444

elemental crystals”. Critical Reviews in Solid State and Materials Science, 39(1)

(2013) 1-24.

Li, C., Lu, X., Ding, W., Feng, L., Gao, Y. and Guo, Z., “Formability of ABX3 (X = F, Cl,

Br, I) halide perovskites”. Acta Crystallographica B, 64(Pt 6) (2008) 702–707.

Li, X., Zhao, H., Xu, N., Zhou, X., Zhang, C. and Chen N., “Electrical conduction behavior

of La, Co co-doped SrTiO3 perovskite as anode material for solid oxide fuel cells”.

International journal of hydrogen energy, 34(15) (2009) 6407–6414.

Li, Y., Gemmen, R. and Liu, X., “Oxygen reduction and transportation mechanisms in solid

oxide fuel cell cathodes”. Journal of Power Sources 195(11) 3345–3358 (2010).

Linz Jr, A. and Herrington, K., “Electrical and optical properties of synthetic calcium

titanate crystal”. The Journal of Chemical Physics, 28(5) (1958) 824-825.

Litimein, F., Bouahafs, B., Dridi, Z. and Ruterana, P., “The electronic structure of wurtzite

and zincblende AlN: ab initio comparative study". New Journal of Physics, 4 (2002)

64.1.

Lombardo, E. A. and Ulla, M. A., “Perovskite oxides in catalysis: past, present and future”.

Research on Chemical Intermediates, 24(5) (1998) 581–592.

Lu, Y., Li, Q. A., Di, N. L. and Cheng, Z. H., “Magnetic properties of Nd0.5Pb0.5-xSrxMnO3

materials”. Chinese Physics, 12 (2003) 789-791.

Luana, V., Costales, A., Pendas, A. M., Florez M. and Fernandez, V. M. G., “Structural and

chemical stability of halide perovskites”. Solid State Communication, 104(1) (1997)

47-50.

Lufaso, M. W. and Woodward, P. M., “Prediction of the crystal structures of perovskites

using the software program SPuDS,” Acta Crystallogr B, 57(Pt 6) (2001) 725–738.

Luo, J., Im J. H., Mayer, M. T., Schreier, M. et al., “Water photolysis at 12.3% efficiency via

perovskite photovoltaics and Earth-abundant catalysts”. Science, 345(6204) (2014)

1593–1596.

References

Page | 445

Mabbs, F. E. and Machin, D. J., “Magnetism and Transition Metal Complexes”. Dover

Publications Incorporation, (2008).

Machin, D. J., Martin, R. L. and Nyholm, R. S., “The preparation and magnetic properties of

some complex fluorides having the perovskite structure”. Journal of the chemical

Society, 0 (1963) 1490-1500.

Madsen, G. K. H., Blaha, P., Schwarz, K., Sjostedt, E. and Nordstrom, L., “Efficient

linearization of the augmented plane-wave method”. Physical Review B, 64

(2001) 195134.

Maeno, Y., Hashimoto, H., Yoshida, K., Nishizaki, S., Fujita, T., Bednorz, J. G., &

Lichtenberg, F. (1994). “Superconductivity in a layered perovskite without copper”.

Nature, 372, 532-534.

Majid, A. and Lee, Y. S., “Correlating lattice constant of cubic perovskites to atomic

parameters using support vector regression”. International journal of Advances in

Information Sciences and Service Sciences, 2(3) (2010) 118-127.

Manivannan, V., Parhi, P. and Kramer, J. W., “Metathesis synthesis and characterization of

complex metal fluoride, KMF3 (M = Mg, Zn, Mn, Ni, Cu and Co) using

mechanochemical activation”. Bulletin of Materials Science, 31(7) (2008) 987–993.

Marc, D. G. and McHenry, M. E., “Structure of materials: an introduction to crystallography,

diffraction and symmetry”. Cambridge University Press, Cambridge, England,

(2007) p.671.

Marezio, M., Remeika, J. P. and Dernier, P. D., “The crystal chemistry of the rare earth

orthoferrites”. Acta Cryst. B, 26 (2008).

Martin, R. M., “Electronic Structure”, Cambridge University Press, (2004).

Martinez-Guerrero, E., Enjalbert, F., Barjon, J., Amalric, E. B., Daudin, B., Ferro, G.,

Jalabert, D., Dang, L. S., Mariette, H., Monteil, Y. and Mula, G., “Optical

Characterization of MBE Grown Zinc Blende AlGaN". physica status solidi (a), 188

(2001) 695-698.

References

Page | 446

Matsuda, T. and others, "Heat capacity measurement of BaUO3". Journal of Alloys and

Compounds, 322(1-2) (2001) 77-81.

Matthias, B. T., “New Ferroelectric Crystals”. Physical Review, 75(11) (1949) 1771.

Mattis, D. C., “The Theory of Magnetism Made Simple”. World Scientific Publishing, 580

(2006).

Mavin, J. Weber, “Handbook of Optical Materials”. CRC Press LLC, (2003).

McHale, J. M. J., “Solution based synthesis of perovskite-type oxide films and powders”.

Los Alamos National Lab., NM (LA-UR--95-440). (United States). Funding

organization: USDOE, Washington, DC (United States) (1995).

Mehonic, A., Cueff, S. B., Wojdak, M., Hudziak, S., Jambois, O., Labbé, C., Garrido, B.,

Rizk, R. and Kenyon, A. J., “Resistive switching in silicon suboxide films”. Journal

of Applied Physics, 111(7) (2012) 074507.

Messiah, A., “Quantum Mechanics”. 1 North-Holland, Amsterdam (1961).

Meziani, A. and Belkhir, H., “First-principles calculations of structural, elastic and electronic

properties of CsCaF3 compound”. Computational Materials Science, 61 (2012) 67-

70.

Miehlich, B., Savin, A., Stoll, H. and Preuss, H., “Results obtained with the correlation

energy density functionals of becke and Lee, Yang and Parr”. Chemical Physics

Letters, 157(3) (1989) 200-208.

Mishra, A. K., Garg, N., Shanavas, K. V., Achary, S. N. and Tyagi, A. K., “High pressure

structural stability of BaLiF3”. Journal of Applied Physics, 110(12) (2011) 123505.

Mitzi, D. B., “In Progress in Inorganic Chemistry”. Ed. by K. D. Karlin, John Wiley & Sons

Inc, New York, 48 (1999) 1–121.

Molenda, J., Świerczek, K. and Zając, W., “Functional materials for the IT-SOFC”.

Journal of Power Sources, 173(2) (2007) 657–670.

References

Page | 447

Monkhorst, H.J. and Pack, J. D., “Special points for Brillouin-zone integrations”. Physical

Review B, 13 (1976) 5188.

Mooser, E. and Pearson, W. B., “On the crystal chemistry of normal valence compounds,”

Acta Cryst. 12 (1959) 015-22.

Moreira, R. L. and A, Dias., “Comment on `Prediction of lattice constant in cubic

perovskites”. Journal of Physics and Chemistry of Solids, 68 (2007) 1617.

Morss, L. R., Edelstein, N. and Fuger, J., “The Chemistry of the Actinide and Transactinide

Elements”. Springer 4 (2010) 185-200.

Moskvin, S., Makhnev, A. A., Nomerovannaya, L. V., Loshkareva, N. N. and Balbashov, A.

M., “Interplay of p-d and d-d charge transfer transitions in rare-earth perovskite

manganites”. Physical Review B: Condense Matter Materials Physics, 82(3) (2010)

035106.

Mousa, A. A., Mamoud, N. T. and Khalifeh, J. M., “The electronic and optical properties of

the fluoroperovskite XLiF3 (X= Ca, Sr, and Ba) compounds”. Computational

Materials Science, 79 (2013) 201–205.

Mubarak, A. A. and Al-Omari, S., “First-principles calculations of two cubic fluoropervskite

compounds: RbFeF3 and RbNiF3”. Journal of Magnetism and Magnetic Materials,

382 (2015) 211–218

Mubarak, A. A. and Mousa, A. A., “The electronic and optical properties of the

fluoroperovskite BaXF3 (X= Li, Na, K, and Rb) compounds”. Computational

Materials Science, 59 (2014) 6-13.

Mubarak, A. A., “Ab initio study of the structural, electronic and optical properties of the

fluoropervskite SrXF3 (X = Li, Na, K and Rb) compounds”. Computational Materials

Science, 81 (2014) 478–482.

Muller, O. and Roy, R., “The Major Ternary Structural Families”. Springer Verlag, New

York, 4 (1974).

References

Page | 448

Muralt, P., Polcawich, R. and Trolier-McKinstry, S., “Piezoelectric thin films for sensors,

actuators, and energy harvesting”. MRS bulletin, 34(9) (2009) 658-664.

Murnaghan, F. D., “The Compressibility of Media under Extreme Pressures”. Proceedings of

the National Academy of Sciences of the United States of America, 30(9) (1944)

244-247.

Murtaza, G. and Ahmad, I., “Shift of indirect to direct bandgap and optical response of

LaAlO3 under pressure”. Journal of Applied Physics, 111(12) (2012) 123116.

Murtaza, G., Ahmad, I., Maqbool, M., Aliabad, H. A. R. and Afaq, A., “Structural and

Optoelectronic Properties of Cubic CsPbF3 for Novel Applications”. Chinese Physics

Letters, 28(11) (2011) 117803.

Murtaza, G., Hayatullah., Khenata, R., Khalid, M. N. and Naeem, S., “Elastic and

optoelectronic properties of RbMF3 (M=Zn, Cd, Hg): A mBJ density functional

calculation”. Physica B, 410 (2013)131-36.

Muta, H., Kurosaki, K. and Yamanaka, S., “Thermoelectric Properties of Lanthanum Doped

Europium Titanate”. Journals of Alloys and Compounds, 368(1-2) (2004) 22-24.

Nakamura, M., “Elastic Properties”, Intermetallic Compounds: Principles”. (1995) Edited by

Westbrook J. H. and Fleischer R. L., John Wiley & Sons Ltd, Chichester, England

873-894.

Narayan, R. and Ramaseshan, S., “Repulsion parameters of ions and radicals—Application to

perovskite structures”. Journal of Physics and Chemistry of Solids, 39(12) (1978)

1287- 1294.

Navrotsky, A., "Energetics and Crystal Chemical Systematics among Ilmenite, Lithium

Niobate, and Perovskite Structures". Chemistry of Materials, 10(10) (1998) 2787.

Nenasheva, E., Kanareykin, A., Kartenko, N., Dedyk, A. and Karmanenko, S., “Ceramics

materials based on (Ba, Sr) TiO3 solid solutions for tunable microwave devices”.

Journal of Electroceramics, 13 (2004) 235-238.

References

Page | 449

Nishimatsu, T., Terakubo, N., Mizuseki, H., Kawazoe, Y., Pawlak, D. A., Shimamuri, K. and

Fukuda, T., “Band structures of perovskite-like fluorides for vacuum-ultraviolet-

transparent lens materials”. Japanese Journal of Applied Physics, 41(4A) (2002) 365.

Obara, H., Yamamoto, A., Lee, C. H., Kobayashi, K., Matsumoto, A. and Funahashi, R.,

“Thermoelectric properties of Y-doped polycrystalline SrTiO3”. Japanese journal of

applied physics, 43(4B) (2004) L540.

Obayashi, H., Sakurai, Y. and Gejo, T., “Perovskite-type oxides as ethanol sensors”.

Journal of Solid State Chemistry, 17(3) (1976) 299-303.

Ogawa, T., “On barium titanate ceramics”. Busseiron Kenkyu, 6 (1947) 1–27 (in Japanese).

Ohno, H., Sher, A., Matsuka, F., Oiwa, A., Eudo, A., “(Ga,Mn)As: A new diluted magnetic

semiconductor based on GaAs”. Applied Physics Letters, 69(3) (1996) 363.

Ohta, S., Nomura, T., Ohta, H. and Koumoto, K. “High-temperature carrier transport and

thermoelectric properties of heavily La- or Nb-doped SrTiO3 single crystals”. Journal

of Applied Physics, 97(3) (2005) 034106.

Ohtaki, M., Koga, H., Tokunaga, T., Eguchi, K. and Arai, H. J., “Metal-Insulator Phase

transition and structural stability in ‘Sb’ Doped CaMnO3 Perovskite”. Journal of

Solid State Chemistry, 120(1) (1995) 105-111.

Okazaki, A. and Suemune, Y., “The Crystal Structures of KMnF3, KFeF3, KCoF3, KNiF3 and

KCuF3 above and below their Néel Temperatures”. Journal of the Physical Society of

Japan, 16 (1961) 671-675.

Oleaga, A., Salazar, A., Skrzypek, D., “Critical behaviour of magnetic transitions in KCoF3

and KNiF3 perovskites”. Journal of Alloys and Compounds, 629 (2015) 178–183.

Onida, G., Reining, L. and Rubio, A., “Electronic Excitations: Density-functional versus

Many- body Green’s-function Approaches”. Review of Modern Physics, 74(2) (2002)

601.

References

Page | 450

Ouenzerfi, R. El., Ono, S., Quema, S. A., Goto, M., Sakai, M. and Sarukura, N., “Design of

wide-gap fluoride heterostructures for deep ultraviolet optical devices”. Japanese

Journal of Applied Physics, 439(12) (2004) L1140.

Pari, G., Mathi, J. S. and Asokamani, R., “Electronic structure and magnetism of KMF3

(M=Mn,Fe,Co,Ni)”. Physical Review B, 50(12) (1994) 8166.

Park, K., Jeong, C. K., Ryu, J., Hwang, G. T. and Lee, K. J., “Flexible and Large-Area

Nanocomposite Generators Based on Lead Zirconate Titanate Particles and Carbon

Nanotubes”. Advanced Energy Materials, 3(12) (2013) 1539–1544.

Parlinski, K. and Kawazoe, Y., “Ab initio study of phonons and structural stabilities of the

perovskite-type MgSiO3”. The European Physical Journal, B 16 (2000) 49-58.

Parr, R. G. and Yang, W., “Density Functional Theory of Atoms and Molecules”. Oxford

University Press, New York, Oxford, (1989).

Patel, J. L., Davies, J. J., Cavenett, B. C., Takeuchi, H. and Horai, K., “Electron spin

resonance of axially symmetric Cr3+ centres in KMgF3 and KZnF3”. Journal of

Physics C: Solid State Physics, 9 (1976) 110.

Patterson, J. and Bailey, B., “Solid-State Physics: Introduction to the theory”. Springer,

(2010).

Penn, D., “Wave-Number-Dependent Dielectric Function of Semiconductors”. Physical

Review, 128(5) (1962) 2093-2097.

Perdew J. P. and Wang, Y., “Accurate and simple analytic representation of the electron-gas

correlation energy”. Physical Review B, 45 (1992) 13244.

Perdew, J. P., “Density-functional approximation for the correlation energy of the

inhomogeneous electron gas”. Physical Review Letters, 33 (1986) 8822.

Perdew, J. P., Burke, K. and Ernzerhof, M., “Generalized Gradient Approximation Made

Simple”. Physical Review Letters, 77(18) (1996) 3865-3868.

References

Page | 451

Pettifor, D. G., “Theoretical predictions of structure and related properties of intermetallics.”

Mater. Sci. Technol. 8(4) (1992) 345-349.

Pisarev, R. V., Siny, I. G. and Smolensky, G. A., “The origin and the magnitude of magneto-

optical effects in ferri- and antiferromagnets”. Solid State Communication, 7(1)

(1969) 23-25.

Piskunov, S., Heifets, E., Eglitis, R. I. and Borstel, G., “Bulk properties and electronic

structure of SrTiO3, BaTiO3, PbTiO3 perovskites: An ab initio HF/DFT study”.

Computational Materials Science, 29(2) (2004) 165–178.

Poirier, J. P., “Introduction to the Physics of the Earth's Interior”. Second edition, Cambridge

university press, United Kingdom, 31 (2000).

Pople, J. A., Gill, P. M. W. and Johnson, B. G., “Kohn-sham density functional theory with a

finite basis set”. Chemical Physics Letters, 199 (1992) 557-560.

Protesescu, L., Yakunin, S., Bodnarchuk, M. I., Krieg, F., Caputo, R., Hendon, C. H., Yang,

R. X., Walsh, A. and Kovalenko, M. V., “Nanocrystals of cesium lead halide

perovskites (CsPbX3, X= Cl, Br, and I): novel optoelectronic materials showing

bright emission with wide color gamut”. Nano letters, 15 (2015) 3692-3696.

Pugh S. F., “Relations between the elastic moduli and plastic properties of polycrystalline

pure metals”. Philosophical Magazine, 45 (1954) 823-843

Punkkinen, M. P. J., “Correlated d-states in the perovskites KFeF3 and KCoF3”. Solid State

Communications, 111(9) (1999) 477–481.

R. Ubic, “Revised Method for the Prediction of Lattice Constants in Cubic and Pseudocubic

Perovskites,” Journal of the American Ceramic Society, 90 (10) (2007) 3326–3330.

Rached, H., Rached, D., Khenata, R., Reshak, A. H. and Rabah, M., “First-principles

calculations of structural, elastic and electronic properties of Ni2MnZ (Z = Al, Ga

and In) Heusler alloy”. Physica Status Solidi B, 246(7) (2009) 1580–1586.

Radiochemie, L. F. and Hochschule, T., “Ternäre und polynäre oxide des protactiniums mit

perowskitstruktur”. Journal of Inorganic and Nuclear Chemistry, 27 (1965) 321-327.

References

Page | 452

Rai, D. P., Ghimire, M. P. and Thapa, R. K., “A DFT study of BeX (X= S, Se, Te),

semiconductor: modified Becke Johnson (mBJ) potential”. Semiconductors, 48(11)

(2014) 1447-1457.

Raja, A., Annadurai, G., Daniel, D. G. and Ramasamy, P., “Synthesis, optical and thermal

properties of novel Tb3+ doped RbCaF3 fluoroperovskite phosphors.” Journal of

Alloys and Compounds, 683 (2016) 654-660.

Ramesh, R. and Spaldin, N. A., Multiferroics: progress and prospects in thin films, Nature

materials, 6(1) (2007) 21-29.

Rao, C. N. R. and Gopalakrishnan, J., “New directions in Solid State Chemistry”. Cambridge

University Press (1997).

Rao, C. N. R. and Raveau, B., (Eds.), “Transition Metal Oxides”. VCH Publishers, New

York, (1995).

Raveau, B., “Transition metal oxides: Promising functional materials”. Journal of the

European Ceramic Society, 25(12) (2005)1965–1969.

Raveau, B., Martin, C. and Maignan, A., “What about the role of B elements in the CMR

properties of ABO3 perovskites?” Journal of Alloys and Compounds, 277 (1998)

461– 467.

Rehr, J. J., Jorissen, K., Ankudinov, A. and Ravel, B., “User's Guide, FEFF version 9.6.4”.

(2013) 213-215.

Reuss A. and Angew, Z., “Berechnung der Fließgrenze von Mischkristallen auf Grund der

Plastizitätsbedingung für Einkristalle”. Journal of Applied Mathematics and

Mechanics, 9(1) (1929) 49-58.

Richard, J. D. T., “Perovskites: Structure-Property Relationships”. John Wiley & Sons

(2013).

Robert, R., Bocher, L., Sipos, B., Döbeli, M. and Weidenkaff, A., “Ni-doped cobaltates as

potential materials for high temperature solar thermoelectric converters”. Progress in

solid state chemistry, 35 (2007) 447–455.

References

Page | 453

Roberts, S., “Dielectric and piezoelectric properties of barium titanate”. Physical Reviews,

71(12) (1947) 890–5.

Rose, S. K., Satpathy, S. and Jepsen, O., “Semiconducting CsSnBr3”. Physical Reviews B, 47

(1993) 4276-4280.

Roth, R. S., “Classification of perovskite and other ABC3 -type compounds”. Series of

selected papers in physics—ferroelectrics, Physical Society of Japan (1961) 11-24.

Roy, A. and Vanderbilt, D., “Theory of prospective tetrahedral perovskite ferroelectrics.

Condensed matter material science”., (2010) arXiv. org., arXiv: 1012.5070v1.

Roza, A. O., Abbasi-Pérez, D. and Luaña, V., “Gibbs2: A new version of the quasiharmonic

model code. II. Models for solid-state thermodynamics, features and

implementation”. Computer Physics Communications, 182(10) (2011) 2232-2248.

Sadd, M. H., “Elasticity Theory, Applications, and Numerics”. Elsevier Academic Press

(2005).

Sagar, R. P., Laguna, H. G. & Guevara, N. L., “Statistical correlation between atomic

electron pairs”. Chemical Physics Letters, 514(4-6) (2011) 352–356.

Sakho, I., Ndao, A. S., Biaye, M. and Wague, A., “Calculation of the ground-state energy,

the first ionization energy and the radial correlation expectation value for He-like

atoms”. Physica Scripta, 74(2) (2006) 180–186.

Salehi, H., “First Principles Studies on the Electronic Structure and Band Structure of

Paraelectric SrTiO3 by different Approximations”. Journal of Modern Physics, 2(9)

(2011) 934-943.

Salmankurt, B. and Duman S., “Investigation of the structural, mechanical, dynamical and

thermal properties of CsCaF3 and CsCdF3”. Materials Research Express 3(4) (2016)

045903.

Sarukura, N., Murakami, H., Estacio, E., Ono, S., Ouenzerfi, R. E., Cadatal, M., Nishimatsu,

T., Terakubo, N., Mizuseki, H., Kawazoe, Y., Yoshikawa, A. and Fukuda, T.,

References

Page | 454

“Proposed design principle of fluoride-based materials for deep ultraviolet light emitting

devices”. Optical Materials, 30(1) (2007) 15-17.

Savrasov, S. Y. “Program LMTART for Electronic Structure Calculations”. Condensed

matter material science, (2004) arXiv. org., arXiv:cond-mat/0409705v1.

Sawada, K. and Ishii, F., “Carrier-induced noncollinear magnetism in perovskite manganites

by first-principles calculations”. Journal of Physics: Condensed Matter, 21(6) (2009)

064246-9.

Schreiber, E., Anderson, O. L. and N. Soga, “Elastic Constants and Their Measurements”.

McGraw-Hill, New York, (1973).

Schrodinger, E., “Uber das Verhaltnis der Heisenberg-Born-Jordanschen Quantenmechanik

zu der meinem”. Annalen der Physik 384(8), 734-756 (1926).

Schwarz, K., Blaha, P. and Trickey, S. B., “Electronic structure of solids with WIEN2k”.

Molecular Physics, 108 (2010) 3147-3166.

Scott, J., “Data storage: Multiferroic memories”. Nature materials, 6(4) (2007) 256-257.

Scullin, M. L., Yu, C., Huijben, M., Mukerjee, S., Seidel, J., Zhan, Q., Moore, J., Majumdar,

A. and Ramesh, R., “Anomalously large measured thermoelectric power factor in

Sr1-xLaxTiO3 thin films due to SrTiO3 substrate reduction”. Applied Physics Letters,

92(20) (2008) 202113-3.

Sebastian, M.T., “Dielectric materials for wireless communication”. Elsevier, 1 (2010).

Segall, M. D., Lindan, P. L. D., Probert, M. J., Pickard, C. J., Hasnip, P. J., Clark, S. J. and

Payne, M. C., “First-principles simulation: ideas, illustrations and the CASTEP

code”. Journal of Physics Condensed Matter, 14 (2002) 2717.

Serrate, D., Teresa, J. M. De. and Ibarra, M. R., “Double perovskites with ferromagnetism

above room temperature”. Journal of physics condensed matter, 19 (2007) 023201.

Shafer, M. W., Mcguire, T. R., Argyle, B. E. and Fan, G., “Magnetic and optical properties

of transparent RbNiF3”. Applied Physics Letters, 10 (1967) 202.

References

Page | 455

Shafiq, M., Arif, S., Ahmad, I., Asadabadi, S. J., Maqbool, M. and Rahnamaye, H. A. A.,

“Elastic and mechanical properties of lanthanide monoxides”. Journal of Alloys and

Compounds, 618 (2015) 292–298.

Shannon, R. D., “Revised effective ionic radii and systematic studies of interatomie

distances in halides and chaleogenides”. Acta Crystallographica, A32 (1976) 751–

767.

Shaw, T. M., Trolier-McKinstry, S. and McIntyre, P.C., “The properties of ferroelectric films

at small dimensions”. Annual Review of Materials Science, 30 (2000) 263.

Shein, I. R., Kozhevnikov, V. L. and Ivanovskii A. L., “First-principles calculations of the

elastic and electronic properties of the cubic perovskites SrMO3 (M= Ti, V, Zr and

Nb) in comparison with SrSnO3”. Solid State Sciences, 10(2) 2008 217-225.

Shein, I. R., Shein, K. I. and Ivanovskii, A. L., “Elastic and electronic properties and

stability of SrThO3, SrZrO3 and ThO2 from first principles”. Journal of Nuclear

Materials, 361(1) (2007) 69-77.

Sinclair, D. C., Adams, T. B., Morrison, F. B. and West, A. R., “CaCu3Ti4O12:

One-step internal barrier layer capacitor.” Applied Physics Letters, 80(12) (2002)

2153-2155.

Slassi, A., “Ab initio study of a cubic perovskite: Structural, electronic, optical and electrical

properties of native, lanthanum and antimony-doped barium tin oxide”. Materials

Science in Semiconductor Processing, 32 (2015) 100–106.

Slater, J. C., “Note on Hartree's Method”. Physical Review, 35(2) (1930) 210.

Slater, J. C., “The Ferromagnetism of Nickel”. Physical Reviews, 49 (1936) 537.

Sleight A.W., Gillson J. L. and Bierstedt P. E., “High-temperature superconductivity in the

BaPb1-xBixO3 systems”. Solid State Communications, 17(1) (1975) 27-28.

Sleight, A. W., “Chemistry of high-temperature superconductors”. Science, 242 (1988)

1519–27.

References

Page | 456

Smith, A. E., Mizoguchi, H., Delaney, K., Spaldin, N., Sleight, A. W. and Subramanian, M.

A., “Mn3+ in trigonal bipyramidal coordination: a new blue chromophore”. Journal of

the American Chemical Society, 131 (2009) 17084–6.

Smith, A. E., Sleight, A. W. and Subramanian, M., “Synthesis and properties of solid

solutions of hexagonal YCu0.5Ti0.5O3 with YMO3 (M=Mn, Cr, Fe, Al, Ga, and In)”.

Materials Research Bulletin, 46(1) (2011) 1–5.

Smith, A. E., Sleight, A. W. and Subramanian, M., “Synthesis and properties of solid

solutions of hexagonal YCu0.5Ti0.5O3 with YMO3 (M=Mn, Cr, Fe, Al, Ga, and In)”.

Materials Research Bulletin, 46(1) (2011) 1–5.

Snaith, H. J., “Perovskites: The emergence of a new era for low-cost, high-efficiency solar

cells”. The Journal of Physical Chemistry Letters, 4(21) (2013) 3623-3630.

Spinicci, R., Faticanti, M., Marini, P., Rossi, S. D. and Porta, P., “Catalytic activity of

LaMnO3 and LaCoO3 perovskites toward VOCs combustion”. Journal of Molecular

Catalysis A: Chemical, 197(1/2) (2003) 147–155.

Stein, D. L. and Newman, C. M., “Spin Glasses: Old and New Complexity”. arXiv preprint

arXiv:1205.3432 11 (2012).at http://arxiv.org/abs/1205.3432.

Stukowski, A., “Structure identification methods for atomistic simulations of crystalline

materials”. Modelling and Simulation in Materials Science and Engineering, 20

(2012) 045021.

Sugano, S., Tanabe, Y. and Kamimura, H., “Multiplets of Transition-Metal Ions in Crystals”.

first ed., Academic Press, New York, London, 70 (1970).

Suryanarayana, P., Gavini, V., Blesgen, T., Bhattacharya, K. and Ortiz, M., “Non-periodic

finite- element formulation of Kohn-Sham density functional theory”. Journal of the

Mechanics and Physics of Solids, 58(2) (2010) 256–280.

Szabo, A. and Ostlund, N. S., “Modern Quantum Chemistry”. Macmillan, New York (1982).

http://arxiv.org/abs/1205.3432

References

Page | 457

Terakubo, N., Mizuseki, H., Kawazoe, Y., Yoshikawa, A. and Fukuda, T., “Proposed design

principle of fluoride-based materials for deep ultraviolet light emitting devices”.

Optical Materials, 30(1) (2007) 15-17.

Torardi, C. C., Subramanian, M. A., Calabrese, J. C. et al., "Crystal structure of

Ti2Ba2Ca2Cu3O10, 125 K superconductor" Science, 240(4852) (1988) 631-4.

Tran, F. and Blaha, P., “Accurate Band Gaps of Semiconductors and Insulators with a

Semilocal Exchange-Correlation Potential”. Physical Review Letters, 102 (2009)

226401.

Tran, F. and Blaha, P., “Accurate Band Gaps of Semiconductors and Insulators with a

Semilocal Exchange-Correlation Potential”. Physical Review Letters, 102(22) (2009)

226401.

Travis, W., Glover E. N. K., Bronstein, H., Scanlon D. O. and Palgrave, R. G., “On the

application of the tolerance factor to inorganic and hybrid halide perovskites: a

revised system” Chemical Science, 7 (2016) 4548.

Travis, W., Glover, E. N. K., Bronstein, H., Scanlon, D. O. and Palgrave R. G., “Electronic

Supplementary Material (ESI) for Chemical Science”. The Royal Society of

Chemistry, (2016).

Turner, C. W. and Greedan, J. E., “Ferrimagnetism in the rare earth titanium (III) oxides,

RTiO3; R= Gd, Tb, Dy, Ho, Er, Tm”. Journal of Solid State Chemistry, 34(2) (1980)

207-213.

Uchino, K., “Piezoelectric actuator renaissance”. Energy Harvesting and Systems. 1(1-2)

(2014) 45–56.

Uchino, K., “The development of piezoelectric materials and the new perspective Advanced

Piezoelectric Materials”. Cambridge: Woodhead, 1 (2010).

Uchino, K., Nomura, S., Cross, L. E., Newnham, R. E. and Jang, S. J., “Electrostrictive

effect in perovskites and its transducer applications”. Journal of Materials Science,

16(3) (1981) 569-578.

References

Page | 458

Vaitheeswaran G., Kanchana V., Svane A. and Delin A., “High-pressure structural study of

fluoro perovskite CsCdF3 up to 60 GPa: A combined experimental and theoretical

study”. Journal of Physics Condensed Matter, 19 (2007) 326214.

Vaitheeswaran, G., Kanchana, V., Ravhi, S., Kumar, A. L., Cornelius, Nicol, M. F., Svane,

A., Christensen, N. E. and Eriksson, O., “High-pressure structural study of

fluoroperovskite CsCdF3 up to 60 GPa: A combined experimental and theoretical

study”. Physical Reviews B, 81 (2010) 075105.

Van, H. R., Wecker, J., Holzapfel, B., Schultz, L. and Samwer, K., “Giant negative

magnetoresistance in perovskite like La2/3Ba1/3MnOx ferromagnetic films”. Physical

review letters, 71 (1993) 2331-2333.

Verma, A. S., Kumar, A. and Bhardwaj, S.R., “Correlation between ionic charge and the

lattice constant of cubic perovskite solids”. Physica status solidi (b), 245(8) (2008)

1520.

Voigt, W., “Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper

Körper”. Annalen der Physik, 38(12) (1889) 573-587.

Wachter, P., Filzmoser, M. and Rebizant, J., “Electronic and elastic properties of the light

actinide tellurides”. Physica B, 293(3-4) (2001) 199-223.

Waghmare, U. V., Rabe, K. M., Demkov, A. A., Navrotsky, A., (Eds.) Materials

Fundamentals of Gate Dielectrics, Springer, (2005) 215.

Wang, J., Yip, S., Phillpot, S. R. and Wolf, D., “Crystal instabilities at finite strain”. Physical

Review Letters, 71(25) (1993) 4182.

Wang, X., Chai, Y., Zhou, L. et al., “Observation of magnetoelectric multiferroicity in a

cubic Perovskite System: LaMn3Cr4O12”. Physical Review Letters, 115 (2015)

087601.

Wang, Z. L. and Kang, Z. C., “Functional and Smart Materials”, Plenum Press, New York, 1

(1998).

References

Page | 459

Weeks, C. and Franz, M. “Topological insulators on the Lieb and perovskite lattices”.

Physical Review B, 82 (2010) 085310.

Weinberg, S., “Lectures in Quantum Mechanics”. Cambridge University Press, Cambridge,

England, 1 (2013).

Wenk, H. R. and Bulakh, A., “Minerals: Their Constitution and Origin. New York, NY:

Cambridge University Press, (2004) p. 413.

Wikimedia Commons., “Band Gap Comparison (Online)”. (2015) Available:

https://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Band_gap_comparis

on.svg/2000px-Band_gap_comparison.svg.png.

Williams, C. W., Morss, L. R. and Choi, I. K., “Stability of tetravalent actinides in

perovskites”. ACS Symposium Series, 17(20) (1984) 323.

Winter, M. and Brodd, R. J., “What are batteries, fuel cells, and supercapacitors?” Chemical

Reviews, 104(10) (2004) 4245–4270.

Woerner, M., Schmising, C. K., Bargheer, M., Zhavoronkov, N., Vrejoiu, I., Hesse, D.,

Alexe, M. and Elsaesser, T., “Ultrafast structural dynamics of perovskite

superlattices”. Applied Physics A, 96 (2009) 83-90.

Woodward, P. M., “Octahedral tilting in perovskites. I. Geometrical considerations”. Acta

Crystallographica Section B structural science, crystal engineering and materials, 53

(1997) 32–43.

Wooten, F., “Optical Properties of Solids”. Academic Press, New York (1972).

Wu, H., and Davies, P. K., “Influence of Non-Stoichiometry on the structure and properties

of Ba(Zn1/3Nb2/3) O3 Microwave Dielectrics: I. Substitution of Ba3W2O9”. Journal of

the American Ceramic Society, 89(7) (2006) 2239-2249.

Wu, Z. and Cohen, R. E., “More accurate generalized gradient approximation for solids”.

Physical Review B, 73 (2006) 235116.

https://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Band_gap_comparison.svg/2000px-Band_gap_comparison.svg.png

https://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Band_gap_comparison.svg/2000px-Band_gap_comparison.svg.png

References

Page | 460

Xiao, Z., “Efficient perovskite light-emitting diodes featuring nanometre-sized crystallites”.

Nature Photonics, 11 (2017) 108–115.

Yalcin, G., Salmankurt, B. and Duman, S., “Investigation of structural, mechanical,

electronic, optical, and dynamical properties of cubic BaLiF3, BaLiH3, and SrLiH3”

Material Research Express, 3(3) (2016) 036301.

Yamada, T., Sandu, C. S., Gureev, M., Sherman, V. O., Noeth, A., Muralt, P., Tagantsev, A.

K. and Setter, N., “Self-Assembled Perovskite-Fluorite Oblique Nanostructures for

Adaptive (Tunable) Electronics”. Advanced materials, 21(13) (2009) 1316-1367.

Yamanaka, S., “Thermal and mechanical properties of zirconium hydride”. Journal of Alloys

and Compounds, 293 (1999) 23–29

Yamanoi, K., Nishi, R., Takeda, K., et al., “Perovskite fluoride crystals as light emitting

materials in vacuum ultra violet region”. Optical materials, 36(4) (2014) 769-772.

Yang, W., Salim, J., Li, S., Sun, C., Chen L., Goodenough J.B. and Kim Y., “Perovskite

Sr0.95Ce0.05CoO3−δ loaded with copper nanoparticles as a bifunctional catalyst for

lithium-air batteries”. Journal of Materials Chemistry, 22 (2012) 18902.

Yi, L., Hai-Jin, L., Qing, Z., Yong, L. and Hou-Tong, L., “Electrical transport and

thermoelectric properties of Ni-doped perovskite-type YCo1-xNixO3 prepared by sol-

gel process”. Chinese Physics B, 22(5) (2013) 057201.

Yu, P., Lee, J. S., Okamoto, S., Rossell, M. D. and Huijben, M. et al., “Interface

ferromagnetism and orbital reconstruction in BiFeO3-La0.7Sr0.3MnO3

heterostructures”. Physical review Letters, 105(2) (2010) 027201.

Zener, C., “Interaction between the d-Shells in the Transition Metals. II. Ferromagnetic

Compounds of Manganese with Perovskite Structure”. Physical Reviews, 82 (1951)

403.

Zhang, H., Li, N., Li, K. and Xue, D., “Structural stability and formability of ABO3-type

perovskite compounds”. Acta Crystallographica Section B structural science, crystal

engineering and materials, 63 (2007) 812–8.

References

Page | 461

Zhang, J., Emelianenko, M. and Du, Q. “Periodic centroidal voronoi tessellations.

International journal of numerical analysis and modeling”, 9(4) (2012) 950-969.

Zhu, X., Liu, Z. and Ming, N. “Perovskite oxide nanotubes: synthesis, structural,

characterization, properties and applications”. Journal of Materials Chemistry, 20

(2010) 4015-4030.

Zieger, T., “Approximate density functional theory as a practical tool in molecular energetics

and dynamics”. Chemical Reviews, 91(5) (1991) 651-667.

List of Publications

Page | 462

LIST OF PUBLICATIONS

This thesis consist of the following Published papers.

1. Erum, N. and Iqbal, M. A., “Effect of hydrostatic pressure on physical properties of

strontium based fluoroperovskites for novel applications”. Materials Research

Express, 5 (2018) 025904.

2. Erum, N. and Iqbal, M. A., “Physical properties of fluorine based perovskites for vacuum-

ultraviolet-transparent lens materials”. Chinese Journal of Physics, 56(4) (2018)

1353-1361.

3. Erum, N. and Iqbal, M. A., “Study of pressure variation effect on structural, opto-

electronic, elastic, mechanical, and thermodynamic properties of SrLiF3”. Physica B,

525 (2017) 60–69.

4. Erum, N. and Iqbal, M. A., “A novel pressure variation study on electronic structure,

mechanical stability and thermodynamic properties of potassium based

fluoroperovskite”. Materials Research Express, 4 (2017) 096302.

5. Erum, N. and Iqbal, M. A., “Mechanical and magneto-opto-electronic investigation of

transition metal based fluoro-perovskites: An ab-initio DFT study”. Solid State

Communications, 264 (2017) 39–48.

6. Erum, N. and Iqbal, M. A., “Ab initio study of high dielectric constant oxide-perovskites:

Perspective for miniaturization technology”. Materials Research Express, 4(2) (2017)

025904.

7. Erum, N. and Iqbal, M. A., “First principles investigation of protactinium-based oxide-

perovskites for flexible opto electronic devices”. Chinese Physics B, 26(4) (2017)

047102.

8. Erum, N. and Iqbal, M. A., “Effect of pressure variation on structural, elastic, mechanical,



List of Publications

Page | 463

9. Erum, N. and Iqbal, M. A., “Physical properties of fluorine based perovskites for

vacuum-ultraviolet-transparent lens materials”. Chinese Journal of Physics, 55(3)

(2017) 893–903.

10. Erum, N. and Iqbal, M. A., “First Principles Investigation of Fluorine Based Strontium

Series of Perovskites”. Communications in Theoretical Physics, 66(5) (2016) 571.

11. Erum, N. and Iqbal, M. A., “Opto-electronic investigation of Rubidium based Fluoro

Perovskite for low birefringent lens materials”. Scientific Inquiry Review,

1(1) (2016) 1- 4.

Computational and Quantum Mechanical Investigations of ...

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