Inorganic Glasses for Photonics

Wiley Series in Materials for Electronic and Optoelectronic Applications

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Series EditorsProfessor Arthur Willoughby, University of Southampton, Southampton, UKDr Peter Capper, formerly of SELEX Galileo Infrared Ltd, Southampton, UKProfessor Safa Kasap, University of Saskatchewan, Saskatoon, Canada

Published TitlesBulk Crystal Growth of Electronic, Optical and Optoelectronic Materials, Edited by

P. CapperProperties of Group-IV, III–V and II–VI Semiconductors, S. AdachiCharge Transport in Disordered Solids with Applications in Electronics, Edited by

S. BaranovskiOptical Properties of Condensed Matter and Applications, Edited by J. SinghThin Film Solar Cells: Fabrication, Characterization, and Applications, Edited by

J. Poortmans and V. ArkhipovDielectric Films for Advanced Microelectronics, Edited by M. R. Baklanov, M. Green,

and K. MaexLiquid Phase Epitaxy of Electronic, Optical and Optoelectronic Materials, Edited by

P. Capper and M. MaukMolecular Electronics: From Principles to Practice, M. PettyCVD Diamond for Electronic Devices and Sensors, Edited by R. S. SussmannProperties of Semiconductor Alloys: Group-IV, III–V, and II–VI Semiconductors,

S. AdachiMercury Cadmium Telluride, Edited by P. Capper and J. GarlandZinc Oxide Materials for Electronic and Optoelectronic Device Applications, Edited by

C. Litton, D. C. Reynolds, and T. C. CollinsLead-Free Solders: Materials Reliability for Electronics, Edited by K. N. SubramanianSilicon Photonics: Fundamentals and Devices, M. Jamal Deen and P. K. BasuNanostructured and Subwavelength Waveguides: Fundamentals and Applications,

M. SkorobogatiyPhotovoltaic Materials: From Crystalline Silicon to Third-Generation Approaches,

G. Conibeer and A. WilloughbyGlancing Angle Deposition of Thin Films: Engineering the Nanoscale, Matthew

M. Hawkeye, Michael T. Taschuk, and Michael J. BrettSpintronics for Next Generation Innovative Devices, Edited by Katsuaki Sato and Eiji

SaitohPhysical Properties of High-Temperature Superconductors, Rainer Wesche

Inorganic Glassesfor Photonics

Fundamentals, Engineeringand Applications

ANIMESH JHAInstitute for Materials Research, University of Leeds, UK

This edition first published 2016 2016 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Names: Jha, Animesh, author.Title: Inorganic glasses for photonics : fundamentals, engineering, andapplications / Animesh Jha.

Other titles: Wiley series in materials for electronic and optoelectronicapplications.

Description: Hoboken, New Jersey : John Wiley & Sons, Inc., [2016] | Series:Wiley series in materials for electronic and optoelectronic applications |Includes bibliographical references and index.

Identifiers: LCCN 2016008930 (print) | LCCN 2016012366 (ebook) | ISBN9780470741702 (cloth) | ISBN 0470741708 (cloth) | ISBN 9781118696101 (pdf) |ISBN 9781118696095 (epub)

Subjects: LCSH: Glass--Optical properties. | Photonics--Materials.Classification: LCC QC375 .J43 2016 (print) | LCC QC375 (ebook) | DDC621.36/50284--dc23

LC record available at http://lccn.loc.gov/2016008930

A catalogue record for this book is available from the British Library.

ISBN: 9780470741702

Cover description: Plumes of plasma generated from the surface of inorganic glass targets during pulsed laser deposition

Set in 10/12pt TimesLTStd-Roman by Thomson Digital, Noida, India

1 2016

To my beloved parents for enthusing me to pursue science and engineering!

My parents’ family for supporting the journey to fulfil my ambitionsin my early career pursued in engineering in India!

To my friends and peers for supporting me in my academic career!

To my work place at the University of Leeds where I exchange knowledge!

To my wife, Aparna and children, Prashant and Govind,for building a home in Leeds and inspiration for life!

Contents

Series Preface xiiiPreface xv

1. Introduction 11.1 Definition of Glassy States 11.2 The Glassy State and Glass Transition Temperature (Tg) 11.3 Kauzmann Paradox and Negative Change in Entropy 41.4 Glass-Forming Characteristics and Thermodynamic Properties 51.5 Glass Formation and Co-ordination Number of Cations 141.6 Ionicity of Bonds of Oxide Constituents in Glass-Forming Systems 201.7 Definitions of Glass Network Formers, Intermediates and Modifiers

and Glass-Forming Systems 231.7.1 Constituents of Inorganic Glass-Forming Systems 241.7.2 Strongly Covalent Inorganic Glass-Forming Networks 261.7.3 Conditional Glass Formers Based on Heavy-Metal Oxide

Glasses 291.7.4 Fluoride and Halide Network Forming and Conditional Glass-

Forming Systems 311.7.5 Silicon Oxynitride Conditional Glass-Forming Systems 361.7.6 Chalcogenide Glass-Forming Systems 371.7.7 Chalcohalide Glasses 45

1.8 Conclusions 46Selected Biography 46References 46

2. Glass Structure, Properties and Characterization 512.1 Introduction 51

2.1.1 Kinetic Theory of Glass Formation and Prediction of CriticalCooling Rates 51

2.1.2 Classical Nucleation Theory 522.1.3 Non-Steady State Nucleation 542.1.4 Heterogeneous Nucleation 552.1.5 Nucleation Studies in Fluoride Glasses 562.1.6 Growth Rate 582.1.7 Combined Growth and Nucleation Rates, Phase Transformation

and Critical Cooling Rate 59

viii Contents

2.2 Thermal Characterization using Differential Scanning Calorimetry(DSC) and Differential Thermal Analysis (DTA) Techniques 622.2.1 General Features of a Thermal Characterization 622.2.2 Methods of Characterization 632.2.3 Determining the Characteristic Temperatures 642.2.4 Determination of Apparent Activation Energy of Devitrification 66

2.3 Coefficients of Thermal Expansion of Inorganic Glasses 682.4 Viscosity Behaviour in the near-Tg, above Tg and in the Liquidus

Temperature Ranges 712.5 Density of Inorganic Glasses 752.6 Specific Heat and its Temperature Dependence in the Glassy State 762.7 Conclusion 77References 77

3. Bulk Glass Fabrication and Properties 793.1 Introduction 793.2 Fabrication Steps for Bulk Glasses 80

3.2.1 Chemical Vapour Technique for Oxide Glasses 803.2.2 Batch Preparation for Melting Glasses 813.2.3 Chemical Treatment Before and During Melting 81

3.3 Chemical Purification Methods for Heavier Oxide (GeO2 and TeO2)Glasses 84

3.4 Drying, Fusion and Melting Techniques for Fluoride Glasses 873.4.1 Raw Materials 883.4.2 Control of Hydroxyl Ions during Drying and Melting of

Fluorides 883.5 Chemistry of Purification and Melting Reactions for Chalcogenide

Materials 913.6 Need for Annealing Glass after Casting 963.7 Fabrication of Transparent Glass Ceramics 973.8 Sol–Gel Technique for Glass Formation 99

3.8.1 Background Theory 993.8.2 Examples of Materials Chemistry and Sol–Gel Forming

Techniques 1033.9 Conclusions 105References 105

4. Optical Fibre Design, Engineering, Fabrication and Characterization 1094.1 Introduction to Geometrical Optics of Fibres: Geometrical Optics of

Fibres and Waveguides (Propagation, Critical and Acceptance Angles,Numerical Aperture) 109

4.2 Solutions for Dielectric Waveguides using Maxwell’s Equation 1144.2.1 Analysis of Mode Field Diameter in Single Mode Fibres 115

4.3 Materials Properties Affecting Degradation of Signal in OpticalWaveguides 1174.3.1 Total Intrinsic Loss 117

Contents ix

4.3.2 Electronic Absorption 1184.3.3 Experimental Aspects of Determining the Short Wavelength

Absorption 1214.3.4 Scattering 1214.3.5 Infrared Absorption 1244.3.6 Characterization of Vibrational Structures using Raman and IR

Spectroscopy 1264.3.7 Experimental Aspects of Raman Spectroscopic Technique 1274.3.8 Fourier Transform Infrared (FTIR) spectroscopy 1284.3.9 Examples of the Analysis of Raman and IR spectra 130

4.4 Fabrication of Core–Clad Structures of Glass Preforms and Fibres andtheir Properties 1414.4.1 Comparison of Fabrication Techniques for Silica Optical Fibres

with Non-silica Optical Fibres 1434.4.2 Fibre Fabrication using Non-silica Glass Core–Clad Structures 1514.4.3 Loss Characterization of Fibres 153

4.5 Refractive Indices and Dispersion Characteristics of Inorganic Glasses 1584.5.1 Experimental Procedure for Measuring Refractive Index of a

Glass or Thin Film 1634.5.2 Dependence of Density on Temperature and Relationship with

Refractive Index 1664.5.3 Effect of Residual Stress on Refractive Index of a Medium and

its Effect 1694.6 Conclusion 170References 170

5. Thin-film Fabrication and Characterization 1785.1 Introduction 1785.2 Physical Techniques for Thick and Thin Film Deposition 1795.3 Evaporation 179

5.3.1 General Description 1795.3.2 Technique, Materials and Process Control 179

5.4 Sputtering 1815.4.1 Principle of Sputtering 181

5.5 Pulsed Laser Deposition 1835.5.1 Introduction and Principle 1835.5.2 Process 1845.5.3 Key Features of PLD process 1865.5.4 Controlling Parameters and Materials Investigated 1875.5.5 Fabrication of Thin Film Structures using PLD and Molecular

Beam Epitaxy 1885.6 Ion Implantation 192

5.6.1 Introduction 1925.6.2 Technique and Structural Changes 1925.6.3 Governing Parameters for Ion Implantation 1935.6.4 Materials Systems Investigated 194

x Contents

5.7 Chemical Techniques 1945.7.1 Characteristics of Chemical Vapour Deposition Processes 1955.7.2 Materials System Studied and Applications 1965.7.3 Molecular Beam Epitaxy (MBE) 196

5.8 Ion-Exchange Technique 1975.9 Chemical Solution or Sol–Gel Deposition (CSD) 200

5.9.1 Introduction 2005.9.2 CSD Technique and Materials Deposited 202

5.10 Conclusion 203References 203

6. Spectroscopic Properties of Lanthanide (Ln3+) and Transition Metal(M3+)-Ion Doped Glasses 2096.1 Introduction 2096.2 Theory of Radiative Transition 2096.3 Classical Model for Dipoles and Decay Process 2126.4 Factors Influencing the Line Shape Broadening of Optical Transitions 2146.5 Characteristics of Dipole and Multi-Poles and Selection Rules for

Optical Transitions: 2186.5.1 Analysis of Dipole Transitions Based on Fermi’s Golden Rule 2196.5.2 Electronic Structure and Some Important Properties of

Lanthanides 2216.5.3 Laporte Selection Rules for Rare-Earth and Transition Metal

Ions 2246.6 Comparison of Oscillator Strength Parameters, Optical Transition

Probabilities and Overall Lifetimes of Excited States 2276.6.1 Radiative and Non-Radiative Rate Equation 2316.6.2 Energy Transfer and Related Non-Radiative Processes 2336.6.3 Upconversion Process 237

6.7 Selected Examples of Spectroscopic Processes in Rare-Earth IonDoped Glasses 2386.7.1 Spectroscopic Properties of Trivalent Lanthanide (Ln3+)-Doped

Inorganic Glasses 2396.7.2 Brief Comparison of Spectroscopic Properties of Er3+-Doped

Glasses 2416.7.3 Spectroscopic Properties of Tm3+-Doped Inorganic Glasses 247

6.8 Conclusions 257References 257

7. Applications of Inorganic Photonic Glasses 2617.1 Introduction 2617.2 Dispersion in Optical Fibres and its Control and Management 261

7.2.1 Intramodal Dispersion 2627.2.2 Intermodal Distortion 2657.2.3 Polarization Mode Dispersion (PMD) 2667.2.4 Methods of Controlling and Managing Dispersion in Fibres 267

Contents xi

7.3 Unconventional Fibre Structures 2697.3.1 Fibres with Periodic Defects and Bandgap 2697.3.2 TIR and Endlessly Single Mode Propagation in PCF with

Positive Core–Cladding Difference 2727.3.3 Negative Core–Cladding Refractive Index Difference 2727.3.4 Control of Group Velocity Dispersion (GVD) 2737.3.5 Birefringence in Microstructured Optical Fibres 274

7.4 Optical Nonlinearity in Glasses, Glass-Ceramics and Optical Fibres 2757.4.1 Theory of Harmonic Generation 2757.4.2 Nonlinear Materials for Harmonic Generations and Parametric

Processes 2797.4.3 Fibre Based Kerr Media and its Application 2857.4.4 Resonant Nonlinearity in Doped Glassy Hosts 2877.4.5 Second Harmonic Generation in Inorganic Glasses 2887.4.6 Electric-Field Poling and Poled Glass 2897.4.7 Raman Gain Medium 2917.4.8 Photo-induced Bragg and Long-Period Gratings in Fibres 292

7.5 Applications of Selected Rare-earth ion and Bi-ion Doped AmplifyingDevices 2947.5.1 Introduction 2947.5.2 Examples of Three-Level or Pseudo-Three-Level Transitions 2967.5.3 Examples of Four-Level Laser Systems 300

7.6 Emerging Opportunities for the Future 3027.7 Conclusions 303References 304

Supplementary References 311Symbols and Notations Used 315Index 317

Series Preface

Wiley Series in Materials for Electronic and Optoelectronic Applications

This book series is devoted to the rapidly developing class of materials used for electronicand optoelectronic applications. It is designed to provide much-needed information on thefundamental scientific principles of these materials, together with how these are employedin technological applications. The books are aimed at (postgraduate) students, researchers,and technologists, engaged in research, development, and the study of materials inelectronics and photonics, and industrial scientists developing new materials, devices,and circuits for the electronic, optoelectronic, and communications industries.

The development of new electronic and optoelectronic materials depends not only onmaterials engineering at a practical level, but also on a clear understanding of the propertiesof materials, and the fundamental science behind these properties. It is the properties of amaterial that eventually determine its usefulness in an application. The series therefore alsoincludes such titles as electrical conduction in solids, optical properties, thermal properties,and so on, all with applications and examples of materials in electronics and optoelectronics.

The characterization of materials is also covered within the series in as much as it isimpossible to develop new materials without the proper characterization of their structureand properties. Structure–property relationships have always been fundamentally andintrinsically important to materials science and engineering.

Materials science is well known for being one of the most interdisciplinary sciences. It isthe interdisciplinary aspect of materials science that has led to many exciting discoveries,new materials, and new applications. It is not unusual to find scientists with a chemicalengineering background working on materials projects with applications in electronics.In selecting titles for the series, we have tried to maintain the interdisciplinary aspect of thefield, and hence its excitement to researchers in this field.

Arthur WilloughbyPeter CapperSafa Kasap

Preface

The pleasure of scientific and philosophical expression or communication prompts deeperthinking, which, as human beings, we share for promoting knowledge. Sharing anddissemination of knowledge is the second greatest charity, after saving and protectinglife and the environment – that is what my parents taught me! Without this beacon ofknowledge in human beings, civilization will remain trapped in the labyrinth of darkness.Can we imagine human civilization without any epic of knowledge – where we would betoday as a civilization? These are very powerful statements and as an academic I sincerelybelieve in the true pursuit of knowledge and, for me, this pursuance became a reality when Icompleted this book.

Several years ago some of my distinguished colleagues asked me whether I would bewilling to write a book on “Inorganic Glasses for Photonics”, to fill a gap in this importantarea of physical and materials science. Perhaps it is appropriate to state at this point theimportance of the subject area without emphasizing it too much. No engineering disciplinecan grow without materials science and vice versa. We chose the title Inorganic Glasses forPhotonics because it bears two key aspects of materials science, the structure of the glassystate and its suitability for functionalizing properties for photonic applications. The study ofstructure–property relationships is an intrinsic part of understanding materials science, andin this book I have attempted to bring out this feature in every chapter in a concise andcontextual manner, and wherever possible with examples.

During the course of writing this book, as expected, I faced many challenges and, in mostcases, I turned these challenges into opportunities for learning new experiences, whichhelped me in forming my thoughts to adopt a different style of expression. This may becomeapparent to those who seek to understand the structure–property relationship of materials.Before writing a complex section, I often felt that my thoughts were in a whirlwind ofthermal and configurational entropy, and that the energy requirement for achieving acoherence of thoughts, as in the manner of a laser cavity, was too high. Consequently the“slope efficiency” for writing each chapter was not the same. Exemplifying the structure–property relationship was not easy, which becomes apparent in some sections of the book,and I am sure this feature will continue to evolve in future. For this, if the readers feel thereare omissions I apologize in advance. However, I have purposely kept away fromincorporating chapters and sections that are well covered in other established text booksin the related subject areas.

Not realizing at the outset of writing this textbook that the year 2015 would be declared bythe United Nations as the Year of Light, in which year I would be able to finish this textbook,the conclusion of this project brought a personal sense of achievement. One hundred yearsago in 1915, Albert Einstein rose to world fame by explaining new properties of light in the

xvi Preface

context of general relativity. Einstein also discovered two other important aspects of lightand matter – the discovery of Brownian motion helped in confirming the value ofAvogadro’s number independently. Einstein’s Nobel Prize winning work on the photo­electric effect is at the genesis of quantum theory. A chance to celebrate the greatachievements of Einstein in the form of a book on “Inorganic Glasses for Photonics” isan infinitesimal contribution to the world community of scientists and engineers.In this book there are seven chapters, which in future may grow into much fuller shape by

incorporating emerging aspects of nonlinear optics, nano-photonics and plasmonics usinginorganic glass as a medium for controlling and manipulating light. Although I have writtena significant section on nonlinear optics in Chapter 7, the aspects of nano-photonics andplasmonics are not discussed because I feel these two areas have not yet reached maturity interms of using glass as a medium for wider device applications. I hope that you would agree!In Chapter 1, the main focus is on the glass science and structures of inorganic glasses that

are commonly used for photonic devices. A range of inorganic glasses are discussed in thischapter, with examples of oxide, fluoride, chalcogenide and mixed anion glasses. I have alsoattempted to explain the thermodynamics of glass-forming liquids in the vicinity of deepeutectic liquid, which is often the composition range for stable glass formation. The theoryof co-ordination number is also discussed in the context of phonon structure.For photonic device applications, a chosen glass composition must be engineered using

the thermal, physical and viscosity properties of a glass. These properties are discussed inChapter 2 by emphasizing the roles of nucleation and crystal growth, e.g. for fibre drawing.Having discussed the important thermal, viscosity and physical properties of glasses in

Chapter 2, in Chapter 3 the fabrication of bulk inorganic glasses using melting and casting isdiscussed for a majority of known inorganic glasses. In this chapter the fabricationprinciples of glass-ceramic materials are also discussed. The theory of sol–gel formationand sol–gel based glass fabrication are also discussed briefly in this chapter.In Chapter 4, I have introduced the standard geometrical optics for fibre optics and briefly

discussed the Maxwell’s equation for modal analysis and its importance in fibre andwaveguide optics. In this chapter I have also brought together the signal degradationmechanism in waveguides and discussed them in some detail, by making comparisons. Inthis approach I have also attempted to bring together the properties of various glasses forfibre and waveguide fabrication. This chapter concludes with a detailed discussion onrefractive index and its dependence on compositions, density, temperature and stress. Therelationship of these properties in controlling bulk optical properties is especiallyemphasized.In Chapter 5, the main emphasis is on the methods of thin-film fabrication using physical

and chemical vapour deposition and pulsed laser deposition including ion implantationtechniques. The pros and cons of each technique are discussed with some examples.I have adopted a different style of presentation in Chapter 6, starting with an introduction

to classical radiative transition theory based on dipole models, and have then explained theconcept of dipoles and electron–phonon coupling in the text. By emphasizing variousquantum mechanical rules, I have then attempted to discuss the radiative, non-radiative,energy transfer and upconversion processes. In view of a wealth of information on rare-earthdoped glass based lasers and amplifiers, my focus has been on exemplifying the significanceof a set of optical transitions for specific rare-earth ions in selected glass based devices forexplaining the structure–property relationships.

Preface xvii

The final chapter 7 is on the photonic device applications of inorganic glasses, fibres andwaveguides. In this context I have discussed the importance of dispersion and dispersioncontrol in optical fibres, unconventional fibres, namely, microstructured fibres, opticalnonlinearity and finally concluding with examples of three- and four-level lasers and theirapplications. The book concludes with a short discussion on the emerging opportunities forinorganic glasses.

To help readers, there is an extensive list of references and supplementary references forfurther reading and in-depth comprehension of topical areas.

Earlier this year, in January 2015, Dr Charles Townes, who discovered masers, passedaway 6 months before reaching his 100th birthday, and in this context the Optical Society ofAmerica’s OPN monthly journal (May 2015 issue, pp. 44–51) published a feature article onthe late Dr Townes. In the inset of this article the “Family Matters” of the Townes–Schawlow were also printed. Here is an excerpt that is quite a profound metaphor, and itgoes like this: “Tiny rabbit and beaver were looking up at the Hoover Dam. The beaver issaying to the rabbit, ‘No I didn’t build it, but it was based on an idea of mine”. Since thediscovery of masers in 1953 and then of lasers in 1960, today we are in the era of ultrafastfemto- and atto-second lasers. The beavers have long gone, but the Hoover Dam continuesto pour out knowledge. For me, it will be truly sensational to produce the most coherent andthe purest form of light. Today, glass-based fibre lasers have been commoditized formanufacturing and materials processing. I hope that this book might help burgeoning mindsto discover new sources of light, perhaps using novel glasses that are not yet discovered.Such engineered materials might make a significant impact in future.

Animesh JhaJuly 2015, University of Leeds, Leeds (UK)

1Introduction

1.1 Definition of Glassy States

A “glassy or vitreous” state is classified as a state of condensedmatter in which there is a clearabsence of a three-dimensional periodic structure. The periodicity is defined by the repetitionof point groups (e.g. atoms or ions) occupying sites in the structure, following a crystallo­graphic symmetry, namely, the mirror, inversion and rotation. A glass is a condensed matterexhibiting elasticity below a phase transition temperature, known as the glass transitiontemperature, which is designated in this text as (Tg). By comparison, an “amorphous” state, asin the “vitreous” state, has an all-pervasive lack of three-dimensional periodicity; it is morecomparablewith a liquid rather than a solid. An amorphous structure lacks elasticity and has apropensity to flow under its own weight more readily than a solid-like vitreous state doesbelow Tg. An amorphous inorganic film also has a glass transition temperature and elasticbehaviour, which varies with that of the corresponding vitreous state of the same material.The recognition of apparent differences in the properties of “vitreous” and “amorphous”structures, will be discussed in subsequent chapters on fabrication and processing and suchcomparative characterizations are essential in developing a deeper understanding of astructure–optical and spectroscopic properties of transparent “inorganic glasses as photonicmaterials” for guiding photons and their interactions with the medium. Such differences instructural and thermal properties between a glassy or amorphous and a crystalline stateexplain why the disordered materials demonstrate unique physical, thermo-mechanical,optical and spectroscopic properties, facilitating light confinement and propagation for long-haul distances better than any other condensed matter.

1.2 The Glassy State and Glass Transition Temperature (Tg)

The liquid-to-solid phase transition at the melting point (Tf) of a solid, for example, ischaracterized as a thermodynamically reversible or an equilibrium transition point, at whichboth the liquid and solid phases co-exist. Since at the melting point both phases are inequilibrium, the resultingGibbs energy change (ΔGf), as shown in Equation 1.1, of the phase

Inorganic Glasses for Photonics – Fundamentals, Engineering and Applications, First Edition. Animesh Jha. 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

2 Inorganic Glasses for Photonics – Fundamentals, Engineering and Applications

transition is zero, which then helps in defining the net entropy change associated with thephase change at Tf:

ΔGf ΔHf T fΔSf 0 (1.1)

In Equation 1.1,ΔHf andΔSf are the enthalpy and entropy changes at themelting point. SinceΔGf equates to a zero value at Tf, from Equation 1.1, the entropy change at Tf consequently isequal to:

ΔHf

ΔSf (1.2a)T f

From Equation 1.2a, for pure solids the magnitude of entropic disorder can thus bedetermined at the melting point by measuring the enthalpy of fusion. This characteristicof a solid–liquid transition will become quite relevant in the examination of glass-formationinmulticomponent systems. In Figure 1.1, the liquid-to-crystal and liquid-to-glass transitionsare shown by identifying the Tf and a range of transition temperatures, T1, T2 and T3,g g grespectively. These glass transition temperatures are dependent on the quenching pathsAA1E, AA2F and AA3G, which differ from the equilibrium route ABCD for liquid-crystaltransition at Tf.In Figure 1.1, the glass experiencing the fastest quenching rate (Q3) has the corresponding

transition temperature at T3, whereas the quenching rates Q2 and Q1 yield glasses havinggtransition temperature at T2 and T1, respectively. The end entropic points thus relateg gto the thermal history of each glass. The slowest cooling rate yields the lowest temperature,as the supercooled liquid state below Tf attains a metastable thermodynamic state, which is

Figure 1.1 Plot of the entropy change (ΔSf in Jmol 1 K 1) in a solid–liquid and liquid-glassy statetransitions, shown schematically to illustrate the respective apparent change in the value ofentropy end point, as a result of various quench rates applied, which are designated by the pathsAA1E, AA2F, and AA3G.

3Introduction

still higher in Gibbs energy than the equilibrium crystalline state designated by line CD inFigure 1.1. When the fastest quenching rate path, AA3G, is followed the liquid has little timeto achieve the thermodynamic equilibrium, as reflected by the transition temperature T3,gwhich is closest to the melting point.

The annealing of the fastest quenched glass in Figure 1.1, having a transition temperatureat T3, provides the driving force for structural relaxation to lower energy states progressively.gWith a prolonged isothermal annealing, the end point entropy state might eventually reachmuch closer to the equilibrium crystalline state (line CD in Figure 1.1). As the annealingallows the quenched glass to dissipate most of the energy in a metastable quenched state, theend point entropy never approaches the line CD,which is consistent with the theory proposedby Boltzmann in the context of the second law of thermodynamics. This conditionmathematically limits the value of viscosity approaching infinity, an impossible value.Considering the thermodynamic state properties, e.g. the molar volume (V) and entropy (S),and their dependence on pressure (P) and temperature (T), any change in the entropy of a statecorresponds to a proportional change in the molar volume, which follows from the differ­entials in Equations 1.2b–d, shown below. It is for this reason that in Figure 1.1 thediscontinuity in fractional change in molar free volume (vf), which is dependent on V, isshown along with the entropy change:

dG SdT VdP (1.2b)

@G

@T P

S (1.2c)

@G

@P T

V (1.2d)

The implication of thermodynamic state analysis in Equations 1.2b–d is that thediscontinuities in glassy states are also observed when their state properties, such asthe enthalpy (H), specific heats at constant pressure (Cp) and volume (Cv), thermalexpansion coefficient (αV) and isothermal compressibility (βT), are plotted againsttemperature. Discontinuities in the thermodynamic state properties for several glass-forming liquids are compared and discussed by Paul [1] and Wong and Angell [2] inpublications that readers may find helpful.From Figure 1.1, the glass transition temperature is represented by the presence of a

discontinuity, which is dependent on the quenching rate (Qi), and the points representingTgs are not sharp or abrupt, as shown in the liquid-to-crystal transition. The range of Tgs inFigure 1.1 is characterized as the “fictive glass temperature” and their position isdependent on the quenching history. Several text books designate the fictive temperatureas Tf, and readers should cautiously interpret this temperature along with the quench rateand associated thermal history, because unlike Tf, the Tgs are not fixed phase transitionpoints. Amajor discrepancy in the property characterizationmight arise if experiments arenot carefully designed to study the sub-Tg and above-Tg structural relaxation phenomena,which are discussed in great detail by Varsheneya [3a] in his text book. Elliott [4] explainsthe exponential relationship between quenching rate and Tg in Equation 1.3, showing thatthe corresponding relaxation time (which is the inverse of the quenching rate) is likely tobe imperceptibly long, since a glass is annealed to achieve a new metastable equilibrium

4 Inorganic Glasses for Photonics – Fundamentals, Engineering and Applications

state above a crystalline phase, corresponding to line CD in Figure 1.1:

Qi Qoexp B1Tg

1T f

(1.3)

In Equation 1.3, the value of Qo for different glasses differs, as observed by Owen [5], andwas found to be of the order of 1023 and 104K s 1 between Se and As2S3 glasses. TheconstantBwas found to be of the order of 3× 10 5K.An analysis of quenching rate and glasstransition temperature implies that near Tg, there is an Arrhenius type activation energybarrier, which is path dependent and can be reached in numerous ways by following differentthermal histories, which is discussed extensively by Varsheneya [3b]. Based on pathdependence analysis and the associated changes in the first order thermodynamic properties,namely the enthalpy of glass transition, the phase transition is a “first-order” transition and,unlike the Curie temperature in a magnetic metallic glass, the glass transition is not a second-order transition. The Curie temperature is a fixed point, dependent upon the electronic-spinrelaxation, the time-scale for which is of the order of 10 15 (femto to sub-femto) seconds,which is six orders of magnitude faster than the molecular relaxation characterized by anArrhenius type of energy barrier. From reaction rate theory, the pre-exponential in the rateequation is equal to kBT/h, where kB and h are the Boltzmann constant and Planck’s constant,respectively and T is the absolute temperature. Applying the reaction rate theory forquenching of a glass, the minimum and maximum values therefore may vary between10 7 and 10 9 s, which leads to an interesting discussion on the interaction of ultrafast lasers(pico- and femtosecond) with a glass and consequential structural changes. It is therefore notunreasonable to expect a dramatic change in the relaxation properties of glassy thinfilms formed in a femtosecond quenching regime when compared with the samecomposition glass produced via splat (106 K s 1) and air quenching (102K s 1) techniques.Such a large difference in the magnitude of quenching glass is likely to yield structuralvariations (molar volume, expansion coefficients, refractive index, electronic edge),which may then be manifested in the corresponding relaxation rate, in accordance withEquation 1.3

1.3 Kauzmann Paradox and Negative Change in Entropy

There has been continued debate on the Kauzmann paradox in the glass literature, in relationto the path dependence of quenching of glass-forming liquids and the attainment of an overallentropy state that is lower than that of the crystalline state (line CD in Figure 1.1), whichimplies that the glass attains a negative entropic state. Based on the entropy change insupercooled glycerol, reported earlier by Jäckle [6] in Figure 1.2, Kauzmann’s data [7] werecritically analysed by Varsheneya [3b], who explained that the laws of thermodynamics arenot exempt within the concept of the “Kauzmann paradox”.In supercooled liquids the structural arrangements are so rapid that the resultant changes

cannot be depicted on the time-scale of measurements. There is, though, a further argumentthat continues to support the nature of thermodynamic laws that the entropy change in a“system” may be negative. However, the “total or universe” entropic change is a sum of theentropies of a “system” and its “surrounding”. Two examples are characterized herein tomakean important point on the negative nature of entropy. The solidification point of quartz is

5Introduction

Figure 1.2 Entropy change with respect to the melting point entropy against temperature,extrapolated to determine the Kauzmann temperature (Tk) after Jäckle [6]. Source: J. Jackel 1986.Reproduced with permission from Elsevier.

2273K, and the entropy of fusion (ΔSf)System, from Equation 1.2a, is 4.52 Jmol 1K 1,yielding an enthalpy (ΔHf) of solidification that is equal to 10278 Jmol 1. Since there is onlya small difference in the entropies of crystalline quartz, liquid silica and the glassy states,with acomparable value of around 4.52 Jmol 1K 1, during quenching as the liquid is solidifying toa lower volume state the sign of entropy “changes to a negative”value below themeltingpoint,indicating more structural order than in the liquid state above its melting point, which isconsistent with Equation 1.1. The enthalpy released into the surrounding at 300K, which isabsorbing the heat (δQ), is+10278 Jmol 1 and therefore the corresponding entropy change inthe surrounding (ΔS)Surrounding, from the second law of thermodynamics, is equal to:

δQ 1027834:26 J mol 1 K 1

TSurrounding 300

which yields a net change in entropy of the universe that is equal to+29.74 Jmol 1K 1. Theformation of a layer of amorphous alumina via the reaction:

2Al solid 3 O2 gas Al2O3 solid2

over the surfaceof aluminium follows an identical argument because of the exothermic nature ofthe enthalpy of formation of alumina, which is 1.8046× 106 Jmol 1 of alumina and the

1Kcorresponding negative value of entropy change is nearly 132 Jmol 1. For an endother­mic reaction, the signs will reverse and the argument still holds, implying that the “exemptionfrom the universal thermodynamic laws” is “impossible” in a physical phenomenon. Theirreversible and path-dependent nature ofmetastable glassy states beautifully follows the laws ofthermodynamics, irrespective of the route by which a glass or an amorphous state is achieved,namely via the quenching, sol–gel, pulsed laser deposition andmechanical grinding techniques.

1.4 Glass-Forming Characteristics and Thermodynamic Properties

In Table 1.1 we compare the enthalpy, entropy, melting and glass transition points for severalglass-forming unary compounds. This comparative exercise of melting, glass-transition andthermodynamic properties at Tf will help us in identifying an important structural feature,

6 Inorganic Glasses for Photonics – Fundamentals, Engineering and Applications

Table 1.1 Comparison of the thermodynamic state properties, enthalpy (ΔHf, J mol 1), lattice energy (ΔHlat kJ mol 1) and entropy (ΔSf, in eu), andmelting points (Tf, K) of glass-forming compounds and their glass transition temperatures (Tg, K) [4,8,9].

Unary compound Tf (K) ΔHf (J mol 1) ΔSf (eu) Tg (K) Ratio Tg/Tf ΔHlat (kJ)

Silica (SiO2) 1996 7700–10 880 1.08–1.30 1453 0.73 911Beryllium fluoride (BeF2) 825 4750 1.37 598 0.73 1028Germanium oxide (GeO2) 1388 8140 1.4 853 0.61 580Boric oxide (B2O3) 723 22 180–24 060 7.33–7.95 530 0.73 1272Zinc chloride (ZnCl2) 591 10250 4.14 380 0.64 415Sulfur (S) 388 1720 4.14 246 0.63 —

Selenium (Se) 494 5860 2.83 318 0.64 —

Arsenic trisulfide (As2S3) 585 28675 11.71 478 0.82 168Arsenic triselenide (As2Se3) 633 40815 15.0 468 0.74 102Arsenic tritelluride (As2Te3) 633 46883 17.28 379 0.60 38Germanium disulfide (GeS2) 1113 765 0.69 157Germanium selenide (GeSe2) 1013 695 0.69 113Si 1685 50570 7.17 —

ZrF4 1205 Sublimes at 1177K — — 1909BaF2 1563 28.5 4.35 1210NaF 1269 33.5 6.30 574ZrF4-BaF2 equimolar glass 823 543CdCl2 842 30.1 8.55 392BaCl2 1195 17.2 2.06 858CdCl2-BaCl2 equimolar glass 583 425

7Introduction

which will aid our understanding of the structure–property relationship. In this process of acomparative analysis of thermodynamic properties, we represent the entropy of melting inEquation 1.2a, which is divided by a factor 4.187, the conversion factor for 1 calorie unit intoa joule unit. The ratio in Equation 1.2a, therefore, is redefined in terms of an “entropy unit(eu)” in Equation 1.4 as a measure of disorder:

ΔHf

ΔSf eu (1.4).4:187T f

From this expression, an eu is a measure of disorder at the melting point, which shifts theentropy of a corresponding liquid at Tf along the line BC in Figure 1.1. In Table 1.1, the lastbut one column gives the ratio Tg over Tf, which is often used to define the glass-formingtendency of a liquid, and is known to vary between 0.60 and 0.80 formost glass formers.Wealso point out to the reader that the literature frequently uses terminologies such the “glass­forming ability” and “glass-forming tendency” which carry analogous meaning. However,neither of these two terminologies should be confused with the “metastable stability” of aglassy state, which can only be quantified by the kinetics of glass formation. Several glass-stability parameters have also been used in the literature, some of which are explained in thecontext of the kinetics of glass formation and the classical theory of crystal nucleation andgrowth.

A comparison of the values of eu for various unary glass formers in Table 1.1 demonstratesthat there are unary compounds, namely SiO2, BeF2 andGeO2, which have a relatively lowervalue of eu (∼1.1 to 1.4) onmelting, suggesting that the extent of structural disorder as a resultof melting at Tf, is comparatively much smaller than any other groups of compounds in thistable. The values of eu for arsenic based chalcogenides (As2S3, As2Se3 and As2Te3) arenaturally the largest, due to their high vapour pressures at themelting points. Unfortunately, asimilar comparison for germanium based chalcogenides cannot be carried out, due to the lackof relevant thermodynamic data in the literature. In Table 1.1 we also find that the values ofthe ratio of Tg to Tf, for glass-formers such as SiO2, GeO2, BeF2, ZnCl2, S and Se arepredominantly in the range 0.65–0.72, which falls in a “critical undercooling range” ofroughly 2/3 of the corresponding melting point of the unary glass-forming compound.Table 1.1 thus shows two important features: (i) there is an entropic disorder associatedwith the glass formation in unary compounds and (ii) each glass-former requires under­cooling,with respect tomelting point. Discussion of the aspects of undercooling and entropicchanges associatedwith glass formationwill be resumed later in this chapter and inChapter 2,to help in explaining the thermal properties and viscosity of various types of glasses.

In Table 1.1 we also compare the lattice energies of commonly known glass-formingcompounds and constituent components of multi-constituent glassy phases. The importanceof the lattice energies of constituents is explained below in the context of eutecticcompositions, at which a vast majority of liquids, when quenched, transform into a glassyphase. The lattice energy of various compounds are best quantified by their heats offormation, which yield the resulting bonds, e.g. in SiO2 the Si O bond, as explained bytheBorn–Haber cycle. InTable 1.1, the values of lattice energies (or the heats of formation) ofpure glass-forming elements, namely Si, Se and S, are zero [8]. To estimate the differences inlattice energies of multi-constituent glasses, say, for example, the equimolar compositionsfor AgI-CsI, the lattice energy difference can be estimated by simply subtracting the value ofCsI from that of AgI and dividing it by 2.

8 Inorganic Glasses for Photonics – Fundamentals, Engineering and Applications

A vast majority of practical inorganic glasses used for engineering applications areconstituted of more than one component. This means that the thermodynamic properties ofliquid mixtures are relevant in the discussion of glass-forming liquids, which may becharacterized using the concepts of classical thermodynamics. Comprehensive essays on theproperties of liquid mixtures with examples of metallic and inorganic oxides are cited in anumber of text books on this subject by the pioneers of applied thermodynamics, Darken andGurry [10], Swalin [11], Richardson [12], Lupis [13], Turkdogan [9b], and also in classicalceramic and halide salt references [14,15]. The properties of liquid mixtures and the use ofphase diagrams for the determination of partial molar quantities of component end membersin a binary mixture are especially discussed in References 9b–13. In a binary liquid, forexample, in which more than one component is required for glass formation, the overallchange in the entropy of a mixture differs significantly. Thermodynamically a binarymixture, for example with XA and XB fractions of constituents A and B, respectively, ismore stable than the pure constituents, A andB. This becomes apparentwhenwe consider thedepression in the melting point of a pure constituent, with respect to a liquidus temperature(Tl)i at a given mole fraction, Xi, which is analogous to Equation 1.2a:

ΔHf ΔHiT l (1.5)i ΔSf ΔSi

whereΔHi andΔSi are the partial molar enthalpy and entropy of mixing of a binary mixture.The value ofΔHi can be assumed to be zero for ideal mixtures. For non-ideal mixtures withnegative enthalpy of mixing, the partial enthalpies are also negative in the numerator ofEquation 1.5. The denominator, however, has a partial entropy of mixingΔSi term, which isalways positive, and for a simple ideal mixture it is equal to R ln Xi. Substituting R ln Xi

in Equation 1.5 yields Equation 1.6, in which the liquidus temperature, (Tl)I can be expressedin terms of composition (Xi) and the melting point and entropy of an end member in a binarymixture. Here R is the universal gas constant, with a value 8.314 Jmol 1K 1 in SI units:

ΔHf T fT l (1.6)i RΔSf R ln Xi 1 ln XiΔSf

From this simple equation we find that for a given value of Xi in a binary or multicomponentliquid the drop in the liquidus temperatures, (Tl)I is large when the value of ΔSf is small. Inaddition, if we consider a non-ideal glass-forming liquid in which the value of ΔHi isnegative, the corresponding reduction in the value of predicted (Tl)I is expected to be muchlarger than when a liquid mixture behaves as an ideal mixture. We can, thereby, criticallyexamine examples of glass formation in inorganic glass-forming liquids. Based on suchcomparisons, the corresponding drop in the liquidus leading to formation of eutecticmixturesis discussed in the context of the partial molar properties of the two binary mixtures. The firstglass-forming system is a monovalent mixture of AgI-CsI, followed by the CdF2-BaCl2system, and finally a series of tetravalent–monovalent and tetravalent–divalent fluorideliquidmixtures, especially in the ZrF4-NaF and ZrF4-BaF2 systems. These liquids are classedas predominantly “ionic liquids”, in which the diffusion of cations and anions is at least 2–3orders of magnitude larger than in covalent liquids of, say, meta silicates, phosphates andborates. The importance of such a discussion on glass-formation in ionic liquids will become

9Introduction

Figure 1.3 Calculated liquidus lines, AC and BC, are compared with the experimentallydetermined data [16]. The experimental data also show the presence of complexes, Ag2CsI3and Cs2AgI3, and two deeper eutectic temperatures in the vicinity of 450–470K. Source: Hulme1989. Adapted with permission from Society of Glass Technology, Sheffield.

apparent when the tendency for polymerization and evolution of glass-forming networks isdiscussed by emphasizing the predominance of complex ordering of structures in liquids, asoften seen in the properties of silicate, phosphate, germanate, and borate glasses. The natureof such ordering is then manifested through the shape and slopes of liquidus curves in theresulting phase diagrams.

Glass-formation in chloride, bromide and iodide systems, e.g. in the AgI-CsI binary, werefirst reported by Ding and co-workers [14]. The diffusion coefficient of Ag+ ions in the AgI-CsI liquids is of the order of 10 1 to 10 2 cm2 s 1 below 100 °C [15]. Hulme and co­workers [16] analysed the phase constitution in AgI-CsI, including the shape of liquiduscurve, leading to the formation of eutectic points. The calculated liquidus lines usingEquation 1.6 and the empirically determined phase equilibrium boundaries are compared inFigure 1.3, in which it is apparent that the ideal solution model, based on Raoult’s law,predicts a eutectic temperature that is ∼180K higher than the actual temperature in thevicinity of 470K in the binaryAgI-CsI system [16]. Evidently, in theAgI-CsImixture there isa significant departure from ideal behaviour, which can bemeasuredwith respect to the valueof partial molar enthalpy,ΔHi, in Equation 1.5. A detailed analysis of the partial enthalpy ofmixing in the binary halide systemmay bemade using theHildebrand’s Regular solution andGuggenheim models both of which are well cited in the literature.

At the eutectic points the liquid solution freezes andyield solids, as shown inFigure 1.3, twoofwhich are based on silver iodide polyanionic complexes [Ag2I3] and [AgI3]

2 , as shown inFigure 1.4, and form AgCs2I3 and Ag2CsI3 crystals, respectively. As explained by Brink and

10 Inorganic Glasses for Photonics – Fundamentals, Engineering and Applications

Figure 1.4 Structures of polyanionic [MX3]2 and [M2X3] are shown in (b) and (c), respectively,

where (a) large grey and small grey circles represent iodine and silver ions, respectively. (b)Corner shared AgI chains are prevalent in the AgCs2I3 complex, (c) whereas double edge-sharedAgI chains dominate the Ag2CsI3 structures. The dotted pyramids in (c) represent the backplaneof the paper, which is why along a shared edge four-iodine ions are shown, (see arrow) and is notrequired in the building of this structure. At other shared edges in (c) this is not apparent [16–18].Source: Hulme 1989. Reproduced with permission from The Society of Glass Technology,Sheffield.

co-workers [17], the [M2X3] and [MX3]2 form via edge-sharing and corner-sharing, which

has been further explained by Wells [18] in his treatise on structural chemistry. The cationicradii of Ag+ and Cs+ are 0.127 and 0.168 nm, respectively, and the corresponding values ofelectronegativity are 1.9 and 0.7,which imply that in the [M2X3] and [MX3]

2 complexes theAg+ cations, due to their smaller size and larger electronegativity than the Cs+ ions, areresponsible for the formation of [M2X3] and [MX3]

2 polyanionic species (Figure 1.4).The second example of polymerization in ionic liquids is illustrated by the examples of

glass formation in CdF2-BaCl2, which was reported by Poulain and Matecki [19]. Anessential aspect of structural analysis in the divalent mixture is the Gibbs energy of mixing,which is a means of identifying how far a glass-forming solution departs from an idealRaoult’s law, as explained above in Equations 1.5 and 1.6. The determination of non-idealityin CdF2-BaCl2 can be explained by introducing a thermodynamic term, which will help lateron in establishing the relationship between the viscosity and glass structure, IR absorptionand spectroscopic properties.The departure from non-ideal Raoult behaviour of a reciprocal salt mixture, CdF2-BaCl2, can

be defined by calculating the entropy of mixture.When a solidmixture of CdF2-BaCl2 is heatedabove the melting points of it constituents, the following ionic exchange reaction occurs:

CdF2 l BaCl2 l CdCl2 l BaF2 l (1.7a)

ΔG1:7a 38650 18:3T J mol 1 (1.7b)

In Equation 1.7a we observe that as a result of ionic exchange CdCl2 and BaF2 are produced.However, if the solution were an ideal one, the difference in the entropies of unmixed solidstates before melting (Sunmix) and after melting (Smix), ΔSmixture, is equal to zero. This meansthat there is no preferential structural association by forming a polyanionic complex in liquids.In reality such ideal behaviour is much rarer than the non-ideal, and as a result of mixing inCdF2 and BaCl2, the large difference in their lattice energies (enthalpy of compoundformation), the heat or enthalpy of mixing is not zero (see Equation 1.6). This means thatin Equation 1.5 the partial thermodynamic quantities in the numerator and denominatorbecome non-zero, and consequently the preferential structural association for complexformation is likely to increase. Thus the “ideal configurational entropy”, which can be